Research ArticleIn-Domain Control of a Heat Equation An ApproachCombining Zero-Dynamics Inverse and Differential Flatness

Jun Zheng1 and Guchuan Zhu2

1Department of Basic Courses Southwest Jiaotong University Emeishan Sichuan 614202 China2Department of Electrical Engineering Polytechnique Montreal PO Box 6079 Station Centre-Ville Montreal QC Canada H3T 1J4

Correspondence should be addressed to Guchuan Zhu guchuanzhupolymtlca

Received 11 November 2015 Revised 17 December 2015 Accepted 21 December 2015

Academic Editor Rafael Morales

Copyright copy 2015 J Zheng and G ZhuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper addresses the set-point control problem of a one-dimensional heat equation with in-domain actuation The proposedscheme is based on the framework of zero-dynamics inverse combined with flat system control Moreover the set-point controlis cast into a motion planning problem of a multiple-input multiple-output system which is solved by a Greenrsquos function-basedreference trajectory decomposition The validity of the proposed method is assessed through the analysis of the invertibility of themap generated by Greenrsquos function and the convergence of the regulation error The performance of the developed control schemeand the viability of the proposed approach are confirmed by numerical simulation of a representative system

1 Introduction

Control of parabolic partial differential equations (PDEs) isa long-standing problem in PDE control theory and practiceThere exists a very rich literature devoted to this topic andit is continuing to draw a great attention for both theoreticalstudies and practical applications In the existing literaturethe majority of work is dedicated to boundary control whichmay be represented as a standard Cauchy problem to whichfunctional analytic setting based on semigroup and otherrelated tools can be applied (see eg [1ndash4]) It is interesting tonote that in recent years some methods that were originallydeveloped for the control of finite-dimensional nonlinearsystems have been successfully extended to the control ofparabolic PDEs such as backstepping (see eg [5ndash7]) flatsystems (see eg [8ndash14]) and their variations (see eg [1516])

This paper deals with the output regulation problem forset-point control of a one-dimensional heat equation viapointwise in-domain (or interior) actuation Notice thatdue to the fact that the regularity of a pointwise controlledinhomogeneous heat equation is qualitatively different fromthat of boundary controlled heat equations the techniquesdeveloped for boundary control may not be directly applied

to the former case This constitutes a motivation for thepresent work The control scheme developed in this paperis based-on the framework of zero-dynamics inverse (ZDI)which was introduced by Byrnes and Gilliam in [17] and hasbeen exploited and developed in a series of works (see eg[18] and the references therein) It is pointed out in [19] thatldquofor certain boundary control systems it is very easy to modelthe systemrsquos zero dynamics which in turn provides a simplesystematic methodology for solving certain problems of outputregulationrdquo Indeed the construction of zero-dynamics foroutput regulation of certain interiorly controlled PDEs is alsostraightforward (see eg [20]) and hence the control designcan be carried out in a systematic manner Nevertheless amain issue related to the application of ZDI method is thatit leads to in general a dynamic control lawThus the imple-mentation of such control schemes requires resolving thecorresponding zero-dynamics which may be very difficultfor generic regulation problems such as set-point controlconsidered in the present work To overcome this difficultywe resort to the theory of flat systems [13 21] We showthat in the context of ZDI design the control can be derivedfrom the so-called flat output without explicitly solving theoriginal dynamic equation Moreover in the framework offlat systems set-point control can be cast into a problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 187284 10 pageshttpdxdoiorg1011552015187284

2 Mathematical Problems in Engineering

of motion planning which can also be carried out in asystematic manner Note that it can be expected that the ZDIdesign is applicable to other systems such as the interiorcontrol of beam and plate equations as an alternative to themethods proposed in for example [11 22 23]

The system model used in this work is taken from [20]In order to perform control design based on the principle ofsuperpositionwe present the original system in a formof par-allel connection As the controlwithmultiple actuators locatedin the domain leads to a multiple-input multiple-output(MIMO) problem we introduce a Greenrsquos function-basedreference trajectory decomposition scheme that enables asimple and computational tractable implementation of theproposed control algorithm

The remainder of the paper is organized as followsSection 2 describes the model of the considered system andits equivalent settings Section 3 presents the detailed controldesign Section 4 deals with motion planning and addressesthe convergence and the solvability of the proposed controlscheme A simulation study is carried out in Section 5 andfinally some concluding remarks are presented in Section 6

2 Problem Setting

In the present work we consider a scalar parabolic equationdescribing one-dimensional heat transfer with in-domaincontrol which is studied in [20] Denote by 119906(119909 119905) thetemperature distribution over the one-dimensional space 119909and the time 119905 The derivatives of 119906(119909 119905) with respect to itsvariables are denoted by 119906

119909and 119906

119905 respectively Consider 119898

points 119909119895 119895 = 1 119898 in the interval (0 1) and assume

without loss of generality that 0 = 1199090lt 1199091lt 1199092lt sdot sdot sdot lt

119909119898lt 119909119898+1

= 1 Let Ω ≐ ⋃119898

119895=0(119909119895 119909119895+1) The considered heat

equation with in-domain control in a normalized coordinateis of the form

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 0 119909 isin Ω 119905 gt 0 (1a)

119906 (119909 0) = 120601 (119909) (1b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(1c)

119906 (119909+

119895) = 119906 (119909

minus

119895) 119895 = 1 119898 (1d)

119861119909119895119906 = [119906

119909]119909119895= V119895(119905)

119895 = 1 2 119898

(1e)

where for a function 119891(sdot) and a point 119909 isin [0 1] we define

[119891]119909= 119891 (119909

+

) minus 119891 (119909minus

) (2)

with 119909minus and 119909

+ denoting respectively the usual meaningof left and right hand limits to 119909 The initial condition isspecified in (1b) with 120601(119909) isin 119871

2

(0 1) It is assumed that insystem ((1a) (1b) (1c) (1d) and (1e)) we can control the heatflow at the points 119909

119895for 119895 = 1 119898 that is

V119895(119905) = [119906

119909]119909119895= 119906119909(119909+

119895 119905) minus 119906

119909(119909minus

119895 119905) (3)

Note that in ((1a) (1b) (1c) (1d) and (1e)) 119861119909119895 119909119895isin (0 1)

represents the pointwise control located in the domainThe space of weak solutions to system ((1a) (1b) (1c) (1d)

and (1e)) is chosen to be1198671(0 1) Note that system ((1a) (1b)(1c) (1d) and (1e)) is exponentially stable in1198671(0 1) if119861

0and

1198611are chosen such that 119896

0gt 0 and 119896

1gt 0 [19]

Denote a set of reference signals corresponding tothe control support points of in-domain actuation by119906119863

119894(119909119894 119905)119898

119894=1 where 119906119863

119894(119909119894 119905) isin 119862

infin 119894 = 1 119898 for all119905 isin (0 119879) and 119879 lt infin Let 119890

119894(119905) = 119906(119909

119894 119905) minus 119906

119863

119894(119909119894 119905) be the

regulation errors Let 119890(119905) = 119890119894(119905)119898

119894=1and V(119905) = V

119894(119905)119898

119894=1

Problem 1 The considered regulation problem for set-pointcontrol is to find a dynamic control V(119905) such that theregulation error satisfies 119890(119905) rarr 0 as 119905 rarr infin

Note that although the model under the form ((1a) (1b)(1c) (1d) and (1e)) allows deducing easily the zero-dynamicsit is not convenient for motion planning and in particularfor establishing the input-output map which is essential forfeedforward control design For this reason we introducean equivalent formulation of the in-domain control problemdescribed in ((1a) (1b) (1c) (1d) and (1e)) by replacing thejump conditions in (1e) by pointwise controls as source termsThe resulting system will be of the following form

119906119905(119909 119905) minus 119906

119909119909(119909 119905) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895(119905)

0 lt 119909 lt 1 119905 gt 0

(4a)

119906 (119909 0) = 120601 (119909) (4b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(4c)

where 120575(119909 minus 119909119895) is the Dirac delta function supported at the

point 119909119895 denoting the position of control support and 120572

119895

119905 997891rarr R 119895 = 1 119898 are the in-domain control signals

Lemma2 Consideringweak solutions in1198671(1198671(0 1) [0 119879])119879 lt infin system ((1a) (1b) (1c) (1d) and (1e)) and system ((4a)(4b) and (4c)) are equivalent if

120572119895(119905) = minusV

119895(119905) = minus [119906

119909]119909119895 119895 = 1 119898 (5)

Proof The proof follows the idea presented in [24] Indeedit suffices to prove ldquosystem ((1a) (1b) (1c) (1d) and (1e))rArr system ((4a) (4b) and (4c))rdquo Let 119883 = 119871

2

(0 1) be aHilbert space equipped with the inner product ⟨V 119908⟩ =

int

1

0

V(119909)119908(119909) d119909 for any V 119908 isin 119883 Let the operator 119860

be defined by 119860V = V119909119909 with domain D(119860) = V isin

1198672

(0 1) 1198610V = 119861

1V = 0 It is easy to see that 119860lowast the

adjoint of 119860 is equal to 119860 Let be an extension of 119860 withthe domain D() = V isin 119883 V isin 119867

2

(⋃119898

119894=0(119909119894 119909119894+1)) 1198610V =

1198611V = 0 V(119909+

119895) = V(119909minus

119895) 119895 = 1 119898 Let V isin D()

Mathematical Problems in Engineering 3

119908 isin D(119860lowast

) = D(119860) Using integration by parts we obtainthat

⟨V 119908⟩ = ⟨V 119860V⟩ +119898

sum

119895=1

(V119909(119909minus

119895) minus V119909(119909+

119895))119908 (119909

119895) (6)

Let 119883minus1

= (D(119860lowast

))1015840 the dual space of D(119860) We need to

define another extension for 119860 Let 1198671

(0 1) rarr 119883minus1

bedefined by

⟨V 119908⟩ = ⟨V 119860lowast119908⟩ forall119908 isin D (119860lowast

) (7)

withD() = 1198671

(0 1) Note that 120575(sdot minus 119909119895) is not in 119883 but in

the larger space119883minus1 It follows from (6) (7) and119860 = 119860

lowast that

V = V +119898

sum

119895=1

(V119909(119909minus

119895) minus V119909(119909+

119895)) 120575 (119909 minus 119909

119895) (8)

in 119883minus1 If V satisfies system ((1a) (1b) (1c) (1d) and (1e))

then V(119905) = V(119905) which yields considering (8) V(119905) = V +sum119898

119895=1(V119909(119909minus

119895)minusV119909(119909+

119895))120575(119909minus119909

119895) Finally we can see that system

((1a) (1b) (1c) (1d) and (1e)) becomes system ((4a) (4b) and(4c)) with 120572

119895(119905) = minusV

119895(119905) = minus[119911

119909]119909119895 119895 = 1 119898 where we

look for generalized solutions V(sdot 119905) isin D() = 1198671

(0 1) suchthat (8) is true in119883

minus1

To establish in-domain control at every actuation pointwe will proceed in the way of parallel connection that is forevery 119909

119895isin (0 1) consider the following two systems

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 0

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(9a)

119906 (119909 0) = 120601119895(119909) (9b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(9c)

119906 (119909+

119895) = 119906 (119909

minus

119895) (9d)

119861119909119895119906 = [119906

119909]119909119895= 119908119895(119905) (9e)

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 120575 (119909 minus 119909

119895) 120573119895(119905)

0 lt 119909 lt 1 119905 gt 0

(10a)

119906 (119909 0) = 120601119895(119909) (10b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(10c)

with sum119898119895=1

120601119895(119909) = 120601(119909) Similarly systems ((9a) (9b) (9c)

(9d) and (9e)) and ((10a) (10b) and (10c)) are equivalentprovided 119906 isin 119867

1

(0 1) and 120573119895= minus119908

119895= minus[119906

119909]119909119895 Let 120572

119895=

minusV119895= 120573119895= minus119908

119895= minus[119906

119895

119909]119909119895

for all 119895 = 1 2 119898 where119906119895 denotes the solution to system ((10a) (10b) and (10c))

It follows that 119906(119909 119905) = sum119898

119895=1119906119895

(119909 119905) is a solution to system((4a) (4b) and (4c)) Moreover

[119906119909]119909119894=

119898

sum

119895=1

[119906119895

119909]119909119894

= [119906119894

119909]119909119894

= V119894 (11)

for all 119894 = 1 2 119898 Hence 119906(119909 119905) = sum119898

119895=1119906119895

(119909 119905) is asolution to system ((1a) (1b) (1c) (1d) and (1e)) Thereforethroughout this paper we assume 120572

119895= minusV119895= 120573119895= minus119908

119895=

minus[119906119895

119909]119909119895

for all 119895 = 1 2 119898 Due to the equivalences ofsystems ((1a) (1b) (1c) (1d) and (1e)) and ((4a) (4b) and(4c)) and systems ((9a) (9b) (9c) (9d) and (9e)) and ((10a)(10b) and (10c)) we may consider ((4a) (4b) and (4c)) andsystem ((9a) (9b) (9c) (9d) and (9e)) in the following parts

It is worth noting that in general the internal pointwisecontrol of the heat equation cannot be allocated at arbitrarypoints In particular the approximate controllability may belost if the support of the control is located on a nodal set ofeigenfunctions (see eg [25 26]) Therefore it is importantto ensure that the controllability property of the system isinsensitive to the location of control support so that theexpected performance can be achieved Indeed the loss ofcontrollability will not happen to the considered system if theparameters 119896

0and 119896

1are chosen appropriately Precisely we

call a point 119909 isin (0 1) strategic if it is not located on a nodalset of eigenfunctions of the corresponding PDE [25 26] Wehave then the following

Proposition 3 For system (4a) with the boundary conditions(4c) all the points 119909 isin (0 1) are strategic for any 119896

0gt 0 and

1198961gt 0 Furthermore the approximate controllability of such

a system is insensitive to the position of control support for all119909 isin (119896

1(1198960+ 1198961) 1)

Proof The eigenfunctions of system (4a) with the boundaryconditions (4c) are given by [27]

120595119899(119909) = cos (120583

119899119909) +

1198960

120583119899

sin (120583119899119909) 119899 = 1 2 (12)

where 120583119899are positive roots of the transcendental equation

tan (120583) =120583 (1198960+ 1198961)

(1205832minus 11989601198961)

1198960gt 0 119896

1gt 0 (13)

The solution of 120595119899(119909) = 0 is given by

tan (120583119899) = minus

120583119899

1199091198960

forall119909 isin (0 1) 1198960gt 0 (14)

Obviously for any fixed 119909 isin (0 1) the solution of (14) cannotbe that of (13) for all 119899 = 1 2 Therefore all the points 119909 isin(0 1) are strategic for any 119896

0gt 0 and 119896

1gt 0 Furthermore

for 120583 to verify simultaneously (13) and (14) it must hold

1205832

= 11989601198961minus 1199091198960(1198960+ 1198961) (15)

Therefore 120583 is real-valued if and only if 119909 le 1198961(1198960+ 1198961) lt

1

Proposition 3 implies that the approximate controllabilityof the considered system could hold in almost the wholedomain by choosing 119896

0≫ 1198961

4 Mathematical Problems in Engineering

3 Control Design and Implementation

In the framework of zero-dynamics inverse the in-domaincontrol is derived from the so-called forced zero-dynamics orzero-dynamics for short To work with the parallel connectedsystem ((9a) (9b) (9c) (9d) and (9e)) we first split thereference signal as

119906119863

(119909 119905) =

119898

sum

119895=1

120574119895(119909 119909119895) 119906119889

119895(119909119895 119905)

119906119889

119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin

(16)

where 120574119895(119909 119909119895) will be determined in Theorem 7 (see

Section 4) Denoting by 120576119895

(119905) = 119906119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905) the

regulation error corresponding to system ((9a) (9b) (9c)(9d) and (9e)) the zero-dynamics can be obtained byreplacing the input constraints in (9e) by the requirementthat the regulation errors vanish identically that is 120576119895(119905) = 0Thus we obtain for a fixed index 119895

120585119905(119909 119905) = 120585

119909119909(119909 119905)

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(17a)

120585 (119909 0) = 0 (17b)

120585119909(0 119905) minus 119896

0120585 (0 119905) = 0

120585119909(1 119905) + 119896

1120585 (1 119905) = 0

(17c)

120585 (119909119895 119905) = 119906

119889

119895(119909119895 119905) (17d)

where 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin It should benoticed that by construction we can always choose suitableinitial data of 119906119889(119909 0) so that the zero-dynamics will have ahomogenous initial conditionThis will significantly facilitatecontrol design Then we can get from (9e)

119908119895= [119906119895

119909]119909119895

= [120585119895

119909]119909119895

(18)

which shows that the in-domain control for system ((9a)(9b) (9c) (9d) and (9e)) can be derived from the solution ofthe zero-dynamics Hence ((17a) (17b) (17c) and (17d)) and(18) form a dynamic control scheme This is indeed the basicidea of zero-dynamic inverse design The convergence ofregulation errorswith ZDI-based control is given in followingtheorem

Theorem 4 The regulation error corresponding to system((9a) (9b) (9c) (9d) and (9e)) 120576119895(119905) = 119906

119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905)

tends to 0 as 119905 tends to infin for any 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) and119879 lt infin

The proof of Theorem 4 can follow the developmentpresented in Section III of [20] for the case of a heat equationwith one in-domain actuator and hence it is omitted Notethat a key fact used in the proof of this theorem is that thesystem given in ((9a) (9b) (9c) (9d) and (9e)) without

interior control is exponentially stable for any initial data120601119895(119909) isin 119871

2

(0 1) if 1198960gt 0 and 119896

1gt 0

To implement the dynamic control scheme composedof ((17a) (17b) (17c) and (17d)) and (18) we resort to thetechnique of flat systems [11 13 28 29] In particular we applya standard procedure of Laplace transform-based method tofind the solution to ((17a) (17b) (17c) and (17d))Henceforthwe denote by

119891(119909 119904) the Laplace transform of a function119891(119909 119905) with respect to the time variable Since ((17a) (17b)(17c) and (17d)) has a homogeneous initial condition thenfor fixed 119909

119895isin (0 1) the transformed equations of ((17a)

(17b) (17c) and (17d)) in the Laplace domain read as

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909 isin (0 119909119895) cup (119909

119895 1) 119904 isin C

(19a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0

120585119909(1 119904) + 119896

1

120585 (1 119904) = 0

(19b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (19c)

We divide ((19a) (19b) and (19c)) into two subsystemsthat is for fixed 119909

119895isin (0 1) considering

119904120585 (119909 119904) =

120585119909119909(119909 119904)

0 lt 119909 lt 119909119895 119904 isin C

(20a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0 (20b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (20c)

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909119895lt 119909 lt 1 119904 isin C

(21a)

120585119909(1 119904) + 119896

1

120585 (119909 119904) = 0 (21b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (21c)

Let 120585119895

minus(119909 119904) and 120585

119895

+(119909 119904) be the general solutions to ((20a)

(20b) and (20c)) and ((21a) (21b) and (21c)) respectivelyand denote their inverse Laplace transforms by 120585119895

minus(119909 119905) and

120585119895

+(119909 119905) The solution to ((17a) (17b) (17c) and (17d)) can be

written as

120585119895

(119909 119905) = 120585119895

minus(119909 119905) 120594

(0119909119895)+ 120585119895

+(119909 119905) 120594

[119909119895 1) (22)

where

120594 (119909)Ω119895

=

1 119909 isin Ω119895sube (0 1)

0 otherwise(23)

Then at each point 119909119894

isin (0 1) by (18) and the argu-ment of ldquoparallel connectionrdquo (see Section 2) we have[119906119909]119909119894= sum119898

119895=1[119906119895

119909]119909119894= [119906119894

119909]119909119894= [120585119894

119909]119909119894 119894 = 1 119898 Hence

Mathematical Problems in Engineering 5

the in-domain control signals of system ((1a) (1b) (1c) (1d)and (1e)) can be computed by

V119894= [119906119909]119909119894= [120585119894

119909]119909119894

119894 = 1 119898 (24)

In the following steps we present the computation ofthe solution to system ((17a) (17b) (17c) and (17d)) 120585119895Issues related to the generation reference trajectory 119906119863(119909 119905)for system ((1a) (1b) (1c) (1d) and (1e)) will be addressed inSection 4

Note that 120585119895

minus(119909 119904) and

120585

119895

+(119909 119904) the general solutions to

((20a) (20b) and (20c)) and ((21a) (21b) and (21c)) aregiven by

120585

119895

minus(119909 119904) = 119862

11206011(119909 119904) + 119862

21206012(119909 119904)

120585

119895

+(119909 119904) = 119862

31206011(119909 119904) + 119862

41206012(119909 119904)

(25)

with

1206011(119909 119904) =

sinh (radic119904119909)radic119904

1206012(119909 119904) = cosh (radic119904119909)

(26)

We obtain by applying (20b) and (20c)

11986211206011(119909119895 119904) + 119862

21206012(119909119895 119904) =

119889

119895(119909119895 119904)

1198621minus 11989601198622= 0

(27)

which can be written as

(

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

)(

1198621

1198622

) = (

119889

119895(119909119895 119904)

0

) (28)

Let

119877119895

minus= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

) (29)

119889

119895(119909119895 119904) = minus det (119877119895

minus) 119895

minus(119909119895 119904) (30)

We obtain

(

1198621

1198622

) =

adj (119877119895minus)

det (119877119895minus)

(

119889

119895(119909119895 119904)

0

) = (

1198960119895

minus(119909119895 119904)

119895

minus(119909119895 119904)

) (31)

Therefore the solution to ((20a) (20b) and (20c)) can beexpressed as

120585

119895

minus(119909 119904) = (119896

01206011(119909) + 120601

2(119909))

119895

minus(119909119895 119904) (32)

Wemay proceed in the sameway to deal with ((21a) (21b)and (21c)) Indeed letting

119877119895

+

= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1206012(1 119904) + 119896

11206011(1 119904) 119904120601

1(1 119904) + 119896

11206012(1 119904)

)

(33)

119889

119895(119909119895 119904) = det (119877119895

+) 119895

+(119909119895 119904) (34)

we get from ((21a) (21b) and (21c))

(

1198623

1198624

) = (

(1199041206011(1 119904) + 119896

11206012(1 119904))

119895

+(119909119895 119904)

minus (1206012(1 119904) + 119896

11206011(1 119904))

119895

+(119909119895 119904)

) (35)

120585

119895

+(119909 119904)

= ((1199041206011(1 119904) + 119896

11206012(1 119904)) 120601

1(119909)

+ (1206012(1 119904) + 119896

11206011(1 119904)) 120601

2(119909))

119895

+(119909119895 119904)

(36)

Applying the results from [30 31] which are based onmodule theory to (30) and (34) we may choose

119895(119909119895 119904) as a

basic output such that

119895

+(119909119895 119904) = minusdet (119877119895

minus) 119895(119909119895 119904)

119895

minus(119909119895 119904) = det (119877119895

+) 119895(119909119895 119904)

(37)

It should be noticed that the concept of basic outputs (or flatoutputs) plays a central role in flat system theory becausethe system trajectory and the input can be directly computedfrom a basic output and its time derivatives [13 21]

Using the property of hyperbolic functions we obtainfrom (32) and (36) that

120585

119895

minus(119909 119904)

= (1198961

sinh (radic119904119909119895minus radic119904)

radic119904

minus cosh (radic119904119909119895minus radic119904))

sdot (1198960

sinh (radic119904119909)radic119904

+ cosh (radic119904119909)) 119895(119909119895 119904)

120585

119895

+(119909 119904) = (119896

1

sinh (radic119904119909 minus radic119904)radic119904

minus cosh (radic119904119909 minus radic119904))

sdot (1198960

sinh (radic119904119909119895)

radic119904

+ cosh (radic119904119909119895)) 119895(119909119895 119904)

(38)

Note that

120585

119895

(119909 119904) =120585

119895

minus(119909 119904) 120594

(0119909119895)+120585

119895

+(119909 119904) 120594

[119909119895 1)(39)

is a solution to ((19a) (19b) and (19c)) Using the fact

sinh119909 =infin

sum

119899=0

1199092119899+1

(2119899 + 1)

cosh 119909 =infin

sum

119899=0

1199092119899

(2119899)

(40)

6 Mathematical Problems in Engineering

we obtain the time-domain solution to ((17a) (17b) (17c) and(17d)) which is given by

120585119895

(119909 119905) =[

[

(11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594(0119909119895)

+ (11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594[119909119895 1)

]

]

(41)

Furthermore by a direct computation we get

[120585

119895

119909]

119909119895

= [(

11989601198961

radic119904

+ radic119904) sinh (radic119904)

+ (1198960+ 1198961) cosh (radic119904)]

119895(119909119895 119904)

= (11989601198961

infin

sum

119899=0

119904119899

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119904119899

(2119899)

+

infin

sum

119899=0

119904119899+1

(2119899 + 1)

) 119895(119909119895 119904)

(42)

It follows from (24) that in time domain the control is givenby

V119895(119905) = [120585

119895

119909]119909119895

= 11989601198961

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899)

+

infin

sum

119899=0

119910(119899+1)

119895(119909119895 119905)

(2119899 + 1)

(43)

Finally provided 119906119889119895(119909119895 119905) = 120585

119895

(119909119895 119905) for 119895 = 1 119898 the

reference trajectory 119906119863(119909 119905) can be determined from (16) and(41)

4 Motion Planning

For control purpose we have to choose appropriate referencetrajectories or equivalently the basic outputs Denote nowby 119906119863

(119909) the desired steady-state profile Without loss ofgenerality we consider a set of basic outputs of the form

119910119895(119905) = 119910 (119909

119895) 120593119895(119905) 119895 = 1 119898 (44)

where 120593119895(119905) is a smooth function evolving from 0 to 1Motion

planning amounts then to deriving 119910(119909119895) from 119906

119863

(119909) and todetermining appropriate functions 120593

119895(119905) for 119895 = 1 119898

To this aim and due to the equivalence of systems ((1a)(1b) (1c) (1d) and (1e)) and ((4a) (4b) and (4c)) we considerthe steady-state heat equation corresponding to system ((4a)(4b) and (4c))

119906119909119909(119909) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895

0 lt 119909 lt 1 119905 gt 0

(45a)

119906119909(0) minus 119896

0119906 (0) = 0

119906119909(1) + 119896

1119906 (1) = 0

(45b)

Based on the principle of superposition for linear systemsthe solution to the steady-state heat equation ((45a) and(45b)) can be expressed as

119906 (119909) = int

1

0

119898

sum

119895=1

119866 (119909 120577) 120575 (120577 minus 119909119895) 120572119895d120577

=

119898

sum

119895=1

119866(119909 119909119895) 120572119895

(46)

where119866(119909 120577) is the Greenrsquos function corresponding to ((45a)and (45b)) which is of the form

119866 (119909 120577) =

(1198961120577 minus 1198961minus 1) (119896

0119909 + 1)

1198960+ 1198961+ 11989601198961

0 le 119909 lt 120577

(1198961119909 minus 1198961minus 1) (119896

0120577 + 1)

1198960+ 1198961+ 11989601198961

120577 le 119909 le 1

(47)

Indeed it is easy to check that119866119909119909(119909 120577) = 120575(119909minus120577) and119866(119909 120577)

satisfies the boundary conditions 119866119909(0 120577) minus 119896

0119866(0 120577) = 0

and 119866119909(1 120577) + 119896

1119866(1 120577) = 0 the joint condition 119866(120577+ 120577) =

119866(120577minus

120577) and the jump condition [119866119909(119909 120577)]

120577= 1

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of motion planning which can also be carried out in asystematic manner Note that it can be expected that the ZDIdesign is applicable to other systems such as the interiorcontrol of beam and plate equations as an alternative to themethods proposed in for example [11 22 23]

The system model used in this work is taken from [20]In order to perform control design based on the principle ofsuperpositionwe present the original system in a formof par-allel connection As the controlwithmultiple actuators locatedin the domain leads to a multiple-input multiple-output(MIMO) problem we introduce a Greenrsquos function-basedreference trajectory decomposition scheme that enables asimple and computational tractable implementation of theproposed control algorithm

The remainder of the paper is organized as followsSection 2 describes the model of the considered system andits equivalent settings Section 3 presents the detailed controldesign Section 4 deals with motion planning and addressesthe convergence and the solvability of the proposed controlscheme A simulation study is carried out in Section 5 andfinally some concluding remarks are presented in Section 6

2 Problem Setting

In the present work we consider a scalar parabolic equationdescribing one-dimensional heat transfer with in-domaincontrol which is studied in [20] Denote by 119906(119909 119905) thetemperature distribution over the one-dimensional space 119909and the time 119905 The derivatives of 119906(119909 119905) with respect to itsvariables are denoted by 119906

119909and 119906

119905 respectively Consider 119898

points 119909119895 119895 = 1 119898 in the interval (0 1) and assume

without loss of generality that 0 = 1199090lt 1199091lt 1199092lt sdot sdot sdot lt

119909119898lt 119909119898+1

= 1 Let Ω ≐ ⋃119898

119895=0(119909119895 119909119895+1) The considered heat

equation with in-domain control in a normalized coordinateis of the form

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 0 119909 isin Ω 119905 gt 0 (1a)

119906 (119909 0) = 120601 (119909) (1b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(1c)

119906 (119909+

119895) = 119906 (119909

minus

119895) 119895 = 1 119898 (1d)

119861119909119895119906 = [119906

119909]119909119895= V119895(119905)

119895 = 1 2 119898

(1e)

where for a function 119891(sdot) and a point 119909 isin [0 1] we define

[119891]119909= 119891 (119909

+

) minus 119891 (119909minus

) (2)

with 119909minus and 119909

+ denoting respectively the usual meaningof left and right hand limits to 119909 The initial condition isspecified in (1b) with 120601(119909) isin 119871

2

(0 1) It is assumed that insystem ((1a) (1b) (1c) (1d) and (1e)) we can control the heatflow at the points 119909

119895for 119895 = 1 119898 that is

V119895(119905) = [119906

119909]119909119895= 119906119909(119909+

119895 119905) minus 119906

119909(119909minus

119895 119905) (3)

Note that in ((1a) (1b) (1c) (1d) and (1e)) 119861119909119895 119909119895isin (0 1)

represents the pointwise control located in the domainThe space of weak solutions to system ((1a) (1b) (1c) (1d)

and (1e)) is chosen to be1198671(0 1) Note that system ((1a) (1b)(1c) (1d) and (1e)) is exponentially stable in1198671(0 1) if119861

0and

1198611are chosen such that 119896

0gt 0 and 119896

1gt 0 [19]

Denote a set of reference signals corresponding tothe control support points of in-domain actuation by119906119863

119894(119909119894 119905)119898

119894=1 where 119906119863

119894(119909119894 119905) isin 119862

infin 119894 = 1 119898 for all119905 isin (0 119879) and 119879 lt infin Let 119890

119894(119905) = 119906(119909

119894 119905) minus 119906

119863

119894(119909119894 119905) be the

regulation errors Let 119890(119905) = 119890119894(119905)119898

119894=1and V(119905) = V

119894(119905)119898

119894=1

Problem 1 The considered regulation problem for set-pointcontrol is to find a dynamic control V(119905) such that theregulation error satisfies 119890(119905) rarr 0 as 119905 rarr infin

Note that although the model under the form ((1a) (1b)(1c) (1d) and (1e)) allows deducing easily the zero-dynamicsit is not convenient for motion planning and in particularfor establishing the input-output map which is essential forfeedforward control design For this reason we introducean equivalent formulation of the in-domain control problemdescribed in ((1a) (1b) (1c) (1d) and (1e)) by replacing thejump conditions in (1e) by pointwise controls as source termsThe resulting system will be of the following form

119906119905(119909 119905) minus 119906

119909119909(119909 119905) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895(119905)

0 lt 119909 lt 1 119905 gt 0

(4a)

119906 (119909 0) = 120601 (119909) (4b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(4c)

where 120575(119909 minus 119909119895) is the Dirac delta function supported at the

point 119909119895 denoting the position of control support and 120572

119895

119905 997891rarr R 119895 = 1 119898 are the in-domain control signals

Lemma2 Consideringweak solutions in1198671(1198671(0 1) [0 119879])119879 lt infin system ((1a) (1b) (1c) (1d) and (1e)) and system ((4a)(4b) and (4c)) are equivalent if

120572119895(119905) = minusV

119895(119905) = minus [119906

119909]119909119895 119895 = 1 119898 (5)

Proof The proof follows the idea presented in [24] Indeedit suffices to prove ldquosystem ((1a) (1b) (1c) (1d) and (1e))rArr system ((4a) (4b) and (4c))rdquo Let 119883 = 119871

2

(0 1) be aHilbert space equipped with the inner product ⟨V 119908⟩ =

int

1

0

V(119909)119908(119909) d119909 for any V 119908 isin 119883 Let the operator 119860

be defined by 119860V = V119909119909 with domain D(119860) = V isin

1198672

(0 1) 1198610V = 119861

1V = 0 It is easy to see that 119860lowast the

adjoint of 119860 is equal to 119860 Let be an extension of 119860 withthe domain D() = V isin 119883 V isin 119867

2

(⋃119898

119894=0(119909119894 119909119894+1)) 1198610V =

1198611V = 0 V(119909+

119895) = V(119909minus

119895) 119895 = 1 119898 Let V isin D()

Mathematical Problems in Engineering 3

119908 isin D(119860lowast

) = D(119860) Using integration by parts we obtainthat

⟨V 119908⟩ = ⟨V 119860V⟩ +119898

sum

119895=1

(V119909(119909minus

119895) minus V119909(119909+

119895))119908 (119909

119895) (6)

Let 119883minus1

= (D(119860lowast

))1015840 the dual space of D(119860) We need to

define another extension for 119860 Let 1198671

(0 1) rarr 119883minus1

bedefined by

⟨V 119908⟩ = ⟨V 119860lowast119908⟩ forall119908 isin D (119860lowast

) (7)

withD() = 1198671

(0 1) Note that 120575(sdot minus 119909119895) is not in 119883 but in

the larger space119883minus1 It follows from (6) (7) and119860 = 119860

lowast that

V = V +119898

sum

119895=1

(V119909(119909minus

119895) minus V119909(119909+

119895)) 120575 (119909 minus 119909

119895) (8)

in 119883minus1 If V satisfies system ((1a) (1b) (1c) (1d) and (1e))

then V(119905) = V(119905) which yields considering (8) V(119905) = V +sum119898

119895=1(V119909(119909minus

119895)minusV119909(119909+

119895))120575(119909minus119909

119895) Finally we can see that system

((1a) (1b) (1c) (1d) and (1e)) becomes system ((4a) (4b) and(4c)) with 120572

119895(119905) = minusV

119895(119905) = minus[119911

119909]119909119895 119895 = 1 119898 where we

look for generalized solutions V(sdot 119905) isin D() = 1198671

(0 1) suchthat (8) is true in119883

minus1

To establish in-domain control at every actuation pointwe will proceed in the way of parallel connection that is forevery 119909

119895isin (0 1) consider the following two systems

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 0

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(9a)

119906 (119909 0) = 120601119895(119909) (9b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(9c)

119906 (119909+

119895) = 119906 (119909

minus

119895) (9d)

119861119909119895119906 = [119906

119909]119909119895= 119908119895(119905) (9e)

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 120575 (119909 minus 119909

119895) 120573119895(119905)

0 lt 119909 lt 1 119905 gt 0

(10a)

119906 (119909 0) = 120601119895(119909) (10b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(10c)

with sum119898119895=1

120601119895(119909) = 120601(119909) Similarly systems ((9a) (9b) (9c)

(9d) and (9e)) and ((10a) (10b) and (10c)) are equivalentprovided 119906 isin 119867

1

(0 1) and 120573119895= minus119908

119895= minus[119906

119909]119909119895 Let 120572

119895=

minusV119895= 120573119895= minus119908

119895= minus[119906

119895

119909]119909119895

for all 119895 = 1 2 119898 where119906119895 denotes the solution to system ((10a) (10b) and (10c))

It follows that 119906(119909 119905) = sum119898

119895=1119906119895

(119909 119905) is a solution to system((4a) (4b) and (4c)) Moreover

[119906119909]119909119894=

119898

sum

119895=1

[119906119895

119909]119909119894

= [119906119894

119909]119909119894

= V119894 (11)

for all 119894 = 1 2 119898 Hence 119906(119909 119905) = sum119898

119895=1119906119895

(119909 119905) is asolution to system ((1a) (1b) (1c) (1d) and (1e)) Thereforethroughout this paper we assume 120572

119895= minusV119895= 120573119895= minus119908

119895=

minus[119906119895

119909]119909119895

for all 119895 = 1 2 119898 Due to the equivalences ofsystems ((1a) (1b) (1c) (1d) and (1e)) and ((4a) (4b) and(4c)) and systems ((9a) (9b) (9c) (9d) and (9e)) and ((10a)(10b) and (10c)) we may consider ((4a) (4b) and (4c)) andsystem ((9a) (9b) (9c) (9d) and (9e)) in the following parts

It is worth noting that in general the internal pointwisecontrol of the heat equation cannot be allocated at arbitrarypoints In particular the approximate controllability may belost if the support of the control is located on a nodal set ofeigenfunctions (see eg [25 26]) Therefore it is importantto ensure that the controllability property of the system isinsensitive to the location of control support so that theexpected performance can be achieved Indeed the loss ofcontrollability will not happen to the considered system if theparameters 119896

0and 119896

1are chosen appropriately Precisely we

call a point 119909 isin (0 1) strategic if it is not located on a nodalset of eigenfunctions of the corresponding PDE [25 26] Wehave then the following

Proposition 3 For system (4a) with the boundary conditions(4c) all the points 119909 isin (0 1) are strategic for any 119896

0gt 0 and

1198961gt 0 Furthermore the approximate controllability of such

a system is insensitive to the position of control support for all119909 isin (119896

1(1198960+ 1198961) 1)

Proof The eigenfunctions of system (4a) with the boundaryconditions (4c) are given by [27]

120595119899(119909) = cos (120583

119899119909) +

1198960

120583119899

sin (120583119899119909) 119899 = 1 2 (12)

where 120583119899are positive roots of the transcendental equation

tan (120583) =120583 (1198960+ 1198961)

(1205832minus 11989601198961)

1198960gt 0 119896

1gt 0 (13)

The solution of 120595119899(119909) = 0 is given by

tan (120583119899) = minus

120583119899

1199091198960

forall119909 isin (0 1) 1198960gt 0 (14)

Obviously for any fixed 119909 isin (0 1) the solution of (14) cannotbe that of (13) for all 119899 = 1 2 Therefore all the points 119909 isin(0 1) are strategic for any 119896

0gt 0 and 119896

1gt 0 Furthermore

for 120583 to verify simultaneously (13) and (14) it must hold

1205832

= 11989601198961minus 1199091198960(1198960+ 1198961) (15)

Therefore 120583 is real-valued if and only if 119909 le 1198961(1198960+ 1198961) lt

1

Proposition 3 implies that the approximate controllabilityof the considered system could hold in almost the wholedomain by choosing 119896

0≫ 1198961

4 Mathematical Problems in Engineering

3 Control Design and Implementation

In the framework of zero-dynamics inverse the in-domaincontrol is derived from the so-called forced zero-dynamics orzero-dynamics for short To work with the parallel connectedsystem ((9a) (9b) (9c) (9d) and (9e)) we first split thereference signal as

119906119863

(119909 119905) =

119898

sum

119895=1

120574119895(119909 119909119895) 119906119889

119895(119909119895 119905)

119906119889

119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin

(16)

where 120574119895(119909 119909119895) will be determined in Theorem 7 (see

Section 4) Denoting by 120576119895

(119905) = 119906119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905) the

regulation error corresponding to system ((9a) (9b) (9c)(9d) and (9e)) the zero-dynamics can be obtained byreplacing the input constraints in (9e) by the requirementthat the regulation errors vanish identically that is 120576119895(119905) = 0Thus we obtain for a fixed index 119895

120585119905(119909 119905) = 120585

119909119909(119909 119905)

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(17a)

120585 (119909 0) = 0 (17b)

120585119909(0 119905) minus 119896

0120585 (0 119905) = 0

120585119909(1 119905) + 119896

1120585 (1 119905) = 0

(17c)

120585 (119909119895 119905) = 119906

119889

119895(119909119895 119905) (17d)

where 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin It should benoticed that by construction we can always choose suitableinitial data of 119906119889(119909 0) so that the zero-dynamics will have ahomogenous initial conditionThis will significantly facilitatecontrol design Then we can get from (9e)

119908119895= [119906119895

119909]119909119895

= [120585119895

119909]119909119895

(18)

which shows that the in-domain control for system ((9a)(9b) (9c) (9d) and (9e)) can be derived from the solution ofthe zero-dynamics Hence ((17a) (17b) (17c) and (17d)) and(18) form a dynamic control scheme This is indeed the basicidea of zero-dynamic inverse design The convergence ofregulation errorswith ZDI-based control is given in followingtheorem

Theorem 4 The regulation error corresponding to system((9a) (9b) (9c) (9d) and (9e)) 120576119895(119905) = 119906

119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905)

tends to 0 as 119905 tends to infin for any 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) and119879 lt infin

The proof of Theorem 4 can follow the developmentpresented in Section III of [20] for the case of a heat equationwith one in-domain actuator and hence it is omitted Notethat a key fact used in the proof of this theorem is that thesystem given in ((9a) (9b) (9c) (9d) and (9e)) without

interior control is exponentially stable for any initial data120601119895(119909) isin 119871

2

(0 1) if 1198960gt 0 and 119896

1gt 0

To implement the dynamic control scheme composedof ((17a) (17b) (17c) and (17d)) and (18) we resort to thetechnique of flat systems [11 13 28 29] In particular we applya standard procedure of Laplace transform-based method tofind the solution to ((17a) (17b) (17c) and (17d))Henceforthwe denote by

119891(119909 119904) the Laplace transform of a function119891(119909 119905) with respect to the time variable Since ((17a) (17b)(17c) and (17d)) has a homogeneous initial condition thenfor fixed 119909

119895isin (0 1) the transformed equations of ((17a)

(17b) (17c) and (17d)) in the Laplace domain read as

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909 isin (0 119909119895) cup (119909

119895 1) 119904 isin C

(19a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0

120585119909(1 119904) + 119896

1

120585 (1 119904) = 0

(19b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (19c)

We divide ((19a) (19b) and (19c)) into two subsystemsthat is for fixed 119909

119895isin (0 1) considering

119904120585 (119909 119904) =

120585119909119909(119909 119904)

0 lt 119909 lt 119909119895 119904 isin C

(20a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0 (20b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (20c)

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909119895lt 119909 lt 1 119904 isin C

(21a)

120585119909(1 119904) + 119896

1

120585 (119909 119904) = 0 (21b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (21c)

Let 120585119895

minus(119909 119904) and 120585

119895

+(119909 119904) be the general solutions to ((20a)

(20b) and (20c)) and ((21a) (21b) and (21c)) respectivelyand denote their inverse Laplace transforms by 120585119895

minus(119909 119905) and

120585119895

+(119909 119905) The solution to ((17a) (17b) (17c) and (17d)) can be

written as

120585119895

(119909 119905) = 120585119895

minus(119909 119905) 120594

(0119909119895)+ 120585119895

+(119909 119905) 120594

[119909119895 1) (22)

where

120594 (119909)Ω119895

=

1 119909 isin Ω119895sube (0 1)

0 otherwise(23)

Then at each point 119909119894

isin (0 1) by (18) and the argu-ment of ldquoparallel connectionrdquo (see Section 2) we have[119906119909]119909119894= sum119898

119895=1[119906119895

119909]119909119894= [119906119894

119909]119909119894= [120585119894

119909]119909119894 119894 = 1 119898 Hence

Mathematical Problems in Engineering 5

the in-domain control signals of system ((1a) (1b) (1c) (1d)and (1e)) can be computed by

V119894= [119906119909]119909119894= [120585119894

119909]119909119894

119894 = 1 119898 (24)

In the following steps we present the computation ofthe solution to system ((17a) (17b) (17c) and (17d)) 120585119895Issues related to the generation reference trajectory 119906119863(119909 119905)for system ((1a) (1b) (1c) (1d) and (1e)) will be addressed inSection 4

Note that 120585119895

minus(119909 119904) and

120585

119895

+(119909 119904) the general solutions to

((20a) (20b) and (20c)) and ((21a) (21b) and (21c)) aregiven by

120585

119895

minus(119909 119904) = 119862

11206011(119909 119904) + 119862

21206012(119909 119904)

120585

119895

+(119909 119904) = 119862

31206011(119909 119904) + 119862

41206012(119909 119904)

(25)

with

1206011(119909 119904) =

sinh (radic119904119909)radic119904

1206012(119909 119904) = cosh (radic119904119909)

(26)

We obtain by applying (20b) and (20c)

11986211206011(119909119895 119904) + 119862

21206012(119909119895 119904) =

119889

119895(119909119895 119904)

1198621minus 11989601198622= 0

(27)

which can be written as

(

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

)(

1198621

1198622

) = (

119889

119895(119909119895 119904)

0

) (28)

Let

119877119895

minus= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

) (29)

119889

119895(119909119895 119904) = minus det (119877119895

minus) 119895

minus(119909119895 119904) (30)

We obtain

(

1198621

1198622

) =

adj (119877119895minus)

det (119877119895minus)

(

119889

119895(119909119895 119904)

0

) = (

1198960119895

minus(119909119895 119904)

119895

minus(119909119895 119904)

) (31)

Therefore the solution to ((20a) (20b) and (20c)) can beexpressed as

120585

119895

minus(119909 119904) = (119896

01206011(119909) + 120601

2(119909))

119895

minus(119909119895 119904) (32)

Wemay proceed in the sameway to deal with ((21a) (21b)and (21c)) Indeed letting

119877119895

+

= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1206012(1 119904) + 119896

11206011(1 119904) 119904120601

1(1 119904) + 119896

11206012(1 119904)

)

(33)

119889

119895(119909119895 119904) = det (119877119895

+) 119895

+(119909119895 119904) (34)

we get from ((21a) (21b) and (21c))

(

1198623

1198624

) = (

(1199041206011(1 119904) + 119896

11206012(1 119904))

119895

+(119909119895 119904)

minus (1206012(1 119904) + 119896

11206011(1 119904))

119895

+(119909119895 119904)

) (35)

120585

119895

+(119909 119904)

= ((1199041206011(1 119904) + 119896

11206012(1 119904)) 120601

1(119909)

+ (1206012(1 119904) + 119896

11206011(1 119904)) 120601

2(119909))

119895

+(119909119895 119904)

(36)

Applying the results from [30 31] which are based onmodule theory to (30) and (34) we may choose

119895(119909119895 119904) as a

basic output such that

119895

+(119909119895 119904) = minusdet (119877119895

minus) 119895(119909119895 119904)

119895

minus(119909119895 119904) = det (119877119895

+) 119895(119909119895 119904)

(37)

It should be noticed that the concept of basic outputs (or flatoutputs) plays a central role in flat system theory becausethe system trajectory and the input can be directly computedfrom a basic output and its time derivatives [13 21]

Using the property of hyperbolic functions we obtainfrom (32) and (36) that

120585

119895

minus(119909 119904)

= (1198961

sinh (radic119904119909119895minus radic119904)

radic119904

minus cosh (radic119904119909119895minus radic119904))

sdot (1198960

sinh (radic119904119909)radic119904

+ cosh (radic119904119909)) 119895(119909119895 119904)

120585

119895

+(119909 119904) = (119896

1

sinh (radic119904119909 minus radic119904)radic119904

minus cosh (radic119904119909 minus radic119904))

sdot (1198960

sinh (radic119904119909119895)

radic119904

+ cosh (radic119904119909119895)) 119895(119909119895 119904)

(38)

Note that

120585

119895

(119909 119904) =120585

119895

minus(119909 119904) 120594

(0119909119895)+120585

119895

+(119909 119904) 120594

[119909119895 1)(39)

is a solution to ((19a) (19b) and (19c)) Using the fact

sinh119909 =infin

sum

119899=0

1199092119899+1

(2119899 + 1)

cosh 119909 =infin

sum

119899=0

1199092119899

(2119899)

(40)

6 Mathematical Problems in Engineering

we obtain the time-domain solution to ((17a) (17b) (17c) and(17d)) which is given by

120585119895

(119909 119905) =[

[

(11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594(0119909119895)

+ (11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594[119909119895 1)

]

]

(41)

Furthermore by a direct computation we get

[120585

119895

119909]

119909119895

= [(

11989601198961

radic119904

+ radic119904) sinh (radic119904)

+ (1198960+ 1198961) cosh (radic119904)]

119895(119909119895 119904)

= (11989601198961

infin

sum

119899=0

119904119899

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119904119899

(2119899)

+

infin

sum

119899=0

119904119899+1

(2119899 + 1)

) 119895(119909119895 119904)

(42)

It follows from (24) that in time domain the control is givenby

V119895(119905) = [120585

119895

119909]119909119895

= 11989601198961

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899)

+

infin

sum

119899=0

119910(119899+1)

119895(119909119895 119905)

(2119899 + 1)

(43)

Finally provided 119906119889119895(119909119895 119905) = 120585

119895

(119909119895 119905) for 119895 = 1 119898 the

reference trajectory 119906119863(119909 119905) can be determined from (16) and(41)

4 Motion Planning

For control purpose we have to choose appropriate referencetrajectories or equivalently the basic outputs Denote nowby 119906119863

(119909) the desired steady-state profile Without loss ofgenerality we consider a set of basic outputs of the form

119910119895(119905) = 119910 (119909

119895) 120593119895(119905) 119895 = 1 119898 (44)

where 120593119895(119905) is a smooth function evolving from 0 to 1Motion

planning amounts then to deriving 119910(119909119895) from 119906

119863

(119909) and todetermining appropriate functions 120593

119895(119905) for 119895 = 1 119898

To this aim and due to the equivalence of systems ((1a)(1b) (1c) (1d) and (1e)) and ((4a) (4b) and (4c)) we considerthe steady-state heat equation corresponding to system ((4a)(4b) and (4c))

119906119909119909(119909) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895

0 lt 119909 lt 1 119905 gt 0

(45a)

119906119909(0) minus 119896

0119906 (0) = 0

119906119909(1) + 119896

1119906 (1) = 0

(45b)

Based on the principle of superposition for linear systemsthe solution to the steady-state heat equation ((45a) and(45b)) can be expressed as

119906 (119909) = int

1

0

119898

sum

119895=1

119866 (119909 120577) 120575 (120577 minus 119909119895) 120572119895d120577

=

119898

sum

119895=1

119866(119909 119909119895) 120572119895

(46)

where119866(119909 120577) is the Greenrsquos function corresponding to ((45a)and (45b)) which is of the form

119866 (119909 120577) =

(1198961120577 minus 1198961minus 1) (119896

0119909 + 1)

1198960+ 1198961+ 11989601198961

0 le 119909 lt 120577

(1198961119909 minus 1198961minus 1) (119896

0120577 + 1)

1198960+ 1198961+ 11989601198961

120577 le 119909 le 1

(47)

Indeed it is easy to check that119866119909119909(119909 120577) = 120575(119909minus120577) and119866(119909 120577)

satisfies the boundary conditions 119866119909(0 120577) minus 119896

0119866(0 120577) = 0

and 119866119909(1 120577) + 119896

1119866(1 120577) = 0 the joint condition 119866(120577+ 120577) =

119866(120577minus

120577) and the jump condition [119866119909(119909 120577)]

120577= 1

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119908 isin D(119860lowast

) = D(119860) Using integration by parts we obtainthat

⟨V 119908⟩ = ⟨V 119860V⟩ +119898

sum

119895=1

(V119909(119909minus

119895) minus V119909(119909+

119895))119908 (119909

119895) (6)

Let 119883minus1

= (D(119860lowast

))1015840 the dual space of D(119860) We need to

define another extension for 119860 Let 1198671

(0 1) rarr 119883minus1

bedefined by

⟨V 119908⟩ = ⟨V 119860lowast119908⟩ forall119908 isin D (119860lowast

) (7)

withD() = 1198671

(0 1) Note that 120575(sdot minus 119909119895) is not in 119883 but in

the larger space119883minus1 It follows from (6) (7) and119860 = 119860

lowast that

V = V +119898

sum

119895=1

(V119909(119909minus

119895) minus V119909(119909+

119895)) 120575 (119909 minus 119909

119895) (8)

in 119883minus1 If V satisfies system ((1a) (1b) (1c) (1d) and (1e))

then V(119905) = V(119905) which yields considering (8) V(119905) = V +sum119898

119895=1(V119909(119909minus

119895)minusV119909(119909+

119895))120575(119909minus119909

119895) Finally we can see that system

((1a) (1b) (1c) (1d) and (1e)) becomes system ((4a) (4b) and(4c)) with 120572

119895(119905) = minusV

119895(119905) = minus[119911

119909]119909119895 119895 = 1 119898 where we

look for generalized solutions V(sdot 119905) isin D() = 1198671

(0 1) suchthat (8) is true in119883

minus1

To establish in-domain control at every actuation pointwe will proceed in the way of parallel connection that is forevery 119909

119895isin (0 1) consider the following two systems

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 0

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(9a)

119906 (119909 0) = 120601119895(119909) (9b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(9c)

119906 (119909+

119895) = 119906 (119909

minus

119895) (9d)

119861119909119895119906 = [119906

119909]119909119895= 119908119895(119905) (9e)

119906119905(119909 119905) minus 119906

119909119909(119909 119905) = 120575 (119909 minus 119909

119895) 120573119895(119905)

0 lt 119909 lt 1 119905 gt 0

(10a)

119906 (119909 0) = 120601119895(119909) (10b)

1198610119906 = 119906119909(0 119905) minus 119896

0119906 (0 119905) = 0

1198611119906 = 119906119909(1 119905) + 119896

1119906 (1 119905) = 0

(10c)

with sum119898119895=1

120601119895(119909) = 120601(119909) Similarly systems ((9a) (9b) (9c)

(9d) and (9e)) and ((10a) (10b) and (10c)) are equivalentprovided 119906 isin 119867

1

(0 1) and 120573119895= minus119908

119895= minus[119906

119909]119909119895 Let 120572

119895=

minusV119895= 120573119895= minus119908

119895= minus[119906

119895

119909]119909119895

for all 119895 = 1 2 119898 where119906119895 denotes the solution to system ((10a) (10b) and (10c))

It follows that 119906(119909 119905) = sum119898

119895=1119906119895

(119909 119905) is a solution to system((4a) (4b) and (4c)) Moreover

[119906119909]119909119894=

119898

sum

119895=1

[119906119895

119909]119909119894

= [119906119894

119909]119909119894

= V119894 (11)

for all 119894 = 1 2 119898 Hence 119906(119909 119905) = sum119898

119895=1119906119895

(119909 119905) is asolution to system ((1a) (1b) (1c) (1d) and (1e)) Thereforethroughout this paper we assume 120572

119895= minusV119895= 120573119895= minus119908

119895=

minus[119906119895

119909]119909119895

for all 119895 = 1 2 119898 Due to the equivalences ofsystems ((1a) (1b) (1c) (1d) and (1e)) and ((4a) (4b) and(4c)) and systems ((9a) (9b) (9c) (9d) and (9e)) and ((10a)(10b) and (10c)) we may consider ((4a) (4b) and (4c)) andsystem ((9a) (9b) (9c) (9d) and (9e)) in the following parts

It is worth noting that in general the internal pointwisecontrol of the heat equation cannot be allocated at arbitrarypoints In particular the approximate controllability may belost if the support of the control is located on a nodal set ofeigenfunctions (see eg [25 26]) Therefore it is importantto ensure that the controllability property of the system isinsensitive to the location of control support so that theexpected performance can be achieved Indeed the loss ofcontrollability will not happen to the considered system if theparameters 119896

0and 119896

1are chosen appropriately Precisely we

call a point 119909 isin (0 1) strategic if it is not located on a nodalset of eigenfunctions of the corresponding PDE [25 26] Wehave then the following

Proposition 3 For system (4a) with the boundary conditions(4c) all the points 119909 isin (0 1) are strategic for any 119896

0gt 0 and

1198961gt 0 Furthermore the approximate controllability of such

a system is insensitive to the position of control support for all119909 isin (119896

1(1198960+ 1198961) 1)

Proof The eigenfunctions of system (4a) with the boundaryconditions (4c) are given by [27]

120595119899(119909) = cos (120583

119899119909) +

1198960

120583119899

sin (120583119899119909) 119899 = 1 2 (12)

where 120583119899are positive roots of the transcendental equation

tan (120583) =120583 (1198960+ 1198961)

(1205832minus 11989601198961)

1198960gt 0 119896

1gt 0 (13)

The solution of 120595119899(119909) = 0 is given by

tan (120583119899) = minus

120583119899

1199091198960

forall119909 isin (0 1) 1198960gt 0 (14)

Obviously for any fixed 119909 isin (0 1) the solution of (14) cannotbe that of (13) for all 119899 = 1 2 Therefore all the points 119909 isin(0 1) are strategic for any 119896

0gt 0 and 119896

1gt 0 Furthermore

for 120583 to verify simultaneously (13) and (14) it must hold

1205832

= 11989601198961minus 1199091198960(1198960+ 1198961) (15)

Therefore 120583 is real-valued if and only if 119909 le 1198961(1198960+ 1198961) lt

1

Proposition 3 implies that the approximate controllabilityof the considered system could hold in almost the wholedomain by choosing 119896

0≫ 1198961

4 Mathematical Problems in Engineering

3 Control Design and Implementation

In the framework of zero-dynamics inverse the in-domaincontrol is derived from the so-called forced zero-dynamics orzero-dynamics for short To work with the parallel connectedsystem ((9a) (9b) (9c) (9d) and (9e)) we first split thereference signal as

119906119863

(119909 119905) =

119898

sum

119895=1

120574119895(119909 119909119895) 119906119889

119895(119909119895 119905)

119906119889

119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin

(16)

where 120574119895(119909 119909119895) will be determined in Theorem 7 (see

Section 4) Denoting by 120576119895

(119905) = 119906119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905) the

regulation error corresponding to system ((9a) (9b) (9c)(9d) and (9e)) the zero-dynamics can be obtained byreplacing the input constraints in (9e) by the requirementthat the regulation errors vanish identically that is 120576119895(119905) = 0Thus we obtain for a fixed index 119895

120585119905(119909 119905) = 120585

119909119909(119909 119905)

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(17a)

120585 (119909 0) = 0 (17b)

120585119909(0 119905) minus 119896

0120585 (0 119905) = 0

120585119909(1 119905) + 119896

1120585 (1 119905) = 0

(17c)

120585 (119909119895 119905) = 119906

119889

119895(119909119895 119905) (17d)

where 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin It should benoticed that by construction we can always choose suitableinitial data of 119906119889(119909 0) so that the zero-dynamics will have ahomogenous initial conditionThis will significantly facilitatecontrol design Then we can get from (9e)

119908119895= [119906119895

119909]119909119895

= [120585119895

119909]119909119895

(18)

which shows that the in-domain control for system ((9a)(9b) (9c) (9d) and (9e)) can be derived from the solution ofthe zero-dynamics Hence ((17a) (17b) (17c) and (17d)) and(18) form a dynamic control scheme This is indeed the basicidea of zero-dynamic inverse design The convergence ofregulation errorswith ZDI-based control is given in followingtheorem

Theorem 4 The regulation error corresponding to system((9a) (9b) (9c) (9d) and (9e)) 120576119895(119905) = 119906

119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905)

tends to 0 as 119905 tends to infin for any 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) and119879 lt infin

The proof of Theorem 4 can follow the developmentpresented in Section III of [20] for the case of a heat equationwith one in-domain actuator and hence it is omitted Notethat a key fact used in the proof of this theorem is that thesystem given in ((9a) (9b) (9c) (9d) and (9e)) without

interior control is exponentially stable for any initial data120601119895(119909) isin 119871

2

(0 1) if 1198960gt 0 and 119896

1gt 0

To implement the dynamic control scheme composedof ((17a) (17b) (17c) and (17d)) and (18) we resort to thetechnique of flat systems [11 13 28 29] In particular we applya standard procedure of Laplace transform-based method tofind the solution to ((17a) (17b) (17c) and (17d))Henceforthwe denote by

119891(119909 119904) the Laplace transform of a function119891(119909 119905) with respect to the time variable Since ((17a) (17b)(17c) and (17d)) has a homogeneous initial condition thenfor fixed 119909

119895isin (0 1) the transformed equations of ((17a)

(17b) (17c) and (17d)) in the Laplace domain read as

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909 isin (0 119909119895) cup (119909

119895 1) 119904 isin C

(19a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0

120585119909(1 119904) + 119896

1

120585 (1 119904) = 0

(19b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (19c)

We divide ((19a) (19b) and (19c)) into two subsystemsthat is for fixed 119909

119895isin (0 1) considering

119904120585 (119909 119904) =

120585119909119909(119909 119904)

0 lt 119909 lt 119909119895 119904 isin C

(20a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0 (20b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (20c)

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909119895lt 119909 lt 1 119904 isin C

(21a)

120585119909(1 119904) + 119896

1

120585 (119909 119904) = 0 (21b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (21c)

Let 120585119895

minus(119909 119904) and 120585

119895

+(119909 119904) be the general solutions to ((20a)

(20b) and (20c)) and ((21a) (21b) and (21c)) respectivelyand denote their inverse Laplace transforms by 120585119895

minus(119909 119905) and

120585119895

+(119909 119905) The solution to ((17a) (17b) (17c) and (17d)) can be

written as

120585119895

(119909 119905) = 120585119895

minus(119909 119905) 120594

(0119909119895)+ 120585119895

+(119909 119905) 120594

[119909119895 1) (22)

where

120594 (119909)Ω119895

=

1 119909 isin Ω119895sube (0 1)

0 otherwise(23)

Then at each point 119909119894

isin (0 1) by (18) and the argu-ment of ldquoparallel connectionrdquo (see Section 2) we have[119906119909]119909119894= sum119898

119895=1[119906119895

119909]119909119894= [119906119894

119909]119909119894= [120585119894

119909]119909119894 119894 = 1 119898 Hence

Mathematical Problems in Engineering 5

the in-domain control signals of system ((1a) (1b) (1c) (1d)and (1e)) can be computed by

V119894= [119906119909]119909119894= [120585119894

119909]119909119894

119894 = 1 119898 (24)

In the following steps we present the computation ofthe solution to system ((17a) (17b) (17c) and (17d)) 120585119895Issues related to the generation reference trajectory 119906119863(119909 119905)for system ((1a) (1b) (1c) (1d) and (1e)) will be addressed inSection 4

Note that 120585119895

minus(119909 119904) and

120585

119895

+(119909 119904) the general solutions to

((20a) (20b) and (20c)) and ((21a) (21b) and (21c)) aregiven by

120585

119895

minus(119909 119904) = 119862

11206011(119909 119904) + 119862

21206012(119909 119904)

120585

119895

+(119909 119904) = 119862

31206011(119909 119904) + 119862

41206012(119909 119904)

(25)

with

1206011(119909 119904) =

sinh (radic119904119909)radic119904

1206012(119909 119904) = cosh (radic119904119909)

(26)

We obtain by applying (20b) and (20c)

11986211206011(119909119895 119904) + 119862

21206012(119909119895 119904) =

119889

119895(119909119895 119904)

1198621minus 11989601198622= 0

(27)

which can be written as

(

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

)(

1198621

1198622

) = (

119889

119895(119909119895 119904)

0

) (28)

Let

119877119895

minus= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

) (29)

119889

119895(119909119895 119904) = minus det (119877119895

minus) 119895

minus(119909119895 119904) (30)

We obtain

(

1198621

1198622

) =

adj (119877119895minus)

det (119877119895minus)

(

119889

119895(119909119895 119904)

0

) = (

1198960119895

minus(119909119895 119904)

119895

minus(119909119895 119904)

) (31)

Therefore the solution to ((20a) (20b) and (20c)) can beexpressed as

120585

119895

minus(119909 119904) = (119896

01206011(119909) + 120601

2(119909))

119895

minus(119909119895 119904) (32)

Wemay proceed in the sameway to deal with ((21a) (21b)and (21c)) Indeed letting

119877119895

+

= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1206012(1 119904) + 119896

11206011(1 119904) 119904120601

1(1 119904) + 119896

11206012(1 119904)

)

(33)

119889

119895(119909119895 119904) = det (119877119895

+) 119895

+(119909119895 119904) (34)

we get from ((21a) (21b) and (21c))

(

1198623

1198624

) = (

(1199041206011(1 119904) + 119896

11206012(1 119904))

119895

+(119909119895 119904)

minus (1206012(1 119904) + 119896

11206011(1 119904))

119895

+(119909119895 119904)

) (35)

120585

119895

+(119909 119904)

= ((1199041206011(1 119904) + 119896

11206012(1 119904)) 120601

1(119909)

+ (1206012(1 119904) + 119896

11206011(1 119904)) 120601

2(119909))

119895

+(119909119895 119904)

(36)

Applying the results from [30 31] which are based onmodule theory to (30) and (34) we may choose

119895(119909119895 119904) as a

basic output such that

119895

+(119909119895 119904) = minusdet (119877119895

minus) 119895(119909119895 119904)

119895

minus(119909119895 119904) = det (119877119895

+) 119895(119909119895 119904)

(37)

It should be noticed that the concept of basic outputs (or flatoutputs) plays a central role in flat system theory becausethe system trajectory and the input can be directly computedfrom a basic output and its time derivatives [13 21]

Using the property of hyperbolic functions we obtainfrom (32) and (36) that

120585

119895

minus(119909 119904)

= (1198961

sinh (radic119904119909119895minus radic119904)

radic119904

minus cosh (radic119904119909119895minus radic119904))

sdot (1198960

sinh (radic119904119909)radic119904

+ cosh (radic119904119909)) 119895(119909119895 119904)

120585

119895

+(119909 119904) = (119896

1

sinh (radic119904119909 minus radic119904)radic119904

minus cosh (radic119904119909 minus radic119904))

sdot (1198960

sinh (radic119904119909119895)

radic119904

+ cosh (radic119904119909119895)) 119895(119909119895 119904)

(38)

Note that

120585

119895

(119909 119904) =120585

119895

minus(119909 119904) 120594

(0119909119895)+120585

119895

+(119909 119904) 120594

[119909119895 1)(39)

is a solution to ((19a) (19b) and (19c)) Using the fact

sinh119909 =infin

sum

119899=0

1199092119899+1

(2119899 + 1)

cosh 119909 =infin

sum

119899=0

1199092119899

(2119899)

(40)

6 Mathematical Problems in Engineering

we obtain the time-domain solution to ((17a) (17b) (17c) and(17d)) which is given by

120585119895

(119909 119905) =[

[

(11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594(0119909119895)

+ (11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594[119909119895 1)

]

]

(41)

Furthermore by a direct computation we get

[120585

119895

119909]

119909119895

= [(

11989601198961

radic119904

+ radic119904) sinh (radic119904)

+ (1198960+ 1198961) cosh (radic119904)]

119895(119909119895 119904)

= (11989601198961

infin

sum

119899=0

119904119899

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119904119899

(2119899)

+

infin

sum

119899=0

119904119899+1

(2119899 + 1)

) 119895(119909119895 119904)

(42)

It follows from (24) that in time domain the control is givenby

V119895(119905) = [120585

119895

119909]119909119895

= 11989601198961

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899)

+

infin

sum

119899=0

119910(119899+1)

119895(119909119895 119905)

(2119899 + 1)

(43)

Finally provided 119906119889119895(119909119895 119905) = 120585

119895

(119909119895 119905) for 119895 = 1 119898 the

reference trajectory 119906119863(119909 119905) can be determined from (16) and(41)

4 Motion Planning

For control purpose we have to choose appropriate referencetrajectories or equivalently the basic outputs Denote nowby 119906119863

(119909) the desired steady-state profile Without loss ofgenerality we consider a set of basic outputs of the form

119910119895(119905) = 119910 (119909

119895) 120593119895(119905) 119895 = 1 119898 (44)

where 120593119895(119905) is a smooth function evolving from 0 to 1Motion

planning amounts then to deriving 119910(119909119895) from 119906

119863

(119909) and todetermining appropriate functions 120593

119895(119905) for 119895 = 1 119898

To this aim and due to the equivalence of systems ((1a)(1b) (1c) (1d) and (1e)) and ((4a) (4b) and (4c)) we considerthe steady-state heat equation corresponding to system ((4a)(4b) and (4c))

119906119909119909(119909) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895

0 lt 119909 lt 1 119905 gt 0

(45a)

119906119909(0) minus 119896

0119906 (0) = 0

119906119909(1) + 119896

1119906 (1) = 0

(45b)

Based on the principle of superposition for linear systemsthe solution to the steady-state heat equation ((45a) and(45b)) can be expressed as

119906 (119909) = int

1

0

119898

sum

119895=1

119866 (119909 120577) 120575 (120577 minus 119909119895) 120572119895d120577

=

119898

sum

119895=1

119866(119909 119909119895) 120572119895

(46)

where119866(119909 120577) is the Greenrsquos function corresponding to ((45a)and (45b)) which is of the form

119866 (119909 120577) =

(1198961120577 minus 1198961minus 1) (119896

0119909 + 1)

1198960+ 1198961+ 11989601198961

0 le 119909 lt 120577

(1198961119909 minus 1198961minus 1) (119896

0120577 + 1)

1198960+ 1198961+ 11989601198961

120577 le 119909 le 1

(47)

Indeed it is easy to check that119866119909119909(119909 120577) = 120575(119909minus120577) and119866(119909 120577)

satisfies the boundary conditions 119866119909(0 120577) minus 119896

0119866(0 120577) = 0

and 119866119909(1 120577) + 119896

1119866(1 120577) = 0 the joint condition 119866(120577+ 120577) =

119866(120577minus

120577) and the jump condition [119866119909(119909 120577)]

120577= 1

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

3 Control Design and Implementation

In the framework of zero-dynamics inverse the in-domaincontrol is derived from the so-called forced zero-dynamics orzero-dynamics for short To work with the parallel connectedsystem ((9a) (9b) (9c) (9d) and (9e)) we first split thereference signal as

119906119863

(119909 119905) =

119898

sum

119895=1

120574119895(119909 119909119895) 119906119889

119895(119909119895 119905)

119906119889

119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin

(16)

where 120574119895(119909 119909119895) will be determined in Theorem 7 (see

Section 4) Denoting by 120576119895

(119905) = 119906119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905) the

regulation error corresponding to system ((9a) (9b) (9c)(9d) and (9e)) the zero-dynamics can be obtained byreplacing the input constraints in (9e) by the requirementthat the regulation errors vanish identically that is 120576119895(119905) = 0Thus we obtain for a fixed index 119895

120585119905(119909 119905) = 120585

119909119909(119909 119905)

119909 isin (0 119909119895) cup (119909

119895 1) 119905 gt 0

(17a)

120585 (119909 0) = 0 (17b)

120585119909(0 119905) minus 119896

0120585 (0 119905) = 0

120585119909(1 119905) + 119896

1120585 (1 119905) = 0

(17c)

120585 (119909119895 119905) = 119906

119889

119895(119909119895 119905) (17d)

where 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) for any 119879 lt infin It should benoticed that by construction we can always choose suitableinitial data of 119906119889(119909 0) so that the zero-dynamics will have ahomogenous initial conditionThis will significantly facilitatecontrol design Then we can get from (9e)

119908119895= [119906119895

119909]119909119895

= [120585119895

119909]119909119895

(18)

which shows that the in-domain control for system ((9a)(9b) (9c) (9d) and (9e)) can be derived from the solution ofthe zero-dynamics Hence ((17a) (17b) (17c) and (17d)) and(18) form a dynamic control scheme This is indeed the basicidea of zero-dynamic inverse design The convergence ofregulation errorswith ZDI-based control is given in followingtheorem

Theorem 4 The regulation error corresponding to system((9a) (9b) (9c) (9d) and (9e)) 120576119895(119905) = 119906

119895

(119909119895 119905) minus 119906

119889

119895(119909119895 119905)

tends to 0 as 119905 tends to infin for any 119906119889119895(119909119895 119905) isin 119867

1

(0 119879) and119879 lt infin

The proof of Theorem 4 can follow the developmentpresented in Section III of [20] for the case of a heat equationwith one in-domain actuator and hence it is omitted Notethat a key fact used in the proof of this theorem is that thesystem given in ((9a) (9b) (9c) (9d) and (9e)) without

interior control is exponentially stable for any initial data120601119895(119909) isin 119871

2

(0 1) if 1198960gt 0 and 119896

1gt 0

To implement the dynamic control scheme composedof ((17a) (17b) (17c) and (17d)) and (18) we resort to thetechnique of flat systems [11 13 28 29] In particular we applya standard procedure of Laplace transform-based method tofind the solution to ((17a) (17b) (17c) and (17d))Henceforthwe denote by

119891(119909 119904) the Laplace transform of a function119891(119909 119905) with respect to the time variable Since ((17a) (17b)(17c) and (17d)) has a homogeneous initial condition thenfor fixed 119909

119895isin (0 1) the transformed equations of ((17a)

(17b) (17c) and (17d)) in the Laplace domain read as

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909 isin (0 119909119895) cup (119909

119895 1) 119904 isin C

(19a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0

120585119909(1 119904) + 119896

1

120585 (1 119904) = 0

(19b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (19c)

We divide ((19a) (19b) and (19c)) into two subsystemsthat is for fixed 119909

119895isin (0 1) considering

119904120585 (119909 119904) =

120585119909119909(119909 119904)

0 lt 119909 lt 119909119895 119904 isin C

(20a)

120585119909(0 119904) minus 119896

0

120585 (0 119904) = 0 (20b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (20c)

119904120585 (119909 119904) =

120585119909119909(119909 119904)

119909119895lt 119909 lt 1 119904 isin C

(21a)

120585119909(1 119904) + 119896

1

120585 (119909 119904) = 0 (21b)

120585 (119909119895 119904) =

119889

119895(119909119895 119904) (21c)

Let 120585119895

minus(119909 119904) and 120585

119895

+(119909 119904) be the general solutions to ((20a)

(20b) and (20c)) and ((21a) (21b) and (21c)) respectivelyand denote their inverse Laplace transforms by 120585119895

minus(119909 119905) and

120585119895

+(119909 119905) The solution to ((17a) (17b) (17c) and (17d)) can be

written as

120585119895

(119909 119905) = 120585119895

minus(119909 119905) 120594

(0119909119895)+ 120585119895

+(119909 119905) 120594

[119909119895 1) (22)

where

120594 (119909)Ω119895

=

1 119909 isin Ω119895sube (0 1)

0 otherwise(23)

Then at each point 119909119894

isin (0 1) by (18) and the argu-ment of ldquoparallel connectionrdquo (see Section 2) we have[119906119909]119909119894= sum119898

119895=1[119906119895

119909]119909119894= [119906119894

119909]119909119894= [120585119894

119909]119909119894 119894 = 1 119898 Hence

Mathematical Problems in Engineering 5

the in-domain control signals of system ((1a) (1b) (1c) (1d)and (1e)) can be computed by

V119894= [119906119909]119909119894= [120585119894

119909]119909119894

119894 = 1 119898 (24)

In the following steps we present the computation ofthe solution to system ((17a) (17b) (17c) and (17d)) 120585119895Issues related to the generation reference trajectory 119906119863(119909 119905)for system ((1a) (1b) (1c) (1d) and (1e)) will be addressed inSection 4

Note that 120585119895

minus(119909 119904) and

120585

119895

+(119909 119904) the general solutions to

((20a) (20b) and (20c)) and ((21a) (21b) and (21c)) aregiven by

120585

119895

minus(119909 119904) = 119862

11206011(119909 119904) + 119862

21206012(119909 119904)

120585

119895

+(119909 119904) = 119862

31206011(119909 119904) + 119862

41206012(119909 119904)

(25)

with

1206011(119909 119904) =

sinh (radic119904119909)radic119904

1206012(119909 119904) = cosh (radic119904119909)

(26)

We obtain by applying (20b) and (20c)

11986211206011(119909119895 119904) + 119862

21206012(119909119895 119904) =

119889

119895(119909119895 119904)

1198621minus 11989601198622= 0

(27)

which can be written as

(

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

)(

1198621

1198622

) = (

119889

119895(119909119895 119904)

0

) (28)

Let

119877119895

minus= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

) (29)

119889

119895(119909119895 119904) = minus det (119877119895

minus) 119895

minus(119909119895 119904) (30)

We obtain

(

1198621

1198622

) =

adj (119877119895minus)

det (119877119895minus)

(

119889

119895(119909119895 119904)

0

) = (

1198960119895

minus(119909119895 119904)

119895

minus(119909119895 119904)

) (31)

Therefore the solution to ((20a) (20b) and (20c)) can beexpressed as

120585

119895

minus(119909 119904) = (119896

01206011(119909) + 120601

2(119909))

119895

minus(119909119895 119904) (32)

Wemay proceed in the sameway to deal with ((21a) (21b)and (21c)) Indeed letting

119877119895

+

= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1206012(1 119904) + 119896

11206011(1 119904) 119904120601

1(1 119904) + 119896

11206012(1 119904)

)

(33)

119889

119895(119909119895 119904) = det (119877119895

+) 119895

+(119909119895 119904) (34)

we get from ((21a) (21b) and (21c))

(

1198623

1198624

) = (

(1199041206011(1 119904) + 119896

11206012(1 119904))

119895

+(119909119895 119904)

minus (1206012(1 119904) + 119896

11206011(1 119904))

119895

+(119909119895 119904)

) (35)

120585

119895

+(119909 119904)

= ((1199041206011(1 119904) + 119896

11206012(1 119904)) 120601

1(119909)

+ (1206012(1 119904) + 119896

11206011(1 119904)) 120601

2(119909))

119895

+(119909119895 119904)

(36)

Applying the results from [30 31] which are based onmodule theory to (30) and (34) we may choose

119895(119909119895 119904) as a

basic output such that

119895

+(119909119895 119904) = minusdet (119877119895

minus) 119895(119909119895 119904)

119895

minus(119909119895 119904) = det (119877119895

+) 119895(119909119895 119904)

(37)

It should be noticed that the concept of basic outputs (or flatoutputs) plays a central role in flat system theory becausethe system trajectory and the input can be directly computedfrom a basic output and its time derivatives [13 21]

Using the property of hyperbolic functions we obtainfrom (32) and (36) that

120585

119895

minus(119909 119904)

= (1198961

sinh (radic119904119909119895minus radic119904)

radic119904

minus cosh (radic119904119909119895minus radic119904))

sdot (1198960

sinh (radic119904119909)radic119904

+ cosh (radic119904119909)) 119895(119909119895 119904)

120585

119895

+(119909 119904) = (119896

1

sinh (radic119904119909 minus radic119904)radic119904

minus cosh (radic119904119909 minus radic119904))

sdot (1198960

sinh (radic119904119909119895)

radic119904

+ cosh (radic119904119909119895)) 119895(119909119895 119904)

(38)

Note that

120585

119895

(119909 119904) =120585

119895

minus(119909 119904) 120594

(0119909119895)+120585

119895

+(119909 119904) 120594

[119909119895 1)(39)

is a solution to ((19a) (19b) and (19c)) Using the fact

sinh119909 =infin

sum

119899=0

1199092119899+1

(2119899 + 1)

cosh 119909 =infin

sum

119899=0

1199092119899

(2119899)

(40)

6 Mathematical Problems in Engineering

we obtain the time-domain solution to ((17a) (17b) (17c) and(17d)) which is given by

120585119895

(119909 119905) =[

[

(11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594(0119909119895)

+ (11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594[119909119895 1)

]

]

(41)

Furthermore by a direct computation we get

[120585

119895

119909]

119909119895

= [(

11989601198961

radic119904

+ radic119904) sinh (radic119904)

+ (1198960+ 1198961) cosh (radic119904)]

119895(119909119895 119904)

= (11989601198961

infin

sum

119899=0

119904119899

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119904119899

(2119899)

+

infin

sum

119899=0

119904119899+1

(2119899 + 1)

) 119895(119909119895 119904)

(42)

It follows from (24) that in time domain the control is givenby

V119895(119905) = [120585

119895

119909]119909119895

= 11989601198961

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899)

+

infin

sum

119899=0

119910(119899+1)

119895(119909119895 119905)

(2119899 + 1)

(43)

Finally provided 119906119889119895(119909119895 119905) = 120585

119895

(119909119895 119905) for 119895 = 1 119898 the

reference trajectory 119906119863(119909 119905) can be determined from (16) and(41)

4 Motion Planning

For control purpose we have to choose appropriate referencetrajectories or equivalently the basic outputs Denote nowby 119906119863

(119909) the desired steady-state profile Without loss ofgenerality we consider a set of basic outputs of the form

119910119895(119905) = 119910 (119909

119895) 120593119895(119905) 119895 = 1 119898 (44)

where 120593119895(119905) is a smooth function evolving from 0 to 1Motion

planning amounts then to deriving 119910(119909119895) from 119906

119863

(119909) and todetermining appropriate functions 120593

119895(119905) for 119895 = 1 119898

To this aim and due to the equivalence of systems ((1a)(1b) (1c) (1d) and (1e)) and ((4a) (4b) and (4c)) we considerthe steady-state heat equation corresponding to system ((4a)(4b) and (4c))

119906119909119909(119909) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895

0 lt 119909 lt 1 119905 gt 0

(45a)

119906119909(0) minus 119896

0119906 (0) = 0

119906119909(1) + 119896

1119906 (1) = 0

(45b)

Based on the principle of superposition for linear systemsthe solution to the steady-state heat equation ((45a) and(45b)) can be expressed as

119906 (119909) = int

1

0

119898

sum

119895=1

119866 (119909 120577) 120575 (120577 minus 119909119895) 120572119895d120577

=

119898

sum

119895=1

119866(119909 119909119895) 120572119895

(46)

where119866(119909 120577) is the Greenrsquos function corresponding to ((45a)and (45b)) which is of the form

119866 (119909 120577) =

(1198961120577 minus 1198961minus 1) (119896

0119909 + 1)

1198960+ 1198961+ 11989601198961

0 le 119909 lt 120577

(1198961119909 minus 1198961minus 1) (119896

0120577 + 1)

1198960+ 1198961+ 11989601198961

120577 le 119909 le 1

(47)

Indeed it is easy to check that119866119909119909(119909 120577) = 120575(119909minus120577) and119866(119909 120577)

satisfies the boundary conditions 119866119909(0 120577) minus 119896

0119866(0 120577) = 0

and 119866119909(1 120577) + 119896

1119866(1 120577) = 0 the joint condition 119866(120577+ 120577) =

119866(120577minus

120577) and the jump condition [119866119909(119909 120577)]

120577= 1

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

the in-domain control signals of system ((1a) (1b) (1c) (1d)and (1e)) can be computed by

V119894= [119906119909]119909119894= [120585119894

119909]119909119894

119894 = 1 119898 (24)

In the following steps we present the computation ofthe solution to system ((17a) (17b) (17c) and (17d)) 120585119895Issues related to the generation reference trajectory 119906119863(119909 119905)for system ((1a) (1b) (1c) (1d) and (1e)) will be addressed inSection 4

Note that 120585119895

minus(119909 119904) and

120585

119895

+(119909 119904) the general solutions to

((20a) (20b) and (20c)) and ((21a) (21b) and (21c)) aregiven by

120585

119895

minus(119909 119904) = 119862

11206011(119909 119904) + 119862

21206012(119909 119904)

120585

119895

+(119909 119904) = 119862

31206011(119909 119904) + 119862

41206012(119909 119904)

(25)

with

1206011(119909 119904) =

sinh (radic119904119909)radic119904

1206012(119909 119904) = cosh (radic119904119909)

(26)

We obtain by applying (20b) and (20c)

11986211206011(119909119895 119904) + 119862

21206012(119909119895 119904) =

119889

119895(119909119895 119904)

1198621minus 11989601198622= 0

(27)

which can be written as

(

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

)(

1198621

1198622

) = (

119889

119895(119909119895 119904)

0

) (28)

Let

119877119895

minus= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1 minus1198960

) (29)

119889

119895(119909119895 119904) = minus det (119877119895

minus) 119895

minus(119909119895 119904) (30)

We obtain

(

1198621

1198622

) =

adj (119877119895minus)

det (119877119895minus)

(

119889

119895(119909119895 119904)

0

) = (

1198960119895

minus(119909119895 119904)

119895

minus(119909119895 119904)

) (31)

Therefore the solution to ((20a) (20b) and (20c)) can beexpressed as

120585

119895

minus(119909 119904) = (119896

01206011(119909) + 120601

2(119909))

119895

minus(119909119895 119904) (32)

Wemay proceed in the sameway to deal with ((21a) (21b)and (21c)) Indeed letting

119877119895

+

= (

1206011(119909119895 119904) 120601

2(119909119895 119904)

1206012(1 119904) + 119896

11206011(1 119904) 119904120601

1(1 119904) + 119896

11206012(1 119904)

)

(33)

119889

119895(119909119895 119904) = det (119877119895

+) 119895

+(119909119895 119904) (34)

we get from ((21a) (21b) and (21c))

(

1198623

1198624

) = (

(1199041206011(1 119904) + 119896

11206012(1 119904))

119895

+(119909119895 119904)

minus (1206012(1 119904) + 119896

11206011(1 119904))

119895

+(119909119895 119904)

) (35)

120585

119895

+(119909 119904)

= ((1199041206011(1 119904) + 119896

11206012(1 119904)) 120601

1(119909)

+ (1206012(1 119904) + 119896

11206011(1 119904)) 120601

2(119909))

119895

+(119909119895 119904)

(36)

Applying the results from [30 31] which are based onmodule theory to (30) and (34) we may choose

119895(119909119895 119904) as a

basic output such that

119895

+(119909119895 119904) = minusdet (119877119895

minus) 119895(119909119895 119904)

119895

minus(119909119895 119904) = det (119877119895

+) 119895(119909119895 119904)

(37)

It should be noticed that the concept of basic outputs (or flatoutputs) plays a central role in flat system theory becausethe system trajectory and the input can be directly computedfrom a basic output and its time derivatives [13 21]

Using the property of hyperbolic functions we obtainfrom (32) and (36) that

120585

119895

minus(119909 119904)

= (1198961

sinh (radic119904119909119895minus radic119904)

radic119904

minus cosh (radic119904119909119895minus radic119904))

sdot (1198960

sinh (radic119904119909)radic119904

+ cosh (radic119904119909)) 119895(119909119895 119904)

120585

119895

+(119909 119904) = (119896

1

sinh (radic119904119909 minus radic119904)radic119904

minus cosh (radic119904119909 minus radic119904))

sdot (1198960

sinh (radic119904119909119895)

radic119904

+ cosh (radic119904119909119895)) 119895(119909119895 119904)

(38)

Note that

120585

119895

(119909 119904) =120585

119895

minus(119909 119904) 120594

(0119909119895)+120585

119895

+(119909 119904) 120594

[119909119895 1)(39)

is a solution to ((19a) (19b) and (19c)) Using the fact

sinh119909 =infin

sum

119899=0

1199092119899+1

(2119899 + 1)

cosh 119909 =infin

sum

119899=0

1199092119899

(2119899)

(40)

6 Mathematical Problems in Engineering

we obtain the time-domain solution to ((17a) (17b) (17c) and(17d)) which is given by

120585119895

(119909 119905) =[

[

(11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594(0119909119895)

+ (11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594[119909119895 1)

]

]

(41)

Furthermore by a direct computation we get

[120585

119895

119909]

119909119895

= [(

11989601198961

radic119904

+ radic119904) sinh (radic119904)

+ (1198960+ 1198961) cosh (radic119904)]

119895(119909119895 119904)

= (11989601198961

infin

sum

119899=0

119904119899

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119904119899

(2119899)

+

infin

sum

119899=0

119904119899+1

(2119899 + 1)

) 119895(119909119895 119904)

(42)

It follows from (24) that in time domain the control is givenby

V119895(119905) = [120585

119895

119909]119909119895

= 11989601198961

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899)

+

infin

sum

119899=0

119910(119899+1)

119895(119909119895 119905)

(2119899 + 1)

(43)

Finally provided 119906119889119895(119909119895 119905) = 120585

119895

(119909119895 119905) for 119895 = 1 119898 the

reference trajectory 119906119863(119909 119905) can be determined from (16) and(41)

4 Motion Planning

For control purpose we have to choose appropriate referencetrajectories or equivalently the basic outputs Denote nowby 119906119863

(119909) the desired steady-state profile Without loss ofgenerality we consider a set of basic outputs of the form

119910119895(119905) = 119910 (119909

119895) 120593119895(119905) 119895 = 1 119898 (44)

where 120593119895(119905) is a smooth function evolving from 0 to 1Motion

planning amounts then to deriving 119910(119909119895) from 119906

119863

(119909) and todetermining appropriate functions 120593

119895(119905) for 119895 = 1 119898

To this aim and due to the equivalence of systems ((1a)(1b) (1c) (1d) and (1e)) and ((4a) (4b) and (4c)) we considerthe steady-state heat equation corresponding to system ((4a)(4b) and (4c))

119906119909119909(119909) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895

0 lt 119909 lt 1 119905 gt 0

(45a)

119906119909(0) minus 119896

0119906 (0) = 0

119906119909(1) + 119896

1119906 (1) = 0

(45b)

Based on the principle of superposition for linear systemsthe solution to the steady-state heat equation ((45a) and(45b)) can be expressed as

119906 (119909) = int

1

0

119898

sum

119895=1

119866 (119909 120577) 120575 (120577 minus 119909119895) 120572119895d120577

=

119898

sum

119895=1

119866(119909 119909119895) 120572119895

(46)

where119866(119909 120577) is the Greenrsquos function corresponding to ((45a)and (45b)) which is of the form

119866 (119909 120577) =

(1198961120577 minus 1198961minus 1) (119896

0119909 + 1)

1198960+ 1198961+ 11989601198961

0 le 119909 lt 120577

(1198961119909 minus 1198961minus 1) (119896

0120577 + 1)

1198960+ 1198961+ 11989601198961

120577 le 119909 le 1

(47)

Indeed it is easy to check that119866119909119909(119909 120577) = 120575(119909minus120577) and119866(119909 120577)

satisfies the boundary conditions 119866119909(0 120577) minus 119896

0119866(0 120577) = 0

and 119866119909(1 120577) + 119896

1119866(1 120577) = 0 the joint condition 119866(120577+ 120577) =

119866(120577minus

120577) and the jump condition [119866119909(119909 120577)]

120577= 1

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

we obtain the time-domain solution to ((17a) (17b) (17c) and(17d)) which is given by

120585119895

(119909 119905) =[

[

(11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

(119909119895minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

(119909119895minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594(0119909119895)

+ (11989601198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)+1

(2119896 + 1) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus 1198960

infin

sum

119899=0

119899

sum

119896=0

1199092119896+1

119895(119909 minus 1)

2(119899minus119896)

(2119896 + 1) [2 (119899 minus 119896)]

119910(119899)

119895

+ 1198961

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)+1

(2119896) [2 (119899 minus 119896) + 1]

119910(119899)

119895

minus

infin

sum

119899=0

119899

sum

119896=0

1199092119896

119895(119909 minus 1)

2(119899minus119896)

(2119896) [2 (119899 minus 119896)]

119910(119899)

119895)120594[119909119895 1)

]

]

(41)

Furthermore by a direct computation we get

[120585

119895

119909]

119909119895

= [(

11989601198961

radic119904

+ radic119904) sinh (radic119904)

+ (1198960+ 1198961) cosh (radic119904)]

119895(119909119895 119904)

= (11989601198961

infin

sum

119899=0

119904119899

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119904119899

(2119899)

+

infin

sum

119899=0

119904119899+1

(2119899 + 1)

) 119895(119909119895 119904)

(42)

It follows from (24) that in time domain the control is givenby

V119895(119905) = [120585

119895

119909]119909119895

= 11989601198961

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899 + 1)

+ (1198960+ 1198961)

infin

sum

119899=0

119910(119899)

119895(119909119895 119905)

(2119899)

+

infin

sum

119899=0

119910(119899+1)

119895(119909119895 119905)

(2119899 + 1)

(43)

Finally provided 119906119889119895(119909119895 119905) = 120585

119895

(119909119895 119905) for 119895 = 1 119898 the

reference trajectory 119906119863(119909 119905) can be determined from (16) and(41)

4 Motion Planning

For control purpose we have to choose appropriate referencetrajectories or equivalently the basic outputs Denote nowby 119906119863

(119909) the desired steady-state profile Without loss ofgenerality we consider a set of basic outputs of the form

119910119895(119905) = 119910 (119909

119895) 120593119895(119905) 119895 = 1 119898 (44)

where 120593119895(119905) is a smooth function evolving from 0 to 1Motion

planning amounts then to deriving 119910(119909119895) from 119906

119863

(119909) and todetermining appropriate functions 120593

119895(119905) for 119895 = 1 119898

To this aim and due to the equivalence of systems ((1a)(1b) (1c) (1d) and (1e)) and ((4a) (4b) and (4c)) we considerthe steady-state heat equation corresponding to system ((4a)(4b) and (4c))

119906119909119909(119909) =

119898

sum

119895=1

120575 (119909 minus 119909119895) 120572119895

0 lt 119909 lt 1 119905 gt 0

(45a)

119906119909(0) minus 119896

0119906 (0) = 0

119906119909(1) + 119896

1119906 (1) = 0

(45b)

Based on the principle of superposition for linear systemsthe solution to the steady-state heat equation ((45a) and(45b)) can be expressed as

119906 (119909) = int

1

0

119898

sum

119895=1

119866 (119909 120577) 120575 (120577 minus 119909119895) 120572119895d120577

=

119898

sum

119895=1

119866(119909 119909119895) 120572119895

(46)

where119866(119909 120577) is the Greenrsquos function corresponding to ((45a)and (45b)) which is of the form

119866 (119909 120577) =

(1198961120577 minus 1198961minus 1) (119896

0119909 + 1)

1198960+ 1198961+ 11989601198961

0 le 119909 lt 120577

(1198961119909 minus 1198961minus 1) (119896

0120577 + 1)

1198960+ 1198961+ 11989601198961

120577 le 119909 le 1

(47)

Indeed it is easy to check that119866119909119909(119909 120577) = 120575(119909minus120577) and119866(119909 120577)

satisfies the boundary conditions 119866119909(0 120577) minus 119896

0119866(0 120577) = 0

and 119866119909(1 120577) + 119896

1119866(1 120577) = 0 the joint condition 119866(120577+ 120577) =

119866(120577minus

120577) and the jump condition [119866119909(119909 120577)]

120577= 1

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Taking119898distinguished points along the solution to ((45a)and (45b)) 119906(119909

1) 119906(119909

119898) we get

(

119906(1199091)

119906 (119909119898)

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)(

1205721

120572119898

)

(48)

Note that in (48) the matrix formed by the Greenrsquos functiondefined an input-output map in steady-state which is alsocalled the influence matrix

Lemma 5 The influence matrix chosen as in (48) is invertibleThus

(

1205721

120572119898

)

=(

119866(1199091 1199091) sdot sdot sdot 119866 (119909

1 119909119898)

d

119866 (119909119898 1199091) sdot sdot sdot 119866 (119909

119898 119909119898)

)

minus1

(

119906(1199091)

119906 (119909119898)

)

(49)

Proof For 119898 = 1 since 1198960gt 0 119896

1gt 0 and 119909

1isin (0 1)

it follows that 119866(1199091 1199091) = (119896

11199091minus 1198961minus 1)(119896

01199091+ 1)(119896

0+

1198961+ 11989601198961) lt 0 Hence it is invertible We prove the claim for

119898 gt 1 by contradiction Suppose that the influence matrix isnot invertible then it is of rank 119898 minus 1 or less Without lossof generality we may assume that for some 119909

119899gt 119909119894with 119894 =

1 119899 minus 1 there exist 119899 minus 1 constants 1198971 1198972 119897119899minus1

such that

119866(119909119895 119909119899) =

119899minus1

sum

119894=1

119897119894119866 (1199091 119909119894) 119895 = 1 119899 (50)

where 1 lt 119899 le 119898 and sum119899minus1119894=1

1198972

119894gt 0 Let

119866 (119909) = 119866 (119909 119909119899)

119865 (119909) =

119899minus1

sum

119894=1

119897119894119866 (119909 119909

119894)

(51)

Equation (50) shows that 119865(119909) = 119866(119909) at every boundarypoint of [119909

1 1199092] [1199092 1199093] [119909

119899minus1 119909119899] Note that 119865(119909) is

a linear function in [1199091 1199092] [1199092 1199093] [119909

119899minus1 119909119899] and that

119866(119909) = (1198961119909119899minus1198961minus1)(119896

0119909+1)(119896

0+1198961+11989601198961) in [119909

1 119909119899] that

is 119866(119909) is a linear function in [1199091 119909119899] Hence 119865(119909) equiv 119866(119909) in

[1199091 119909119899]

By 119865(1199091) = 119866(119909

1) we get

1198961119909119899minus 1198961minus 1 =

119899minus1

sum

119894=1

119897119894(1198961119909119894minus 1198961minus 1) (52)

By 119865(119909119899) = 119866(119909

119899) we get

1198960119909119899+ 1 =

119899minus1

sum

119894=1

119897119894(1198960119909119894+ 1) (53)

Therefore119899minus1

sum

119894=1

119897119894= 1 (54)

By 119865119909(119909+

1) = 119866119909(119909+

1) and 119865

119909(119909minus

119899) = 119866119909(119909minus

119899) we get

1198960(1198961119909119899minus 1198961minus 1) = 119896

01198961

119899minus1

sum

119894=1

119897119894119909119894minus 1198960(1198961+ 1)

119899minus1

sum

119894=2

119897119894

+ 11989711198961= 11989601198961

119899minus1

sum

119894=1

119897119894119909119894+ 1198961

119899minus1

sum

119894=1

119897119894

(55)

It follows thatsum119899minus1119894=2

119897119894= 0 which yields considering (54) 119897

1=

1 By 119865119909(119909+

2) = 119866119909(1199092) = 119865119909(119909minus

2) we deduce

11989711198961(11989601199091+ 1) + 119897

21198961(11989601199092+ 1)

+ 1198960

119899minus1

sum

119894=3

119897119894(1198961119909119894minus 1198961minus 1)

= 11989711198961(11989601199091+ 1) + 119896

0

119899minus1

sum

119894=2

119897119894(1198961119909119894minus 1198961minus 1)

(56)

which gives 1198972

= 0 Similarly by 119865119909(119909+

119895) = 119866

119909(119909119895) =

119865119909(119909minus

119895) (119895 = 3 4 119899 minus 2) we obtain 119897

3= 1198974= sdot sdot sdot = 119897

119899minus2= 0

Hence 119897119899minus1

= 0 Then we deduce from (55) that

1198960(1198961119909119899minus 1198961minus 1) = 119896

1(11989601199091+ 1) (57)

It follows from (53) that 11989601199091+ 1 = 119896

0119909119899+ 1 We conclude

then by (57) that 1198960+ 1198961+ 11989601198961= 0 which is a contradiction

to 1198960gt 0 and 119896

1gt 0

In steady-state we can obtain from (43) that

V119895= (11989601198961+ 1198960+ 1198961) 119910 (119909

119895) = minus120572

119895 (58)

Finally 119910(119909119895) can be computed by (49) and (58) for a given

119906119863

(119909)It is worth noting that (49) provides a simple and straight-

forwardway to compute the static control from the prescribedsteady-state profile Indeed a direct computation can showthat applying (49) will result in the same static controlobtained in [20] where a serially connectedmodel is used

To ensure the convergence of (41) and (43) we choose thefollowing smooth function as 120593

119895(119905)

120593119895(119905) =

0 if 119905 le 0

int

119905

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591

int

119879

0

exp (minus1 (120591 (1 minus 120591)))120576 d120591 if 119905 isin (0 119879)

1 if 119905 ge 119879

(59)

which is known asGevrey function of order 120590 = 1+1120576 120576 gt 0

(see eg [13])

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

025

05

075

1uD(x)

01 02 03 04 05 06 07 08 09 10x

(a)

minus10minus75minus5

minus250

255

7510

01 02 03 04 05 06 07 08 09 10x

1205721minus12

(b)

Figure 1 Controller characteristics (a) desired profile (b) static control signals

Lemma 6 If the basic outputs 120593119895(119905) 119895 = 1 119898 are chosen

as Gevrey functions of order 1 lt 120590 lt 2 then the infinite series(41) and (43) are convergent

The claim of Lemma 6 can be proved by following astandard procedure using the bounds of Gevrey functions(see eg [13])10038161003816100381610038161003816120593(119896+1)

(119905)

10038161003816100381610038161003816le 119872

(119896)120590

119870119896

exist119870119872 gt 0 forall119896 isin Zge0 forall119905 isin [119905

0 119879]

(60)

Therefore the details of proof are omitted

Theorem 7 Assume 1198960gt 0 and 119896

1gt 0 Let the basic outputs

120593119895(119905) 119895 = 1 119898 be chosen as (59) with an order 1 lt 120590 lt 2

Let the reference trajectory of system ((1a) (1b) (1c) (1d) and(1e)) be given by (16) with

120574119895(119909 119909119895) = minus

(11989601198961+ 1198960+ 1198961) 119866 (119909 119909

119895)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

119895 = 1 119898

(61)

where 119866(119909 120577) is the Greenrsquos function defined in (47) Then theregulation error of system ((1a) (1b) (1c) (1d) and (1e)) withthe control given in (43) tends to zero that is 119890

119894(119905) = 119906(119909

119894 119905) minus

119906119863

119894(119909119894 119905) rarr 0 as 119905 rarr infin for 119894 = 1 2 119898

Proof By a direct computation we have

1003816100381610038161003816119890119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816(119909119894 119905) minus 119906

119863

119894(119909119894 119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 119906119889

119895(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894 119905)

+

119898

sum

119895=1

(11989601198961+ 1198960+ 1198961) 119866 (119909

119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816119906 (119909119894 119905) minus 119906 (119909

119894)1003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119906 (119909119894) + (119896

01198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11989601198961+ 1198960+ 1198961)

sdot

119898

sum

119895=1

119866(119909119894 119909119895) 120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

minus (11989601198961+ 1198960

+ 1198961)

119898

sum

119895=1

119866(119909119894 119909119895) 119910 (119909

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(62)

By (58) and (46) it follows 119906(119909119894) = minus(119896

01198961+ 1198960+

1198961) sum119898

119895=1119866(119909119894 119909119895)119910(119909119895) Based on (41) (44) and the property

of 120593119895(119905) we have

120585119895

(119909119895 119905)

(1198960119909119895+ 1) (119896

0(119909119895minus 1) minus 1)

997888rarr 119910 (119909119895)

as 119905 rarr infin

(63)

Note that 119906(119909119894 119905) rarr 119906(119909

119894) as 119905 rarr infinTherefore |119890

119894(119905)| rarr 0

as 119905 rarr infin

Remark 8 For any 119909 isin (0 1) replacing 119909119894by 119909 in the proof

ofTheorem 7 we can get |119906(119909 119905) minus 119906119863(119909 119905)| rarr 0 as 119905 rarr infinwhich shows that the solution119906(119909 119905) of system ((1a) (1b) (1c)(1d) and (1e)) converges to the reference trajectory 119906119863(119909 119905) atevery point 119909 isin (0 1)

5 Simulation StudyIn the simulation we implement system ((4a) (4b) and(4c)) with 12 actuators evenly distributed in the domainat the spot points 113 213 1213 The numericalimplementation is based on a PDE solver pdepe in MatlabPDE Toolbox In numerical simulation 200 points in spaceand 100 points in time are used for the region [0 1] times [0 05]The basic outputs 120593

119895(119905) used in the simulation are Gevrey

functions of the same order In order tomeet the convergencecondition given in Lemma 6 the parameter of the Gevreyfunction is set to 120576 = 11 The feedback boundary controlgains are chosen as 119896

0= 10 and 119896

1= 1 A perturbation of the

form 119906(119909 0) = cos(120587119909) is applied at 119905 = 0 in the simulationThe desired steady-state temperature distribution is a

piecewise linear curve depicted in Figure 1(a) which is

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0025050751 00102030405

minus1

minus05

0

05

1

u(xt)

Position x Time t

(a)

0025050751 00102030405

minus1

minus05

0

05

1

Position x Time t

e(xt)

(b)

0025050751 00102030405

minus05

0

05

1

minus1

u(xt)

Position x Time t

(c)

0025050751 00102030405

minus1

minus05

0

05

1

e(xt)

Position x Time t

(d)

Figure 2 Evolution of temperature distribution (a) response with static control (b) regulation error with static control (c) response withdynamic control (d) regulation error with dynamic control

a solution to ((45a) and (45b)) The corresponding staticcontrols 120572

1 120572

12 are shown in Figure 1(b) Note that

the dynamic control signals 120572119894(119905) are smooth functions

connecting 0 to 120572119894for 119894 = 1 12 The evolution of

temperature distribution with static and dynamic controlas well as the corresponding regulation errors with respectto the static profile defined as 119890(119909 119905) = 119906(119909 119905) minus 119906

119863

(119909) isdepicted in Figure 2 The simulation results show that thesystem performs well with the developed control scheme Itcan also be seen that the dynamic control provides a fasterresponse time compared to the static one

6 Conclusion

This paper presented a solution to the problem of set-pointcontrol of temperature distributionwith in-domain actuationdescribed by an inhomogeneous parabolic PDE To applythe principle of superposition the system is presented ina parallel connection form The dynamic control problemintroduced by the ZDI design is solved by using the techniqueof flat systems motion planning As the control with multiplein-domain actuators results in a MIMO problem a Greenrsquosfunction-based reference trajectory decomposition is intro-duced which considerably simplifies the control design andimplementation Convergence and solvability analysis con-firms the validity of the control algorithm and the simulationresults demonstrate the viability of the proposed approachFinally as both ZDI design and flatness-based control canbe carried out in a systematic manner we can expect thatthe approach developed in this work may be applicable to abroader class of distributed parameter systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by a NSERC (Natural Scienceand Engineering Research Council of Canada) DiscoveryGrant The first author is also supported in part by theFundamental Research Funds for the Central Universities(682014CX002EM) The authors would like to thank theanonymous reviewer for the constructive comments that helpimprove the paper

References

[1] A Bensoussan G Da Prato M Delfour and S K MitterRepresentation and Control of Infinite-Dimensional SystemsBirkhauser Boston Mass USA 2nd edition 2007

[2] J M Coron Control and Nonlinearity American MathematicalSociety Providence RI USA 2007

[3] R F Curtain and H J Zwart An Introduction to Infinite-Dimensional Linear System Theory vol 12 of Text in AppliedMathematic Springer New York NY USA 1995

[4] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Birkhauser Basel Switzerland 2009

[5] M Krstic and A Smyshyaev Boundary Control of PDEs ACourse on Backstepping Designs SIAM New York NY USA2008

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[6] A Smyshyaev and M Krstic Adaptive Control of ParabolicPDEs Princeton University PressPrinceton University PressPrinceton NJ USA 2010

[7] D Tsubakino M Krstic and S Hara ldquoBackstepping control forparabolic PDEs with in-domain actuationrdquo in Proceedings of theAmerican Control Conference (ACC rsquo12) pp 2226ndash2231 IEEEMontreal Canada June 2012

[8] B Laroche PMartin and P Rouchon ldquoMotion planning for theheat equationrdquo International Journal of Robust and NonlinearControl vol 10 no 8 pp 629ndash643 2000

[9] A F Lynch and J Rudolph ldquoFlatness-based boundary control ofa class of quasilinear parabolic distributed parameter systemsrdquoInternational Journal of Control vol 75 no 15 pp 1219ndash12302002

[10] P Martin L Rosier and P Rouchon ldquoNull controllability of theheat equation using flatnessrdquo Automatica vol 50 no 12 pp3067ndash3076 2014

[11] T Meurer Control of Higher-Dimensional PDEs Flatness andBackstepping Designs Springer Berlin Germany 2013

[12] N Petit P Rouchon J-M Boueilh F Guerin and P PinvidicldquoControl of an industrial polymerization reactor using flatnessrdquoJournal of Process Control vol 12 no 5 pp 659ndash665 2002

[13] J Rudolph Flatness Based Control of Distributed ParameterSystems Shaker Aachen Germany 2003

[14] B Schorkhuber T Meurer and A Jungel ldquoFlatness ofsemilinear parabolic PDEsmdasha generalized Cauchy-Kowalevskiapproachrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2277ndash2291 2013

[15] A Kharitonov and O Sawodny ldquoFlatness-based feedforwardcontrol for parabolic distributed parameter systems with dis-tributed controlrdquo International Journal of Control vol 79 no 7pp 677ndash687 2006

[16] T Meurer ldquoFlatness-based trajectory planning for diffusion-reaction systems in a parallelepipedonmdasha spectral approachrdquoAutomatica vol 47 no 5 pp 935ndash949 2011

[17] C I Byrnes and D S Gilliam ldquoAsymptotic properties of rootlocus for distributed parameter systemsrdquo in Proceedings ofthe 27th IEEE Conference on Decision and Control pp 45ndash51Austin Tex USA December 1988

[18] C I Byrnes D S Gilliam C Hu and V I Shubov ldquoAsymptoticregulation for distributed parameter systems via zero dynamicsinverse designrdquo International Journal of Robust and NonlinearControl vol 23 no 3 pp 305ndash333 2013

[19] C I Byrnes D S Gilliam A Isidori and V I ShubovldquoZero dynamics modeling and boundary feedback design forparabolic systemsrdquoMathematical and Computer Modelling vol44 no 9-10 pp 857ndash869 2006

[20] C I Byrnes D S Gilliam A Isidori and V I Shubov ldquoInteriorpoint control of a heat equation using zero dynamics designrdquo inProceedings of the IEEE American Control Conference pp 1138ndash1143 Minneapolis Minn USA June 2006

[21] M Fliess J Levineb P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[22] J Rudolph and F Woittennek ldquoFlachheitsbasierte rands-teuerung von elastischen balkenmit piezoaktuatorenrdquoAutoma-tisierungstechnik Methoden und Anwendungen der Steuerungs-Regelungs-und Informationstechnik vol 50 no 9 pp 412ndash4212002

[23] J Schrock T Meurer and A Kugi ldquoMotion planning for piezo-actuated flexible structures modeling design and experimentrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 807ndash819 2013

[24] R Rebarber ldquoExponential stability of coupled beams withdissipative joints a frequency domain approachrdquo SIAM Journalon Control and Optimization vol 33 no 1 pp 1ndash28 1995

[25] M Berggren Optimal control of time evolution systems control-lability investigations and numerical algorithms [PhD thesis]Rice University Houston Tex USA 1995

[26] C Castro and E Zuazua ldquoUnique continuation and controlfor the heat equation from an oscillating lower dimensionalmanifoldrdquo SIAM Journal on Control and Optimization vol 43no 4 pp 1400ndash1434 2004

[27] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp HallCRC NewYork NY USA 2002

[28] Y Aoustin M Fliess H Mounier P Rouchon and J RudolphldquoTheory and practice in the motion planning and control of aflexible robot armusingMikusinski operatorsrdquo inProceedings ofthe 5th IFAC Symposium on Robot Control pp 287ndash293 NantesFrance September 1997

[29] M Fliess PMartinN Petit andP Rouchon ldquoActive restorationof a signal by precompensation techniquerdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 1007ndash1011Phoenix Ariz USA December 1999

[30] H Mounier J Rudolph M Petitot and M Fliess ldquoA flexiblerod as a linear delay systemrdquo in Proceedings of the 3rd EuropeanControl Conference (ECC rsquo95) Rome Italy September 1995

[31] P Rouchon ldquoMotion planning equivalence and infinitedimensional systemsrdquo International Journal of Applied Mathe-matics and Computer Science vol 11 no 1 pp 165ndash180 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of