Robust switching adaptive control of multi-input …ioannou/2003update/d66.pdf610 IEEE TRANSACTIONS...

610 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002

Robust Switching Adaptive Control of Multi-InputNonlinear Systems

Elias B. Kosmatopoulos and Petros A. Ioannou, Fellow, IEEE

Abstract—During the last decade a considerable progress hasbeen made in the design of stabilizing controllers for nonlinear sys-tems with known and unknown constant parameters. New designtools such as adaptive feedback linearization, adaptive back-stepping, control Lyapunov functions (CLFs) and robust controlLyapunov functions (RCLFs), nonlinear damping and switchingadaptive control have been introduced. Most of the results devel-oped are applicable to single-input feedback-linearizable systemsand parametric-strict-feedback systems. These results, however,cannot be applied to multi-input feedback-linearizable systems,parametric-pure-feedback systems and systems that admit alinear-in-the-parameters CLF. In this paper, we develop a generalprocedure for designing robust adaptive controllers for a largeclass of multi-input nonlinear systems. This class of nonlinear sys-tems includes as a special case multi-input feedback-linearizablesystems, parametric-pure-feedback systems and systems thatadmit a linear-in-the-parameters CLF. The proposed approachuses tools from the theory of RCLF and the switching adaptivecontrollers proposed by the authors for overcoming the problemof computing the feedback control law when the estimation modelbecomes uncontrollable. The proposed control approach has alsobeen shown to be robust with respect to exogenous bounded inputdisturbances.

Index Terms—Feedback linearizable systems, robust adaptivecontrol, switching control.

I. INTRODUCTION

DURING the last decade a considerable progress hasbeen made in the design of stabilizing controllers for

nonlinear systems with known and unknown constant parame-ters. New design tools such as adaptive feedback linearization[1], [5], [19], adaptive backstepping [6], [12], [20], controlLyapunov functions (CLFs) and robust control Lyapunovfunctions (RCLFs) [2], [13], [21], [22], nonlinear damping andswapping [11], [12] and switching adaptive control [8], [9]have been introduced. Using these new design tools, globallystabilizing controllers have been constructed for various classesof nonlinear systems such as single-input feedback-linearizablesystems [1], [5], [9] and parametric-strict-feedback systems [6],[12], [20]. Despite the success of the aforementioned designtools to resolve a variety of adaptive control problems fornonlinear systems, the problem of adaptive control of nonlinear

Manuscript received July 13, 1998; revised January 20, 2000 and July 10,2001. Recommended by Asociate Editor M. Krstic. This work was supportedin part by the National Aeronautics and Space Administration (NASA) underGrant Number NAGW-4103, and in part by the National Science Foundationunder Grant Number ECS 9877193.

E. B. Kosmatopoulos is with the Department of Production Engineering andManagement, Technical University of Crete, Chania 73100, Greece.

P. A. Ioannou is with the Department of Electrical Engineering-Systems, Uni-versity of Southern California, Los Angeles, CA 90089-2563 USA.

Publisher Item Identifier S 0018-9286(02)03739-X.

systems is still very much unexplored. For example, there existsno procedure for designing a globally stable feedback controlsystem for multi-input feedback linearizable systems of theform

(1.1)

where , denote the state and control inputvectors of the system, respectively,, , are con-stant unknown matrices and, are continuous matrix func-tions satisfying and is nonsingular for all

. The existing adaptive control designs guarantee [1],[5], [19] closed-loop stability only for the case where the con-stant matrices and are known; an exception is the casewhere (i.e., the system (1.1) is single-input) and the pair

is in a special canonical form [9].

Another example is the system of the form (parametric-pure-feedback system)

(1.2)

where is a vector of unknown constant parameters,denotes the state vector of the system

and , , are continuous functions. The procedures pro-posed in [6], [11], [20] are applicable to this system if both

and ,, where denotes the esti-

mate of ; moreover, these procedures guarantee global stabilityonly in the case where the input vector-fieldis independent of , i.e., in the case where and thefunctions are independent of .

In this paper, we develop a general procedure for designingrobust adaptive controllers for a large class of multi-input non-linear systems with exogenous bounded input disturbances. Theclass of systems for which the proposed approach is applicableis characterized by the assumption that the function de-pends linearly on unknown constant parameters, wherede-notes the input vector field, is a CLF (RCLF) for the systemand denotes the Lie derivative of with respect to . Thisclass of nonlinear systems includes as a special case the systems(1.1) and (1.2). The proposed approach combines the theory ofCLF (RCLF) and the switching adaptive controller proposed bythe authors [9] for overcoming the problem of computing thecontrol law in the case where the estimation model becomes un-controllable.

Contrary to the classical adaptive approach where the con-trol law depends on estimates of the system vector-fields, in ourcase, the control law depends on estimates of the “RCLF term”

0018-9286/02$17.00 © 2002 IEEE

KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS 611

which depends both on the system vector-fields and theRCLF function . The advantage of such an approach is thatthe Lyapunov inequalities relating the parameter estimation er-rors and the time-derivative of the RCLF are easy to handle. Thedisadvantage is that the controllers that are designed based onthe RCLF theory critically depend on the knowledge of .Adaptive versions of such controllers may fail due to the fact thatthe estimate of may have different sign, at certain times,than the sign of the actual . Even worse, we may have thecase where the estimate of is close to zero and the actualvalue of is far from zero, which implies that the estima-tion model becomes uncontrollable while the actual model isnot. This problem is overcome by using a switching adaptivecontrol law. This control law is a modified version of the oneoriginally proposed by the authors for overcoming the problemof computing the control law in the case where the estimationmodel becomes uncontrollable for the case of single-input feed-back-linearizable systems in canonical form [9]. Such a controllaw appropriately switches between two adaptive controllerswhich have the following properties: (i) both controllers behaveapproximately the same in the nonadaptive case (i.e., in the caseof known system parameters), and (ii) when the one of thesecontrollers becomes nonimplementable, the other one is imple-mentable.

The significance of the proposed approach is that it is the firstto overcome the problem of constructing globally stabilizingcontrollers for systems of the form (1.1) and (1.2). In addition,the proposed approach can be used to solve control problems fora wider class of plants than those described by (1.1) and (1.2).

The paper is organized as follows. In Section II, we presentthe problem formulation and some results from the theory ofRobust Control Lyapunov Functions. In Section III, we presentand theoretically analyze the proposed approach. In Section IV,we present various classes of systems for which the proposedapproach is applicable, and, finally, Section V is the conclu-sion section. We close this section by mentioning that in [23] anadaptive controller for muliple-input–multiple-output (MIMO)nonlinear systems is proposed where, similar to this paper, in-stead of linearly parameterizing the state equations the con-troller design is based on a linearly parameterized positive func-tion. The class of systems dealt in [23] is not as broad as theones dealt in this paper and more restrictive assumptions on thesystem dynamics are imposed. Also, in [3], a switching con-troller is proposed for nonlinear systems with unknown param-eters that obey a CLF that may also depend on the unknownparameters. The assumptions made in [3] are that the vector ofunknown parameters belong to a finite set and that the controllerfor the case where the system parameters are known satisfiessome robustness conditions.

A. Notation and Preliminaries

If is a vector, denotes the Euclidean norm. In this paper,we use the following version of the signum function:

ifif

We say a function is of class when is con-tinuous, strictly increasing, and . We say is of class

when is of class and satisfies as .We say a function is of class when

is of class for each fixed and decreasesto zero as for each fixed . We say that a contin-uous function belongs to when it has continuous first deriva-tive, and is smooth when it has continuous derivatives of anyorder. If is a positive constant, then the set-Ball is definedto be the set . If is a subset of then

. Finally, if is a -dimensionalsquare matrix, then denotes the -dimensional vectorwhose first entries are the entries of the first column of, thenext entries are the ones of the second column of, etc.

II. PROBLEM FORMULATION

In this paper, we consider nonlinear systems of the form

(2.1)

where , , denote the vectors ofsystem states, control inputs and disturbances and, , are

vector-fields of appropriate dimensions. We assume that thedisturbance vector is bounded, i.e., where isa positive constant. The control objective is to find the controlinput as a function of such that all closed-loop signals arebounded and as . Since and is as-sumed to be any general unknown bounded continuous functionof time, in many cases, the best that we can hope is closed-loopsignal boundedness and convergence ofto a residual boundedset whose size is of the order of.

Definition 1: [2], [14] Let be a compact subset ofsuch that . Moreover, suppose that the control inputis chosen as , where is an appropriate feedback.Then, the solutions of the closed-loop system are robustly glob-ally uniformly asymptotically stable (RGUAS) with respect to

(RGUAS- ) when there exists a class function suchthat for any initial condition and any admissible dis-turbance , all solutions of the closed-loopsystem starting from exist for all and satisfy

for all . The solutions of the closed-loop systemare RGUAS when they are RGUAS-{0}.

Definition 2: [2], [14] The system (2.1) is robustly asymp-totically stabilizable (RAS) when there exists a control law

such that the closed-loop system solutions are RGUAS.The system (2.1) is robustly practically stabilizable (RPS) whenfor every there exists a control law and a com-pact set satisfying -Ball such that the so-lutions of the closed-loop system are RGUAS-. The systemis Robustly Stabilizable (RS) when there exists a control law

and a compact set satisfying such thatthe solutions of the closed-loop system are RGUAS-.

Let be a function. We say thatif , for all and there exists aclass function such that for all .Let denote the Lie derivative ofwith respect to , and be defined as follows:


where is a nonnegative constant. Note that since ,then is compact for all . Let us now recall thedefinition of RCLF as defined in [2], [13] as well as the rela-tion between the RCLF and the robust stabilization problem ofsystem (2.1).

Definition 3: A function is an RCLF for the system(2.1) if and, moreover, there exists a positive constant

such that satisfies

if

(2.2)

We let denote the smallest value offor which (2.2) is satis-fied. Let also be defined as follows: if , and

is an arbitrarily small positive constant if . Finally, letbe defined as

The next theorem [2] states that the existence of an RCLF forsystem (2.1) is a necessary and sufficient condition for robuststabilization of system (2.1).

Theorem 1: If system (2.1) is RAS or RPS or RS via a locallyLipschitz control law , then there is a smooth RCLFfor system (2.1). On the other hand, if there exists a RCLF forsystem (2.1), then (2.1) is RS. If furthermore , then (2.1)is RPS.

The proof of Theorem 1 can be found in [2].

III. T HE ADAPTIVE CONTROLLER

Our first assumption for system (2.1) is the following.

A1) System (2.1) is RS (or RPS). Since system (2.1) isRS or RPS, from Theorem 1 we have that there exists asmooth RCLF for system (2.1). The time-derivative ofthis RCLF is given by

(3.1)

Let us assume that the function can be written aslinear combination of known functions and unknown con-stant parameters. That is, we assume the following.A2) The function satisfies

(3.2)

where is a constant unknown matrix and theregressor vector is a known nonlinear vectorfunction.

In Section IV, we give examples of classes of systems thatsatisfy Assumption A2).

Let us define the sets

where and is a compact subset.andwill be defined explicitly later. Moreover, let be apositive design constant satisfying

(3.3)

Now, let , denote the following sets:

Obviously, for all . Thefollowing Lemma holds.

Lemma 1: There exists a scalar (that depends on thecompact set ) such that for all the following holds:

Proof: Since and are continuous, we have thatis continuous, too. Let be the set

. Also, for each let be the largestnumber such that for alland let be defined as

From the continuity of and the fact that is compact wehave that (a continuous function is uniformly continuouson a compact set, [18, Th. 4.19]). Then,is given by

where

where denotes the boundary of. From the above definitionsand the fact that is continuous, it is obvious thatand that implies that ,which concludes the proof.

We are now ready to present the proposed controller. The con-trol input is chosen as follows:

(3.4)

where

•

(3.5)

•

(3.6)


• denotes the estimate of the-th entry of thevector , generated as

(3.7)

where is the estimate of [ denotes thethcolumn of the matrix defined in (3.2)];

• is a design function chosen to satisfy

(3.8)

• is a continuous-switching signal which is used to con-tinuously switch from control to control , and viceversa

if

if

if

(3.9)

The parameter estimates are updated using the followingsmooth projection update law [17] (here, de-note the parameter estimation errors)

(3.10)where is a symmetric positive–definite design matrix and

is defined as [17] (3.11), shown at the bottom of the page,where

where is positive design constant. The variableis a hysteresis-switchingsignal to be defined

explicitly later in this section. Finally, is a discontinuousfunction defined as follows:

ifotherwise

(3.12)

where is a positive design constant. From the definition of, we have that the adaptive law (3.10) becomes inactive

when is smaller than . The role of will be made clearduring the proof of the main result of this paper.

Let us analyze the proposed control law. More precisely, con-sider the following Lemma.

Lemma 2: Consider the variables

(3.13)

Note that

(3.14)

Moreover, define

The following statements hold:

a)

where

(3.15)

b)

Proof: The proof can be found in Appendix I.Let

(3.16)

From Lemma A.1 part a) (see Appendix I), we have that, for, and therefore the term can be

made arbitrarily close to 1 by increasing the design constant.Therefore, from Lemma 2 we have that for

(3.17)

where (note that). Using now the facts that is close

to one and is bounded from above by we can seefrom the above inequality that both control laws andhave approximately the same effect on the term . Moreprecisely, the both have the effect of replacing the unknown term

by the negative definite term , plus a term thatdepends on the estimation error .

If the term was absent in the right-hand side(RHS) of (3.17) then, using standard arguments in adaptive con-trol, we could show that the use of the parameter projection law(3.10) with takes care of the effect of the estimation error

ifif and

otherwise(3.11)


term . However, although the term hassmaller magnitude than , it may have destabilizing ef-fects to the closed-loop system due to the fact that it depends onthe unknown function .

In order to overcome the problem where the termmay have a destabilizing effect, we pro-

ceed as follows: As it will be shown in the proof of Theorem 2,the following term appears in the RHS of the time-derivative ofan appropriately defined Lyapunov-like function

The term appears in the Lyapunov equation dueto the projection law (3.10).

Define

(3.18)where and is a continuous func-tion, satisfying where

and . Ob-viously for all .

, are continuous functions for all, where is a convex set which is defined

explicitly in Theorem 2 later in this section. Therefore, usingstandard arguments from the theory of approximation ofnonlinear functions (see, e.g., [7], and the references therein)we can show that for any , there exist1 two convex sets

, two constant vectors , anda nonlinear function2 such that

(3.19)

and

(3.20)

It is no loss of generality to assume that the entries of,as well as the entries of all vectors or are3

nonnegative. Also, since we assumethat and that for all suchthat (again this is true provided we chooseappropriately; for instance the entries of could be chosen sotheir magnitude is bounded by ). Using (3.19) and (3.20)and the fact that we obtain

(3.21)

where

1Note that, in general," ! 0 asn ! 1.2For notational simplicity we assume that both approximators have the same

regressor terms.3Such an assumption holds if we augment the regressor vector� so that it

contains both the entries� (�) and�� (�).

and

Using (3.20) and (3.21), we have that for

(3.22)

where denotes the estimate of anddenotes the parameter estimation error. The first two

terms in the RHS of inequality (3.22) can be taken care of byusing adaptive update laws. Thus we have to choosein sucha way that the rest three terms remain bounded by a small designconstant. We will use the following hysteresis-switching adap-tive scheme for :

if

otherwise(3.23)

where ,

, where .The problem now is in the design of the parameter estimationlaws for . There are two issues regarding thedesign of such estimation laws. The first is the issue of keepingthe parameter estimates bounded; such a problem can beeasily addressed using projection laws similar to (3.10) and(3.11). The second issue has to do with the problem when,at certain time-instants, is very close to zerowhile is large positive. In such a case, the term

may not be boundedby a small design constant.

To round this problem, we proceed as follows. First, observethat from the definitions of , , we have that4

(3.24)

Therefore, using the properties and, we have that

(3.25)

where is the convex set defined as

(3.26)

4The integerj ? k is defined asj ? k = n (k � 1) + j.


where is a large positive constant such that .The first two inequalities define the constraint that the entries of

are positive, the third one defines the constraint thatand the last inequality defines the constraint

that is bounded from above by .Relation (3.25) motivates us to propose the following adap-

tive laws for :

(3.27)

where , is a positive constant, ,and is the projection law which

keeps and is defined as follows:

(3.28)

where if orand and

, otherwise. Similarly, ifor and

and , otherwise.We are now ready to prove closed-loop stability under the the

control law (3.4)–(3.11), (3.23), and (3.27).Theorem 2: Consider the unknown system (2.1) and the con-

trol law (3.4)–(3.11), (3.23) and (3.27). Assume that A1) andA2) hold, that , are vector-fields, and the disturbance

is continuous and bounded. Moreover, assume that the fol-lowing hold.

C1) satisfies (3.8) and , where is definedin Lemma 1.C2) is large enough so that where

Moreover, is large enough so thatand where is a positive

design constant.Then, for any compact set and for any positive con-stant the following holds: there exist positive con-stants , , (dependent on ) and such that, for anyinitial state , the control law (3.4)–(3.11) with

guarantees that all theclosed-loop signals are bounded andconverges to the residualset

where the constantsatisfies wherewhen .

Proof: Since is continuous and bounded, issmooth with respect to its arguments [see Lemma A.1, part(c)], is a continuous signal, we have that the source ofdiscontinuities in the closed-loop system dynamics are thehysteresis-switching variables and the parameter projectionlaws (3.10) and (3.27). Parameter projection laws guarantee

the existence and uniqueness of solutions in the sense ofCaratheodory [16] (the proof of [16] can be easily revised toinclude the case where the projection law is inactive when

). Therefore, using similar arguments as those in[15] (see also [24, Lemma A.1]) we can establish that thereexists an interval of maximal length on which thehysteresis-switching closed-loop system possesses a uniquesolution with ’s piecewise constant on and, moreover,that on each strictly proper subinterval canswitch at most finite times. Note also that , arebounded due to the projection update laws (see Lemma A.1,part (c) in Appendix I).

Now, let denote the smallest positive constant satisfying

Also, let denote the set

where is a positive constant. From the above definitions, wehave that

Since , we have that there exists a positive realsuch that for all . Let us analyze the

closed-loop system for . Define the Lyapunov-likefunction

In Appendix I, we show that for

(3.29)

where , are positive terms that can be made arbitrarilysmall by increasing (or, equivalently, decreasing).

Using now the fact that if and, the definition of and the definition of , it

is straightforward to show that, there exist positive constantsand a positive–definite function such that for

(3.30)

provided that remains on where is the compactset

In (3.30), is a positive constant. Such a constant exists, pro-vided that are large enough. and defined inTheorem 2 are related as follows: The variableis a positiveconstant such that . For the time being let us assumethat is the smallest constant for which . Using now the


definition of the function , part (c) of Lemma A.1, and the factthat is decreasing as long as does not enter , wecan see that remains in the interior of for allprovided that

(3.31)

Given any and independent of we have thatthere exists such that the above inequality is valid for all

(note that the term depends on the dimensionof the nonlinear approximators and thus it may be increasing

as becomes larger, i.e., as increases; note also thatmaybecome arbitrarily close to by increasing ).

Since remains in the interior of for all ,we have that we can extend all the way to . On the otherhand, the fact that is decreasing outside together withthe boundedness of due to the projection updatelaws implies that . Finally, since outside

for all implies that enters in finite time .We will show that converges to , that is, there is a time

instant such that . Let us define

, i.e., . While isin , the parameter adjustment laws become inactive, whichin turn implies that the term is constant while .Consider the following situation: enters at time-instant

, enters at , exits at and exits at . Since outside, we have that will enter again at time-in-

stant and at , where . Whenthe parameter adjustment laws are inactive, and thus

. Moreover, from the definition of we havethat .Finally, is decreasing for and .Using the above, we can see that . This is since:(a) and (b) is decreasing at and

and thus is decreasing in . There-fore, at the time enters , is strictly smaller than itsvalue when entered previously. It can be seen that, byappropriately defining , the following also holds:

. If the above claim is not true, we can always increasethe value of to make5 it true. Therefore, each timeentersthe term is smaller than the one during the previous visitof in which in turn, implies that eventually stays in

foreover. This claim can be proven by contradiction: ifwould not stay in then we would have that converges tozero, which in turn implies that converges to , andthus, would be—at the limit—decreasing outside.

5This can be done as follows: suppose that�(t ) � �(t ) which in turnimplies that�(t) increased att 2 [t ; t ) by an amount that is larger than�(t ) ��(t ). In other words, we have the case where� has changed moreduring the trip ofx(t) from the boundary of to the boundary of than it haschanged whilex(t) was outside. In that case, we can increase�c (i.e., increasethe size of) so thatx(t) never left. Note that the amount by which�c has tobe increased should be very small.

Remark 1: As it is seen from (3.31) the reason we chooseis to make , and thus , independent of the di-

mension of the nonlinear approximators.Remark 2: In the case where does not satisfy the linear-

in-the-parameters assumption A2), the proposed approach isstill applicable, by assuming that the function can beapproximated by linear-in-the-weights nonlinear functions. In[10], we show how we can apply the proposed strategy in thecase where does not satisfy assumption A2) or, for thecase where the system dynamics are completely unknown.

IV. A PPLICATION TOVARIOUS CLASSES OFSYSTEMS

In this section, we present some examples of classes of non-linear systems which satisfy assumption A2). For simplicity, weconsider the case where the external disturbance . Theresults can be easily extended to the case where external inputdisturbances are present.

A. Multi-Input Feedback-Linearizable Systems

One class of systems that satisfy assumption A2) is the classof multi-input feedback-linearizable systems. For this classthere exists no general methodology for designing adaptivecontrollers that guarantee global stability for the closed-loopsystem.

Let us consider the class of multi-input feedback linearizablesystems whose dynamics can be described as follows:

(4.1)

where are unknown constant matrices, areknown nonlinear continuous vector functions and, moreover, thematrix pair is stabilizable and the functions aresuch that and is nonsingular for all .Let be a stabilizing gain matrix for the pair . Then, aRCLF for system (4.1) is the function [2] where

is the symmetric positive–definite solution of the Lyapunovequation

(4.2)

where is a symmetric positive–definite matrix. Then, thefunctions and are given by

(4.3)

and

(4.4)

It is not difficult to see that if is a vector whose entries are theelements and , is a vector whose entries are theelements , is a vector whose entries are the elements of

and and is a matrix whose entries are the elementsof , then, and can be written in the form (3.2). Inother words, systems of the form (4.1) satisfy assumption A2).

In a similar way, we can show that the feedback linearizablesystems of the form

(4.5)

satisfy assumption A2). Here, are unknownconstant matrices, are known nonlinear continuous


vector functions and, moreover, the unknown matrix pairis stabilizable and the continuous functions

are such that and is nonsingular forall . For the case of the system (4.5) the functions

are given as follows:

(4.6)

and

(4.7)

where is defined in (4.2). Next we show how to chosethe controller parameters: First, let us examine the equality

is valid. Since we have thatwhich, due to the nonsingularity of

, implies that andtherefore implies . On theother hand, from the Lyapunov equality (4.2) it can be easilyseen that implies that

, and therefore,we conclude that implies that .Thus, we have that where is defined in Definition 3and therefore can be any arbitrarily small constant. Once

is chosen then can be chosen as follows: First note thatsince is nonsingular, we have thatfor some positive number . Assuming that a lower boundof is known we have that implies

(4.8)

By adding and subtracting the terms andin the RHS of (4.6) and using (4.2) we obtain after

some algebraic manipulations

Using (4.8), we have that implies

where is a known compact set satisfying . From theabove inequality, we have thatcould be set equal to the largest

that satisfies . Note that an upperbound on is needed in order to solve the above problem.The constant can be then set equal to where

is the minimum value of that satisfies , whereis defined in the proof of Theorem 2.

Finally, the proposed controller function can be chosenas follows:

where are positive constants satisfying.

It is worth noticing that the existing adaptive control designsguarantee global stability only in the case where the matrixfor the case of system (4.1) or the matricesand for the caseof system (4.5) are known.

B. Linear-in-the-Parameters Nonlinear Systems WithLinear-in-the-Parameters RCLF

Consider now the class of nonlinear systems of the form (2.1)whose vector-fields and are linear combinations of knownfunctions and unknown constant parameters, i.e.,

and

where are known nonlinear vector functions andare unknown constant matrices (vectors). Moreover, as-

sume that an RCLF for the system can be written as a linearcombination of known functions and unknown constant param-eters, i.e.,

where is a known nonlinear vector function and is anunknown constant vector. Then, it can be easily seen that theabove system satisfies assumption A2) where the entries ofare the elements of , the entries of are the elementsof , the entries of are the elements of ,and the entries of are the elements of .

C. Parametric-Pure-Feedback Systems

Let us now try to apply the results of Section III to nonlinearsystems that take the form

(4.9)

where , are smooth known functions andis the vector of constant but unknown system parameters.

Let us rewrite (4.9) as

(4.10)

where . Systems of the form (4.10) arecalled parametric-pure-feedback (PPF) systems [6], [12], [20].Although the problem of constructing globally stable adaptivecontrollers for the simplest case of parametric-strict-feedback(PSF) systems6 has been completely solved [6], [12], [20], theproblem of constructing globally stable adaptive controllers forPPF systems remains an open problem and is the subject of thissubsection.

Although the results developed in Sections I–III are for statestabilization, for the particular case of PPF systems, they can be

6The class of PSF systems refers to the subclass of PPF systems (4.10) whichsatisfy� f (z ; . . . ; z ) = 0; � g (z ; . . . ; z ) = 0.


readily extended to the case of asymptotic tracking as well. Thecontrol objective is to force the system outputto asymptoti-cally track a reference signal . We assume that the firsttime derivatives of are known. Also it is assumed that aswell as its first time derivatives are bounded and smoothsignals. Before we design the feedback law, we will transformthe system (4.9) into a suitable form. The procedure we followis that of [6] and [12], and it is based on the backstepping inte-grator principle [22].

Step 0): Let . Let also be positiveconstants to be chosen later.Step 1): Using the “chain of integrators” method, we seethat, if was the control input in the -part of (4.10) and

was known, then the “control law”

(4.11)

would result in a globally asymptotically stable tracking,since such a control law would transform the-part of(4.10) as follows

However, the state is not the control. Therefore, we de-fine to be the difference between the actualand itsdesired expression (4.11):

(4.12)

Using the above definition of , the definition of andthe -part of (4.10), we found that

(4.13)

Step 2): Using the above definitions of , we have that

(4.14)

where is a -dimensional vector that consists ofall elements that are either of the form or of the form

where by we denote the-th entry of the vector. In the system (4.14), we will think of as our control

input. Therefore, as in Step 1), we define the new stateas

(4.15)

Substituting (4.15) into (4.14) yields

(4.16)

Step 3): Using the above definitions of , we havethat after some algebraic manipulations

(4.17)

where is a vector that consists of all elements that areeither of the form or of the form or of the form

.In the system (4.17), we will think of as our control

input. Therefore, as in Step 2), we define the new stateas

(4.18)

Substituting (4.18) into (4.17) yields

(4.19)

Step : Using the definitions ofand working as in the previous steps we may express thederivative of as

(4.20)

where the vector contains all the terms of the formwith . Defining now as

follows:

(4.21)

we obtain that

(4.22)

Step : Using the definitions of and workingas in the previous steps we may express the derivative of

as follows:

(4.23)

where the vector contains all the terms of theform with ,

and

is given by

(4.24)

Using the definitions of , and by rearrangingterms, we may rewrite (4.23) as follows:

(4.25)


Therefore, using the aforementioned methodology, we havetransformed system (4.10) into the following:

......

. . .. . .

......

... (4.26)

or

(4.27)

where . Let us define the outputof the previous system as

where . Obviously, the above system isfeedback-linearizable, if the following assumption holds.

A1’) for all .Note that the variables are not available for the

control design since they depend on the unknown vector.We will show that, if the constants are chosen so that

for all , then the function

is an RCLF for system (4.9) and, moreover, that the resultingfunctions and satisfy assumption A2). By differenti-ating with respect to time, we obtain that

(4.28)

and, therefore

(4.29)

and

(4.30)

From the assumption A1’), it can be easily seen that

. The quantity isnegative definite provided that for all , and thus, isan RCLF for system (4.9) and moreover . Note now thatusing the definitions of we can rewrite ’s as follows:

where and are appropriately defined known functions(note that ) Therefore, we have that

and

From the above two equations, it can be easily seen that thefunctions and satisfy assumption A2), by defining thevectors and as the vectors whose entries are the elements

and defining the functions and appropriately.The proposed controller parameters can be chosen as follows:

since the constant , the constant can be chosen tobe any arbitrarily small positive number. Working similar tothe case of feedback linearizable systems in Section IV-A wecan show that can be set equal to the largestthat satisfies

where is defined as fol-lows

where is a positive constant satisfyingand is a known compact set satisfying .

The constant must be chosen so that andwhere is defined in the proof of Theorem 2 .

Regarding the design of the function , one possible wayis to choose as follows:

where is a positive constant satisfying .Remark 3: Theorem 2 is not directly applicable to the class

of PPF systems (4.9), since the terms and in (4.29)and (4.30) are explicit functions of. However, Theorem 2 canbe easily modified to be applicable to such a case. The only mod-ification needed is that the function as well as the regressorterms in the adaptive laws should be explicit functions of(thus,for instance the regressor terms of the nonlinear approximators

should be replaced by . If we carry outthe same analysis as in the proof of Theorem 2, by incorporatingthe above modifications, it can be seen that results of Theorem2 are applicable to the class of systems (4.9).


V. CONCLUSION

In this paper, we proposed a switching adaptive controller formulti-input nonlinear systems whose dynamics are nonlinearlyaffected by external input disturbances. The proposed approachis applicable to nonlinear systems for which the functioncan be written as a linear combination of unknown constantsand known nonlinear functions. By making use of the notionof RCLF [2] and a modified version of the switching adaptivecontroller of [9] we showed that the proposed controller guaran-tees bounded closed-loop signals and convergence of the state toa residual set. The proposed control approach is used to designstabilizing controllers for multi-input feedback-linearizable sys-tems, PPF systems and linear-in-the-parameters nonlinear sys-tems which admit a linearly parameterized RCLF.

APPENDIX

Lemma A.1:Assume that the conditions of Theorem 2 hold.Then the following statements are true.

a) For all we have that

and

b) ,.

c) is smooth with respect to its arguments. Moreover,

and

where is a finite positive constant, independent of.Similarly, for

d)

e)

and

Proof:

a) Since we have that . There-fore, by taking into account (3.3) we conclude that

forall , i.e., the denominator of never becomeszero provided that . Since , we have that

and therefore, from (3.3) we obtain that, for all

and thus, is bounded from above by for all. Using now the fact that is bounded from above

by for all and (3.16), it is straightforward toverify the second inequality of part a).

b) The proof is straightforward.c) The proofs are similar to those of [17] and [4, Th. 4.4.1].d) The proof is straightforward.e) Since from part c) of this Lemma

, we have that andtherefore, there exists with suchthat . Therefore

We have that (sinceand the design assumption for all

), which concludes the proof.

Proof of Lemma 2:

a) From (3.5), we have that

Therefore


(A.1)

b) Using (3.6), we obtain

(A.2)

Proof of (3.29): Differentiating with respect to timeand using (3.4)–(3.11) and (3.18)–(3.27), Lemma 2, LemmaA.1, and the fact that we obtain thatfor , the equation at the bottom of thepage holds true,


where

From (3.3), it can be seen that for any there exists ansuch that .

Therefore

and, thus, we finally obtain that

(A.3)

REFERENCES

[1] G. Cambion and G. Bastin, “Indirect adaptive state feedback control oflinearly parameterized nonlinear systems,”Int. J. Adapt. Control SignalProcessing, vol. 4, pp. 345–358, Sept. 1990.

[2] R. A. Freeman and P. V. Kokotovic, “Inverse optimality in robust stabi-lization,” SIAM J. Control Optim., vol. 34, no. 4, pp. 1365–1391, July1996.

[3] H. Hespana and A. S. Morse, “Supervision of families of nonlinear con-trollers,” in Proc. 35th IEEE Conf. Decision Control, vol. 4, Dec. 1996,pp. 3771–3773.

[4] P. A. Ioannou and J. Sun,Stable and Robust Adaptive Control. UpperSaddle River, NJ: Prentice-Hall, 1996.

[5] I. Kanellakopoulos, P. V. Kokotovic, and R. Marino, “An extended directscheme for robust adaptive nonlinear control,”Automatica, vol. 27, pp.247–255, Mar. 1991.

[6] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic de-sign of adaptive controllers for feedback linearizable systems,”IEEETrans. Automat. Contr., vol. 36, pp. 1241–1253, November 1991.

[7] E. B. Kosmatopoulos and M. A. Christodoulou, “Techniques and appli-cations of recurrent high order neural networks in the identification ofdynamical systems,” inNeural Network Systems Techniques and Appli-cations, C. Leondes, Ed. New York: Academic, 1999.

[8] E. B. Kosmatopoulos, “Universal stabilization using control Lyapunovfunctions, adaptive derivative feedback and neural network approxima-tors,” IEEE Trans. Syst., Man, Cybern. B, vol. 28B, pp. 472–477, June1998.

[9] E. B. Kosmatopoulos and P. A. Ioannou, “A switching adaptive con-troller for feedback linearizable systems,”IEEE Trans. Automat. Contr.,vol. 44, pp. 742–750, Apr. 1999.

[10] E. B. Kosmatopoulos, “Robust neural stabilizers for unknown systems,”IEEE Trans. Neural Networks, submitted for publication.

[11] M. Krstic and P. V. Kokotovic, “Adaptive nonlinear design withcontroller-identifier separation and swapping,”IEEE Trans. Automat.Contr., vol. 40, pp. 426–440, Mar. 1995.

[12] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic,Nonlinear and Adap-tive Control Design. New York: Wiley, 1995.

[13] Y. Lin and E. D. Sontag, “Control Lyapunov universal formulae for re-stricted inputs,”Control: Theory Adv. Technol., vol. 10, pp. 1981–2004,1995.

[14] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov the-orem for robust stability,”SIAM J. Control Optim., vol. 34, pp. 124–160,1996.

[15] A. S. Morse, D. Q. Mayne, and G. C. Goodwin, “Applications of hys-teresis switching in parameter adaptive control,”IEEE Trans. Automat.Contr., vol. 34, pp. 1343–1354, Sept. 1992.

[16] M. M. Polycarpou and P. A. Ioannou, “On the existence and uniquenessof solutions in adaptive control systems,”IEEE Trans. Automat. Contr.,vol. 38, pp. 474–480, Mar. 1993.

[17] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimationfrom the Lyapunov equation,”IEEE Trans. Automat. Contr., vol. 37, pp.729–740, 1992.

[18] W. Rudin,Principles of Mathematical Analysis. New York: McGraw-Hill, 1984.

[19] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,”IEEE Trans. Automat. Contr., vol. 34, pp. 405–412, Apr. 1989.

[20] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of non-linear systems with a triangular structure,”IEEE Trans. Automat. Contr.,vol. 39, pp. 1411–1428, July 1994.

[21] E. D. Sontag, “A universal construction of Arstein’s theorem on non-linear stabilization,”Syst. Control Lett., vol. 13, no. 2, pp. 117–123,1989.

[22] J. Tsinias, “Sufficient Lyapunov-like conditions for stabilization,”Math.Control, Signals, Syst., vol. 2, no. 4, pp. 343–357, 1989.

[23] B. Yao and M. Tomizuka, “Adaptive robust control of a class of multi-variable nonlinear systems,”IFAC World Congress, vol. F, pp. 335–340,1996.

[24] S. R. Weller and G. C. Goodwin, “Hysteresis switching adaptive controlof linear multivariable systems,”IEEE Trans. Automat. Contr., vol. 39,pp. 1360–1375, July 1994.


Elias B. Kosmatopoulosreceived the Diploma in production and managementengineering, and the M.Sc. and Ph.D. degrees in electronics and computer en-gineering, all from the Technical University of Crete (TUC), Greece, in 1990,1993, and 1995, respectively.

He is currently a tenure-track Lecturer in the Department of Production andManagement Engineering, and Deputy Director of the Dynamic Systems andSimulation Laboratory, at TUC. Prior to joining TUC, he was a Research Assis-tant Professor with the Department of Electrical Engineering-Systems, Univer-sity of Southern California (USC), Los Angeles, and a Postdoctoral Fellow withthe Department of Electrical and Computer Engineering, University of Victoria,Vistoria, BC, Canada. He is the author or coauthor of more than 20 journal ar-ticles and book chapters, and more than 40 conference publications in the areasof neural networks, adaptive and neural control, fuzzy systems, and intelligenttransportation systems. He has been involved in various research projects fundedby the European Community, the National Sciences of Engineering ResearchCouncil (NSERC) of Canada, the National Aeromautics and Space Administra-tion (NASA), the Air Force, and the Department of Transportation involving vir-tual reality, fault detection and identification, manufacturing systems, robotics,fuzzy controllers, telecommunications, design and control of flexible and spacestructures, active isolation techniques for civil structures, control of hypersonicvehicles, automated highway systems, agile port technologies, and intelligenttransportation systems. He has served as a reviewer for various journals andconferences, and has served as the session chairman or cochairman for variousinternational conferences.

Petros A. Ioannou(S’80–M’83–SM’89–F’94) received the B.Sc. degree (withFirst Class Honors) from University College, London, U.K., and the M.S. andPh.D. degrees from the University of Illinois, Urbana, in 1978, 1980, and 1982,respectively.

From 1975 to 1978, he held a Commonwealth Scolarship from the Associationof Commonwealth Universities, London, U.K. From 1979 to 1982, he was a Re-search Assistant at the Coordinated Science Laboratory at the University of Illi-nois. In 1982, he joined the Department of Electrical Engineering-Systems, Uni-versity of Southern California, Los Angeles, where he is currently a Professor inthe same Department and the Director of the Center of Advanced TransportationTechnologies. His research interests are in the areas of adaptive control, neuralnetworks, vehicle dynamics, and control and intelligent vehicle and highway sys-tems. He has been an Associate Editor forThe International Journal of ControlandAutomatica. He has published five books and over 100 technical papers. Heis a member of the AVCS Committee of Intelligent Transportation Systems (ITS)America, and a Control Systems Society member of the IEEE Technical Activ-ities Board Committee on Intelligent Transportation Systems.

Dr. Ioannou was awarded several prizes, including the Goldsmid Prize and theA.P. Head prize from University College, London, U.K. In 1984, he was a recip-ient of the Outstanding Transactions Paper Award for his paper “An AsymptoticError Analysis of Identifiers and Adaptive Observers in the Presence of Para-sitics, ” which appeared in the IEEE TRANSACTIONS ONAUTOMATIC CONTROL

in August 1982. He was also the recipient of a 1985 Presidential Young Inves-tigator Award for his research in Adaptive Control. He has been an AssociateEditor for the IEEE TRANSACTIONS ONAUTOMATIC CONTROL

Robust switching adaptive control of multi-input …ioannou/2003update/d66.pdf610 IEEE TRANSACTIONS...

Documents

Transcript of Robust switching adaptive control of multi-input …ioannou/2003update/d66.pdf610 IEEE TRANSACTIONS...