Controlled and Conditioned In Variants in Linear Systems Theory

7/31/2019 Controlled and Conditioned In Variants in Linear Systems Theory

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Controlled and ConditionedInvariants in Linear

Systems Theory

G. Basile and G. Marro

Department of Electronics, Systems and Computer Science

University of Bologna, Italy

e-mail: gbasile, [email protected]

October 21, 2003


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iv Glossary

x n the n-norm of vector xx, y the inner or scalar product of vectors x and y

grad f the gradient of function f (x)sp{xi} tha span of vectors {xi}dimX the dimension of subspace X X the orthogonal complement of subspace X O(x, ) the -neighborhood of xintX the interior of set X cloX the closure of set X A, X matrices or linear transformationsO a null matrixI an identity matrixI n the n

n identity matrix

AT the transpose of AA the conjugate transpose of AA 1 the inverse of A (A square nonsingular)A+ the pseudoinverse of A (A nonsquare or singular)adj A the adjoint of Adet A the determinant of Atr A the trace of A(A) the rank of Aim A the image of A (A) the nullity of A

kerA the kernel of AA n the n-norm of A

A| I the restriction of the linear map A to the A-invariant I A|X / I the linear map induced by A on the quotient space X / I end of discussion

Let x be a real number, the signum fuction of x is dened as

sign x := 1 for x 01 for x < 0

and can be used, for instance, for a correct computation of the argument of the complexnumber z = u + jv :

|z| := u2 + v2 ,arg z := arcsin vu2 + v2 sign u + 2 1 sign u sign v ,where the co-domain of function arcsin has been assumed to be ( / 2, / 2].


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Glossary v

b) Specic symbols and abbreviations

J a generic invariantV a generic controlled invariantS a generic conditioned invariantmax J (A, C) the maximal A-invariant contained in Cmin J (A, B) the minimal A-invariant containing Bmax V (A, B, E ) the maximal ( A, B)-controlled invariantcontained in E min S (A, C, D) the minimal ( A, C)-conditioned invariantcontaining Dmax V R (A( p), B( p), E ) the maximal robust ( A( p), B( p))-controlledinvariant contained in E R the reachable set of pair ( A, B ):

R=min J (A, B), B:= im BQ the unobservable set of pair ( A, C ):

Q=max J (A, C), C:= ker C RE the reachable set on E :

RE = V min S (A, E , B), where V := max V (A, B, E ), E := ker E QD the unobservable set containing D:

QD = S

+max V (A, D, C), where S

:=min S (A, C, D), D:=im D(B+ D ,E ) the lattice of all ( A, B+ D)-controlled invariantsself-bounded with respect to E and containing D:(B+ D ,E ) := {V : AVV + B, DVE , VV B} (CE ,D) the lattice of all ( A, C)-conditioned invariantsself-hidden with respect to Dand contained in E : (CE ,D) := {S : A(S C)S , DSE , SS + C}V m the inmum of (B+ D ,E ) :

V m =

V

S 1 ,

S 1 :=min

S (A,

E ,

B+

D)

S M the supremum of (CE ,D ) :(S M = S + V 1 , V 1 =max V (A, D, CE ) )V M a special element of (B+ D ,E ) , dened as V M := V (V 1 + S 1 )S m a special element of (CE ,D ) , dened as S m := S + V 1 S 1


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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction to Systems 11.1 Basic Concepts and Terms . . . . . . . . . . . . . . . . . . . . . 11.2 Some Examples of Dynamic Systems . . . . . . . . . . . . . . . 31.3 General Denitions and Properties . . . . . . . . . . . . . . . . 101.4 Controlling and Observing the State . . . . . . . . . . . . . . . 201.5 Interconnecting Systems . . . . . . . . . . . . . . . . . . . . . . 24

1.5.1 Graphic Representations of Interconnected Systems . . . 241.5.2 Cascade, Parallel, and Feedback Interconnections . . . . 28

1.6 A Review of System and Control Theory Problems . . . . . . . 301.7 Finite-State Systems . . . . . . . . . . . . . . . . . . . . . . . . 35

1.7.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . 381.7.2 Reduction to the Minimal Form . . . . . . . . . . . . . . 431.7.3 Diagnosis and State Observation . . . . . . . . . . . . . 461.7.4 Homing and State Reconstruction . . . . . . . . . . . . . 491.7.5 Finite-Memory Systems . . . . . . . . . . . . . . . . . . 51

2 General Properties of Linear Systems 572.1 The Free State Evolution of Linear Systems . . . . . . . . . . . 57

2.1.1 Linear Time-Varying Continuous Systems . . . . . . . . 572.1.2 Linear Time-Varying Discrete Systems . . . . . . . . . . 61

2.1.3 Function of a Matrix . . . . . . . . . . . . . . . . . . . . 622.1.4 Linear Time-Invariant Continuous Systems . . . . . . . . 662.1.5 Linear Time-Invariant Discrete Systems . . . . . . . . . . 73

2.2 The Forced State Evolution of Linear Systems . . . . . . . . . . 762.2.1 Linear Time-Varying Continuous Systems . . . . . . . . 762.2.2 Linear Time-Varying Discrete Systems . . . . . . . . . . 792.2.3 Linear Time-Invariant Systems . . . . . . . . . . . . . . 802.2.4 Computation of the Matrix Exponential Integral . . . . . 842.2.5 Approximating Continuous with Discrete . . . . . . . . . 87

2.3 IO Representations of Linear Constant Systems . . . . . . . . . 892.4 Relations Between IO and ISO Representations . . . . . . . . . 92

2.4.1 The Realization Problem . . . . . . . . . . . . . . . . . . 94

vii


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Contents ix

4.5 Extensions to Quadruples . . . . . . . . . . . . . . . . . . . . . 2394.5.1 On Zero Assignment . . . . . . . . . . . . . . . . . . . . 243

5 The Geometric Approach: Synthesis 2475.1 The Five-Map System . . . . . . . . . . . . . . . . . . . . . . . 247

5.1.1 Some Properties of the Extended State Space . . . . . . 2505.1.2 Some Computational Aspects . . . . . . . . . . . . . . . 2555.1.3 The Dual-Lattice Structures . . . . . . . . . . . . . . . . 260

5.2 The Dynamic Disturbance Localization and the Regulator Problem2675.2.1 Proof of the Nonconstructive Conditions . . . . . . . . . 2705.2.2 Proof of the Constructive Conditions . . . . . . . . . . . 2745.2.3 General Remarks and Computational Recipes . . . . . . 2805.2.4 Sufficient Conditions in Terms of Zeros . . . . . . . . . . 285

5.3 Reduced-Order Devices . . . . . . . . . . . . . . . . . . . . . . . 2865.3.1 Reduced-Order Observers . . . . . . . . . . . . . . . . . 2905.3.2 Reduced-Order Compensators and Regulators . . . . . . 292

5.4 Accessible Disturbance Localization and Model-Following Control 2955.5 Noninteracting Controllers . . . . . . . . . . . . . . . . . . . . . 298

6 The Robust Regulator 3076.1 The Single-Variable Feedback Regulation Scheme . . . . . . . . 3076.2 The Autonomous Regulator: A General Synthesis Procedure . . 312

6.2.1 On the Separation Property of Regulation . . . . . . . . 321

6.2.2 The Internal Model Principle . . . . . . . . . . . . . . . 3236.3 The Robust Regulator: Some Synthesis Procedures . . . . . . . 3246.4 The Minimal-Order Robust Regulator . . . . . . . . . . . . . . . 3356.5 The Robust Controlled Invariant . . . . . . . . . . . . . . . . . 338

6.5.1 The Hyper-Robust Disturbance Localization Problem . . 3436.5.2 Some Remarks on Hyper-Robust Regulation . . . . . . . 345

A Mathematical Background 349A.1 Sets, Relations, Functions . . . . . . . . . . . . . . . . . . . . . 349

A.1.1 Equivalence Relations and Partitions . . . . . . . . . . . 357A.1.2 Partial Orderings and Lattices . . . . . . . . . . . . . . . 359

A.2 Fields, Vector Spaces, Linear Functions . . . . . . . . . . . . . . 363A.2.1 Bases, Isomorphisms, Linearity . . . . . . . . . . . . . . 367A.2.2 Projections, Matrices, Similarity . . . . . . . . . . . . . . 372A.2.3 A Brief Survey of Matrix Algebra . . . . . . . . . . . . . 376

A.3 Inner Product, Orthogonality . . . . . . . . . . . . . . . . . . . 380A.3.1 Orthogonal Projections, Pseudoinverse of a Linear Map . 384

A.4 Eigenvalues, Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 387A.4.1 The Schur Decomposition . . . . . . . . . . . . . . . . . 391A.4.2 The Jordan Canonical Form. Part I . . . . . . . . . . . . 392A.4.3 Some Properties of Polynomials . . . . . . . . . . . . . . 397A.4.4 Cyclic Invariant Subspaces, Minimal Polynomial . . . . . 398


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Chapter 1

Introduction to Systems

1.1 Basic Concepts and Terms

In this chapter standard system theory terminology is introduced and explainedin terms that are as simple and self-contained as possible, with some representa-tive examples. Then, the basic properties of systems are analyzed, and conceptssuch as state, linearity, time-invariance, minimality, equilibrium, controllability,and observability are briey discussed. Finally, as a rst application, nite-statesystems are presented.

Terms like system, system theory, system science, and system engi-neering have come into common use in the last three decades from various elds(process control, data processing, biology, ecology, economics, traffic-planning,

electricity systems, management, etc.), so that they have now come to assumevarious shades of meaning. Therefore, before beginning our treatment of sys-tems, we shall try to exactly dene the object of our study and outline the classof problems, relatively restricted, to which we shall refer in this book.

The word system denotes an object, device, or phenomenon whose timeevolution appears through the variation of a certain number of measurableattributes as with, for example, a machine tool, an electric motor, a computer,an articial satellite, the economy of a nation.

A measurable attribute is a characteristic that can be correlated with one ormore numbers, either integer, real or complex, or simply a set of symbols. Ex-amples include the rotation of a shaft (a real number), the voltage or impedancebetween two given points of an electric circuit (a real or complex number), anycolor belonging to a set of eight well-dened colors (an element of a set of eightsymbols; for instance, digits ranging from 1 to 8 or letters from a to h), theposition of a push button (a symbol equal to 0 or 1, depending on whether it isreleased or pressed). In dealing with distributed-parameter systems, attributescan be represented by real or complex-valued functions of space coordinates.Examples include the temperature along a continuous furnace (a real functionof space), the voltage of a given frequency along a transmission line (a complexfunction of space coordinates).

In order to reproduce and analyze the behavior of a system, it is necessaryto refer to a mathematical model which, generally with a certain approxima-

1


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1.3. General Denitions and Properties 11

4. a state set X 5. an output set Y The values and functions belonging to the above sets are called admissible .

Only two possibilities will be considered for the time set: T = R (time ismeasured by a real number) and T = Z (time is measured by an integer number).It is worth noting that the properties required for a set to be a time set from astrict mathematical viewpoint are fewer than the properties of either R or Z ; forinstance, multiplication does not need to be dened in a time set. Nevertheless,since the familiar R and Z t our needs, it is convenient to adopt them as theonly possible time sets and avoid any subtle investigation in order to nd outwhat is strictly required for a set to be a time set. On the basis of this decision,the following denitions are given.

Denition 1.3.1 (continuous-time and discrete-time system) A system is said to be continuous-time if T = R , discrete-time if T = Z .Denition 1.3.2 (purely algebraic system) A memoryless or purely algebraicsystem is composed of sets T , U , Y , and an input-output function or input-output map:

y(t) = g(u(t), t ) (1.3.1)

Denition 1.3.3 (dynamic continuous-time system) A dynamic continuous-time system is composed of sets

T ( = R ),

U ,

U f ,

X ,

Y of a state velocity

function indexstate velocity function

x(t) = f (x(t), u(t), t) (1.3.2)

having a unique solution for any admissible initial state and input function and of an output function or output map

y(t) = g(x(t), u(t), t ) (1.3.3)

Denition 1.3.4 (dynamic discrete-time system) A dynamic discrete-timesystem is composed of sets T ( = Z ), U , U f , X , Y of a next-state function 4

x(i + 1) = f (x(i ), u(i ), i ) (1.3.4)and of an output function or output map

y(i ) = g(x(i ), u(i ), i ) (1.3.5)

The following denition refers to a specialization of dynamic systems thatoccurs very frequently in practice.

Denition 1.3.5 (purely dynamic system) A purely dynamic system is onein which the output map reduces to

y(t) = g(x(t), t ) (1.3.6)


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20 Chapter 1. Introduction to Systems

Property 1.3.4 (decomposability of state transition and response functions)In linear systems the state transition (response) function corresponding to theinitial state x0 and the input function u(

) can be expressed as the sum of the

zero-input state transition (response) function corresponding to the initial statex0 and the zero-state state transition (response) function corresponding to theinput function u().Proof. Given any admissible initial state x0 and any admissible input functionu|[t0 ,t ], assume in equations (1 .3.22, 1.3.23), :=1, :=1, x01 := x0, x02 :=0,u1():=0, u2() := u(); it follows that

(t, t 0, x0, u()) = (t, t 0, x0, 0) + (t, t 0, 0, u()) (1.3.24) (t, t 0, x0, u()) = (t, t 0, x0, 0) + (t, t 0, 0, u()) (1.3.25)

The former term of the above decomposition is usually referred to as the freemotion ( free response), the latter as the forced motion ( forced response ). Thefollowing properties are immediate consequences of the response decompositionproperty.

Property 1.3.5 Two states of a linear system are indistinguishable in [t0, t1]if and only if they generate the same free response in [t0, t1].

Property 1.3.6 A linear system is in minimal form if and only if for any initial instant of time t0 no different states generate the same free response.

1.4 Controlling and Observing the StateThe term controllability denotes the possibility of inuencing the motion x()or the response y() of a dynamical system by means of the input function(or control function) u()U f .In particular, one may be required to steer a system from a state x0 to x1or from an event ( t0, x0) to (t1, x1): if this is possible, the system is said to becontrollable from x0 to x1 or from (t0, x0) to (t1, x1). Equivalent statements are:the state x0 (or the event ( t0, x0)) is controllable to x1 (or to ( t1, x1)) andthe state x1 (or the event ( t1, x1)) is reachable from x0 (or from (t0, x0)).

Example. Suppose the electric motor in Fig. 1.5 is in a given state x0 att = 0: a typical controllability problem is to reach the zero state (i.e., to nullthe armature current, the angular velocity, and the angular position) at a timeinstant t1, (which may be specied in advance or not), by an appropriate choiceof the input function segment u|[0,t 1 ]; if this problem has a solution, state x0 issaid to be controllable to the zero state (in the time interval [ t0, t1]).

Controllability analysis is strictly connected to the denition of particularsubsets of the state space X , that is:1. the reachable set at the nal time t = t1 from the event (t0, x0)

R+ (t0, t1, x0) := {x1 : x1 = (t1, t0, x0, u()), u()U f } (1.4.1)


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62 Chapter 2. General Properties of Linear Systems

3. separation (if for at least one k, (i) := (i, k ) is nonsingular for all i):

(i, j ) = (i ) 1( j ) (2.1.20)

4. time evolution of the determinant:

det (i, j ) =i 1

k= j

det Ad(k) (2.1.21)

Also, the adjoint system concept and related properties can easily be ex-tended as follows. Proofs are omitted, since they are trivial extensions of thosefor the continuous-time case.

Denition 2.1.3 (discrete-time adjoint system) The linear time-varying sys-

tem p(i ) = AT d (i ) p(i + 1) ( Ad() real) (2.1.22)

or p(i ) = Ad(i ) p(i + 1) ( Ad() complex) (2.1.23)

if A( ) is complex, is called the adjoint system of system (2.1.16).Property 2.1.5 The inner product of a solution x(i ) of equation (2.1.16) and a solution p(i ) of equation (2.1.22) or (2.1.23) is a constant.

Property 2.1.6 Let (i, j ) be the state transition matrix of system (2.1.16)

and (i, j ) that of the adjoint system (2.1.22) or (2.1.23). Then T (i, j ) (i, j ) = I ( A() real) (2.1.24)

or (i, j ) (i, j ) = I ( A() complex) (2.1.25)

2.1.3 Function of a Matrix

Consider a function f : F F (F := R or F := C ) that can be expressed as aninnite series of powers, i.e.,f (x) =

i=0

ci xi (2.1.26)

The argument of such a function can be extended to become a matrix insteadof a scalar through the following denition.

Denition 2.1.4 (function of a matrix) Let A be an n n matrix with ele-ments in F and f a function that can be expressed by power series (2.1.26); function f (A) of matrix A is dened by f (A) :=

i=0

ci Ai (2.1.27)


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k et , > 0 k et , < 0

k et , = 0 k t2 et , > 0

k t2 et , ,< 0 k t2 et , = 0

(a) (b)

(c) (d)

(e) (f)

Figure 2.1. Modes corresponding to real eigenvalues.


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Proof. It has already been proved that im P is an A-invariant. Since all thecolumns of matrix P clearly belong to all other A-invariants containing B, imP coincides with min

J (A,

B).

Property 2.6.9 Refer to system (2.2.15, 2.2.16). The controllable set to theorigin is equal to the reachable set from the origin, i.e.,

Rt1 (0) = R+t1 (0) (2.6.38)Consider (2.6.5), which in this case can be written as

Rt1 (0) = e At 1 R+t1 (0)and note that any A-invariant is also an invariant with respect to eAt and e At , asimmediately follows from the power series expansion of the matrix exponential;equality (2.6.38) is implied by e At 1 being nonsingular.

It is remarkable that the expressions for R+t1 (0) and Rt1 (0) derived earlier areindependent of t1, i.e., in the case of linear time-invariant continuous systemsthe reachable subspace and the controllable subspace do not depend on thelength of time for control, provided it is nonzero.

From now on, the simple symbol Rwill be used for many sets referring tocontrollability of linear time-invariant continuous systems:

R:= min

J (A,

B) =

R+

t1(0) =

R

t1(0) =

W +

t1(0) =

W

t1(0) (2.6.39)

The subspace B:=im B will be called the forcing actions subspace and Rthe controllability set or controllability subspace . System (2.2.15, 2.2.16) will besaid to be completely controllable if R= X . In this case it is also customary tosay that the pair (A, B ) is controllable.

Observability of linear time-invariant systems is now approached in a similarway. Denote by Qt1 (u(), y()) and Q+t1 (u(), y()) the sets of all initial and nalstates compatible with input function u() and output function y() in [0, t1],also called the unobservable set and the unreconstructable set in [0 , t1] withrespect to the given input and output functions. By Property 2.6.5

Qt1 (0, 0)

and Q+t1 (0, 0) are subspaces of X , while by Property 2.6.6 Qt1 (u(), y()) andQ+t1 (u(), y()) are linear varieties contained in X .Property 2.6.10 In the case of linear time-invariant systems the following equalities hold:

Qt1 (0, 0) Qt2 (0, 0) for t1 t2 (2.6.40)Q+t1 (0, 0) Q+t2 (0, 0) for t1 t2 (2.6.41)

Proof. Initial states belonging to Qt2 (0, 0) cause zero free response in the timeinterval [0 , t2], which contains [0, t1], so that they also belong toQ

t1 (0, 0) and

(2.6.40) is proved. A similar argument proves (2.6.41).


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146 Chapter 3. The Geometric Approach: Classic Foundations

Since the columns of the transformed matrix A1 coincide with those of AT 1 ex-pressed in the new basis, and b is the rst vector of the new basis, A1 := T 11 A T 2and b1 := T 1

1b have the structures

A1 =

0 0 0 . . . 0 01 0 0 . . . 0 10 1 0 . . . 0 2... ... ... . . . ... ...0 0 0 . . . 0 n 20 0 0 . . . 1 n 1

b1 =

100...00

(3.3.18)

Let us now derive for A, b another structure related to the controllabilityassumption, called the controller canonical form . Assume the coordinate trans-

formation matrix T 2 := [q1 q2 . . . qn ] with

qn = bqn 1 = Aqn + n 1qn = Ab + n 1b. . . . . .q2 = Aq3 + 2qn = An 2b + n 1An 3b + . . . + 2bq1 = Aq2 + 1qn = An 1b + n 1An 2b + . . . + 1b

(3.3.19)

This can be obtained from the previous one by means of the transformationT 2 = T 1 Q, with

Q :=

1 . . . n 2 n 1 1 2 . . . n 1 1 0 3 . . . 1 0 0... . . .

......

...1 . . . 0 0 0

(3.3.20)

which is clearly nonsingular (the absolute value of its determinant is equal toone). Columns of Q express the components of basis T 2 with respect to basisT 1. Since

Aq1 = An b +n 1i=1 iA

ib = 0b = 0qnAqi = qi 1 i 1qn (i = 2 , . . . , n )matrix AT 2, partitioned columnwise, is

AT 2 = [ 0qn | q1 1qn | . . . | qn 2 n 2qn | qn 1 n 1qn ]so that, in the new basis, A2 := T 12 A T 2 and b2 := T

12 b have the structures

A2 =

0 1 0 . . . 0 00 0 1 . . . 0 00 0 0 . . . 0 0...

...... . . .

......

0 0 0 . . . 0 1

0 1 2 . . . n 2 n 1

b2 =

000...01

(3.3.21)


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3.3. Controllability and Observability 147

Matrices A1 and A2 in (3.3.18) and (3.3.21) are said to be in companion form .The controllability and the controller canonical form can easily be dualized, if pair ( A, c) is observable, to obtain the canonical forms relative to output , whichare the observability canonical form and the observer canonical form .

Four SISO Canonical Realizations. All the preceding canonical forms canbe used to derive SISO canonical realizations whose coefficients are directlyrelated to those of the corresponding transfer function. Refer to a SISO systemdescribed by the transfer function

G(s) = n sn + n 1sn 1 + . . . + 0sn + n 1sn 1 + . . . + 0

(3.3.22)

and consider the problem of deriving a realization ( A,b,c,d) with ( A, b) con-trollable. To simplify notation, assume n = 4. According to the above derivedcontrollability canonical form, the realization can be expressed as

z(t) = A1 z(t) + b1 u(t)y(t) = c1 z(t) + d u(t)

with

A1 =

0 0 0 01 0 0 10 1 0 20 0 1 3

b1 =

100

0c1 = [ g0 g1 g2 g3 ] d = 4

(3.3.23)

u1111 s 1s 1s 1s 1

z4

yd = 4

g0 g1 g2 g3

3 2 1

0

z1 z1 z2 z2 z3 z3 z4

Figure 3.4. The controllability canonical realization.

The realization is called controllability canonical realization . The corre-sponding signal-ow graph is represented in Fig. 3.4. Coefficients gi are relatedto i , i (i = 0 , . . . , 3) by simple linear relations, as will be shown. By applying


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with

A4 =

0 0 0 01 0 0

10 1 0 20 0 1 3

b4 =

0 1 2 3

c4 = [ 0 0 0 1 ] d = 4

(3.3.27)

where i = i 4 i (i = 0 , . . . , 3) are the same as in the controller canonicalrelization.The identity of i (i = 0 , . . . , 3) (the components of b4) with the correspond-

ing is on the left of (3.3.22) is proved again by using the Masons formula.Coefficients f i (i = 0 , . . . , 3) of the observer realization are consequently derivedfrom b3 = Q 1b4.

3.3.4 Structural Indices and MIMO Canonical Forms

The concepts of controllability and controller canonical forms will now be ex-tended to multi-input systems. Consider a controllable pair ( A, B ), where Aand B are assumed to be respectively n n and n p. We shall denote byb1, . . . , b p the columns of B and by p its rank. Vectors b1, . . . , b are as-sumed to be linearly independent. This does not imply any loss of generality,since, if not, inputs can be suitably renumbered. Consider the table

b1A b1

A2 b1. . .

b2 . . .

A b2 . . .A2 b2 . . .

bA b

A2 b(3.3.28)

which is assumed to be constructed by rows: each column ends when a vector isobtained that can be expressed as a linear combination of all the previous ones.This vector is not included in the table and the corresponding column is notcontinued because, as will be shown herein, also all subsequent vectors wouldbe linear combinations of the previous ones. By the controllability assumption,a table with exactly n linearly independent elements is obtained. Denote byr i (i = 1 , . . . , ) the numbers of the elements of the i-th column: the abovecriterion to end columns implies that vector Ar i bi is a linear combination of allprevious ones, i.e.,

Ar i bi =

j =1

r i 1

h=0

ijh Ah b j j


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3.4. State Feedback and Output Injection 159

which has a structure equal to that of A2, but with all signicant elements(i.e., those of the second, third, and ninth row) arbitrarily assignable. It is noweasily shown that the eigenvalues are also arbitrarily assignable. In fact, it ispossible to set equal to zero all the elements external to the 2 2, 44, and33 submatrices on the main diagonal, so that the union of their eigenvaluesclearly coincides with the spectrum of the overall matrix. On the other hand,the eigenvalues of these submatrices are easily assignable by a suitable choiceof the elements in the last rows (which are the coefficients, with sign changed,of the corresponding characteristic polynomials in monic form).

Only if. Suppose that ( A, B ) is not controllable, so that the dimension of

R=min J (A, B) is less than n. The coordinate transformation T = [ T 1 T 2] withimT 1 = RyieldsA := T 1A T = A11 A12O A22

B := T 1B = B1O (3.4.6)

where, in particular, the structure of B depends on Bbeing contained in R.State feedback matrix F corresponds, in the new basis, toF := F T 1 = [F 1 F 2] (3.4.7)

This inuences only submatrices on the rst row of A , so that the eigenvaluesof A22 cannot be varied.

If there exists a state feedback matrix F such that A + BF is stable, thepair (A, B ) is said to be stabilizable. Owing to Theorem 3.4.2 a completelycontrollable pair is always stabilizable. Nevertheless, the converse is not true:complete controllability is not necessary for ( A, B ) to be stabilizable, as thefollowing corollary states.

Corollary 3.4.1 Pair (A, B ) is stabilizable if and only if R:=min J (A, B) isexternally stable.Proof. Refer to relations (3 .4.6, 3.4.7) in the only if part of the proof of Theorem 3.4.2: since (A11 , B1) is controllable, by a suitable choice of F 1 it is

possible to obtain A11 + B1 F 1 having arbitrary eigenvalues, but it is impossibleto inuence the second row of A . Therefore, the stability of A22 is necessaryand sufficient to make A + B F (hence A + BF ) stable.

Similar results concerning output injection are easily derived by duality.Refer to the system represented in Fig. 3.8(b), described by

x(t) = ( A + G C ) x(t) + B v(t) (3.4.8)y(t) = C x(t) (3.4.9)

The more general result on pole assignment by state feedback (Theorem3.4.2) is dualized as follows.


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Proof. The proof of Theorem 3.4.2 has shown that by a suitable choice of the feedback matrix F a system matrix can be obtained in the block-companion form , i.e., with companion matrices on the main diagonal and the remainingelements equal to zero: the dimension of each matrix can be made equalto but not less than the value of the corresponding input structural index.The eigenvalues of these matrices can be arbitrarily assigned: if they are allmade equal to one another, owing to Lemma 3.4.1 a Jordan block of equaldimension corresponds to every companion matrix. Since multiplicity of aneigenvalue as a zero of the minimal polynomial coincides with the dimension of the corresponding greatest Jordan block (see the proof of Theorem 2.5.5), themultiplicity of the unique zero of the minimal polynomial is equal to the greatestinput structural index, i.e., to the controllability index. On the other hand, if the assigned eigenvalues were not equal to each other, the degree of the minimalpolynomial, which has all the eigenvalues as zeros, could not be less, since theeigenvalues of any companion matrix (hence of that with greatest dimension)have a multiplicity at least equal to the dimension of the corresponding Jordanblock in this matrix, which is unique owing to Lemma 3.4.1. In other words, inany case the degree of the minimal polynomial of a companion matrix is equalto its dimension.

The theorem just presented is very useful for synthesis as a complementon structure assignment of Theorem 3.4.2 on pole assignment: in fact, itstates a lower bound on the eigenvalue multiplicity in the minimal polynomial.

For instance, in the case of discrete-time systems, by a suitable state feedback(which sets all the eigenvalues to zero) the free motion can be made to convergeto zero in a nite time. The minimal achievable transient time is specied inthe following corollary.

Corollary 3.4.3 Let (A, B ) be controllable. By a suitable choice of F , matrix A + BF can be made nilpotent of order equal, at least, to the controllability index of (A, B ).

Proof. Apply the procedure described in the proof of Theorem 3.4.4 to obtainan A + BF similar to a block companion matrix with all eigenvalues zeroand with blocks having dimensions equal to the values of the input structuralindices. This matrix coincides with the Jordan form: note that a Jordanblock corresponding to a zero eigenvalue is nilpotent of order equal to itsdimension.

Theorem 3.4.4 and Corollary 3.4.3 are dualized as follows.

Theorem 3.4.5 Let (A, C ) be observable. A suitable choice of G allows, besidesthe eigenvalues to be arbitrarily assigned, the degree of the minimal polynomial of A + GC to be made equal, at least, to the observability index of (A, C ).


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As a consequence of Theorems 3.4.2, 3.4.3, and 3.4.6, it follows that, if the triple ( A,B,C ) is completely controllable and completely observable, theeigenvalues of the overall system represented in Fig. 3.11 are all arbitrarilyassignable. In other words, any completely controllable and observable dynamicsystem of order n is stabilizable with an output-to-input dynamic feedback (or,simply, output dynamic feedback ), i.e., through a suitable dynamic system, alsoof order n.

The duality between control and observation, a characteristic feature of lin-ear time-invariant systems, leads to the introduction of the so-called dual ob-servers or dynamic precompensators , which are also very important to charac-terize numerous control system synthesis procedures.

To introduce dynamic precompensators, it is convenient to refer to the blockdiagram represented in Fig. 3.12(a), which, like that in Fig. 3.9, represents theconnection of the observed system with a model; here in the model the purelyalgebraic operators have been represented as separated from the dynamic part,pointing out the three-map structure. The identity observer represented inFig. 3.10 is obtained through the connections shown in Fig. 3.12(b), in whichsignals obtained by applying the same linear transformation G to the outputsof both the model and the system are added to and subtracted from the forcingaction. These signals, of course, have no effect if the observer is trackingthe system, but inuence time behavior and convergence to zero of a possibleestimate error.

The identity dynamic precompensator is, on the contrary, obtained by exe-

cuting the connections shown in Fig. 3.12(c), from the model state to both themodel and system inputs. Also in this case, since contributions to inputs areidentical, if the system and model states are equal at the initial time, their sub-sequent evolutions in time will also be equal. The overall system, representedin Fig. 3.12(c), is described by the equations

x(t) = A x(t) + B F z (t) + B v(t) (3.4.27)z(t) = ( A + B F ) z(t) + B v(t) (3.4.28)

from which, by difference, it follows that

e(t) = A e(t) (3.4.29)

If the triple ( A,B,C ) is asymptotically stable (this assumption is not veryrestrictive because, as previously shown, under complete controllability andobservability assumption eigenvalues are arbitrarily assignable by means of adynamic feedback), once the transient due to the possible difference in theinitial states is nished, system and precompensator states will be identical atevery instant of time. If the considered system is completely controllable, thedynamic behavior of the precompensator can be inuenced through a suitablechoice of matrix F . For instance, an arbitrarily fast response can be obtainedby aptly assigning the eigenvalues of A + BF .


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3.4. State Feedback and Output Injection 169

+

++

_

+

+

+

+

u

u

u

x = A x + B uy = C x



y

y

y

z = A z +

z = A z +

z = A z +

z

z

z

C

C

C

B

B

B

G

G

F

F

v

(a)

(b)

(c)

Figure 3.12. Model, asymptotic observer and dynamic pre-compensator (dual observer).


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4.1. Controlled and Conditioned Invariants 217

The unassignable internal eigenvalues of V 0 are those of A22: since theyclearly coincide with the unassignable external eigenvalues of S 0 , the followingproperty holds.Property 4.1.20 V 0 is internally stabilizable if and only if S 0 is externally stabilizable.

The preceding argument reveals the existence of two interesting one-to-onecorrespondences between the elements of the lattices (B,C) and (C,B) and the in-variants of the linear transformation corresponding to A22 (which, as remarked,expresses (A + BF )|V 0 / (V 0 S 0 ) or (A + GC )|(V 0 + S 0 )/ S 0 ). More precisely, the twoone-to-one correspondences are set as follows: let r := dim( V 0 S 0 ), k :=dim S 0 ,and X be a basis matrix of a generic A22-invariant. The subspaces

V := im T I r OO X O OO O

S := im T I r O OO X OO O I k rO O O

(4.1.43)

are generic elements of (B, C) and (C, B) respectively.We shall now consider the extension of the concept of complementability,introduced for simple invariants in (Subsection 3.2.5), to controlled and condi-tioned invariants.

Denition 4.1.9 (complementable controlled invariant) Let

V ,

V 1, and

V 2 be

three controlled invariants such that V 1VV 2. V is said to be complementablewith respect to (V 1, V 2) if there exists at least one controlled invariant V c such that V V c = V 1V + V c = V 2

Denition 4.1.10 (complementable conditioned invariant) Let S , S 1, and S 2be three conditioned invariants such that S 1SS 2. S is said to be comple-mentable with respect to (S 1, S 2) if there exists at least one conditioned invariant S c such that S S c = S 1

S + S c = S 2In the particular case of self-bounded controlled and self-hidden conditioned

invariants, the complementability condition can still be checked by means of the Sylvester equation. In fact, they correspond to simple A22-invariants instructure (4.1.42).

The Sylvester equation can also be used in the general case. It is worthnoting that the complementability condition can be inuenced by the feedbackmatrices which transform controlled and conditioned invariants into simple(A + BF )-invariants or ( A + GC )-invariants.


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246 Chapter 4. The Geometric Approach: Analysis

39. , On a conjecture of Basile and Marro, J. Optimiz. Th. Applicat. , vol. 41,no. 2, pp. 371376, 1983.

40. Silverman, L.M., Inversion of multivariable linear systems, IEEE Trans. Au-tom. Contr. , vol. AC-14, no. 3, pp. 270276, 1969.

41. Wen, J.T., Time domain and frequency domain conditions for strict positiverealness, IEEE Trans. Autom. Contr. , vol. 33, no. 10, pp. 988992, 1988.

42. Wonham, W.M., Algebraic methods in linear multivariable control, System Structure , ed. A. Morse, IEEE Cat. n. 71C61, New York, 1971.

43. , Linear Multivariable Control: A Geometric Approach , Springer-Verlag, NewYork, 1974.

44. Wonham, W.M., and Morse, A.S., Decoupling and pole assignment in linearmultivariable systems: a geometric approach, SIAM J. Control , vol. 8, no. 1,

pp. 118, 1970.45. , Feedback invariants of linear multivariable systems, Automatica , vol. 8,

pp. 93100, 1972.


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5.1. The Five-Map System 249

u y

x2 = A2 x2

x1 = A1 x1 + A3 x2 +B1 u + D1 d

y = C 1 x1 + C 2 x2e = E 1 x1 + E 2 x2

x2

d e

Figure 5.2. Controlled system including an exosystem.Summing up, the following assumptions are introduced:

1. the pair (A1, B1) is stabilizable2. yje pair (A, C ) is detectable

Note that the plant is a well-dened geometric object, namely the A-invariantdened by

P := {x : x2 = 0} (5.1.8)The overall system represented in Fig. 5.1 is purely dynamic with two inputs,d and r , and one output, e. In fact, by denoting with

x := xz (5.1.9)

the extended state (controlled system and regulator state), the overall systemequations can be written in compact form as

x(t) = A x(t) + D d(t) + R r (t) (5.1.10)e(t) = E x(t) (5.1.11)

whereA := A + BKC BL

MC N D := D

O

R := BS R E := [ E O ]

(5.1.12)

Note that, while the quintuple ( A,B,C,D,E ) that denes the controlledsystem is given, the order m of the regulator and matrices K,L,M,N,R,S area priori unknown: the object of synthesis is precisely to derive them. Thus, theoverall system matrices A, R are also a priori unknown.

In some important synthesis problems, like the disturbance localization bydynamic compensator, and the regulator problem (which will both be stated inthe next section), input r is not present, so that the reference block diagramsimplies as in Fig. 5.3.


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250 Chapter 5. The Geometric Approach: Synthesis

y

x = A x + B u + D d

y = C xe = E x

d e

u

x = N z + M yu = L z + K y

Figure 5.3. Reference block diagram for the disturbance

localization problem by dynamic compensator, and the reg-ulator problem.

5.1.1 Some Properties of the Extended State Space

We shall now show that geometric properties referring to A-invariants in theextended state space reect into properties of ( A, B)-controlled and ( A, C)-conditioned invariants, regarding the controlled system alone. This makes itpossible to state necessary and sufficient conditions for solvability of the mostimportant synthesis problems in terms of the given quintuple ( A,B,C,D,E ).

The following property, concerning algebraic output-to-input feedback, isuseful to derive the basic necessary structural condition for dynamic compen-sator and regulator design. 1

Property 5.1.1 Refer to the triple (A,B,C ). There exists a matrix K such that a given subspace V is an (A + BKC )-invariant if and only if V is both an (A, B)-controlled and an (A, C)-conditioned invariant.Proof. Only if. This part of the proof is trivial because if there exists amatrix K such that ( A + BKC )

VV clearly there exist matrices F := KC and

G := BK such that V is both an ( A + BF )-invariant and an ( A + GC )-invariant,hence an (A, B)-controlled and an ( A, C)-conditioned invariant.If. Consider a nonsingular matrix T := [ T 1 T 2 T 3 T 4], with imT 1 = V C,im [T 1 T 2] = V , im [T 1 T 3] = C, and set the following equation in K :

K C [T 2 T 4] = F [T 2 T 4] (5.1.13)

Assume that C has maximal rank (if not, it is possible to ignore some outputvariables to meet this requirement and insert corresponding zero columns in thederived matrix). On this assumption C [T 2 T 4] is clearly a nonsingular square

1 See Basile and Marro [4.6], Hamano and Furuta [18].


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5.1. The Five-Map System 261

as a reference to derive more complex structures, like those that are used inconnection with quintuple ( A,B,C,D,E ) to solve synthesis problems. Thebasic property that sets a one-to-one correspondence between the lattice of all(A, B)-controlled invariants self-bounded with respect to Cand that of ( A, C)-conditioned invariants self-hidden with respect to B, is stated as follows.Property 5.1.4 Let V be any (A, B)-controlled invariant contained in C, and S any (A, C)-conditioned invariant containing B: then

1. V S is an (A, B)-controlled invariant;2. V + S is an (A, C)-conditioned invariant.

Proof. From

A (S C)S S B (5.1.42)A V V + B V C (5.1.43)

it follows that

A (V S ) = A (VSC)A V A (S C)(V + B) S = V S + BA ((V + S ) C) = A (V + S C) = A V + A (S C)V + B+ S = V + S

The fundamental lattices are dened as

(B,C) := {V : A V V + B, V C, V V 0 B} (5.1.44) (C,B) := {S : A (S C)S , S B, S S 0 + C} (5.1.45)

with

V 0 := max V (A, B, C) (5.1.46)S 0 := min S (A, C, B) (5.1.47)

Referring to these elements, we can state the following basic theorem.

Theorem 5.1.1 Relations

S = V + S 0 (5.1.48)V = S V 0 (5.1.49)

state a one-to-one function and its inverse between (B, C) and (C, B). Sumsand intersections are preserved in these functions.Proof. V + S 0 is an (A, C)-conditioned invariant owing to Property 5.1.4, self-hidden with respect to Bsince it is contained in S 0 + C. Furthermore

(V + S 0 ) V 0 = V + S 0 V 0 = V


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Conditions (5.2.32) and (5.2.33) imply the particular structures of matricesB and C . As far as the structure of A is concerned, note that the zerosubmatrices in the third row are due to the particular structure of B and to

V m being an (A, B)-controlled invariant, while those in the third column aredue to the particular structure of C and to S M being an (A, C)-conditionedinvariant. If the structural zeros in A , B , and C are taken into account, from(5.2.27, 5.2.28) it follows that all the possible pairs S , V can be expressed as

S = S m + im ( T 3X S ) and V = V m + im ( T 3X V ) (5.2.35)where X S , X V are basis matrices of an externally stable and an internallystable A33-invariant subspace respectively. These stability properties followfrom S being externally stabilizable and V internally stabilizable. Condition(5.2.29) clearly implies imX S

imX V , so that A33

is stable. Since, as hasbeen previously pointed out, S M is externally stabilizable and V m internallystabilizable, the stability of A33 implies the external stabilizability of S m andthe internal stabilizability of V M .8If. The problem admits a solution owing to Theorem 5.2.1 with S := S M and V := V M .Proof of Theorem 5.2.4. Only if. We shall rst present some generalproperties and remarks on which the proof will be based.

(a ) The existence of a resolvent pair ( S , V ) induces the existence of a secondpair (

S ,

V ) whose elements, respectively self-hidden and self-bounded, satisfyS R and VE , with R dened by (5.1.69) andE := {V : V m V V } (5.2.36)

Assume

S := ( S + S m ) S M = S S M + S m (5.2.37)V := V + V m (5.2.38)Note that S S M is externally stabilizable since S and S M are, respectively byassumption and owing to Lemma 4.2.1, so that

S is externally stabilizable as

the sum of two self-hidden conditioned invariants, one of which is externallystabilizable. V P contains Dand is internally stabilizable by assumption andV m is internally stabilizable owing to Lemma 4.2.1. Furthermore, V mP : infact, refer to the dening expression of V m - (5.1.70) - and note that

S 1 := min S (A, E , B+ D)min J (A, B+ D)P The intersection V P = V P + V m is internally stabilizable since both con-trolled invariants on the right are. On the other hand, V being externallystabilizable implies that also V is so because of the inclusion VV .

8 External stabilizability of

S m , which is more restrictive than that of

S M , is not considered

in the statement, since it is a consequence of the other conditions.


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5.2. The Dynamic Disturbance Localization and the Regulator Problem 279

We shall now review all the points in the statement, and prove the neces-sity of the given conditions. Condition (5.2.12) is implied by Theorem 5.2.2,in particular by the existence of a resolvent pair which satises

DSVE .

Let V be a resolvent, i.e., an externally stabilizable ( A, B)-controlled invariant:owing to Property 4.1.13 V + R is externally stable; since VV , V + R isalso externally stable, hence V is externally stabilizable and (5.2.13) holds. If there exists a resolvent S , i.e., an (A, C)-conditioned invariant contained in E ,containing Dand externally stabilizable, S M is externally stabilizable owing toLemma 4.2.2. Thus, the necessity of (5.2.14) is proved. To prove the necessityof (5.2.15), consider a resolvent pair ( S , V ) with S R and VE , whose ex-istence has been previously proved in point a . Clearly, V L := V P belongs toR . On the other hand, S P , which is a conditioned invariant as the inter-section of two conditioned invariants, belongs to R (remember that

S m

V m

and V mP ); furthermore, subspaces V L P and V M P clearly belong to R .From SV it follows that SPV L P ; moreover, clearly V L PV M P .Owing to points b and c, S P , V L P , V M P , S , correspond to invariantsJ 1, J 2, J 3, J 4 of matrix P such that

J 1 J 2 J 3 and J 1 J 4S being externally stabilizable J 4 and, consequently, J 3 + J 4, is externallystable. Therefore, considering that all the eigenvalues external to J 3 are theunassignable ones between V M P and V M , J 3 + J 4 must be the whole spaceupon which the linear transformation expressed by P is dened. This featurecan also be pointed out with the relation

V M P + S = V M Matrix P is similar to

P 11 O P 22 OO O P 33 OO O O P 44

where the partitioning is inferred by a change of coordinates such that the rstgroup corresponds to J 1, the rst and second to J 2, the rst three to J 3 andthe rst and fourth to J 4. The external stabilizability of S implies the stabilityof P 22 and P 33 , while the internal stabilizability of V L P implies the stabilityof P 11 and P 22: hence J 3 is internally stable, that is to say, V M P is internallystabilizable. A rst step toward the proof of complementability condition(5.2.16) is to show that any resolvent VE is such that V p := V + V M P isalso a resolvent and contains V M . Indeed, V p is externally stabilizable becauseof the external stabilizability of V ; furthermore V p P is internally stabilizablesince

(V + V M P ) P = V P + V m P


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From SV and (5.2.46) it follows that

V M =

S +

V M

P

V +

V M

P Owing to point f , considering that all the exogenous modes are unstable, andV p and V are externally stabilizable, it follows that

V p + P = V + P = X where X denotes the whole state space of the controlled system, hence

V p + V P = V (5.2.47)Owing to d , the controlled invariants V M + V p P , V M + V P and V p corre-spond to Q-invariants

K1,

K2,

K3, such that

K1K2,

K1K3. Note also that,

owing to (5.2.47), K2 + K3 is the whole space on which the linear transformationexpressed by Q is dened. Therefore, matrix Q is similar toQ11 Q13O OO O Q33

where the partitioning is inferred by a change of coordinates such that the rstgroup corresponds to K1, the rst and second to K2, the rst and third to K3.Submatrix Q11 is stable since its eigenvalues are the unassignable ones internalto

V p P , while Q

33has all its eigenvalues unstable since, by the above point

e, they correspond to the unassignable ones between V P and V . Therefore,K1 is complementable with respect to ( {0}, K3); this clearly implies that K2is complementable with respect to ( {0}, K2 + K3), hence (5.2.16) holds for thecorresponding controlled invariants.

If. Owing to the complementability condition (5.2.16), there exists a con-trolled invariant V c satisfying

V c (V M + V P ) = V M (5.2.48)V c + ( V M + V P ) = V (5.2.49)

We will show that ( S M , V c) is a resolvent pair. Indeed, by (5.2.49) V M V c, andby (5.2.12) and (5.2.49) DS M V cE , since S M V M . Adding P to bothmembers of (5.2.48) yields V c + P = V + P : owing to f , (5.2.13) implies theexternal stabilizability of V c. By intersecting both members of (5.2.48) with P and considering that V cV , it follows that V c P = V M P , hence by (5.2.15)V c P is internally stabilizable.

5.2.3 General Remarks and Computational Recipes

The preceding results are the most general state-space formulations on theregulation of multivariable linear systems. Their statements are quite simple


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and elegant, completely in coordinate-free form. The constructive conditionsmake an automatic feasibility analysis possible by means of a computer, havingthe ve matrices of the controlled system as the only data for both problemsconsidered (disturbance localization by means of a dynamic output feedbackand asymptotic regulation). This automatic check is particularly interesting inthe multivariable case, where loss of structural features for implementability of agiven control action may arise from parameter or structure changes (a structurechange may be due, for instance, to interruption of communication channels inthe overall system).

However, a completely automatic synthesis procedure based on the afore-mentioned constructive conditions is not in general satisfactory for the followingreasons:

1. When the existence conditions are met, in general the problem admits severalsolutions, which are not equivalent to each other; for instance, if the plantis stable, regulation can be obtained both by means of a feedforward or afeedback controller since either device satises the conditions of Problem5.1.2 but, in general, feedback is preferable since it is more robust againstparameter variation or uncertainty;

2. The order of the regulator derived in the constructive proofs of Theorems5.2.2 and 5.2.4 is quite high (the plant plus the exosystem order); however,it is worth noting that this is the maximal order that may be needed, sincethe regulator has both the asymptotic tracking and stabilization functions,and is actually needed only if the controlled plant is strongly intrinsicallyunstable.

Both points 1 and 2 will be reconsidered in the next chapter, where a newformulation and solution for the regulator problem will be presented. It isa particular case of that discussed in this chapter, but specically orientedtoward the achievement of robustness and order reduction, hence more similar tothe formulations of synthesis problems for standard single-input, single-outputautomatic control systems.

An Example. To illustrate these arguments, a simple example is in order.Consider the single time constant plant with input u and output c described by

G(s) =K 1

1 + 1 s(5.2.50)

and suppose that a controller is to be designed such that the output of the plantasymptotically tracks a reference signal r consisting of an arbitrary step plus anarbitrary ramp. Thus, the exosystem is modeled by two integrators in cascadehaving arbitrary initial conditions, and the tracking error, which is required toconverge to zero as t approaches innity, is dened as

e(t) = r (t) c(t) (5.2.51)


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The solution presented in Fig. 5.13 is a typical feedforward control systemand corresponds to

N := 1 2

M := 1 1 2L :=

1K v

1 T 2

K := 0 T K r 2In this case, to achieve the prescribed behavior the structural conditions

K r = K 1 and T = 1 + 2 (5.2.53)

must be satised. Note that (5.2.52) is expressed by an inequality so that, if the choice of T is sufficiently conservative, it continues to be satised also in

the presence of small parameter changes, and the feedback scheme maintainsthe regulation property or is robust with respect to parameter variation oruncertainty. On the other hand, the strict equalities in (5.2.53) are bothnecessary for the regulation requirement to be met, so that the feedforwardscheme is not robust. In the feedback case robustness is achieved through twosignicant, basic features: the controlled variable e (i.e., the tracking errorwhich must be asymptotically nulled) is measured or computed without anyerror (the summing junction is assumed to be perfect), and an internal model of the exosystem is embodied in the regulator (the double pole at the origin).In this way a replica of the signal to be asymptotically tracked is internallygenerated in the extended plant and automatically trimmed in the presenceof stability of the overall system to null the effects of any parameter variationon the asymptotic behavior of e and exactly track the reference input r . Inboth schemes the regulator has a relatively low order (two in the feedbackcase, one in feedforward), but the plant is per se stable. Of course, a purelyfeedforward control cannot be implemented when the plant is unstable: in thiscase a dynamic feedback must be added to it, at least to achieve stability,independently of regulation.

We shall now consider compensator and regulator synthesis in greater detailand show that the synthesis procedures presented to prove sufficiency of the

conditions stated in Theorems 5.2.1 and 5.2.3 can be framed in a more generalcontext, which will be used in the next section for complete treatment of reduced-order observers, compensators, and regulators.

The procedures used in the proofs of the preceding theorems lead to synthesisof compensators and regulators called full-order since their state dimensioncoincides with that of the controlled system (including the exosystem) andobserver-based since they are realized according to the scheme of Fig. 5.5(a),in which state feedback is obtained through an observer. They are exactlydualizable: thus full-order compensators and regulators dual observer-based are obtained. These are realized according to the scheme of Fig. 5.5(b), inwhich output injection is obtained through a dual observer. We shall nowbriey present the algebraic manipulations concerning this dualization.


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Controlled and Conditioned In Variants in Linear Systems Theory

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Transcript of Controlled and Conditioned In Variants in Linear Systems Theory