Realizing the Action of a Cascade Compensator by State Feedback

Copyright © IFA C 11th Trie nnial World Congress, Tallinn , Esto nia, lISSR , 1990

REALIZING THE ACTION OF A CASCADE COMPENSATOR BY STATE FEEDBACK

v.

Kucera

I nstitute of Information Theory and Automation, Czechoslovak A cademy of S ciences, 182 08 Prague 8, Czechoslovakia

Abstract , The class of dynamic cascade compensators is characterized whose effect on a given linear generalized state-space system can be represented by a linear static state feedback . Internal properness and stability of the resulting closed-loop system are investigated. A simple procedure is proposed for the calculation of any and all feedback gains given the system and the compensator.

Keywords . Linear systems; Generalized state-space systems; State feedback ; Cascade compensation; System theory.

C(s) = [Im - LT(sJrl M .

INTRODUCTION

The aim of the paper is to study the converse problem: Given a compensator with rational transfer function C(s), can its action on the given system be realized by applying an admissible state feedback?

Consider a generalized state-space system governed by the equation Ex = Fx + Gu (1) where x ERn , u E R m and E , F and G are constant matrices with entries in R, the field of real numbers.

The answer is a qualified affirmative. We shall give conditions which single out precisely those C(s) whose action can be so realized . Furthermore, it will be shown which additional conditions imposed on C(s) correspond to internal properness and / or stability of the feedback system. The proofs are constructive and lead to a simple calculation of any and all feedback gains L, M given T( s) and

When sE - F is square and non-singular, the system (1) is said to be regular and gives rise to the transfer function

T(s) = (sE - F)-lG,

(2)

C(s) .

which is a rational n X m matrix. For background , motivation and elements of the theory of such systems the reader is referred to Lewis (1986) .

These results generalize those obtained by Hautus and Heymann (1978) for controllable state-space systems

The application of state feedback u =

Lx+Mv,

x= (3)

+ GL)x + GMv.

(4)

We say the feedback (3) is admissible if both Im - LT(s) and M are non-singular matrices. The resulting system is then regular and gives rise to the transfer function

TL ,M( S) = T(s) [Im - LT(s Jr l M .

Fx

+ Gu ,

(7)

namely that the action of C(s) on (7) can be implemented by an admissible state feedback (3) if and only if C(s) is biproper and for every polynomial vector u(s) such that T( s)u(s) is polynomial, C-l(s)u(s) is also polynomial. Results like these provide a vehicle for the study of state feedback control problems by means of algebraic properties of transfer functions .

where v E Rm and L,M are constant matrices with entries in R, results in the closed-loop system

Ex = (F

(6)

(5)

ADMISSIBLE FEEDBACK

Obviously, the effect of admissible state feedback on the given system can be represented by a dynamic compensator cascaded with the system and having the transfer function

The application of state feedback (3) may result in a nonregular closed-loop system (4), no matter whether the orig-

247

inal system (1) is regular or not. An admissible state feedback (3), however, preserves the regularity.

T(s) '" B(s)A - I(S)

(8)

where A(s) and B(s) are relatively right prime polynomial matrices such that the composite matrix

Lemma 1. Let sE - F be non-singular and T(s) defined by (2) . Then, for any compatible matrix L with entries in R, sE - F - CL is non-singular if and only if Im - LT(s) is non-singular.

A(s) ] [ B(s)

(9)

is column reduced .

Proof: We write (2) as In order to prove our main result, we shall need the following lemma, adapted from Kucera and Zagalak (1989), which provides a condition for a polynomial equation to have a constant solution .

(sE - F)T(s) '" C and compose it with CLT(s) to obtain

(sE - F - GL)T(s) '" G[Im - LT(s) l. If Im - LT(s) is non-singular, then

Lemma 2 . The equation

X(s) '" T(s)[Im - LT(sJrI is a rational solution of the equation

XP(s)

(sE - F - CL)X(s) '" G .

Cl

rank [sE - F - GL rank [sE - F

GI [

rank [sE - F

GI

In

-L and only if the rows of the matrix [

n R-linear space as those of [ so that sE - F - GL is non-singular.

yts) '" (sE - F - GLtlC

[

is a rational solution of the equation

X

o

span the same

Y ] [ P(S)]

In

Q(s)

-

[ R(s) ] Q(s) .

P(s)] [R(S)] [ Q(s) , Q(s)

Hence, over R(s)'

LT(s)1

].

]

If X is non-singular, the constant transformation matrix is also non-singular and the rows of the matrices

Y(s)[Im - LT(s) 1 '" T(s) .

-

~i:l

~i:l

Proof: For necessity, let X, Y be a solution pair of (10) with entries in R. Then

On the other hand, let sE - F - GL be non-singular. Then

rank[Im

(10)

for matrices P(s),Q(s) and R(s) of size m x m, n x m and m x m, respectively, with entries in R [sJ, the ring of polynomials in the indeterminate s over R , has a solution pair X, Y with entries in R such that X is non-singular if

Hence, over the field R(s) of rational functions,

rank(sE - F - CL)

+ YQ(s) '" R(s)

ran k [

Im - LT( s) ] T(s)

Im ran k [ 0

rank [

:~)

-L] [ Im

In

T(s)

span the same R-linear space . ]

To prove sufficiency, let TI and T2 be non-singular matrices with entries in R such that

]

m

so that Im - LT(s) is non-singular.

0

where the rows of [ We are now in a position to interpret the notion of admissible state feedback. The non-singularity of Im - LT( s) preserves the regularity of the system while the non-singularity of M preserves the control abilities of the input. Therefore, no loss of these crucial properties occurs through the application of an admissible state feedback.

~~i:l

]

as well as those of [

~~i:ll

are R-linearly independent, and where the rows of Pds and Rds) do not belong to the R-linear row space of Q(s) while those of Pd s) and R2(S) do. If the rows of [ and [

~i:l

]

~i:l

]

generate the same R-linear space, then so do

the rows of PI(S) and RI(s) . Let the number of these roW5 be p. Then there exists a non-singular p x p matrix Xl having entries in R such that

FEEDBACK IMPLEMENTATION We have seen that the action of an admissible state feedback (3) on a regular system (1) can be represented by a dynamic precompensator with transfer function (6). To study the converse problem, we start by writing the transfer function (2) of system (1) in the matrix fraction form

and matrices X 2 , Y2 with entries in R such that

248

which is equivalent to

So we have

PI(S) P2 (s)

1

C(s)

o

= [Im -

LT(s) ]-1 M.

D

In case the transfer function (8) of system (1) is strictly proper, the column degrees of A(s) dominate in (9),

Q(s) It follows that the pair

degcoli A(s) = 1 + degcoli B(s),

i = 1,2, ... ,m

so that the highest-column-degree coefficient matrices satisfy

solves the equation (10) and X is non-singular .

A (S)] = [ A ( s ) h, [ B(s) h, 0

D

].

Hence the condition (c) of Theorem 1 holds if and only if

Theorem 1. Let (1) be a regular system , T(s) in (8) its transfer function and C(s) a rational m x m matrix with entries in R( s) . Then there exists an admissible state feedback (3) such that

degcoli A(s)

(a) C(s) is non-singular;

polynomial. We have thus obtained the result of Hautus and Heymann (1978) for ordinary state-space systems.

(b) C-I(s)A(s) is polynomial;

those of [

]

span the same R-linear space as

C-I(s)A(s) ] B(s) .

PROPERNESS AND STABILITY

Proof: To prove necessity, let constant m.atrices L, M defining an admissible feedback (3) exist and satisfy

C(s)

The feedback implementation of a cascade compensator has numerous practical advantages over the feedforward scheme. These include the possibility of designing a system which is internally proper and / or stable. It is therefore natural to ask the following question: Which additional properties of C(s) will result in an internally proper and/ or stable feedback system (4)?

[Im - LT(sJrI M A(s) [A(s) - LB(s) ]- IM .

The condition (a) is an immediate consequence of admissibility. Write

C- l A(s) = M-I [A(s) - LB(s) ],

We say with Kucera (1986) that a regular, generalized state-space system like (1) is internally proper if it has no infinite eigenvalue, i.e., if its free response x(t) is devoid of impulses at t = 0 for every initial state x(O - ).

which implies (b) . We then have

[

C-I(s)A(s) ] = B(~

= 1,2, ... , m

for a non-singular matrix U with entries in R . This is the case if and only if C(s) is a biproper matrix. Furthermore, the condition (b) is equivalent to C-l(s)u(s) being polynomial for every polynomial u(s) such that T(s)u(s) is also

if and only it the following conditions all hold:

;~:l

i

and

C(s) = [Im - LT(s) ]-IM

(c) the rows of [

= degcoli C-I(s)A(s),

[M0

l

l -ML- ] [ A(s) ] ~

B(~

We say that a regular, generalized-state space system like (1) is internally stable if it has no finite unstable eigenvalues, i.e., if its free response x(t) tends to zero when t -> 00 for every initial state x( -0) .

which shows that the polynomial matrices

A(S)] [C - I(S)A(S)] [ B(s)' B(s) are related by a contant non-singular matrix. This proves

Only controllable eigenvalues can be altered by feedback . A finite eigenvalue So of (1) is said to be controllable if

(c) . For sufficiciency, let (a), (b) hold and consider the equation

XA(s) + Y B(s) = C-I(s)A(s)

rank [sE - F

(11)

C] = n

for s = So while the infinite eigenvalue of (1) is said to be controllable if

in polynomial matrices. In view of (c), the hypotheses of Le=a 1 are satisfied. It follows that (11) has a constant soluton pair X, Y such that X is non-singular. Put

rank [(l / s)E - F

C] = n

for s = 0, see Verghese, Levy, and Kailath (1981) and Cobb (1984). Then (11) yields

M - I [A(s) - LB(s) ] = C- I(s)A(s)

(12)

Let k denote the total column degree of the polynomial

249

matrix (9), that is ,

k

=

~degcoli m

[

Thus (4) has no finite unstable controllable eigenvalues if and only if det C-l(s)A(s) is a strictly Hurwitz polynomial.

A(s) ] B(s) .

Now the finite non-controllable eigenvalues of (4), if any, are those of (1). By assumption, they are all stable . Thus (4) yields no unstable behaviour at all if and only if detC - l(s)A(s) is a strictly Hurwitz polynomial. 0

Then we have the following result. Theorem 2. Let (1) be a regular system whose infinite eigenvalue is controllable, let T(s) in (8) be its transfer function and C(s) a rational m x m matrix. Suppose that there exists an admissible state feedback (3) such that

CONSTRUCTION

C(s) = IIm - LT(sJrl M. Then the resulting closed-loop system (4) is internally proper if and only if degdetC - l(s)A(s) = k .

The sufficiency proof of Theorem 1 is constructive and provides a procedure for the calculation of L and M given T(s) and C(s). The procedure can be summarized as follows.

(13)

(a) Factorize T(s) as in (8),

T(s) = B(s)A-l(S),

Proof: It follows from (4) and (5) that

(sE - F - GLtlG = B(sJlA(s) - LB(sJrl .

where A(s) and B(s) are relatively right prime polynomial matrices such that

Therefore the number of controllable eigenvalues of (4), counted with multiplicities, is equal to

A(s) ] [ B( s)

~d ~

is column reduced.

1. [ A(s) - LB(s) ] _ egco. B(s) -

(b) Calculate C-l(s)A(s); if it fails to be polynomial, stop.

~degCOli [~m ~L] [ ~~:~ ]

(c) Solve the equation (U),

~degCOli [ ~~:~ ]

XA(s)

+ Y B(s)

= C- l(s)A(s),

for a constant solution pair X, Y with X non-singular; if it fails to exist, stop. (d) Put L = _X-ly, M = X - l.

k.

Theorem 1 implies that C-l(s)A(s) is a polynomial matrix. The number of finite controllable eigenvalues of (4), multiplicities included, is given by (12) as

In general, there are more that one pair of matrices L, M that implement the action of the given C(s) upon T(s). Any and all such matrices are obtained by this procedure.

degdetIA(s) - LB(s)] = degdetC - l(s)A(s) . Hence (4) has no infinite controllable eigenvalue if and only if (13) holds. Now the non-controllable eigenvalues of (4), if any, are those of (1) . By assumption they are all finite . Thus (4) has no impulsive behaviour at all if and only if (13) holds. 0

EXAMPLES Two illustrative examples are included.

Theorem 3 . Let (1) be a regular system whose finite unstable eigenvalues are all controllable, let T(s) in (8) be its transfer function and C(s) a rational m x m matrix. Suppose that there exists an admissible state feedback (3) such that C(s) = IIm - LT(s)]-lM.

Example 1 . Consider a system (1) h- --ing the coefficient matrices

E=

Then the resulting closed-loop system (4) is internally stable if and only if detC - l(s)A(s) is a strictly Hurwitz poly-

0 0 0 0

1 0 0 0

0 0 0 0 1 0 0

,F=

1 0 0 0

0 1 0 0

0 0 0 0

and the transfer function

nomial.

T(s) =

Proof: It follows from (4) and (5) that

(sE - F - GL)-lG = B(s)IA(s) - LB(s)]-l . Then the compensator

Hence, the finite controllable eigenvalues of (4) are given by det[A(s) - LB(s)] . Theorem 1 implies that C - l(s)A(s) is a polynomial matrix. Using (12), det C-l(s)A(s) is equal to det[A(s) - LB(s)] up to a non-zero constant multiplier.

C(s) _

- s2

250

s

+ 2s + 1

0

: l'

-1

G=

0 -1 1 0

satisfies the conditions (a) through (c) of Theorem 1 and its effect upon (1) can be realized by any state feedback (3) having the gains

L

= [-a

with arbitrary a

#

1 - 2a 0 and

-

Cl

13], M =

CONCLUSIONS Given a regular, generalized state-space system (1), the transfer function C(s) of all dynamic cascade compensators whose effect on (1) can be realized by an admissible state feedback (3) has been described in Theorem 1. The properties of C(s) which correspond to internal properness and

Cl

13 .

stability of the closed-loop system (4) have been given in Theorems 2 and 3, respectively.

Applying Theorems 2 and 3, the resulting closed-loop system is seen to be internally proper and stable; it has one non-controllable eigenvalue equal to -1. A direct implementation of the feedforward compensator would result in an unstable system. 0

A major role is played by the matrix C - 1(s)A(s). It must be a polynomial matrix for a feedback implementation of C(s) to exist, and its determinant is the characteristic polynomial of the controllable part of the closed-loop system.

Example 2 . Consider a single-input regular system (1) over R with transfer function

T(s)

Theorem 1 extends Theorem 5.10 of Hautus and Heymann (1978) to the realm of generalized, not necessarily controllable, state-space systems. The result is constructive and provides a useful tool for the study of state feedback control problems by means of algebraic properties of transfer function.

= B(s) a(s)

where a(s) is a polynomial that is relatively right prime with the polynomial vector B(s). We wish to characterize, in an explidt manner, all cascade compensator transfer functions C(s) whose effect on (1) is realizable by an admissible state feedback (3) .

REFERENCES

The conditions (a) and (b) of Theorem 1 imply that

C(s) = a(s)

Cobb, J. D . (1984). Controllability, observability, and duality in singular systems. IEEE Trans. Automat. Contr ., AC-29, 1076-1082 .

p(s) for a polynomial p(s)

# o.

By (c), the rows of the matrices

a( s)] [ B(s) ,

[p( s)

Hautus, M. L. J . and M. Heymann (1978). Linear feedback - an algebraic approach . SIAM J. Contr . Optimiz., 16, 83-105.

]

B(s)

Kulera, V. (1986) . Internal properness and stability in linear systems. Kybernetika, 22, 1-18.

span the same R-linear space. Hence if we let k denote the highest degree occurring in a(s) and B(s), then degp(s)

Kulera, V. and P. Zagalak (1989) . Constant solutions of polynomial equations . Preprints IFAC Workshop on System Structure and Control, Prague, Czechoslovakia. pp. 33-36.

:s k

with equality holding when T(s) is strictly proper. The integer k is known as the controllability index of (1), see Kulera and Zagalak (1989). 0

Lewis, F . L. (1986). A survey of linear singular systems. J. Circuits, Systems, Signal Proc ., Special Issue on Singular Systems,S, 3-36. Verghese, G. C., B. C. Levy and T. Kailath (1981) . A generalized state-space for singular systems. IEEE Trans. Automat. Contr., AC-26, 811-831.

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