Fundamental theorem of state feedback for singular systems

24, No. 5, pp. 653-658, 1988 Printedin Great Britain.

0005-1098/88$3.00+ 0.00 PergamonPresspie (~ 1988InternationalFederationof AutomaticControl

Autoraatica, Vol.

Fundamental Theorem of State Feedback for Singular Systems* V. K U ~ E R A t $ and P. Z A G A L A K t The dynamics o f all p r o p e r systems obtainable by state f e e d b a c k f r o m a singular system are completely characterized. Key Words--Linear systems; singular systems; pole placement; state feedback; controllability; matrix polynomial equation.

of the Kronecker form of (1)

Abstract--The limits of state feedback in altering the dynamics of singular systems are studied. A necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback from the singular system. The condition consists of inequalities which involve the controllability indices of the singular system and the degrees of the invariant polynomials. The development is based on the properties of polynomial matrices. A procedure is given for the calculation of a feedback gain which achieves the desired dynamics.

(2)

E2.~2 = Xx + G2u

(3)

with x: • R",, x2 • R"~ where n: = deg det (sE F) and E 2 is nilpotent of rank rE. Then (2) is a strictly proper system having n: finite poles and no infinite ones, while (3) is a polynomial system having r2 infinite poles and no finite ones. For more background material see Verghese et al.

INTRODUCTION

RECENT PAPERS by numerous researchers have

sought to generalize many of the fundamental results of linear system theory to the realm of singular (descriptor, generalized state-space, semi-state, differential-algebraic, implicit) linear systems EYe = Fx + Gu

icl = F:xl + Glu

(1981). It is our intent to study the dynamics of the systems which can be obtained from (1) by applying state feedback u = -Kx,

(1)

(4)

where K is a constant matrix. In particular, we are interested in determining the limits of state feedback (4) in altering the dynamical behaviour of (1).

where x • R n, u • R " and E, F, G are constant matrices with E possibly singular of rank r. The dynamics of (1) are completely characterized by the pencil s E - F. We assume that system (1) is regular, i.e. s E - F is invertible, and say it is strictly proper, proper and polynomial if the rational matrix ( s E - F)-: is respectively, strictly proper, proper and polynomial. Although our results are in terms of (1), proofs are more conveniently coached in terms

For the strictly proper system (2), whose dynamics can be fully described by the invariant polynomials of s/,, - F~, the question was settled by Rosenbrock (1970, Chapter 5, Theorem 4.2) in terms of inequalities which involve the controllability indices of (2) and the degrees of the desired invariant polynomials. This result is referred to as the fundamental theorem of state feedback. For a general system (1), the first question which arises is whether a state feedback (4) exists which makes (1) proper, that is to say, which brings all of its infinite poles to finite positions. An affirmative answer was given by Cobb (1981) who actually showed that under the condition of impulse controllability on (1) these poles can be assigned to arbitrary positions. Armentano (1984) and Ozqaldiran and Lewis (1984) then showed how to calculate the

* Received 12 February 1987; revised 15 February 1988; received in final form 10 March 1988. The original version of this paper was presented at the 10th IFAC World Congress which was held in Munich, F.R.G. during July 1987. The Published Proceedings of this IFAC Meeting may be ordered from: Pergamon Press pie, Headington Hill Hall, Oxford OX30BW, England. This paper was recommended for publication in revised form by Associate Editor E. Kreindler under the direction of Editor H. Kwakernaak. t Institute of Information Theory and Automation, Czechoslovak Academy of Sciences, 182 08 Prague, Czechoslovakia. $ Author to whom correspondence should be addressed. 653

654

V. KLI('ERA and P.

feedback gain K which results in the specified pole positions. These results, however, give only a partial picture as to what can be accomplished by state feedback (4) with regard to altering the dynamics of system (1). The complete answer is given by the analogue of Rosenbrock's inequalities to be obtained later in the paper. These inequalities provide necessary and sufficient conditions for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback (4) from singular system (1). This is a stronger result than those of Cobb, Armentano, and Ozgaldiran and Lewis in that complete structure of repeated poles is considered, as implied by the invariant polynomials. PROBLEM STATEMENT Consider a singular system governed by equation (1) with rank E = r and rank G = m. Let cl(s), cz(s) . . . . , Cm(S) be monic polynomials with coefficients in R which satisfy the conditions

c~+t(s) dividesG(s),

i=l, 2,...,m-1

deg ci(s) = r.

(5) (6)

i=1

The problem considered in the paper can be stated as follows. Does there exist a state feedback (4) such that the matrix s E - F + GK has invariant polynomials cl(s), c2(s) . . . . , c,,(s) completed by c,,+1(s) . . . . . c,(s) = 1? If so, give conditions for existence and a procedure to calculate K. The motivation of the problem is to investigate the limits of state feedback (4) in making system (1) proper and modifying its dynamics, which in this case can be described by a list of invariant polynomials. The specification of all n invariant polynomials then guarantees the regularity of the closed loop system.

ZAGAI_AK

where wl = degic P(s), i = 1, 2 . . . . . q has rank q. If the rows of P(s) are arranged st) that degir P(s) >~degj, P(s), i -deg/~. P(s) for i < j, P(s) is column degree ordered. Two polynomial matrices P(s) and Q(s) are left coprirne if there are polynomial matrices XL(S) and YL(s) such that P(s)X,.(s)+ Q(s)YL(s) is unimodular, i.e. has a polynomial inverse. Similarly P(s) and Q(s) are right coprime if there are polynomial matrices XR(s) and YR(s) such that XR(s)P(s)+ YR(S)Q(s) is unimodular.

Definition 1. Polynomial matrices D(s), N(s) such that (1) (sE - F)-IG = N(s)D-~(s) (2) D(s), N(s) are right coprime

[O(s)]

(3) [N(s)

is column reduced and column

degree ordered are said to form a normalized right matt& fraction description (MFD) of system (1). CONTROLLABILITY The problem of pole assignment by state feedback in singular systems is closely related to the notions of controllability (Rosenbrock, 1970) and impulse controllability (Cobb, 1984). We shall use the following terminology proposed by Verghese et al. (1981).

Definition 2. System (1) is said to be controllable if system (2) is controllable and system (3) is impulse controllable, i.e. if either of the following two conditions holds: nl--I

(1)

Z

Im F',G, = R',

i=O

and rt2--1

~

Im E~G. + Ker E~ = R"~,

i=(I

MATRIX FRACTION DESCRIPTION For any p x q polynomial matrix P(s), write deg/r P(s) for the degree of row i of P(s) and deg,~ P(s) for the degree of column i of P(s). We say that P(s) is row reduced if its highest row-degree coefficient matrix Phr = lim diag [s- ~'~. . . . .

s-~qP(s)

where v~ = degs~ P(s), i = 1, 2 . . . . , p has rank p, and is column reduced if its highest columndegree coefficient matrix

P,c = lim P(s) diag [s -~, . . . . .

s .... ]

(2) [ s E - F zeros.

G]

has

no

finite

and

infinite

For use in our development, however, we need a more detailed structure of system (1) with respect to controllability.

Definition 3. Let D(s), N(s) be a normalized right MFD of system (1) and let nl, n 2 , . . . , n,, D(s)] be the column degrees of N(s)J' Then the integers n~, n 2. . . . .

indices of (1).

n m are called controllability

State feedback for singular systems These indices play a similar role for (1) as the ordinary controllability indices do for strictly proper systems. In particular, we have the following

655

the type

[-G

sE-F]=P[-G

(sE'-F')Q]

where P, Q are constant nonsingular matrices; hence (1) is controllable as well.

Theorem 1. System (1) is controllable if and only if

FUNDAMENTAL

~

rli=r.

i=l

Proof. To prove the necessity we note from Definition 1 that

s _ lE l:ll_-0

where D(s), N(s) is a normalized right MFD of (1). The controllability of (1) implies the existence of a minor of order n in [ - G sE - F] whose degree is r = r a n k E . Observe that the rows of [ - G s E - F ] form a basis for an n - d i m e n s i o n a l subspace of the ( m + n ) dimensional rational vector space over R and

[z)(s)]

that the columns of I.N(s)J form a basis for its

Theorem 2. Let (1) be a controllable system and n 1 1 > n 2 ~ > ' " t > n m the ordered list of controllability indices of (1). Further, let cl(s), c 2 ( s ) , . . . , Cm(S) be arbitrary monic polynomials subject, however, to conditions (5) and (6). Then there exists a constant matrix K such that s E - F + GK has invariant polynomials cl(s), c2(s) . . . . . cm(s) and cm+l(s) . . . . . cn(s) = 1 if and only if k

orthogonal complement. By Forney (1975, Section 6, Theorem 3) there exists a minor of

Fv(s)7

order m in I_N(s)J whose degree is also r. Since the

column

h i , n2, . . .

, n m

degrees w e have

•

of

[D(s)] k N(S) J

are

ni=r.

i=1

Conversely, let [ - G ' minimal basis (Forney,

sE'-F'] 1975)

for

define a a space

iv(s)]

THEOREM

The limits of state feedback in assigning finite poles to the singular system are given here in a form which parallels the analogous result of Rosenbrock (1970, Chapter 5, Theorem 4.2) for strictly proper systems.

k

degc,(s)>/~ni, i=1

k = l, 2, . . . , m.

(7)

i=1

Proof. To establish necessity, let O(s), N(s) be a normalized right MFD of system (1). Suppose there exists K such that s E - F + GK has invariant polynomials cl(s), c2(s) . . . . . cm(s) and c,,+l(s) . . . . . cn(s) = 1. Then, by Lemma 1 of the Appendix, the matrix D ( s ) + K N ( s ) has invariant polynomials q(s), c2(s) . . . . . c,,(s). Let na>~n2>~...>~nm be the controllability indices of (1). Then hi, i = 1, 2 . . . . . m, are

of r.~D(s)] N(s)J

orthogonal to that spanned by !_N(s)J' where

the column degrees

G' is n x m and E', F ' are n x n matrices over R. Since

Now it is easy to see that inequalities (7) must be satisfied. The product ck+~(s)'" cm(s) is the greatest common divisor of all minors of order m - k in D(s) + KN(s). It follows that

~ 1 2 i -~ r, i=l

D(s) + KN(s).

and hence of

we can write

[-G'

sE' -

-G; F'] = rL - G ;

s E ; - F;] -F; J

where E'I is an r x r matrix of rank r. It follows from the minimality of the basis that s E ' F', G' are left coprime and that rank

[ o

-G;

e;]

- F ; J = n.

This means that [ - G ' s E ' - F ' ] has no finite and infinite zeros. Hence E', F ' , G ' define a controllable system by Definition 2. The system is related to system (1) by a transformation of

degci(s)<~ 2 i=k+l

ni,

k=O, 1. . . . .

m-1.

i=k+l

(81 By (6) and by Theorem 1,

degci(s) = r = ~ ni. i=I

(9)

i=1

Therefore equality holds in (8) when k = 0 and the inequalities can be reordered to give (7). Sufficiency is proved by construction following Ku~era (1981). If Q(s), c2(s) . . . . . c,,(s) are monic polynomials which satisfy (7) as well as the intrinsic properties (5) and (6), form rn x m

656

V. KU(2ERA and P. ZAGALAK

matrix C'(s) = diag [cl(s) . . . . .

c,,,(s)].

(10)

If degicC'(s)=ni, i = 1 , 2 . . . . . m, we define C(s) by C(s) = C(s). If there is a column i for which degi~ C ( s ) > ni there must be a column j for which degjc C ( s ) < n j , for (9) holds. Then Lemma 2 of the Appendix can be applied, several times if necessary, to bring C'(s) to a matrix C(s) whose column degrees are precisely nl, n2, • . . , n,,, and whose invariant polynomials remain cl(s), c2(s) . . . . . cm(s). Now let X, Y be a constant solution pair of the equation

XD(s) + YU(s) = C(s)

(11)

with X nonsingular. The existence of X, Y is guaranteed by Lemma 3 of the Appendix. Then

K = X-IY

(12)

qualifies as one possible feedback gain which assigns the invariant polynomials cl(s), c2(s) . . . . . c,,(s) and cm+l(s) . . . . . c,(s) = 1 to s E - F + GK. This is an immediate consequence of Lemma 1. CONSTRUCTION

The sufficiency part of the proof of Theorem 2 provides a construction of K which achieves the desired dynamics. The major steps of the procedure are summarized below. (1) Given E, F and G calculate a normalized right MFD D(s), N(s) with column degrees nl,

n2~

. . . , rim.

(2) Calculate a matrix C(s) having invariant polynomials q(s), c 2 ( s ) , . . . , c,,(s) and column degrees n l,n~_ . . . . . n,, using Lemma 2 repeatedly. (3) Calculate a constant solution pair X, Y with X nonsingular of the equation

XD(s) + rN(s) = C(s) using Lemma 3. (4) Put K = X-IY. It is to be noted that the feedback gain K provided by this procedure is by no means unique. The degrees of freedom in choosing K are embodied in Step 2, that is to say, in the variety of ways in which unimodular transformations can be performed on matrix (10) to achieve the desired distribution of its column degrees.

characterized in Theorem 2. This is called a fundamental theorem of state feedback for singular systems. The proof of the result is constructive and furnishes a procedure for the actual calculation of the feedback gain which yields the desired dynamics. The fundamental theorem was derived here for regular systems only. The situation is more complicated for non-regular systems, i.e. when det ( s E - F ) = 0. Only partial results are available in this case. Another open problem is the characterization of the dynamics of all regular polynomial systems that can be obtained by state feedback from a given singular system. Such a complement of the fundamental theorem would make complete the picture of the possibilities the state feedback has in altering the dynamics of singular linear systems. REFERENCES Armentano, V. A. (1984). Eigenvalue placement for generalized linear systems. Syst. Control Lett., 4, 199-202. Cobb, D. (1981). Feedback and pole placement in descriptor variable systems. Int. J. Control, 6, 1135-1146. Cobb, D. (1984). Controllability, observability, and duality in singular systems. IEEE Trans. Aut. Control, AC-29, 1076-1082. Forney, G. D. (1975). Minimal bases of rational vector spaces with applications to multivariable linear systems. SIAM J. Control, 13, 493-520. Ku~era, V. (1979). Discrete Linear Control: The Polynomial Equation Approach. Wiley, Chichester. Ku~era, V. (1981). Assigning the invariant factors by feedback. Kybernetika, 17, I18-127. Oz~aldiran, K. and F. L. Lewis (1984). A result on the placement of infinite eigenvalues in descriptor systems. Proc. American Control Conf., pp. 366-371, San Diego, CA. Rosenbrock, H. H. (1970). State-space and Multivariable Theory. Wiley, New York. Verghese, G. C., B. C. L6vy and T. Kailath (1981). A generalized state-space for singular systems. IEEE Trans. Aut. Control, AC-26, 811-831. APPENDIX: LEMMAS FOR THE PROOF OF THEOREM 2 The Appendix contains a series of lemmas which are needed to prove Theorem 2 and at the same time seem to be of independent interest.

Lemma 1. Let D(s), N(s) be a normalized right MFD of a controllable system (1). Then, for any m × n constant matrix K such that either of the matrices D(s)+KN(s) or s E - F + G K is nonsingular, the other matrix is also nonsingular and both have the same nonunit invariant polynomials. Proof. By Definition 1. r

[-G sE-F]LN(s) J=o. Postmultiply

CONCLUSION

The dynamics of all proper systems obtainable by applying state feedback to a given singular system with s E - F invertible have been

[o(~)l N(s)J

[-G

s E - F]

by

by V-' where -K

T

and

premultiply

State feedback for singular systems to get .......

[-G

[ O ( s ) + KN(s) ] , N(s) J=U.

sr.-r.t~,IL

Now we suppose that D(s)+ KN(s) is nonsingular and show that so is sE - F + GK. To this end we rewrite the last relation as

(sE - F + GK)H(s) = G,

H(s) = N(s)[D(s) + KN(s)]-L Clearly (13) represents a system of n linear equations over the field of real rational functions. Hence rank [ - G s E F + GK] = rank (sE - F + GK). By Definition 2, controllability implies

sE-F]=rank[-G

with X, Y constant and X nonsingular, we shall have proven the theorem. Without any loss of generality, suppose that system (1) has been put in Kronecker form (2), (3) with Ez in the Jordan form consisting of p blocks

(13)

where

n=rank[-G

657

of size qi, i = 1, 2 . . . . . p, ordered so that q~ <~q2 <~" " " <~qp. We set qo = 0. Let k be the number of blocks having size 1. An additional change of bases in R n and R m brings G2 to a column echelon form. The form consists, from left to right, of d zero columns where d is the rank deficiency of [E2 G2], 0 <~d <~k, then of p - d columns

c2liI

sE-F]T

=rank[-G

sE-F+GK]

so that sE - F + GK is nonsingular and (13) can he given the form

N(s){D(s) + KN(s)] -l = (sE - V + GK)-1G. Observe that sE - F + GK, G is a left coprime MFD and that D(s) + KN(s), N(s) is a fight coprime MFD of the same rational matrix H(s). It then follows from Kueera (1979, Chapter 5.1, Theorem 11) that s E - F + G K and D(s) + KN(s) have the same set of nonunit invariant polynomials.

where 1 is at position q t + q 2 + ' " + q l , i=d+l ..... p and finally of m - p columns whose elements in rows q~+qz+'"+ql, i=d+l,...,p, are allzero and whose remaining elements are irrelevant. Note that the units of Gzi sitting at the positions qt + qz + " • " + q, for i = k + 1 , . . . , p are guaranteed by impulse controllability of (3). Under these assumptions, (15) can be written as

Lemma 2 (Rosenbrock, 1970, Chapter 5, Lemma 4.1). Let C(s) be a column reduced, polynomial m x m matrix. Suppose degjc C(s) < deg~,c C(s) for some/" and k. Then C(s) can be transformed by unimodular transformations to a column reduced matrix C'(s) with column degrees degic C'(s) = deg/¢ C(s),

//N'(s)|

-z-GGt sI,,--O Ft

0 = sEz - 1,2 J L N2(s) J

.

(17)

We observe that sl, t - Ft is column reduced while sE 2 - 1,2 is not. Denote by T the transformation which interchanges column ql + q2 + " " " + qi-t + 1 of sE z - 1, 2 with column i of (32 for i = k + 1. . . . . p. Then

i ~j, k

degic C'(s) = degjc C(s) + 1 degk~ C'(s) = degk~ C(s) - 1.

Proof. We shall proceed by construction. First postmultiply C(s) by C~-). Then add s times row j to row k in C(s). This leaves the degree of each column but ] and k unchanged. It also leaves unchanged the degrees of the elements in positions (i,j), i--/=k and (i, k), i ~ j ; places a monic polynomial of degree deg/~ C(s)+ 1 in position (k, ]); and does not increase the degree of the element in position (k, k). Let h be the coefficient of s a~s*~cC`) in the element in position (k, k). Put d = degkc C(s) - degi~ C(s) - 1 and subtract hs a times column j from column k. This reduces the degree of column k below degk~ C(s). The resulting matrix C'(s) then satisfies the conditions of the lemma.

L:G~VX(s) sl.,Yt(s)-F,

Y"(oS) ] X T

0

sE - 1,2 j

=

-

sl.,-

L - G;

Fl

E~(s) J

o

where E~(s) is a column reduced polynomial matrix with column degrees deg~c E-;(s) = 0,

Lemma 3. Let D(s), N(s) be a normalized right MFD of a controllable system (1) and nl, n z . . . . . nm be the column

i=qt+qz+...+qj_l+l, j=l,2 ..... p-1

= 1, otherwise

(18)

degrees of LN(s)J" Suppose C(s) is a polynomial m x m matrix which is column reduced with column degrees nt, n2 . . . . , nm. Then the equation

XD(s) + YN(s) = C(s)

and H't is a constant matrix. Write [ D(s)

r D'(s)]

T-1I NI(s) --/N,(O/

(14)

L Nz(s) has a constant solution pair X, Y such that X is nonsingular.

Proof. Since D(s), N(s) are right coprime, it follows from Ku&ra (1979, Chapter 3.2, Theorem 11) that (14) has a polynomial solution pair X(s), Y(s) and

observe

that

D'(s)

LN~(s) J is

column

reduced

If we bring (15) by unimodular transformations to the form

D(s) ] N t ( s ) / is so and T -1 brings the highest degree rows of

N2(s)J

deg~cNl(s ) < n i,

i=1,2 .....

m

while for the polynomial part LN(s)J

since

N2(s ) to D(s). Clearly D'(s) has column degrees n l, n 2. . . . , n,,. For the strictly proper part we have

sE - FJ LN(s) J

[%

and

(16,

deg/c N~(s) <~ni,

1, 2 . . . . .

m.

658

V. KU(2ERA a n d P. ZAGALAK

Having applied T, the identity (17) becomes

1

-

o

E'(s)JLN~(s)J

Uc,,] .

(19)

L o J

Now we can divide Yi(s) by sl,,~ - F1 to obtain YI(s) = Q1(s)(sl.,

- F I) + Y',

for the Ql(s) polynomial and Y'~(sl,,, - FI) L strictly proper, and then divide Y;(s) - Qt(s)H'l by E~(s) to obtain

for some G~o and E" o nonsingular of size n. - k. Hence the last n z - k rows of N" o are zero. Since the first k columns of are zero by virtue of (18), we conclude from i21) that X = Ch~D~c ~ is a constant nonsingular matrix. Now apply the inverse transformation T-~ to restore the original position of columns and rows. One has

[

2 Y~ -G'~ s G - F~ E;(s) -Gi 0

H]•

xT

Y'2(s) - Q,(s)H~ = Q2(s)E;(s) + ~'.

=

for the Q2(s) polynomial and f'2E~-1(s) strictly proper. Hence both Y'a and 112 are constant matrices and the premultiplication of (19) by the unimodular matrix

[l,.0

-Q,(S)l.~

0

sl,,~ - I~ t)0 sE2 - I,, z

and

F

-Q2(s)]

T/N,(s) / = IN,(s) LN,'_(s)J L N~(s)

1._. J

0

~

]

so that (20) reads

yields

-~i

st.,-o F, B', //N'(s)/=

-G2

where f((s) = X'(s) + Qx(s)G~ + Qz(s)G~. Now we postmultiply both sides of (20) by diag I s - " . . . . . s -""] and pass to the limit for s---,~°. This brings the highest column-degree coefficient matrices into the picture, namely

[f(

Y't

_ F "(s)l

Y2]IN,(s)|

= G~,

| N,(s) | =

(20)

E~(s) J LN~(s) J

(21)

LN;_(s)J,,~

0

(22)

s E 2 - I. 2 LN=(s)J

where f( and 17"~ are clearly constant matrices. This transformation replaces column i of X by the zero column ql+qz+...+q,_~+l ofYzfori=k+l . . . . . p. H e n c e X looses rank to m - p + k. To restore the rank of f(, we pick rows q~ + q2 + " " " + qi, i = k + 1. . . . . p, of [ - G 2 0 s g 2 - I~z ]. These rows are constant and their portions sitting m G, are linearly independent of f(. Hence there exists a constant matrix P such that the transformation

0:1

where 2 = lim f((s). Hence f((s) = 2 is a constant matrix. s ~

Furthermore, write

0

l.:

applied to (22) gives NI(S) |

=

N~(s) J~

t_N;_oj

for some N:~0 and D;,~ nonsingular. It follows from (20) that [G~

Ei(s)]h.[D)~] = 0.

-G, sz~,° r, 0 /tN,(s) = -G= sE:- t°~_J L N2(s) where X ' is now nonsingular. On comparing (23) with (16). it is seen that

x=x',

In more detail 0

0

]ro ,l

E~o J LN:~oJ = 0

(23)

Y=[Y',

Y']

is a solution pair of equation (14) having the required properties.