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case inversion has been solved for only the scalar case [SI. and applications which are largely in the area of image processing are given in [4]. [9]. [IO].
1
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1983
1
11. IMFULSERESPONSE INTERSION
In h s section we mill give the essential results starting with the following. Theorem 2.1: Z has an M-delay inverse if and only if there exist m X p R-matrices KO. K,.. .,K,+,such that E!=o K, 5 -,= 0. j = 0. I. . ..M - I. andX;yOK,Fb,-,= J,,,. Proofc a) .Vecexsi[v: Let Q = x , a o Q , z - ' be an M-delay left inverse of G,. Then the necessity tri\
Ki2-'.
pt2-'=_-.tf.p
<;_-l=z-z'f
r>O
i=O
,OUT
Fig. I
Proof.. If either of the two conditions is violated, then as is clear from Lemmas 2.1 and 2.2, no inverse exists for any !M 0. On the other hand. if therc is an R-matrjx K such that KFu = I",. then by Lemma 2.1 qn,( F,) = R . And if KF, = 0. i > 0 then the free !-module generated by the columns of [PI . . . ;,$,I falls in the Kernel of K and since the nonzero elements of the-free R-module generated by the columns of Fo fall outside the Kernel ofK.we have that those two modules intersect trivially. This completes the theorem.
120 rv.
where .I
Po= I,,
and P,=
KIF,id,-l.
i>O.
J=O
Since P,, > 0 are m X m R-matrices P is an element of the ring R"'x"' (the ring of formal power series over RntX"'). Also, since Po = I,,, we have P i s a unit in R n i x m [ [ z - ' ] (see ] [Ill). or in other words. P - ' exists and isitself a formal power series overR"""'. Therefore. P E.;tj.oK,z-' can be taken as Gg if it is realizable. This part is trivial since
[ [ z - 'I]
CoNCLUsIoN
In [4] it was shown that if R is a p.i.d. then ( I ) serves as the first-level realization model of a 2-D system. Therefore. our inversion method can give an inverse of a 2-D system nith delay in one direction. Similarly. an inverse in the other direction can be obtained. If both exist. an inverse in both directions (Le.. along both coordinates of the indexing set of a 2-D system) can be obtained. Delaydifferential systems [2] are another area of application. REFERENCES
Y Rouchaleau. B. F. Wyman. and R. E. Kalman. "Algebraic stmcture of linear
P - ' = [,FOPi:-.]-'= [Irn+
P,C' I,
I
1
dynamicalsystems 111. Realization theon, overa Bcud. S o U . S . . vol. 69, pp. 3401-3106.1972. -I
which, using the fact that F; = CA' ' B ( i 3, 0) and simple manipulation, where ] - ' C,$i=Z:L,K,C..i.w-' comes to P - ' = [ ~ " ~ + C , ~ f ( ~ J - A ) - ' B and. hence. can be realized as in Fig. I. Therefore. G i = P r = O K I -- - ' is an ,Wdelay inverse of G P (the second part is trivially realizable as a feed-fomard circuit). It is interesting to note that the structure is similar to the one obtained by Massey and Sain [6] for the case R = field. T h s completes the theorem.
'.x.''
111. COWTITATIONOF KO.K,: . ..K,,
Let us denote the HZ X ( M + I)p R-matrix [ K,+,K, - I . . . KO]by k and the ( M 7 I ) p X m R-matrices [O"Fi . . . F& - ,] 0' denotes the Aong column of p X m block null matrices and the prime denotes the transpose) by f,: i = 0. I: . Then theorem 2.1 implies that for an !M-delay inverse K F o = I , a n d K F , = O . i = l :... M. Lemma 2.1: Fohas a left R-inverse if and only if 6,(io) = R (the ideal generated by m-minors of F,). Proofc See [12]. Also. the inverse can be computed using the invariant factor decomposition theorem [13. Appendix IOA]. Lemma 2.2: Z has an M-delay inverse only if the freq R-modules generated by the columns of Fo and the columns of [ F , . . . F,%,]intersect trivially. Proot Suppose thata nonzerc R-vector exists in the intersection. Thenit easilyfollows that both the all-zero sequence anda nonzero sequence, when applied at the input of Z. give an all-zero sequence at the output. Therefore, Z cannot have any M-delay inverse. We now have the following. Theorem 2.3: Z has an M delay inverse if and qnly if $(, F,) = R and the free R:modules generated by the columns of F, and the columns of [ FlF2 . . ..F,bf] intersect trivially (this . - is equivalent to the condition that r ~ [ i ; o ~ , . . - F , ~ l = n z ~ r ~ [ ~ , ~ ~ ~ - . ~ , , , l ) .
..jv.
commutatix ring." Proc.
NUI.
E D. Sontag. "Lmear systemsovercommutative rings: A survey." Rwerche Dl Auro,,~urru.vol. 7. no. I. pp 1-34.1976. S. Ellenberg. .4uIomara Lm~guugesund !Waocltrnes. i.b/. A . NewYork:Academic, 1974. approach." Eindhoven Inst Technol.. R. Eising. "2-D systems: An algebraic Amsterdam. The Netherlands. Math Cent. Tract 125. 1980. J L Massey and M. K. Sain. "Inverses of linear sequentialnrcuits." IEEE Trun.. Contplrr , vol. C-17. pp 330-337. Apr. 1968. -. "lnvertibdity of linear rime-invariant dynanucal systems." IEEE Tram. A u m mar. Conrr.. vol AC-14. pp. 141-149. Apr 1969. L M . Sdverman. "Inversion of multivariable linear s)stems." IEEE Tram A U I O ~ I U I . Conrr , vol. AC-14. pp. 210-276. June 1969. R. Eismg. "State-spacerealization and inversion of 2-D systems." I E E E Trum. C t r w r n S u r . . vol. CAS-27. pp. 612-619. Jul) 1980. X. h.1. Sondhi. "lmaoe restoration: The removal of spatiallv . . invariant deeradation%" Pmr I E E E . vol. 60. pp 842-853. Jul) 1972. B R Hunt. "Digital image processing." Proc. IEEE. vol. 63. pp 693-708. Apr. 1975. N McCov.Theon of R i w x . NewYork:Macmillan. 1961. E D. Sontag. "&the &verses of polynomial andother matrices." I E E E Trum. Auronxar Conrr.. vol AC-25. July 1980. R. E Kalman. P. L. Falb.and M Arbib. Topics rn .Murhentuma/ Srsrenz Theon.. Neu York: McGraw-Hill. 1969.
On the Structure of Balanced and Other Principal Representations of SISO Systems K. V. FERNANDO AND H. NICHOLSON Abstract -A new matrix IV<,which can be considered as a cross-Gramian matrix which contains information about both controllability and observability is defined for single-input, single-output, linear systems. Using this Xlanuscnpt received October 21. 1981: rerised March 74. 1987 and August 9. 1982. T h e autherc are ulth the Department of Control Engineering. Universltg of Sheffield. Shclfield SI 3JD. England
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ON AUTOMATIC CONTROL, VOL.
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matrix, the structural properties of linear systems are studied in the context of principal component analysis. The matrix kVc, can be used in obtaining balanced and other principal representations without computation of the controllabilit?. and the observability Gramians. The importance of this matrix in model-order reduction is highlighted. I. INTRODUCTION Moore [ I ] used concepts from the principal component analysis of Hotelling to investigate the controllability and observability of linear systems, and also as a tool for model-order reduction. The technique is based. essentially, on simultaneous diagonalization of the controllability Gramian W: and the observability Gramian WO2using appropriate similarity transformations. It was shown by lMoore thatit is somewhat inadequate and sometimes misleading to study controllability or observability individually, and combined investigations are required. In this note, we define a new matrix W,, which can be considered as a cross-Gramian matrix and whch carries information pertaining to controllability and observability, and whichis directly connected to both controllability and observability Gramians. Thus, this new matrix is a natural candidate for the study of combined investigations of controllability and observability. and it is used to expound the structure of SISO linear systems in the framework of principal component analysis. Our main results are based on the absolute value symmetry of the state matrix under balanced conditions. However, the analysis could be carried out using more general principal (axis) representations [3] in which more specific balanced representations also belong. The role of principal representations in model-order reduction is also investigated. The spectral structure of the matrix Wcois paramount in our analysis and, in fact, the absolute values of the spectrum are given by the singular values of the system. We also show the relationship between the singular values and the dc gain of the system, and the importance of that result as an alternative criterion for model-order reduction.
229
1983 Principal (Axis) Represenration:
where C; and E: are positive diagonal matrices. We note that one of the diagonal matrices. Zf or X:, is arbitrary, but notboth; We deno_e-in~ernallybalanced and principal representations by S ( A I b c) , and S ( A , b, c), respectively. 111. THE SYMMETRY IN INTERNALLY BALANCED SYSTEMS Moore [ 11 referred to the absolute value symmetry of the state matrix A in single-input, single-output, internally balanced systems. We present that property as a lemma. Lenfrna 1: If the system S( b. E) is internally balanced, then I) b,=+i, 2) either hiJ = ~i1:. or u' = u,' if i,iJ= ;,tJ* o 3) i. = - i i f b b =-;;.to 'J / I I J ' J .. 4) either Ci,j = - Cij, and u: = u,' or C ,i = Cij, 7 0 if b, b, = C,2J = 0 where a,, denotes the (i.j)th element of the matrix A . Proo) Since the Gramian matrices Wc2(P ) . W:( P) are diagonal and equal to Z2. the diagonal elements of (1) and (2) are of the form.
A,
-
and part 1) of the Lemma is true. The (i. j)th elements of (1) and (2) are given by
and the difference and the sum of (3) and (4) are of the form ^
.
(u~-u~)(Ci,,-ij,)=-bIbj+C,i;
.^
11.PRELIMINARIES
(~,'+~,')(ri,,+Ci,,)=-b,bj-C,CJ.
(5)
(6)
For the linear nth-order single-input. single-output, asymptotically stable, time-invariant system S( A , b , c) described by
If bibJ = &ijt 0, then from (5). the element Ci,j appears symmetrically in the matrix A or u: = u,.' If the second possibility b,b, = - iti,* 0 is satisfied, then from (6),the element Ci . appears skew-symmetrically. i(t)=Ax(t)+bu(t), y(t)=cx(r) : J" If the remaining possibility, b,b, = cicJ = 0 is true, then from (3) and (4) the controllability Gramian matrix W: is given by one possibility is 6; = is,. However, since u: and u; are positive, the condition a,, = ijiis not admissible, and hence, Li,j = - CiJi and u: = u:. Wc'AT+ A W,' = - bb'. (1) other The possibility is C ,i = Ci, = 0. The appearance of nonsymmetrical or nonskervsymmetrical elements We assume that the system is controllable and, thus. Wc2is a positive according to Part 2) of Lemma 1 is nongeneric. If this symmetry is not definite matrix. Similarly, the observability Gramian matrix W: can be present, then it can be achieved using a rotational (orthogonal) transfordefined by mation (see [4] for details). In the remainder of this note, we assume that the internally balanced Wo2A+ A'W; = - c'c. (2) representation possesses the absolute value symmetry. We also assume that the system is observable, resulting in a positive IV. THE CROSS-GWANMATRIX W,, definite observable Gramian. In the principal component analysis approach of Moore [I]! the system Using the impulse responses of the controllable system and the obsenrS( A . b c, ) is transformed into the internally balanced form defined by able system, we define a matrix W,, as W:( P )
=
W;( P ) = z2.
The matrix P denotes the similarity transformation
S(A.b,c)-,S(P-'AP,P-'b.cP) required to bring the Gramian matrices to the balanced format.The matrix C2 is diagonal and the &agonal positive elements are called the singular values of the system. E2=diag(uf,...,u,:) .
which we call the cross-Gramian matrix of the system. To our knowledge, this matrix W,, has not appeared previously in the control literature. It is easily seen that the matrix W,, can be computed by solving the linear matrix equation W,,A
+ A W,,
= - bc.
(8)
Since the state matrix A is assumed to be stable, a unique solution matrix W,, exists. Standard algorithms are available for obtaining this solution
PI. We assume that they are ordered in the nonincreasing order of magnitude. A more general format which encompasses the balanced forms can be '
defined in the following manner, which is called a principal axis representation [3].
It is intuitively clear that the matrix W,, carries information about both controllability and observability. This contrasts with theGramian W,' which contains controllability data only and the Gramian W: which contains observability data only.
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It =’as shown by Moore [ I ] that it is inadequate and sometimes misleading to study controllability and obsenrability indikidually. and a combined approach is required in the analysis of dynamical systems. We will demonstrate that this cross-Gramian matrix W;.,is consistent xvith the philosophy advocated by Moore. and also fundamental to it. ” invariant It iseasily seen that the eigenvalues of the matrix I+;are under similarity transformations of the system. We denote these eigenvalues by A,. I = I. and form the diagonal eigenvalue matrix ,I as
n.
,1= diag( X,: . ..A,!) The following result showsthe importance of the matrix W c o in the principal component analysis of SISO linear svstems. Theorem I ; If the system S ( A , h, i.) is internally balanced with absolute value symmetry in the state matrix A . then I ) the corresponding cross-Gramian matrix W,,(P) for the balanced system is diagonal and the diagonal elements are given by A,=a:
,
X,
- 0:
if
c,=h,*O
if
e, = - h,;
w0(
M/,2(jS)=-diag(...
.6,2/2ii,;:..)
ct.b’(P)=-diag(... .~~/2ii,,,-..) forii,,*O
v.
W,’(
byo’
Proof: We observe that any principal representation differs from the internally balanced format by a diagonal similarity transformation. However.the matrix yo(P) is equalto A when the system is internally balanced. and it does not vary under diagonal similarity transformations. 0 The rule for obtaining the signatures are obvious. The following result confirms the validity of the converse of Theorem 1 for principal representations. Corollan 2; If the matrix f i is diagonal, then the corresponding system S( AT 6. F) = Sf 7- ‘AF F- ’h. c f i is a principal representation. Proof: If the matrix bi$o( P) is diagonal. then it is equal to the eigenvalue matrix h (assuming that the diagonal elements are ordered in the nonincreasing order of absolute value magnitude). It can be shown by substitution that
thus satishlng the conditions for principal representations.
2) the square of the matrix Wco(Fj is equal to the product P)W:( p) under any arbitrary similarity transformation P. That is.
W A = r+;2
AC-28,NO. 2, FEBRUARY 1983
.
Proof: Form the &agonal signature matrix L‘ such that
0
MODEL-ORDERREDUCTION USING PRINCIPAL REPRESEhTATIONS
Moore [ I ] used internally balanced representations in model-order reduction based on subsystem elimination. However, as the following result indicates. we may use principal representations instead of internally balanced representations in model-order reduction and obtain the same reduced-order model. . Theoren! 2; Let S ( A , h. .‘, be the internally balanced representation and S ( A , h, F) be a principal representation with I
u, = 1 11, = -
1
if
i,= h, * 0
if
E, = - h,.
s(~,~,~.)=s(D-’~D.D-~~,ED)
If the system is internally balanced. (2) can be written in the form
where D is an arbitrary diagonal matrix which defines all possible principal representations for the system (assuming that the matrix A is ordered in the nonincreasing order of absolute value magnitude). If the internally balanced representation and the principal representation are partitioned in the format
~ 2 + kA 5 2 = -
and by premultiplqing the above by C’, we obtain
((;X.:)k+(~.~r(;)((;X’)=-bE. The (i. j ) t h element of the matrix Uk‘U consider all four possibilities.
(9)
is equal to u,uJci,, and we
w i $ order of k,I = order
of
i ,,, etc., then
the balanced representation
S ( A , , . h,,;,j and the principal representation S ( A , , , h , . E , ) describe the
same reduced-order model. froof: If the matrix D is partitioned in the same format as
D = D l @D2 Thus. UkrU = k. and by comparing (9) to (8). we obtain the required result [part I)]
then
To prove part 2). from (IO) W A ( P ) = X 4 . However, w z ( P ) b t ; 2 ( P ) = X4. and hence
0 which completes the proof. If we decompose the matrices X’ and 11 conforming to the partitions in Theorem 2. then
z2=X:ez;.
if
sign ( F , )
X,= - a,’
if
sign (;;I =sign
= sign
=
11,eA’.
-
Moore [ I ] used the trace of the diagonal Gramian matrix X: as a measure of error in model reduction. The trace of the matrix 1’.which is equal to the sum of the singular values. can be considered as the total “energy” of the system and relative error ratios can be computed in conjunction with this value. The follo-ring result indicates that the trace of the matrix is related to the dc gain of the system. Theorem 3: The sum of the eigenvalues X,. i = 1. n gives half the dc gain of the system. That is.
( - i,).
trace h = - TcA-Ib.
It is easily seen that this result is true even under any arbitrary similarity 0 transformation F, whch completes the proof. The follo\+ing result extends the above theorem concerning balanced systems to more general princip4 representations. Corolluv I : If the system S(A. h. F) is a principal representation. then the corresponding matrix W c o ( P) is diagonal and the diagonal elements are given by
X, = a,’
11
-
(b,),
b, = S,* 0
1
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1983
23 1
Proo) W,,
=jEeAfbceA1dt 0
1 trace W,, = k E c e Z A f b d=t - -cA -Ib. 2 Fig. I
Since, trace W,, = traceh, the result follows. 0 An alternative criterion for model reduction can be stated using the above Theorem. Instead of the requirement that the trace of the matrix 2 ; is “small,” we may specifythat the dcgain of the subsystem S ( A22. b 2 ,c 2 ) , given by twice the trace of the matrix A 2, should be small. The smallness of the singular values guarantees low dc gains; however, the converse is not necessarily true. This is a significant result since, in most established model-order reduction methods, the dc gain is one of the criteria for obtaining reduced order models. Often, it is specified that the reduced order model and the original model should have the same dc gain and, dueto the direct relationship between the singular values and the dc gain, it may be possible to take such constraints into account in this approach. VI. CONCLUSIONS We have defined a new matrix W,, which can be considered as a cross-Gramian matrix, and which contains information pertaining to both controllability and observability. Using this matrix, the structure of SISO linear systems in the context of principal component analysis has been studled. It was shown that its properties can beused in model-order reduction in the framework of more general principal representations without computing the more specific balanced representations. However, both principal and balanced representations give the same reduced-order model. Due to inherent signatures, the cross-Gramian matrix W‘, contains more information than the controllability and observability Gramian matrices. In fact, it can be shown that W,, and the Hankel matrix associated with the same system, share common properties including the Cauchy index [5]. We have also proved the relationship between the singular values of the system and the dc gain. It was explained how this property can be used as alternative criterion in model-order reduction.
existence of such a controller. Actuator failure is also eonsidered at the end as a dual pmblem.
I. INTRODUCTION In this correspondence we let R ( s ) denote the set of matrices of rational functions with real coefficients, and S(s) the subset of R ( s ) consisting of proper functions whose poles lie in the open left half-plane. the closed Then a matrix A belongs to S(s) if it has no poles in C,, right-half s-plane together with the infinity point. A square matrix Ci in S(s) is said to be + n o d u l a r if its inverse belongs also to S(s). This requires that detU should have no zeros in A square matrix D is said to be a right (left) divisor of a matrix A if there exists a matrix C such that A = CD ( A = DC). Then, two matrices A and B in S(s) with the same number of columns are right (left) coprime if and only if every common right (left) divisor is unimodular. Alternatively,
[As1
matrices A and B d l be coprime if and only if has full column rank for all s E C+e. Consider a plant E R ( s ) , which we assume to be strictly proper. Let ( N o , Do)and ( D o , X,)be a right-coprime and left-coprime factorjzation of Po over S(s).Then there exist stable proper matrices X,, Yo, X,, and Yo such that
c,
- -
+ YODO= I ioio 4- DOYO= I . (1) Let ( N c , 0,)and (0,. kc)be a right-coprime and left-coprime factorizaX,No
tion of a proper compensator C E R (s) over S ( s ) . This compensator is to be designed so that the feedback system of Fig. I achieves internal stability. meaning stability of the system with input ( r , d )and output ( e . u ) , where r , d , e , u are the reference, disturbance, error, and input, respectively. Then C achieves internal stability if and only if [ I ]
~ , D ~ + ~ ~ N or ~ 5Ec %~
REFERENCES B. C. Moore. “Principal wmponent analysisin linearsystems:Controllability, obsemabilit),and model reduction.” I E E E Tram. A u r o m f . Conrr.. vol. AC-26, pp. 17-32. Feb. 1981. R. H. Banels and G. W. Stewart, “Solution of the matrix equation A X + X B = C.” Conlnlun. Ass. Cornput. .Mach., vol. 15, pp. 820-826, Sept. 1972. C. T.Mullis and R. A. Roberts. “Synthesis of minimum roundoff noise fixed point digital filters.” I E E E Trum. Circurrs Xvsl., vol. CAS-23, pp. 551-561. Sept. 1976. L. Pernebo and L. .M. Silverman. “Model reduction ria balanced state space representations,” I E E E Tram. Aufomul. Conlr., vol. AC-27. pp. 382-387. Apr. 1982. K.V. Fernando and H. Nicbolson,.”On the Cauchy index of linear systems.” t h l s issue. pp. 222-224.
where % is the set of unimodular matrices. If condition (2) is satisfied, then the set of proper controllers which achieves internal stability is given by
A plant is said to bestrongly stabilizable [2] if there exists a stable proper compensator which stabilizes the plant.It can easily be seen that if C E S(s), then ( I , C) is a left-coprime factorization of C over S ( s ) and we can write (2) as
Do+ CN, E %.
Stabilization of a Class of Plants with Possible Loss of Outputs or Actuator Failures ALBERT J. ALOS Abstruct -The loss of one or more outputs of a plant is reflected by one or more rows of zeros in the plant’s transfer function. In this note, the simultaneous stabilization of a plant and the same plant with loss of outputs is considered. Necessary and sufficient conditions for simultaneous stabilization of a class of plants are given as well as a simple test to determine the Manuscript received September 15. 1981; revised April 19, 1982. The author is with the Department of Electrical Engineering, Universityof Lagos, Lagos, Nigeria
o+ic~o~% (2)
(4)
Therefore, a plant Po is strongly stabilizable if and only if there exists C E S(s) such that (4) holds. A practical test for the existence of such a matrix C was proposed by Youla et a/. [3] in the following theorem: Theorem I (Youla et al., 1974): Let ( Np, D p ) be a right-coprime factorization of a plant P over S ( s ) . Let a,. . . a, be the real nonnegative values of s such that P ( s ) = 0. Then P is strongly stabilizable if and only if
are all of the same sign. The problem of simultaneous stabilization consists in finding a conthat is, troller that stabilizes two plants, Po and PI,simultaneously [4];
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