A matrix-pencil-based approach is presented to interpret transition matrices, inconsistent initial conditions, and systems properties of regular generalized state-.
IMA Journal of Mathematical Control & Information (1998) 15, 73-91
A matrix-pencil-based interpretation of inconsistent initial conditions and system properties of generalized state-space systems G. KALOGEROPOULOS
Section of Mathematical Analysis, Department of Mathematics, University of Athens, Panepistemiopolis, 15784, Athens, Greece K. G. ARVANITIS
Division of Computer Science, Department of Electrical and Computer Engineering, National Technical University of Athens, Zographou, 15773, Athens, Greece
A matrix-pencil-based approach is presented to interpret transition matrices, inconsistent initial conditions, and systems properties of regular generalized statespace (GSS) systems. On the basis of the well known Weierstrass canonical form of a regular pencil, several definitions of transition matrices for GSS systems are given. Convolution forms of the forced state evolution of GSS systems are also established, both for the case of consistent and of inconsistent initial conditions. Moreover, a fundamental interpretation of inconsistent initial conditions of GSS systems is outlined. Finally, the notion of several types of controllability and observability Gramians of GSS systems is introduced. Relations of these Gramians to the respective controllability and observability properties of GSS systems are examined, and simple and easily checked algebraic criteria based on these Gramians, are established. It is pointed out that these results appear to befirstin thefieldof GSS systems.
1. Introduction Since the mid 1970s, many reports have been published of research focused on generalized state-space (GSS) systems (also called singular systems, descriptor variable systems, implicit systems). The interest in GSS systems is mainly due to the extensive applications of this kind of linear system to large-scale systems, singular perturbation theory, economics, demography, circuit theory, robotics, aeronautics, medicine, and other areas (see Ref. 11 and the references therein for an extensive overview of the applications of GSS systems). Up to now, several approaches have been developed to study issues of analysis and synthesis for GSS systems, in the frequency domain as well as in the time domain [1, 3-8, 10-20]. Both continuous-time and discrete-time GSS systems have been studied. From the analysis point of view, the reported results include studies on the structure of matrix pencils associated with a GSS system [8, 10, 11], solvability [3,4, 6, 12, 13, 17,19, 20], inconsistent initial conditions [5], controllability and observability [1,6, 14, 15, 17, 19,20], decomposition [11], equivalence [11, 16], generic properties [18], inversion, canonical forms, stability, realization theory, etc. [11]. From the synthesis point of view, the design problems of GSS systems which have been treated successfully in the past, include pole 73 i, Orfofd Uirivtruly Prcu 1998
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G. KALOGEROPOULOS AND K. G. ARVANITIS
assignment, optimal control, decoupling control, model matching, observer design, etc. [7, 11]. Overall, from the so-far reported results, it is well recognized that there has been considerable success in extending many results from system and control theory of linear state-variable systems to GSS systems. However, it appears that some important aspects of linear system theory have not been analysed in the case of GSS systems. In particular, even though state response of GSS systems is well known [3, 4, 6, 19, 20], explicit forms of the solution of state-variable systems involving convolution integrals do not as yet have an analogue in the case of GSS systems. This is mainly due to the fact that the 'transition matrix' of GSS systems has not hitherto been defined. As a consequence, studies on system properties of statevariable systems, such as controllability and observability, which take advantage of the notion of controllability and observability Gramians [2, 9] are completely deficient in GSS system theory, even though (as mentioned above) system properties of GSS systems have been studied by numerous methods [11]. Another motivation for studying GSS systems on the basis of transition matrices and Gramians stems from the fact these objects are crucially involved in the study of sampled-data statevariable systems. So, it appears that transition matrices and Gramians of GSS systems will make a beneficial contribution to a consistent theory for sampled-data GSS systems. Finally, even though it is well recognized that inconsistent initial conditions, which cause impulsive behaviour of GSS systems, are accepted, there are no results in the literature concerning the possible relation between consistent and inconsistent initial conditions of GSS systems. In the present paper, our purpose is to address the problems mentioned above. Our attention is focused on GSS systems whose associated matrix pencils are regular. For this class of GSS systems, we exhibit a matrix-pencil approach, based on appropriate matrix transformations and on the well known Weierstrass canonical form of a regular matrix pencil [8]. On the basis of this approach, two different definitions of the transition matrix of GSS systems are introduced. The independence of the forms of these matrices on the particular choice of the transformations used is established. Fundamental relations between these two transition matrices are also examined. On the basis of these transition matrices, several explicit expressions of the state response of GSS systems in convolution forms are obtained. Both consistent and inconsistent initial conditions cases are considered. Furthermore, an important interpretation of inconsistent initial conditions of GSS systems is made, and a fundamental relation between consistent and inconsistent initial conditions of GSS systems is established. This relation suggests that the impulsive behaviour of GSS systems supplied by an inconsistent initial condition can be viewed as the result of the attempts of the system to reach a consistent initial condition in an infinitesimal time. Finally, the notion of R-, FS-, and C-controllability Gramians (resp. observability Gramians) of GSS systems is introduced. Relations of these Gramians to the controllability and observability properties of GSS systems are examined. In particular, simple and easily checked algebraic criteria based on these Gramians are established for R- and C-controllability (resp. observability), of GSS systems. It is pointed out that most of the results of the present paper appear to be first in the field of GSS systems.
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75
2. Some preliminaries on matrix pencils Throughout the paper, we address GSS systems having the state-space representation Fx(t) = Gx(t) + Bu(l),
y{t) = Cx(t),
*(/„-) =: *o given, m
(2.1) r
where x(t) € R" is the generalized state vector, n(/) e R and y(t) 6 R are the control and the output vectors respectively, and x0 is the initial state vector, and where all matrices have real entries and compatible dimensions. Furthermore, is assumed that det F = 0. As is well known, we can readily associate with system (2.1) a matrix pencil {sF-G: s € C}; we shall denote this by sF-G, where s represents a (complex) indeterminate. In the present paper, we study inconsistent initial conditions and system properties of GSS systems of the form (2.1), on the basis of an approach relying on matrix pencils. For this reason, in this section, we give some useful definitions and relations regarding this subject. 2.1 Given F,G € R mx " (or Cmxn) and an indeterminate s, the matrix pencil sF - G is said regular when m = n and det (sF —G)^0. In any other case, the pencil will be called singular. O DEFINITION
In the present paper, our attention is focused on regular pencils. Let LJ, „ be the set of n x n regular pencils, i.e.
Cxn and sF-G regular}.
Lj,,n :={sF-G:F,Ge
On this set we define an equivalence relation given by the following definition. DEFINITION 2.2 The pencil sF— G € LJ, n is said to be strictly equivalent to the pencil sFi - G\ e Vn>n if and only if
where P, Q € Cx", det P / 0, and det Q ^ 0. D This strict equivalence relation can be defined rigorously, in the sense of transformation groups, as follows. Consider the set Q ••= {(P, Q):P€ C""\ Q 6 C"x",del J V 0, and a composition rule * defined on Q as • : Q x Q-* g : (PuQt) * (P2,Q2) := It may readily be verified that (g, *) forms a nonabelian group. Furthermore, an action o of the group (g, *) on the set LJ, „ of regular pencils is defined by o : g x U,fl -> U, n : ((/», Q),sF-G)~ (P, Q) ° (sF - G) := P(sF - G)Q. This group has the following properties.
(a) (PuQi)°l(P2,Q2)°(sF-G)]
= [(Pl,Qi)*(P2,Qz)}°(sF-G)
for every
P\,P2,Qi,Q2 e C M , such that det/»,/O, d e t P 2 / 0 , d e t d / O , d e t & / 0 , forallsF-GeLJi,,,.
and
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G. KALOGEROPOULOS AND K. G. ARVANIT1S
(b) eGo(sF-G) =sF-G for all sF-GeV^n, identity element of the group (Q, *).
where e c = (I n) I n ) is the
The action o of the group (Q, *) on the set Lj, n defines a transformation group denoted byM. DEFINITION 2.3
For sF-G
Go(sF-
S Lj, n , the subset
G) := {(P, Q) o (sF- G) : (P, Q) e Q) C L ^
will be called the orWf oisF-G
at £.
D
Also M defines an equivalence relation on Lj, n , which is called the strict-equivalence relation, denoted by = s . So, for sF-G and sF{ - Gx in Lj, „, we define sF-G =, sF, - G, «• there exists (P, Q)eQ such that (P, C) o (sF - G) = sF, - (?i. The equivalence class E , ( s F - C) := {P(sF- G)Q : (P, g) € S} is characterized by a uniquely defined element, known as the complex Weierstrass canonical form [8], specified by the complete set of invariants of Es(sF - G). This is the set of elementary divisors obtained by factorizing the invariant polynomials fi(s,s) of the homogeneous matrix pencil sF—sG into powers of homogeneous polynomials irreducible over C. In the case where sF - G e L|, „ and det F = det G = 0, we have elementary divisors of the following types: (a) zero elementary divisors are those of the type s*"; (b) nonzero finite elementary divisors are those of the type (5 — a)T, with a / 0 ; (c) infinite elementary divisors are those of the type sp. The complex Weierstrass canonical form sFw — Gm of the regular pencil sF - G, where det F = 0, is defined by [8]: sFw - C w = block-diag(5l/, -J;sH-
I,).
(2.2)
The first normal Jordan-type block slp — J is defined by the set of nonzero finite elementary divisors (s-aj)Pj (j = l,...,u) of the pencil sF—G, where s\p-J=
block-diag [s\Pj - JPj (a,)] •
Also the a blocks of the second uniquely defined block sH - \q in (2.2), corresponds to the infinite elementary divisors sq> (j = I,...,a) of the pencil sF — G associated with the blocks sH ?/ - \qj (j = I,...,a), where Y^=\aj-a with qx < ••• < qa, given by sH - \q = block-diag (sH - 1 ), >=I
a
and H is a nilpotent matrix of index q* = max {^ :_/ = 1,..., a } , while the matrices
MATRIX PENCILS AND GENERALIZED STATE-SPACE SYSTEMS
77
Jp,(a) and Hq are defined as a
1 0
0 a
•V") =
1
... o •.
0
"-.
1
0
a
0 0 '-.
0
0
0
1 0
•••
0 0 1
0 0
••
0
\
0
•.
1
0
0
'
3. State transition matrices of GSS systems
Our purpose in this section is to define transition matrices of GSS systems of the form (2.1), in analogy to the transition matrix of state-variable systems (also called regular systems, i.e. of systems of the form (2.1) with F = In), and to present their explicit forms in terms of the Weierstrass canonical form of the pencil sF —G associated with (2.1). We next present two different forms of transition matrices for GSS systems. The first form represents the relation between x(t) and x(t0-), while the second represents the relation between x(t) and Fx(t0-). The transition matrix from x(/ 0 -) to each x(/) can be defined through the solution of the homogeneous version of (2.1), which has the form Fx(t) = Gx(t),
*(/„-) given.
(3.1)
Consider now the state-variable transformation where detg ^ 0. On the basis of (3.2), relation (3.1) can be rewritten as FQi(t) = GQz(t), x(t0-) = Qz(t0-).
(3.3)
Premultiplying both sides of (3.3) by the matrix P (with detP ^ 0), we may easily obtain Fwz(t) = GwZ(t),
z(h-) = Q-'x(t0-).
(3.4)
Clearly, according to (2.2), relation (3.4) can be decomposed as z P (0 = Jzp(t), = Zq('),
z p (/ 0 -) = Zpo,
(3.5a)
Zq('O-) - ZqO-
(3-5b)
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G. KALOGEROPOULOS AND K. G. ARVANITIS
The solution of (3.5a) is well known to be while the solution of (3.5b) has the form 0
if ZqQ = 0,
where 5 (t) (•) is the kth derivative of the unit-measure impulse at the origin. Relation (3.7) can be easily obtained by appropriately manipulating (3.5b) on the basis of the Laplace transform (see Ref. 20 for details). Combining relations (3.6) and (3.7) we obtain = ae , - i ^Zp0
where O{t - t0) := blockdiag exp/(/ - /„); - £ &&'-%)
'="
V
.
(3.8)
/
Taking into account (3.2), (3.4), and (3.5), wefinallyobtain x(t)=4(t-to)x(io-),
(3.9)
#(/ - t0) := Qfi(t - to)Q~l.
(3.10)
where
We now give the following definition. DEFINITION 3.1 The transition matrix of the GSS system (2.1) from the initial state x(t0-) to the state x(t), is the n x n matrix
with n(t ~ t0) given in (3.8b).
•
At this point, we emphasize the interesting fact that the form of the matrix #(/ - /0) does not depend on the particular form of the transformation matrices P and Q, used. This observation is summarized in the following theorem. 3.1 The matrix S(t - t0) is unique, in the sense that its form is independent of the choice of the matrices P and Q. THEOREM
Proof. According to the results in Ref. 7, the matrix pair (P, Q) is related to every other transformation pair (P*, Q*) that satisfies P'FQ* = PFQ = Fw,
P*GQ' = PGQ = GW1
(3.11)
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79
through the relations P=TP\
Q = Q*T\
(3.12)
where T is an n x n nonsingular matrix of the form r=blockdiag(J,;r2), with 7", € Rpxp
and T2 € Rqxq.
(3.13)
Let &*(t - t0) be the n x n matrix
4*(t-to):=Q*n(t-to)Q*-1.
(3.14)
Substituting (3.8b), (3.12), and (3.13) into (3.14), we obtain
= o-
(4-9)
Multiplying both sides of (4.9) from the left by g j , we obtain
Qlk = GjlGpCGjGpr'Gj -1,]*5 = o.
(4.10)
AGkergJ.
(4.11)
klimQp
(4.12)
Therefore Relation (4.11) implies that
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G. KALOGEROPOULOS AND K. G. ARVANITIS
Note also that x 0 = QpZpo, which implies that (4.13)
xoeimQp. Since xj$ = x0 + k, we can readily conclude from (4.12) and (4.13) that This completes the proof of the theorem.
•
On the basis of Theorem 4.1, the impulsive behaviour of a GSS system in the case where it is supplied by inconsistent initial conditions can be viewed as the result of the efforts of the system to reach a consistent initial condition of the form (4.2), in an infinitesimal time. 5. The forced state evolution of GSS systems and its convolution form
In this section, our attention is focused on the general solution of (2.1). In particular, we are interested in deriving its explicit expression in the form of a summation of the solution of (3.1) and some convolution integrals. The reason for addressing this problem is that such expressions greatly facilitate the study of system properties of GSS systems, as exhibited in Section 6. To this end, let PB =
(5.1)
B=
where Pp is the pxn matrix of full row rank whose rows are the first p rows of the matrix P, while Pq is the qxn matrix of full row rank whose rows are the last q rows of the matrix P, and where the submatrices Bp and Bq are of dimension p x n and qxn respectively. Under the transformation defined by the matrices P and Q, we can easily obtain
zp(t) = Jzp(t) + Bpu{t),
zp(t0
"iq(') = Z q ( ' ) + V ( ' ) -
Zq('0
(5.2a)
The general solution of (5.2a) is well known to be zp(t) = [exp/(/ - , 0 )] Zp0 + [' [exp/(/ -
(5.3)
The solution of (5.2b) can be obtained using the Laplace transform, and has the form 9-1
-£»'*••
i f ZqO = 0,
(5.4) if 1=0
0,
MATRIX PENCILS AND GENERALIZED STATE-SPACE SYSTEMS
83
where, by definition, H° = I,. Combining relations (5.3) and (5.4), we obtain
'qxm
9-1
/=o Taking into account (3.2) and (3.4), we finally obtain
f l
[ o, xm j or, in an alternate form,
x(0 = #(/ - to)x(to-) + Qp [' [exp/(* - 0]V(0d£ - E (5.6) Clearly, relations (5.5) and (5.6) constitute alternative forms of the general solution of (2.1). We are now able to establish the following theorem. THEOREM
5.1 If the pencil sF—G is regular, then the solution x(t) of (2.1) is
unique. Proof. Let (P, Q) and {P\ Q") be two matrix pairs such that (3.11) holds. These pairs are related through (3.12) and (3.13). Let x*(t) be the general solution of (2.1) in the case where the transformation pair (P*, Q*) is used. Then,
f
J'o-
(5.7) According to Theorem 3.1, we have &(t-to) = 4>(t-to).
(5.8)
On the other hand, due to the properties of the block matrix J defined by (3.13), it is easy to check that the following relations hold *•('-) = *('o-). Q; = QPTU
Q^ = Q^T2,
B; = P;B= rr'Ppfl = rr'Bp,
x
p; = n pp,
(5.9a)
/»* = r2-'#»q,
B* = P^B = T^P^B = 72'flq.
(5.9b)
(5.9c)
Substituting (5.9a-c) into (5.7), and taking into account (3.16) and (3.17), we finally conclude that x*(t) = x(t), and the proof of the theorem is completed. • Thus far we have established that the general solution of (2.1) is given by relations (5.5) or (5.6). Let us try now to present this solution in an convolution form, in
84
G. KALOGEROPOULOS AND K. G. ARVANITIS
analogy to the solution of state-variable systems. This is done in the following theorem. THEOREM
5.2 The general solution of (2.1) can be written in the convolution form
x(t) = #(r - to)x(to-) +\
#(r - 0B"{0 d£ +
J'o-
where
#(' - 0*«(0
(5-10)
J'o-
5(0,
(5.11a) (5.11b)
B=Q Proof. In order to prove (5.10), it suffices to show that
