A tutorial on the geometric analysis of linear time-invariant implicit systems

0005-1098192 $5.00+ 0.00 PergamonPresspie ~) 1991InternationalFederationof AutomaticControl

Automatica, Vol.28, No. 1, pp. 119-137, 1992 Printedin GreatBritain.

A Tutorial on the Geometric Analysis of Linear Time-invariant Implicit Systems* F. L. LEWISY

A tutorial o f the geometric analysis of both regular and nonregular (possibly nonsquare) implicit systems examines the inner, outer, and composite subspaces, and their roles in the analysis of implicit systems are shown. Key Words--Implicit systems; generalized state-space systems; linear systems; geometric theory; invariance; duality; controllability and observability.

Abstract--A background of recent work in nonregular and

equals the number of factories, an uncommon case. In network analysis, there are several effects that cannot be modeled using state-space systems. One example is hysteresis (Newcomb, 1982). Moreover, it is not uncommon to obtain rectangular systems of differential equations (Dziurla and Newcomb, 1987; Lovass-Nagy and Powers, 1975), which are easily handled using a nonsquare E. Recent work (Rassai and Newcomb, 1988; Syrmos et al., 1989) has shown that semistate equations may be used to describe solid-holed torus knots in 3-dimensional space. For an excellent review of these applications see Newcomb and Dziurla (1989). Aplevich (1981, 1985, 1989) has shown that many design problems, including matrix-fraction design, can be cast as nonoriented or implicit systems. These design equations are generally nonsquare. Stable algorithms that solve a derived output-feedback problem may then be used to solve the design problem. Both the input/output (i/o) decoupling problem and the disturbance decoupling problem have recently been examined for singular systems, using both proportional and proportional-plus-derivative (PD) feedback (Armentano, 1985; Dai, 1988, 1989; Ailon, 1988; Banaszuk et al., 1988d]. The conditions involved offer increased understanding of decoupling, as well as some simplified decoupling schemes. Indeed, in contrast to the complicated solution for the input-output decoupling problem (i.e. Morgan's problem) for state-space systems, a necessary and sufficient condition for i/o decoupling using PD feedback is simply the nonsingularity of the transfer function (Dai, 1988)! State-space systems are closed under neither

regular implicit systems of the form E Y c = A x + B u is provided. Then, the output-nulling, unknown-input and composite subspaces are defined for singular systems, with recursions given for their computation. Inner subspaces, that is those in the domain of E and A, are distinguished from outer subspaces, those in the codomain of E and A. The notions of composite preimage and composite image are introduced. The result is a framework that unifies some of the recent work in the geometric theory of implicit systems. System properties are discussed for both regular and nonregular systems, and duality is investigated.

Implicit systems

1. INTRODUCTION

Tr~ERE ARE many arguments for using the implicit, singular, semistate, or descriptor system description

E~ = Ax + Bu

(1.1)

with E generally singular or even nonsquare, instead of the state-space description which has E = L Among these are the fact that in largescale systems, economics, networks, power, neural systems and elsewhere, the singular formulation appears more conveniently and naturally than the state-space formulation. Particularly in economics, nonsquare implicit systems are common. Indeed, square systems arise only when the number of goods produced * Received 10 October 1989; revised 25 June 1990; revised 28 March 1991; received in final form 6 June 1991. The original version of this paper was presented at the IFAC workshop on System Structure and Control which was held in Prague, Czechoslovakia during September 1989. The published proceedings of this IFAC Meeting may be ordered from Pergamon Press pie, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by Associate Editor V. Ku~era under the direction of Editor H. Kwakernaak. ? Automation and Robotics Research Institute, The University of Texas at Arlington, 7300 Jack Newell Blvd S, Ft Worth, TX 76118, U.S.A. ~ o z~:~ ,

119

120

F . L . LEw~s

inversion nor PD feedback. Note that the inverse system generally contains derivatives of the system outputs and that the derivative feedback u = - K ~ yields a system like ( I + BK)~ = Ax, where (I + BK) is generally singular. Singular systems overcome these limitations. It has moreover, been demonstrated (Cobb, 1986) that a sequence of state-space systems may not converge, although its solution sequence may. Singular systems are the natural closure of state-space systems with respect to convergence of solutions. The geometric approach In this paper we take a geometric approach. As is well known, this has deficiencies, among them an inability to address robustness issues. However, it does provide a great deal of insight into the structure of a system, which may then be taken into account during the control's design phase. Moreover, the geometric techniques yield numerically stable design techniques. In fact, the geometric conditions in this paper can be recast as Lyapunov design equations (Lewis and 0z~aldiran, 1989; Lewis, 1990; Syrmos, 1991), which may then be solved using the generalized Hessenberg form (Van Dooren, 1979; Lewis et al., 1989) to obtain feedback controllers and observers. Two subspace recursions which are fundamental in geometric system theory (Wonham, 1979) are Sk+t =A-~(Xk + R(B)) fqN(C)

(1.2)

Yk+l = (AYk + R(B)) f3 N(C),

(1.3)

whose relationship may or may not be clear. However, the corresponding recursions for singular systems are Xk+ , = A - I ( E X k + R(B)) UIN(C)

(1.4)

Yk+l

(1.5)

----

E-'(AY~ + R(B)) N N(C),

whence it is clear that the two arc nothing but the anticausal versions of each other; that is, they are obtained from each other by interchanging E and A (note that writing Axk = Ex,+t amounts to a backwards recursion in time). In this paper we provide a background in both regular and nonregular singular systems. Then, we introduce the output-hulling, unknown-input, and composite subspaccs for singular systems (cf. Molinari, 1976), thus providing a complete framework to relate some recent results in geometric system theory. We discuss system properties for both regular and nonregular systems, and give a theory of duality. An important aspect of singular systems is that, in contrast to the state-space case, the

domain and the codomain of E and A may not generally be considered as the same space. Thus, the geometric theory of singular systems provides a far richer structure than that of state-space systems. We defined the composite preimage and the composite image in order to show more clearly the relations between subspaces. 2. BACKGROUND: NONREGULAR SYSTEMS The range and nullspace of a linear operator A are denoted by A [or sometimes R(A)], N(A), respectively. A -t denotes the inverse image of A, or, if it exists, the usual inverse of its matrix representation. The (Moore-Penrose) pseudoinverse of A is denoted by A ÷. Given two matrices A and B, by [A/B] we shall mean [A r Br] r. Subspaces will be denoted by S, X and so on. The annihilator of S will be written S±; it resides in the dual space to that in which S resides. The dimension of S will be denoted by dim (S). Three singular systems We shall be interested in the time-invariant continuous singular system E:~ = A x + B u ,

x(0) given

y = Cx

(2.1) (2.2)

with x ~ X =- R n, u ~ U =- R m, y ~ Y =- R ~', and E generally singular and even nonsquare. Thus, we are not making the regularity assumption IsE-AI4=O a.e. (where "a.e." denotes for almost every s; that is, the normal rank of a polynomial matrix). Let us take E as a q x n matrix. Let us consider inputs and solutions of (2.1), (2.2) which are Laplace-transformable. Then, to determine a discrete equivalent for this continuous system over the Laplace transformable functions we may proceed as follows. Laplace transformation of (2.1), (2.2) yields (sE - A ) X ( s ) = Ex(O) + BU(s)

(2.3)

Y(s) = CX(s).

(2.4)

Define the Laurent series expansions of the Laplace transforms X(s), U(s), and Y(s) as X ( s ) = ~ XiS-i, i=--ix U(S) = ~ Uis-i, i ~ ~iu

(2.5)

Y(s) = ~, yis -~. i= --i~

Selecting the lower indices as the largest values for which these expansions exist renders the

Geometric analysis of implicit systems expansions unique. Due to (2.2), ix>-iy. Assuming that B has full column rank ensures that iu -< ix + 1 (Malabre, 1987). Substituting (2.5) into (2.3), (2.4) yields Exk+ 1 ----Axk + BUk -b 6kEx(O)

Yk = CXk,

--ix -- 1 <--k,

(2.6) (2.7)

with 6k the Kronecker delta. See Schumacher (1983). Since discrete-time systems often arise directly and are of interest in their own right, we shall be concerned also with the discrete singular system Ex~+I = AXk + Bu~

(2.8)

y~ = Cx~,

(2.9)

where x e X = R ~, u e U=- R '~, Y e Y=- R p, and

E is generally singular and nonsquare. In connection with (2.8), there are several possibilities for specifying the boundary conditions as well as the region of interest of the index k. In case x0 is prescribed, we take k e Z~, the nonnegative integers, and call the resulting trajectory xk the forward solution. In case the region of interest is k = 0, 1. . . . . L - 1 for some fixed L and the boundary condition Woxo + Wt.x~. = w • R ~

(2.10)

is prescribed, with the vector w and matrices W0, WL given, we call the resulting trajectory x~ a symmetric solution. Situations like this arise in economics and elsewhere. A special case of (2.10) is Uoxo = Wo,

ULx~.= w~

(2.11)

with dim (Wo) + dim (w~) = n. This amounts to (2.10) with W0=[Uo/0], W~=[0/U~], w =

[Wo/Wd. In any case there are certain restrictions on the selection of the boundary conditions, which must be admissible in a sense to be discussed. We shall symbolize systems of any of these three types by (E, A, B, C). For simplicity we shall take B of full column rank and C of full row rank. The dual system to ( E , A , B, C) will be defined as (E r, A r, C r, B r) and the anticausal system as (A, E, B, C). Examining (2.8), we may interchange the roles of E and A and write A x k =EXk+ 1 - - B u k. Then, it is clear that the anticausal system and the system are related by time reversal (cf. Willems, 1989).

121

and B + the pseudoinverse of B (Karcanias and Hayton, 1981). Then, defining

and performing the equivalence transformation ( W - 1 E , W - 1 A , W - x B , C) on (2.1), for instance, yields NE2 = N A x (2.13) u = B+(E~ - A x ) .

(2.14)

The equivalence between (2.1) and (2.13)/(2.14) is as follows. For a given u(t), a solution x(t) of (2.1) is also a solution of (2.13). For a given solution x(t) of (2.13), there exists a control u(t) such that x(t) is also a solution of (2.1); namely, the control given by (2.14). Since (2.13) is not affected by any feedback of the form u = - K x , it is termed the feedback-free representation of the corresponding singular system. See Karcanias (1987), Karcanias and Hayton (1981) and Karcanias and Kalogeropoulos (1987) for an extensive analysis of the algebraic properties of singular systems in terms of the algebraic properties of (2.13). The upshot is that the reachability properties of (E, A, B, 0) may be studied in terms of the properties of the pencil ( s N E - N A ) . [However, some care is required in distinguishing between the effects of proportional feedback and proportional-plusderivate feedback in extending results for (sNE-NA) back to the original system (see Loiseau et al., 1989, cf. Ku~era and Zagalak, 1988).] This simplified problem results if we are not insistent on dealing only with square pencils [note that (sNE - N A ) has dimension (n - m) x n]. It is not ditficult to determine a state-variable equivalent of the feedback-free system, and hence of the singular system (Ailon, 1987; Lovas-Nagy et al., 1986). Indeed, suppose that there exist F and G so that NEF = NA,

NEG = 0

(2.15)

with G a full rank annihilator of NE. Consider then the state-space equation Yc = Fx + Gv.

(2.16)

Premultiplication of this equation by N E yields nothing but (2.13). Note that the (semi)state vector x(t) has not changed in all these transformations.

Equivalent system descriptions

Let us now mention some simplified system formulations that are equivalent to (E, A, B, C). To simplify the dynamical equations (2.1), (2.6), or (2.8) we may take N a left annihilator

Theorem 1. Let

(1) I s E - A I 4~0 (2) rank [E

a.e.

B] -- n

(2.17) (2.18)

122

F . L . LEWIS

(3) rank [sE - A

B] = n for all finite s.

(2.19)

Then, the state-space equivalent (2.16) exists. Moreover, (F, G) is a controllable pair. Proof. From condition 1, E is square n x n. From condition 2,

[;+][E B]=[BN+EE Ol]in and since W [see (2.12)] is nonsingular rank (NE)= n - m . Since NE has full row rank, R(NA) c R(NE) and a matrix F satisfying (2.15) exists. Now

injection applied to the singular system, we may as well call this the output-injection-free representation. Again, the observability properties of ( E , A , O, C) may be studied by considering the simplified problem of investigating the nonsquare n × n - p pencil ( s E M AM). Let us note here merely that (sEM - A M ) expresses the properties of (sE - A ) restricted to N(C). To study the trajectories of a system restricted to a general subspace S c X, we may select a basis S for S and note that if x ( t ) E S , then x = S w for some w(t). Thus, the system restricted to S is

0]

[sB÷E - BA

G]=[ sNE-NA LSGE - G+A

0] lm "

NE is a full rank left annihilator of G, so [NE/G +] is nonsingular. Therefore, rank ( s N E - N A ) = n - m for all finite s implies that rank [sI - F G] = n for all finite s. ~ Rank condition 1 is the regularity condition, 2 is the condition for reachability at infinity, and 3 is the condition for finite reachability. All of these will be discussed later. An appealing design procedure for the control of (2.1), then, is to determine a v(t) that yields a desired x(t) in (2.16) using standard state-space techniques. Then, the required control u(t) for (2.1) that generates the same x(t) is given by (2.14) (Ailon, 1987). A related procedure is given by Lovass-Nagy et al. (1986), and an alternative technique for associating a statespace system with a singular system in Banaszuk (1989). A corresponding approach may be used to study the observability properties of (E, A, 0, C) (Karcanias, 1987; Shafai and Carroll, 1987). Thus, select M as a right annihilator and C ÷ the pseudoinverse of C and define V = [M

C+].

(2.20)

Then, a semistate transformation 2 = V - i x on (2.1), (2.2) yields [EM

EC+I[;:]=[AM Y=[0

AC+][;:]

I][;~].

(2.23)

y = CSw.

(2.24)

I,~

and the nonsingularity of W and condition 3 show that rank (sNE - NA ) = n - m for all finite s. Nnally [~][sl-F

ES~ = ASw + Bu

(2.21) (2.22)

Since (sEM - A M ) is independent of any output

A study of the properties of the nonsquare pencil ( s E S - A S ) reveals the properties of (E, A, B, C) restricted to S. We call (2.23) the trace of (2.1) restricted to S (Banaszuk et al., 1988b; Karcanias, 1987). Let us note that all of these various representations are in the singular formulation, though they may not even be square. Thus, if we are willing to think in terms of singular and not state-space systems considerable freedom is gained in terms of system analysis and design. Input-output pairing Laplace transformation of (2.1), (2.2) yields (sE - A ) X ( s ) = Ex(O) + BU(s)

(2.25)

Y(s) = CX(s).

(2.26)

Even if I s E - A I =-0 or ( s E - A ) is nonsquare, there exists in any singular system an inputoutput pairing relation between Laplacetransformable u(t) and y(t) when E x ( 0 ) = 0 (Newcomb, 1966; Dziurla and Newcomb, 1987; Grimm, 1988; Willems, 1981, 1982, 1989). To find such a relation, write

V(s)][,e-A LW(s)

Z(s)JL

-c

~ M(s)[SE_-c A ]

1J (2.27)

with M(s) a unimodular polynomial matrix and D(s) a full row rank greatest common right divisor (gcrd) of (sE - A) and C (Kailath, 1980). Then, M(s)[SE-c A --

:lOg,

BI[

X(s) 1 0JL-U(s)J

x(s 1

W(s)BJ L - U(s) J

[0

=M(s)-Y(s)

]=

L-Z(s)Y(s)J'

(2.28)

Geometric analysis of implicit systems so that

Z(s)Y(s) = W(s)BU(s)

(2.29)

Theorem 2. A solution to (2.1) exists for all admissible u(t) if and only if rank[sE-A

D(s)X(s) = r ( s ) B U ( s ) - V(s)Y(s).

(2.30)

The first of these equations is the input-output pairing relation sought. The second equation always has a solution X(s) [since D(s) has full row rank] and shows how the semistate is determined from the system inputs and outputs. It illustrates the fact [Aplevich, 1981, 1985, 1989] that a singular system is nonoriented, with the inputs and outputs playing similar roles. We should note that if

rank[SEpAl=n,

a.e.

(2.31)

then D(s) is nonsingular (Kailath, 1980) so that (2.30) has a unique solution X(s) for all U(s), Y(s) [which must satisfy the compatibility condition (2.29)]. D-~(s) is in this event generally rational. However, if the rank condition (2.31) holds for all finite s (i.e. the system is observable at finite s), then D(s) is unimodular so that X(s) is proper whenever U(s) and Y(s) are proper. It should be noted that there generally exist some restrictions on the allowable inputs U(s), since X(s) in (2.30) should satisfy Ex(O)= O. A detailed analysis should take these restrictions into account. A similar development holds, of course, for discrete systems.

123

B]=rank[sE-A]

a.e.

(2.32)

If it exists, the solution to (2.1), (2.2) is unique with respect to y(t) if and only if

rank[sE~m]=rank[sE-A]

a.e.

(2.33)

Let E be square so that q = n. Then a solution exists for all Bu(t) and is unique with respect to y for all C if and only if rank [sE - A] = n

a.e.

(2.34) []

Note that the regularity condition (2.34) is a sufficient condition for both (2.32) and (2.33) if

q=n. 3. B A C K G R O U N D : R E G U L A R SYSTEMS

If E is square and (2.34) holds, we call (E, A) regular. A simple test for regularity is Luenberger's Shuttte Algorithm.

Algorithm 1. For k = 0, 1 . . . . .

n - 1 perform A~,+I] Ak+lJ

T~[AE: ~ k ] = [ E ~ + l

(3.1)

with E0 = E, A0 = A, Ao = 0, Tk any nonsingular row compression, and E~÷~ of full row rank. [] The system is regular if and only if rank (En) = n.

Existence and uniqueness of solutions We now pass on to an analysis of the existence and uniqueness of solutions to singular systems. For convenience, consider the continuous-time system (2.1), (2.2). We shall say that a solution to (2.1) exists if (2.25) has a solution X(s) for all rational U(s) when Ex(O)= 0. This, of course, restricts our consideration to inputs u(t) and solutions x(t) that have rational Laplace transforms. In contrast to the state-space case, however, the transforms may not be proper. If the solution x(t) exists, it may not be unique. The solution is said to be unique with respect to y(t) if y(t) is unique. The next result is well known and follows directly by considering (2.25) and (2.26) when E x ( 0 ) = 0 , which have a solution when B c R(sE - A ) , and a unique solution with respect to y(t) when N ( s E - A ) c N ( C ) [Lewis, 1983, 1986]. We should remark that, if Ex(O)--0 then the set of inputs is generally restricted. We call admissible those Laplace-transformable inputs that yield an x(t) satisfying Ex(O) = O.

Solutions of regular systems If (sE - A ) is regular, then (2.25), (2.26) yield X (s) = (sE - A )-' Ex(O) + (sE - A )-I BU(s) Y(s) = C(sE - A)-IEx(O) + n(s)U(s),

(3.2) (3.3)

where the transfer function is

H(s) = C(sE - A)-IB. Thus,

in this case

(2.29)

becomes

(3.4) Y(s)=

H(s)U(s). For regular systems, there exists the Laurent series expansion about infinity of the resolvent matrix given by

( s E - A ) - ~ = s -~ ~

k=--~

dPkS-k,

(3.5)

which is unique for some minimum /~, which is called the index of (sE - A). If E is nonsingular, then /~ = 0 . The sequence tp~ is termed the fundamental matrix sequence of ( s E - A ) . An algorithm for the computation of q~, based on

124

F . L . LEwIs

first taking ( s E - A) to triangular form, is given in Mertzios and Lewis (1990). Alternatively, q~k may be computed using the Drazin inverse (Campbell, 1980). By using (3.5) and (2.5) in (2.25) and identifying the highest power of s that occurs, it can be seen that, when ( s E - A) is regular ix - iu = ~ - 1.

by the solution to

Ix0l LXLI [(q,o + ,/,_,_)B(q,,

1_

.

whence the discrete version of (3.5) and an inverse Z-transform yield the forward solution k+/~--I

k • Z.

(3.8)

i=0

In contrast to the state-space case, x~ depends not only on ui, i < k but also on at most / ~ - 1 future values of the input. Thus, singular systems exhibit anticipatory behavior. We should note that the standard LQ regulator exhibits such behavior; that is the state/costate equations comprise a nonoriented system (due to the dependence of the solution on initial as well as final conditions). In the case where the symmetric boundary condition (2.10) is prescribed and the region of interest is k e [0, L - 1], the solution can be shown to be

xk = q)~Exo - dp_~+~Ex~ L-1

+ ~, dp,_i_~Bu,,

k=l .....

L-1.

(3.9)

i=0

The initial and final semistates Xo and x~ are computed using the next result, which also tells when there is a unique solution (Lewis and Mertzios, 1990).

Theorem 2. Let (2.8) be regular. Then, the input sequence u~ together with boundary condition (2.10) define a unique solution xk if and only if the 2n x 2n matrix

M-=[ -(4)*--*+~-0A (~)°+4)-L)E] (3.10) L

Wo

w~

+ q,-,~+OB .

.

.

• • •

.

+

J

is nonsingular. In this event, Xo and XL are given

,3.,,

where Uo.L = [UL-1/ " " /ul/uo]. The nonsingularity of M is equivalent to the admissibility of the symmetric boundary condition (2.10); that is, to the uniqueness of the solution xk. If (sE - A ) is regular, then define P. = (cE - A ) - I E ,

X ( z ) = (zE - A )-~zExo + (zE - A)-~BU(z), (3.7)

~k-i-lBUi,

.

(3.6)

In terms of the fundamental matrix, explicit solutions for continuous and discrete singular system may be presented. Let us consider (2.8), for instance (Lewis and Mertzios, 1990). Considering first the forward solution, where x0 is specified and k • Z, the discrete version of (3.2) is

xg = CPkEXo+ ~

.

(3.12)

A = (ce - A)-IA,

B = (cE - A)-~B, for some c such that I c E - A I ~0. Then, ~ and ~ commute and (E, A, B, C) is said to be in standard f o ~ . In this form, the admissibility test for the symmetric b o u n d a ~ condition simplifies (Nikoukhah et al., 1987). Indeed, (2.10) is admissible if and only if

(3. ~3)

~ = Wo~ ~ + w ~ A ~

is nonsingular. Defining ~ o = ~ - ~ W o , ~ = ~ - l w z , ff = ~ - ~ w , the system with symmetric boundary conditions is put into "normalized form". For systems in normalized form, the symmetric solution is given by L--1

x~ = A ~ - ~

+ ~ G(~, i)Bu.

(3.~4)

i=0

with G(k, i) the Green's function which depends on powers of ~ and ~ as well as on ~o and ~ . A computational technique for solving for x~ given (2.8) and (2.10) for general regular systems (i.e. those in "nonstandard" form) is given by Luenberger's boundary recursion algorithm (Luenberger, 1989).

Algorithm 3. Set Zo(0, 1) = - A , Z l ( 0 , 1) = E. For k = 1, 2 . . . . . L - 1 perform [HF~ jGkk][Zo(00,k ) Zk(O,k)_A 0El =[Zo(0, k + l ) X/, where the nonsingular.

0 Zk+~(O,k+l)] I Yk

transformation

on

the

(3.15) left

is []

Theorem 4. Boundary condition (2.10) is admis-

Geometric analysis of implicit systems sible if and only if [Zo(O,/~) Wo

Z-=L

Z,~(O,/4] W/~ _1

is nonsingular. Then, the initial semistates may be found using

(3.16) and

z[X°]:[V~w-' ]. LXLJ

k=L-1

2,1

.....

(3.18)

Vk = Fkvk-~ + GkBuk,

L - 1,

Vo = Buo.

(3.19)

[]

Weierstrass f o r m If ( s E - A ) is regular, then by selecting the

finite and infinite eigenvectors as a basis for X, and their images under A and E, respectively, as a basis for the codomain of E and A, the singular system may be brought to Weierstrass form. Illustrating for the continuous systm (2.1), ~1 = JXl -F BlU

N~ 2 = x 2 + B2u ,

Xk+l=A-IEXk, Yk+~ = E - ~ A Y k ,

<3.17)

where the intermediate input sequence is defined by

k = 1, 2 . . . . .

is computationally more convenient to work in terms of the original system matrices ( E , A , B, C). Thus, defining the recursions (Wong, 1974)

final

Moreover, the intermediate semistates are given by xk = --XkXo - YkXk+~ + H k v k - 1 + JkUk,

125

(3.20) (3.21)

where xa e R"', xz e R% J is a Jordan matrix representing the finite structure of (sE - A ) , and N is a nilpotent Jordan matrix, with index of nilpotence /~, representing the infinite structure of (sE - A ) . Note that/~ also appears in (3.5). It is sometimes necessary to separate out the effect of the zero eigenvalues of (sE - A ) . Then, we may write the Weierstrass form more explicitly as ~o = Noxo + Bou

(3.22)

Ycr= Jrrt + Bru

(3.23)

N=~= = x= + B=u,

(3.24)

where x= corresponds to x2 in (3.21), No is a nilpotent Jordan matrix representing the structure of ( s E - A ) at s = 0, Jr is a nonsingular Jordan matrix representing its structure at finite nonzero values of s, and x0 e R n°, xr e R + with no+n t =n I. Extending Wong (1974) and Lewis (1984) we shall call Ho = R "° the zero manifold, H~ = R"' the initial manifold, H® = R "~ the infinite manifold, and HF = R+ ~ R"~ the final manifold. While the Weierstrass form is very useful for use in proofs, as well as for the insight it gives, it

X0

(3.25)

Yo

(3.26)

we have the next well-known result describing the various manifolds in the original basis. Theorem 5. 1. Ho = X , if Xo = O 2. H t = X n i f X o = R n 3. n = = Y n i f Yo=O 4. H F = Y, if Yo= R ~.

[] We are now at the point where the following may be demonstrated. Denote subspaces of the anticausal system by superscript " a " and subspaces of the dual system by superscript " d " . Theorem 6. With regard to the anticausal system: 1. (a) n g = H ~ (b) n~ = HF (c) HA = Ho (d) n,~ = Hr. 2. With regard to the dual system: (a) Ho~ = ( A H v ) ± (b) H~ = (AH=) ±

(c)

=

(EH,) l

(d) H~ = (EHo) ±. Proof. With regard to the causal system, simply exchange E and A in (3.25), (3.26). With regard to the anti-causal system to prove the first two equalities, write

Xdk+I=A-TETX~,

Xdo

and use standard identities (Wonham, 1979) to determine the annihilator sequence (X~+O±=AE-I(X~)±,

(X~) -~.

Define Sk = E-~(X~,) ± so that (X~,+I) + = ASk and Sk+~ = E-~ASk,

So = E-'(X~o) ±.

(1)

(a) In the case of Ho~, set X0~ = 0 so that yields HF. Therefore,

S o = R ". Then, (1) H~ = X~ = (AHF) ±.

(b) In the case of H/~, set Xo~ = R " so that We claim that (1) now yields S, = H , . Indeed, consider (3.26) with Yo=0. Then Y0 c So, and if Yk c Sk, then SO=N(E).

Yk +~ = E - 1 A Yk c E - I A S k = Sk +~.

On the other hand, SOc Y1, and if Sk = Yk+l, then Sk+~ = E - ~ A S k ~ E-~AYk+~ = Yk+z.

126

F . L . LEwis

That is, H= = S~ = H=. Equalities (c) and demonstrated.

(d)

are

similarly []

The equalities in part 1 reveal nothing but the relation between the pencil ( s E - A ) and the anticausal pencil ( E - t A ) . This relation is explored in more detail from the point of view of Kronecker invariants in Karcanias (1981). Duality results 2b and 2c were first demonstrated in Cobb (1984). It is important to note that spaces in the domain have annihilators residing in the dual space of the codomain of E and A. Thus, as we shall see in the next sections, duality for singular systems has a far richer structure than it does for state-space systems, where the domain and codomain of s l - A may always be considered as the same space. It is interesting to consider the solutions for singular systems in Weierstrass form. Let us look at the solution for (2.1) and the forward solution of (2.8). First, consider the continuous-time implicit system. Since, in the Weierstrass form ( s E A)-~=(SI-J) -~(sN-I) -~, we may write (3.2) as

X~(s) = (sl - J)-~x~(O) + (sl - J)-~B~ U(s) (3.27)

We say x(0) is admissible if there exists an input u ( t ) • C " - I ( u ) [i.e. u(t) is # - 1 times differentiable] such that x(t) is continuously differentiable. To select an input that avoids impulses in x(t), (3.32) says the initial condition must satisfy

X2(0 ) • N - a R [ N B z

N2B2 ' ' ' Nu-aB2].

(3.33)

Defining the reachable subspace at infinity as

R ~ = R[B2

N B z . . . N"-IB~]

(3.34)

and noting that any value of xl(0) is admissible, we see that the subspace of admissible initial conditions is given by

x(O) • R"' ~ (R~ + N ( N ) ) .

(3.35)

It can be shown that the subspace of admissible conditions for discrete-time systems is also given by this expression. An analysis of the discrete symmetric solution in Weierstrass form is given in Luenberger (1989). Condition (3.10)/(3.13)/(3.16) for admissibility of the boundary condition on [0, L - 1] reduces to the nonsingularity of

Mw¢=[Wo~ +Jt~WL~

N~Wo2+ Wm], (3.36)

where W0=[W01 W02], W L = [ W m Weierstrass form. Moreover,

Win]

in

X2(s) = (sU - l)-'Uxz(O) + (sU - I)-~BzU(s) (3.28) so that x~(t) =e~x~(0) +

e~('-~B~u(~) d~

(3.29)

_0,],

N--1 X:~(1) = -- Z

(~(i-li(t)Nix~(O)

i=1 .u-1 - ~ N~B~u~°(t), i=0

(3.30)

with 6 ( 0 the Dirac delta and superscript (i) the ith distributional derivative. Using the identity

u ti) = u lil+ 6uli-q(0) + . • • + O
~-1

~-2

~ u i o 2 u l i l ( t ) - ~ ~(i)Ni i=0 i=0

I ~--2--i ] × Nx2(O)+ ~ NJ÷lB2ut~l(o) .

o,0,

zk(0,

U0k].

(3.31)

where [i] denotes the ith ("regular") derivative, we may write (3.30) as Ozqaldiran (1985)

Xz(t ) = -

(3.37)

(3.32)

j=O

An alternative expression for xl(t) in terms of the Moore-Penrose inverse is given in Dias and Mesquita (1990).

4. OUTPUT-NULLING AND COMPOSITE SUBSPACES The motivation for this section is to provide a general framework to tie together the recent work by several researchers in the geometric theory and properties of generally nonregular singular systems. The material here is inspired by Basile and Marro (1969), Wonham (1979) and Molinari (1976) and is a special case of the material in Beauchamp (1989). Wherever proofs are omitted they may be found in Beauchamp (1989) and Beauchamp et al. (1991). Also important in this context is the work by

Geometric analysis of implicit systems Rosenbrock (1974), Willems (1981, 1982), Ber- " nhard (1982), Schumacher (1983), Armentano (1986) and Malabre. (1987a, b, 1988, 1989a, b). Output-nulling To develop firmly on the systems, let us

and unknown-input subspaces a geometric theory that is based dynamical properties of singular consider the discrete system Ex~+~ = AXk + BUk

(4.1)

Yk = CXk.

(4.2)

The corresponding theory for continuous systems of the form (2.1), (2.2) will then be retrieved through (2.6). Let x • R ~ = - X , u•Rn~==-U, y • R P = - Y , and E be a q × n matrix. That is, we shall not assume regularity. Following Banaszuk et al. (1988a, b,c, 1990), we shall call X the inner space, and R q, the codomain of E and A, the outer space _X. Subspaces of _X will be called inner subspaces, and subspaces of _X will be underlined (cf. Malabre (1987a, b, 1989a, b)) and called outer subspaces. The semistate (or inner variable) is Xk while Ex~ and A x k a r e outer variables. In economics, the inner and outer variables are traditionally distinguished from one another and both play an important role. In fact xk represents intensity of production, gxk represents goods consumed, and Axk represents goods produced. The terminology "inner" and " o u t e r " arises from a need to distinguish between objects in the domain and the codomain for implicit systems. It should not be confused with the terminology of Willems (1981, 1982, 1989) and Aplevich (1981, 1985, 1989) where Xk is called the "internal" variable and Uk and yk are "external" variables. Much thought and discussion have failed to produce more suitable terms, though as an alternative all objects in the codomain could be prefixed by "co-". A general sort of design problem may be attacked by requiring the output y~ to be zero over an interval. For instance, if it is desired to maintain the state trajectory xk within a specified subspace S c X , we may select C such that N(C) = S; then, yk = 0 is equivalent to Xk • S. By _Xk we mean the proper sequence {x0, x~ . . . . . xk_~}. Similarly defined are Yk and _uk. By X'k we mean the generally nonproper sequence { . . . . x_~,Xo, Xl . . . . . Xk-1}. By a trajectory we shall mean a solution sequence Xk, generally nonproper, to (4.1) for some Uk. Define four subspace sequences: Unknown-input (UI) unobservable subspace Vk = {Xo • X [ ~ a trajectory _X~+1 for some _uk w i t h y ~ = 0 } , k - > l , Vo=R ~. - -

127

Null-output (NO) controllable subspace Ck = { x 0 • X [ 3 a proper trajectory _Xk+l for some _Ukwith _Yk= 0 and Xk = 0}, k -> 1, Co = 0. Unknown-input (UI) unconstructible subspace Uk = {X~ • X ] ~ a trajectory _x~,+~for some _Uk with Yk = 0}, k >-- 1, Uo = R n. Unknown-input (UI) unconstructible subspace U~ = {Xk • X [ ~ a trajectory _x~,+~ fo some _Uk with y~ = 0}, k -> 1, Uo = R ~. Note that the definitions are prespecified for k = 0 independently from the dynamics of the system. This is due to the fact that in computing, for instance, Vk, only outputs through Yk-1 are considered. In the case of Vo, Y-1 is not even defined, so that no restrictions should be placed on x0. It should be noted that these definitions generalize those of Molinari in several ways. In particular, in the definition of Rk we do not require x0 = 0, but instead that the semistate trajectory be proper (i.e. E x 0 = 0 so that Ax_ 1 -[- Bu_ ~= 0). By using standard techniques (Aoki and Li, 1973) the next result may be shown. -

Theorem 7. Define the subspace recursions Xk+~ =A-~(EXk + B) f~ N(C), Xo

(4.3)

Yk+, = E-'[A(Y~, fq N(C)) + B], Yo. (4.4) Then 1. Vk = Xt, when 2. R~ = Yk when 3. Ck = Xk when 4. Ug = Yk when

Xo = R n Yo = N ( E ) X0 = 0 Y0 = R n.

[]

Another subspace of importance is almost reachability subspace (ars): RE = {Xk • X ] 3 a proper trajectory xk÷l for some Uk with Yk÷l=O}, k-->l, Roc = N ( E ) fq

N(C). Note that Rk and Rkc are identical except that in the latter it is required also that y, = 0. Standard techniques show that the ars is given by R~+I = E - ' ( A R ~ + B) f3 N(C), R~ = N ( E ) N N(C).

(4.5)

This is identical to the ars (Ra.) used in Willems (1981) when E = I . (Thus, our terminology "almost teachability subspace". In point of fact ars may not be a suitable name for RE, but it is traditional.) Clearly, for k > 0 RE = Rk N N(C),

(4.6)

whence derives our notation superscript " C " . In the remainder of this paper we shall consider only V~,, Rk, and R~c, leaving Ck and Ut

128

F.L.

for future reside in subspaces. sequences

investigation. These three sequences the domain X and so are inner Define three related outer subspace of _X by

Yk = EVk

(4.7)

V_~ = y~ + B = EV~ + B

(4.8)

_Rk+l = aRc~ + B

(4.9)

Vk+, = A-'V_~ (q U(C)

(4.10)

R k = E-I_Rk

(4.11)

RE-- Rk f~ N(c) = E-~_Rk f~ N(C).

(4.12)

At this point it will be useful to examine Fig. 1, which shows the relations between these inner and outer subspaces, as well as the recursions that generate them. In the figure, N(C) is denoted by N. This figure extends the notions in Malabre (1987). It is straightforward to give all of the outer subspaces sensible dynamical definitions in terms of outer variables. See the work of Banaszuk et al. (1988a, b, c, 1989a, b, 1990). The next result is a main duality result. It may he viewed as the extension of Theorem 6 part 2 to the case of a nonhomogeneous system with an output. The proofs are obtained by using the standard identities in Wonham (1979) on the generating recursions (see also Malabre, 1987).

Theorem 8. 1.

=

=

To unify these various subspaces and streamline the relations among them, define the composite image of an inner subspace S c X as

S = ES + A S + B

(4.13)

and the composite preimage of an outer subspace Tc_Xas

T=E-ITnA-1TC~N(C).

(4.14)

Then

and note that

2.

LEWIS

3. y~O = R~ "±. [] It is important to note that the duals of inner subspaces are outer subspaces. These duality relations are denoted by a double bar in Fig. 1.

Theorem 9. Let S = T'. Then _S= T ±. That is, the composite preimage and composite image are dual concepts.

We shall now consider some properties of the limiting subspaces, which are denoted using "*". First, note from the recursions and the properties of the inverse image that

A V , c E V , + B,

(4.15)

ERc, c ARC, + B

(4.16)

that is, V, is an (A, E, B)-controlled invariant subspace (Verghese, 1981; cf. Wonham, 1979) and RC, is an ( E , A , B)-controlled invariant subspace. Also

A-1Va, fq N(C) ~ E --1 V ,B

(4.17)

E - 1 R , f~ N(C)

(4.18)

=

A-'R,,

that is _V, n is an (E, A, N(C))-conditioned invariant subspace while _R, is an (A, E, N(C))conditioned invariant subspace (cf. Basile and Marro, 1969; Malabre, 1987). It should be noted that controlled-invariant subspaces reside in the inner space while conditioned-invariant subspaces reside in the outer space. In the limiting case, relations (4.7)-(4.12) hold with subscripts k or k + 1 replaced by *

t V~+~= A'~(EVt+B) nN, Vo= R"

R~.~= ~ ( E - ~ N )

IEV~

[

[

Rk~N

=E-~ ~ N

_ E(A-~VS~N) -~ B~+,+ B, ~0= E+B ~

~-~

[

Rk~.,= E'~(A(R~nN) +B)~ ~ = N(~)

_V~+ B =EV~ + B

k=k+1

+ B, Ro= 0

E'~_~

V~+,= E(A'~(_V~+B) nN), ~'o= E

the []

n~.~= E'~(~a~+S) n . , ~o= ~(S)~n. ~

~

k=k+l ARC~ + B

~N FIG. 1. I n n e r a n d o u t e r s u b s p a c e s .

Geometric analysis of implicit systems These relations, as well as some others, are formalized next.

Theorem 10. 1. V_S, = E V , + B = E V , + A V, + B 2. V, = a --1 y ,B n N ( C ) = E - 1 y .B n A

--1

y , Bn

N(C) 3. _R, = ARC, + B = ERC, + ARC, + B 4. RC,=E-'_R, riM(C) = E-'_R, nA-'_R, n

N(C) 5. Let N ( E ) n N ( C ) = O . Then d i m ( E V , ) = dim (V,). Let N ( A ) n N(C) = 0. Then dim (ARC*) = dim (RC*).

Proof. 1-4 The first equalities are by definition. The second equalities follow using (4.15)-(4.18). 5(a) dim ( E V , ) = dim (V,) - dim (N(E) n V,), but V, c N(C). (b) dim (ARC,) = dim (RC,) - dim (N(A) n RC,), but RC, c N(C) [] Thus, 1-4 show that the subspaces V, and _V, n (and also RC* and _R,) are respectively the composite preimage and composite image of each other. When ( s E - A ) is regular, the rank conditions in part 5 are the conditions for observability at infinity and at zero, respectively.

Composite subspaces In the state-space case E = I, if C = 0 then R, is the reachable subspace, while if B = 0 then V, is the unobservable subspace. However, due to the nonoriented or noncausal nature of singular systems, things are not so simple in case E is singular or nonsquare. Therefore, let us define the composite subspaces ~k = V, n Rkc

(4.19)

_~ = y,~ n _R,

(4.20)

#k = V~ + RC*

(4.21)

_V~-~- _R,.

(4.22)

-~k =

Note which of these are inner and which outer subspaces. Let us discuss ~k. Although x ~ R k guarantees that there exists a proper trajectory x; inside N(C) on [0, k] with a final value of Xk =X, it does not guarantee that the trajectory remains inside N(C) for i > k, or even that a solution to (4.1) exists for i > k . However, X e ~ k guarantees both that there exists a proper trajectory xi inside N(C) on [0, k] with a final value of Xk = X, and that the trajectory exists for i > k and remains within N(C). In the case E = I , ~ , is the supremal

129

reachability subspace contained in N(C), ~ , is the infimal almost conditionally-invariant subspace containing B, # , is the supremal almost controlled-invariant subspace contained in N(C) (i.e. Willems' Va,), and _~, is the infimal complementary observability subspace containing B (or, alternatively, Willems' _Vb,). See Wonham (1979), and Willems (1981, 1982). We shall call ~ , (_~,) the supremal inner (outer) null-output reachability subspace. We shall call # , ( # , ) the infinal inner (outer) unknown-input unobservability subspace. These names will be justified as we proceed. The next theorem summarizes some of the properties of the composite subspaces. Some of these properties have appeared in the work of Banaszuk et al. (1988a, b, c, 1989a, b, 1990), as well as, when E = I, in the work of Willems (1981, 1982). We should recall that no assumption of regularity has been made here.

Theorem 11. 1. ~ = V, n Rk

_~ =

( y , n _R~) + B

•gk = (Vk + R , ) n N(C)

_~k= Y~+ _~, 2. ~ kd = ~ ,.l_

~

=

y~

3. ~ , = E - ~ , n A - ~ , n N(C) = E - ~ , n A - ~ y ~ n N(C) ~, = E~, +A~, + B = E~, + B =A~, + B ~ , = E - ~ , n A - ~ , n N(C) = E - ~ , n

N(C) = A - ' ~, n N(C) ~, = E~, + A~, + B = EV, + AR~ + B 4. E ~ , c A ~ , + B and A ~ , c E ~ , + B E - I ~ , n N ( C ) c A - ~ , and A - ~ , n N(C) c E - ~ , . Remarks. Part 1 shows some alternative techniques for computing the composite subspaces. Part 2 demonstrates the duality between the composite subspaces. Part 3 demonstrates that the inner and outer versions of the subspaces are the composite image and preimage of each other, as well as giving simplified expressions for the image of ~ , and the preimage of _~,. Part 4 says that ~ , and # , each enjoy two invariance properties. Proof of Theorem 11. (Note: the simplest way to follow the proofs is by reference to Fig. 1.) 1.a. ~k = V, ('~ RCk = V , n Rk fq N ( C ) = V , n Rk since V, ~ N(C). b. -~k = y,~ n _Rk = ( y , + B) n _R~ = y , n _R,~ +

B since B c _Rk. c and d follow similarly.

130

F.L.

LEWlS

2.a. (~)~- = (V~, ~ R~a)" = (V~,)" + (R~Ca)± = _R, + _V~ by Theorem 8. b. similarly. 3.a. E-a_~, IqA-I_~, ~ N ( C ) = E - I ( y ~ , n _R,) n A - I ( y ~ , n _R,) n N(C) = E-'_R, f l A-'y~, n N(C) by (4.17), (4.18) = R ,c n V, by Theorem 10. b. E~t, + B = E ( V , n R c , ) + B = E[V, I-1 N(C) n E-'(ARC, + B)] + B by (4.5) = [E(V, n N(C)) n (ARC. + B)] + B = [EV, f3 (ARC. + B)] + B = (EV, + B) n (ARC.+ B) since B =ARC,+ B = _V, n n _R, = _~, by Theorem 10. A ~ , + B = A ( V , n RC,) + B = A [ A - ' ( E V , + B) n N(C) n RC*] by (4.3), and then similar to the previous proof. Now, note that

Theorem 12.

1. ¢ , = E v ,

5. ¢~,

1. ¢ , = e ( V , n Rc*) = E[V, n N(C) ~ E-'(ARC, + B)] by (4.5)

= E ( V , n N(C)) n (ARC, + B) = EV, n _R, 2. _~, = (_V, + B) ~ _R, = (_V, n _R,) + B since B c _R,.

3. U, = E-I~_,= E-~(V_ n, + _R,) = E-'[E(A-'V_~, n N(C)) + B + _R,] = A-I(V_B, n N(C)) + N ( E ) + E-I_R, = V, + E-I_R..

[]

4. O, n N(C) = N(C) n ( v , + R , )

In Oz$aldiran and Lewis (1989), a controllability subspace was defined as C = ER where R is a reachability subspace. Thus, define the kth null-output controllability subspace as ¢~ = E ~ ,

= V, + N(C) n R , since V, = N(C) = V , + RC, =

(4.24)

5. REACHABILITY, OBSERVABILITY AND DUALITY FOR NONREGULAR SYSTEMS In this section we shall discuss reachability and observability for general systems which may be nonregular. A duality theory is presented. This work is motivated by the point of view of

and note that 0h is an inner subspace. The next result shows further relations between ~ , and 0 , and previously defined subspaces, as well as giving another duality result.

~v~l

l Vk+B

yBk,~= E (A'I~BknN) +B /

~_.=_v.~_R. ~//

~ //

R~**= A(E-I_RInN)+B

/

~

.. _ ~

g~,~= ~[~-~(g~+~)n~]~

k=k+l

.

~..~.=v.÷~.

I~ ~

~ ~W ~'~[~(~n~)+~] ~

[]

All the subspaces we have mentioned so far are shown in Fig. 2, which should clarify the relations between them.

and note that ¢ , is an outer subspace. Likewise, define the kth unknown-input unconstructible subspace as

Vi,i= A ~(EV~+B)ON

.,~,.

5. ( ~ , ) ~ = (v_~, n _~,)" = R , + V, = 0 , .

(4.23)

0, = E - ' _ ~

hi = U~.

Proof.

E ~ , + A ~ , + B = ( E ~ , + B) + ( A ~ , + B) = _~, c and d similarly. 4. Follows directly from 3.

n_a,=v_,ns,

2. _~, = ¢ , + B 3. O , = V, + E-I_R, = V, + R , 4. o¢, = 0 , M N(C)

~ RknN

.

N Rck+l= E -1(ARCk+B) nN

~

~

A-ly~i n N

k=k+l ARck + B

FIG. 2. Compositesubspaces.

Geometric analysis of implicit systems Banaszuk et al. (1988a, b,c, 1989a, b, 1990) 0zqaldiran (1985, 1986, 1987), and Baser and 0zqaldiran (1989). We shall also discuss the minimality (Aplevich, 1985), regularizability (Ozqaldiran, 1985, 1986, 1987), and regularity (Armentano, 1986) of singular systems. There are many definitions of reachability and observability properties in the literature. This plethora of terms is due to the fact that there are two sorts of reachability involved with the infinite modes, and two sorts of observability involved with the infinite modes. In addition, reachability and observability properties in the domain and codomain are different. Here, we clarify this situation. Inner and outer system properties Recall the composite subspaces that were defined in Section 4 as ~,~-V, nRc,--E-'_R, nA-'V_~,nN(C)

(5.1)

_~, -= _V, ~ VI _R, = E ~ , + A ~ , + B

(5.2)

`9,=-V,+RC,=E-~,fqA-I_#,fqN(C)

(5.3)

#_, =- V_n, + _R, = E V , + ARC, + B.

(5.4)

We called ~ , (_~,) the supremal inner (outer) null-output reachability subspace, and `9, (_#,) the infimal inner (outer) unknown-input unobservability subspace. A general theory of reachability and observability may be developed that allows both outputs and nonzero inputs using the output-nulling and unknown-input notions (Beauchamp et al., 1991); however, in this section we shall follow the current work by considering C = 0 for reachability and B = 0 for observability. The importance of ~ , and ,9, in reachability and observability respectively, has been demonstrated in the literature. The next two theorems collect some of these results, while giving some additional ones that show the importance of the outer composite subspaces. Various sorts of reachability and observability are defined in the theorems. We mention both the continuous system (2.1), (2.2) and the discrete forward and symmetric systems described by (2.8), (2.9). Note that regularity of ( E , A ) is not assumed; indeed, E may be nonsquare. We use the notation for sequences introduced in Section 4, adding the notation _x'~ (_x~) to denote nonproper (proper) sequences with unbounded support on the right. Theorem 13: Inner composite subspaces. I. Reachability properties. Let C = 0. Then 1. Forward discrete system. _R, = {x e X [ 3 a proper trajectory _X~ for some _u~ with xg = x for some k}.

131

2. Symmetric discrete system. ~ , = X iff for every pair ( x , z ) e X x X there exists a trajectory _Xk+~ ' with x0 = x and Xk = Z for some k e Z and some _ug. II. Observability properties. Let B = 0. Then 1. Forward discrete system. `9, = 0 iff yg = 0, k -> 0, implies x0 = 0 and yg = 0, k - 0, implies x0=0. 2. Symmetric discrete system. ,9, = 0 iff, for all k ~ Z , _yg+~=O implies that _Xk+l=0, or equivalently that x0 = 0 and xg = 0. Proof. 1.1. Recall that ~ , = V, t~ R ,c. Now, x e R ,c iff there exists k and _Uk SO that the trajectory is proper and Xk = X. The trajectory exists for i > k for some ui, i >- k, iff Xk = X ~ V,.

The rest of the result has been shown by Banaszuk et al. (1988a, b, c, 1989a, b, 1990). [] In connection with ~ , we shall need to consider the systems with disturbances d defined by Ex = A x + Bu + d

(5.5)

EXk+~ = AXk + Bug +dg.

(5.6)

Theorem 14: Outer composite subspaces I. Reachability properties. Let C = 0. Then 1. Symmetric discrete system. _~, = _X iff, for every pair (x, z ) e X x X and every disturbance sequence _dk, there exists a trajectory _x~,+~ of (5.6) with x0 = x and Xk = Z for some k 6 Z and some _Ug. 2. Continuous system. _~, = _X iff, for every x e X and every d(t) ~ C~-I(_X), there exists a continuously differentiable trajectory x(t) of (5.5) with x ( 0 ) = 0 and x ( T ) = x for some T and u(t) ~ C"-~( U). II. Observability properties. Let B = 0. Then 1. Symmetric discrete system. _#, = 0 iff, for all k ¢ Z , ),g÷~=0 implies that E_~x+~= A__~x+~= 0 , or equivalently that E x o = 0 and Axg = O. 2. Continuous system. _#, = {Ex(0) e _X t 3 a solution x(t) with y(t) = 0}. [] Remarks. We call the system reachable if ~ , = X (with C = 0) and observable if # , = 0 (with B = 0). Because of Theorem 14, if _~, = _X we shall call the system strongly reachable. If # , = 0 we shall call the system weakly observable. The first two properties are defined with respect to the domain X and the last two with respect to the codomain _X.

132

F.L.

Proof of Theorem 14. 1.1 and I1.1 have been shown by Banaszuk et al. (1988a, b, c, 1989a, b, 1990). For 1.2, note that if x ( 0 ) = 0 then the discrete equivalent (2.6) is exactly (5.6). 11.2 has been shown by Baser and Ozqaldiran (1989). [] Regular systems We assume now (E, A ) is regular. Then, we may give explicit expressions for some of the subspaces in terms of familiar subspaces in the Weierstrass form (3.20), (3.21). In the Weierstrass basis, partition the output matrix C so that Y =Yl+Y~,

Yl= Clx~,

y~= C~x~, (5.7)

with xl ~ R "~, x~ ~ R ~, y ~ R p. The initial manifold is Ht = R n~ and the infinite manifold is H ~ = R "2. Define the finite and infinite reachable subspaces in the standard fashion (Yip and Sincovec, 1981) as, respectively, RF= R[B1

JB, . . . Jnt-lB1].

(5.8)

R~ = R[B2

gB2"

(5.9)

" N"-IB2]

LEWIS

R~=, = (E-1AR~) (') N(C), R~o= U ( e ) ~ N(C).

Theorem 16 (Observability properties). Let (E, A ) be regular and B = 0. Then 1. The limit V, of (5.13) is independent of Vo as long as H, ~ N ( C ) ~ Vo.

2. V , = U,~ 3. RC.=u~ 4. 5 , = U ~ U , ~ .

N

.

,

U= = N

[c21 C2.N .

•

Proof of Theorem 16. The proofs are in the style of Ozqaldiran (1985). 1. Let Vk be given b_y (5.13) and 17,+1 = A-IE~'k fq N ( C ) with Vo -- H~ fq N(C), where H~ = H, in Hk+I = A-1EHk, with Ho = R". Then I2oc Vo. If ("k ~ Vk, then

~/,+, = A-1E~Zk f"l N(C) ~ A - a E V , n N ( C ) = vk+l.

Therefore I~, c V,. On the other hand, Ho ~ V,. If Hk ~ V,, then (5.10)

Hk+l = A-1EHk ~ A - ~ E V , = A - a E V , tq N ( C ) = V,.

L C2N u- 1 J

Lcd.,-lj

[]

Remarks. Result 1 is a technical result required to prove results 2 and 3. A comparison of this theorem to Theorem 15 presages the duality results which will soon be formalized.

and the finite and infinite unobservable subspaces, respectively as

U~=

(5.14)

The next two theorems give additional insight on some of our subspaces.

Thus, ~'o = Ht fq N ( C ) ~ V,. If I2~ ~ V,, then

Theorem 15 (Reachability properties). Let (E, A) be regular and C = 0. Then 1. V , = H ~ R = 2. R C,= R F ~DH~ . 3. ~ , = R v ~ R ~

Therefore, 17",= V.. Now suppose that l?k+l = (A-1Ef"k) fq N(C), with "('0= H1 fq N(C). Then Vo = 12o= ~o- If Wk :~ ~-[k = ~ k , then

~rk+ 1 = A-'E(Zk fq N ( C ) = A - ~ E V , fq N ( C ) = V,.

f"k÷l = A-1E~'k fq N ( C ) c A-1EVk Remark. In Weierstrass form, therefore, we have V, = R " ' ~ R ~

(5.11)

RC, = Rv • R "~.

(5.12)

Proof. Parts 1 and 2 are proven by performing the recursions on the Weierstrass form. In proving 2, use is made of the fact that the limiting value of R~ is independent of the initial value Roc as long as RCo~ H~ ~ N ( C ) (0zqaldiran, 1985). Part 3 follows directly. [] In showing the next result, we use the fact (See Fig. 1), that when B = 0,

Vk+~=(A-XEVk)~N(C),

Vo=R"

(5.13)

n N ( C ) = vk+~

and (/k÷, = A - ' E ( / k N N ( C ) = A-'E("k n N ( C ) = f4÷,.

Therefore, lT"n= V,. 2. Consider (5.13) with initial condition Vo = Ht in the Weierstrass basis. According to part 1, the limiting value V, will be unaffected by the change in Vo. By li, i = 1, 2 denote the ni × ni identity matrix. Then Vo = R n' • 0 and V, = (J ~ I2)-'[(I, ~) N)(R"' ~ 0)] fq N[C, = q ~ I2)-l[R n' ~ 01 n NIC,

= (n", ~ O) n N[C,

Cd C:] = N(C,) ~ O.

Ca]

Geometric analysis of implicit systems If V~ = Ni=O k--1 N(CIJ i) (3 0 =-- UFk (3 O, then

Vk+l = (J (3 12)-a[(I1 (3 N)(UFk (3 0)] f~ N[Cl = (J (3 I2)-l[VFk (3 0)] f~ N[Ca C2] = [i~-~=~N(ClJi)(3Ol f'~ N[Ca

c2]

Theorem 18. 1. Let C = 0 and rank [E A B] = n . ~ , = X iff _~, =_X. 2. Let B = 0 and rank [E/A/C]=n. # , = 0 iff _#, = O. The

Re

= fq N(Cd') (30. i=0

Therefore, V, = Ue (30. 3. Considering (5.14) in the Weierstrass basis yields Roc = 0 (3 (N(N) f3 N(C2)) and

Then []

~ N[C1 C2] = (Ia (3 N)-I[O (3 (N(N) VIN(C2))] C2]

= [0 (3 ( N ( N 2) f3 N(C2N))] C2l

= 0 (3 [N(N 2) ~ N(C~) ~ N(C2N)]. Now, an inductive proof like the one in part 3 shows that, for all k -> 0,

R f = 0(3 N(N~'+a)(] N(C2N ~ , i=0

whence RC, = 0 (3 U® since N" = 0. 4. By pans 2 and 3, V. + R .c = UF (3 U~.

system

is called

finite

= R nl and finite observable if

reachable

if

UF = O.

There are two sorts of reachability and observability associated with the infinite modes. We say (E, A, B) is reachable at infinity if all the modes at infinity, including trivial chains (i.e. those of length one), are reachable. This is equivalent to R~ R hE. On the other hand, we say (E, A, B) is controllable at infinity if the nontrivial modes at infinity are reachable. This is equivalent to R~ + N(E) = g hE. Ozqaldiran (1985, 1986) has shown that for continuous-time systems, reachability at infinity is equivalent to the ability to reach any state in R "2 from x2(0)--0, while controllability at infinity is equivalent to the ability to drive any x2(0) to the origin in R hE. Likewise, we call the system observable at infinity if all the modes are observable. This is equivalent to U~ = 0 (with B = 0). We say the system is detectable at infinity if the nontrivial modes are observable. This is equivalent to =

Rxc = (I1 (3 N)-a[(J (312)[0 (3 (N(N) ~ N(C2))]

f'3 N [ C l

Then

GI

k

CI N [ C 1

133

U~ ~ R(E) = O. []

Under certain conditions reachability is equivalent to strong reachability, and observability is equivalent to weak observability.

Theorem 17. 1. Let C = 0 and (E,A) be regular. Then ~ , = X i f f _~, =_X. 2. Let B = 0 and (E, A) be regular, then # , = 0 iff ~ , = O.

Proof.

Reaehability and observability at infinity are in the spirit of Rosenbrock (1974) and Yip and Sincovec (1981), while controllability and detectability at infinity are in the spirit of Verghese et al. (1981).

Inner~outer duality At this point a convenient duality theory may easily be presented between the inner composite subspaces of Theorem 13 and the outer composite subspaces of Theorem 14. Indeed, this theory has already been presented in Theorem 11, part 2, which stated

1. In Weierstrass form

~ , = V_B,A _R, = (EV, + B) VI (ARC, + B)

~kd D-- -#kA-

(5.15)

= [(Ia (3 N)(R'*' (3 R®) + R[B1/B2]]

_~ = ,~-

(5.16)

N [(] (3 Iz)(RF (3 R "2) + R[B1/B2]] = [(R"' (3 NR=) + R[B1/Bz]]

N [(JRF (3 R "2) + R[B1/Bz]] = [R"' (3 R~o] N [RF (3 R"2], which is isomorphic to ~ , . 2. Similar.

[]

In fact Banaszuk et al. (1989a) have shown the following milder result.

That is, reachability and weak observability are duals, as are strong reachability and observability (Banaszuk et al., 1989a). The convenience of this approach to duality arises from the fact that both inner and outer subspaces are involved in the theory. It should be contrasted with the somewhat clumsy approach to duality taken in Cobb (1984), where only inner subspaces are involved and the Weierstrass basis is employed.

134

F.L.

Rank tests It would be desirable to extend the state-space Hautus-like rank tests to the case of implicit systems, generally nonregular. The next theorem provides this extension, as well as some increased insight on the structure of nonregular systems (Aplevich, 1985; Banaszuk et al., 1989b). Theorem 19. I. Let C = 0. Then 1. _R, = _X iff RIsE - A

B] = _,_,_Xfor all finite

S.

2. RC,=X i f / E c R [ s E - A

B] for all finite

S.

3. V , = X i f f A c R [ E B]. 4. V f = _Xiff R[E BI=_X. 5. _~, = _X iff R[sE - A B] = _X for almost all finite s. 6. ~ , = X iff the conditions in parts 2 and 3 hold, or equivalently i f / R [ E A] ~ R[sE tA B] for all (s, t) # (0, 0). 7. _~, = _X i f / t h e conditions in parts 1 and 4 hold, or equivalently i f / R I s E - tA B] = _,_X for all (s, t) :# (0, 0). II. Let B = 0. Then 1. V, = 0 iff N [ s E - A/C] = 0 for all finite s. 2. _V, ~ = 0 iff N ( E ) ~ N [ s E - A / C ] for all finite s. 3. _R, = 0 iff N(A) = N[E/C]. 4. R~, = 0 i f / N [ E / C ]

LEWIS where x(t) is the internal variable and the external variable w(t) is comprised of the system inputs and outputs. Regularity is not assumed. A realization (F, G, H, 0) with external behavior w(t) is said to be (externally) minimal if: 1. The rows of [ s F - G H] are independent over the field of rational functions of s. 2. The length of x(t) is minimal over the set of pencil realizations of w(t). The next result uses HI and Ho~ as defined in Theorem 5 as well as _V, a and _R, from Fig. 1. For clarity, let us denote HI associated with (sF - G) as H~(F, G, O, 0), and so on.

Theorem 20. System (5.17) is minimal if and only if: 1. F has full column rank, or equivalently H,(F, G, O, O) = O. 2. (sF - G) has full column rank for all finite s, or equivalently HI(F, G, O, O) = O. 3. IF HI has full row rank, or equivalently _V,B(F, G, H, 0) = _X. 4. [sF - G H] has full row rank for all finite s, or equivalently _R,(F, G, H, 0) = _X. Proof. The first statement in each part follows from Aplevich (1985). The second condition follows from Theorem 19. [] In the special case of

= O.

5. ~ , = 0 iff N [ s E - A / C ] = O for almost all finite s. 6. _~, = 0 if/ the conditions in parts 2 and 3 hold, or equivalently i f / N [ E / A ] ~ N[sE tA/C] for all (s, t) #: (0, 0). 7. 3~, = 0 iff the conditions in parts 1 and 4 hold, or equivalently i f / N [ s E - tA/C] = 0 for all (s, t) :# (0, 0).

Remarks. In both I and II, conditions 5 are milder than conditions 1. In fact 1.1 is the condition for finite reachability, while 1.5 is a condition for the existence of a solution for all inputs. On the other hand II.1 is the condition for finite observability, while 11.5 is a condition for the uniqueness of solutions with respect to the output. See Section 2. Proof. The proofs are either shown in Banaszuk et al. (1989a, b) and Aplevich (1985) or follow directly from results there. [] Minimality Willems (1981, 1982, 1989) and Aplevich (1981, 1985, 1989) have considered systems of the form FY¢= Gx + Hw, (5.17)

Ek = Ax + Bu

(5.18)

y = Cx

(5.19)

(5.17) becomes

so that the theorem becomes

Corollary 21. System (5.18), (5.19) is minimal if and only if 1. E has full column rank, or equivalently H.(E, A, 0, 0) = 0. 2. [ s E - A/C] has full column rank for all finite s, or equivalently V,(E, A, O, C) = 0. 3. [E B] has full row rank, or equivalently _V,~(E, A, B, 0) = _X. 4. [ s E - ,4 B] has full row rank for all finite s, or equivalently _R,(E, A, B, 0) = _X. Proof. Parts 1 and 2 are immediate. 3. Note that [E B] has full row rank iff 0

has full row rank. 4. Note that [ s E - A

B] has full row rank for

Geometric analysis of implicit systems all finite s iff

[sE~AIB 0

~]

has full row rank for all finite s. At first sight, it may appear that observability at infinity is not required by this theorem. However, note that condition 1 indeed implies the full rank of [E/C]. In general, externally minimal realizations are not square. Grimm (1988) defines (5.18), (5.19) as minimal if the number of rows and columns of E are minimal among systems having the same input-output relation. This results in more symmetric conditions for minimality where condition 1 of the corollary is replaced by the full column rank of [E/C], or equivalently R,C=0. It also requires the absence of trivial chains at infinity, that is AN(E) c R(E). Verghese et al. (1981) consider the regular case and define a system to be (internally) minimal if it is finite reachable, finite observable, infinite controllable, and infinite detectable. Thus, they are not concerned with the reachability or observability of the trivial chains at infinity. This is equivalent to conditions 2 and 4 of the Corollary as well as the absence of infinite zeros in [sE - A B] and [sE - A/C].

Regularizability and regularity Even if an implicit system is not regular, it may become regular when either semistate feedback or output injection are applied. This, we call the regularizability property. The next results have been shown in Oz~aldiran and Lewis (1990) and Baser and Oz~aldiran (1989).

Theorem 22. I. Let E be square and C = 0 . Then the following are equivalent. 1. (E, A, B, 0) is regularizable by semistate feedback, that is, there exists K such that (E, A - BK) is regular. o _S~ ~ _So

II. Let E be square and B = 0 . Then the following are equivalent. 1. (E,A,O, C) is regularizable by output injection, that is, there exists L such that (E, A - LC) is regular. 2. 3~, = O. [] We conclude with the next and final result, which shows that subspaces are useful in testing the regularity of the homogeneous pencil ( s E - A) (Armentano, 1986).

135

Theorem 23. Let B = 0 and C = 0. Then 1. (sE - A ) has full row rank iff -5, = _,_X. 2. (sE - A) has full column rank iff ~ , = 0. 3. If E is square, then ( s E - A) is regular iff _5, = _X, or equivalently iff ~ , = 0.

[]

6. CONCLUSIONS

We have provided a background of some of the recent results in the geometric theory of nonregular and regular implicit systems of the forms E~ = Ax + Bu and E x k + 1 = Axk + Buk. To provide a framework that unifies the recent geometric results of several researchers, we defined the unknown-input, output-nulling, and composite subspaces. The distinction was emphasized between inner subspaces, which reside in the domain of E and A, and outer subspaces, which reside in the codomain of E and A. It was shown that controlled-invariant subspaces are inner subspaces, while conditioned-invariant subspaces are outer subspaces. The duals of inner subspaces were shown to be outer subspaces, and vice versa. Recursions were provided to compute the various subspaces of interest. The notions of composite preimage and composite image were introduced. The result was a general framework that encompasses various subspaces important in the analysis of implicit systems, as well as the familiar "almost invariant" subspaces from state-variable theory. System properties and duality were examined for both regular and nonregular systems. The recently derived "Hautus-like" rank tests for implicit systems were summarized, showing that they provide more structural information than is usually realized. Acknowledgements--I should like to thank the reviewers for their critical and incisive comments which have done much to help improve the quality of this paper. This research was supported by NSF grant ECS-8805932.

REFERENCES Aiion, A. (1987). Controllability of generalized linear time-invariant systems. I E E E Trans. Aut. Control, AC-32, 429-432. Ailon, A. (1988). Decoupling of a singular system via a proportional state feedback. Internal Report, Dept. of Electr. and Comp. Eng., BenoGurian University of the Negev, Israel. Aoki, M. and M. T. Li (1973). Partial reconstruction of state vectors in decentralized dynamic systems. I E E E Trans. Aut. Control, A~2-18, 289-292. Aplevich, J. D. (1981). Time-domain input-output representations of linear systems. Automatica, 17, 509-522. Aplevich, J. D. (1985). Minimal representations of implicit linear systems. Automatica, 21, 259-269. Aplevich, J. D. (1989). Recursive controller design using non-oriented implicit systems. Proc. American Control Conf., pp. 2134-2139. Armentano, V. A. (1985). Exact disturbance decoupling by a

136

F.L.

proportional-derivative state feedback law. Proc. IEEE Conf. Decision and Control, pp. 533-538. Armentano, V. A. (1986). The pencil ( s E - A ) and controilability-observabilityfor generalized linear systems: A geometric approach. SIAM J. Control and Optimiz., 24, 616-638. Banaszuk, A. (1989). A new approach to regular descriptor systems. Internal Report, Inst. of Mathematics, Polish Academy of Sciences, Warsaw, Poland. Banaszuk, A., M. Koci~cki and K. M. Przyluski (1988a). Remarks on controllability of implicit linear discrete-time systems. Syst. Control Lett., 10, 67-70. Banaszuk, A., M. Koci~cki and K. M. Przyluski (1988b). On almost invariant subspaces for implicit linear discrete-time systems. Syst. Control Lett., 11, 289-297. Banaszuk, A., M. Koci~cki and K. M. Przytuski (1988c). Duality of various notions of observability and controllability for implicit linear discrete-time systems. Institute of Mathematics, Polish Academy of Sciences, Warsaw, preprint 417. Banaszuk, A., M. Koci~cki and K. M. Przyluski (1990). The disturbance decoupling problem for implicit linear discrete-time systems. SIAM J. Control Optimiz. 28, 1270-1293. Banaszuk, A,, M. Koci~cki and K. M. Przyluski (1989a). Remarks on duality between observation and control for implicit linear discrete-time systems. Proc. IFAC Workshop Systems Structure and Control. Prague, pp. 257-260. Banaszuk, A., M. Kocic~ckiand K. M. Przyluski (1989b). On Hautus-type conditions for controllability of implicit linear discrete-time systems. Circuits, Systems Signal Proc., 8, 289-298. Baser, U. and K. Oz~aldiran (1989) Observability and Regularizability by Output Injection of the Descriptor Systems, Circuits, Systems, Signal Proc. Internal Report, Dept. of Electrical and Electronic Eng., Bogazi~i University, Istanbul, Turkey. Basile, G. and G. Marro (1969). Controlled and conditioned invariant subspaces in linear system theory. J. Optimiz. Theory Appl., 3, 306-315. Beauchamp, G. (1989). Algorithms for singular systems. Ph.D. Thesis, School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA. Beauchamp, G., A. Banaszuk, M. Koci~cki and F. L. Lewis 0991). Inner and outer geometry for singular systems with computation of subspaces. Int. J. Control, 53, 661-687. Bender, D. J. (1987). Lyapunov-like equations and reachability/observability gramians for descriptor systems. IEEE Trans. Aut. Control, AC-32, 343-348. Bender, D. J. and A. J. Laub (1985). Controllability and observability at infinity of multivariable linear secondorder models. IEEE Trans. Aut. Control, AC-30, 1234-1237. Bernhard, P. (1982). On singular implicit linear dynamical systems. SIAM J. Control Optimiz., 20, 612-633. Campbell, S. L. (1980). Singular Systems of Differential Equations. Pitman, San Francisco. Campbell, S. L. and K. D. Yeomans (1987). Solving singular systems using orthogonal functions. CRSC Technical Report 100587-01, Dept. of Mathematics, North Carolina State University, Raleigh, North Carolina. Cobb, D. (1984). Controllability, observability, and duality in singular systems. 1EEE Trans. Aut. Control, AC-26, 1076-1082. Cobb, J. D. (1986). Fundamental properties of the manifold of singular and regular linear systems. J. Math. Anal. Appl., 120, 328-353. Dai, L. (1988). Observers for discrete singular systems. 1EEE Trans. Aut. Control, AG33, 187-191. Dai, L. (1988). Decoupling control for singular systems. Internal Report, Inst. Systems Sci., Academia Sinica, Beijing. Dai, L. (1989). Singular Control Systems. Lecture Notes in Control and Information Sciences vol. 118, Springer, New York.

LEWIS Dias, R. J. and A. Mesquita (1990). A closed form solution for regular descriptor systems using the Moore-Penrose generalized inverse. Automatica, 26, 417-420. Dziurla, B. and R. W. Newcomb (1989). Nonregular semistate systems: Examples and input-output pairing. Proc. 26th Conf. on Decision and Control. Los Angeles, CA, pp. 1125-1126. EI-Tohami, M., V. Lovass-Nagy and R. Mukundan (1983). On the design of observers for generalized state space systems using singular value decomposition. Int. J. Control, 38, 673-683. EI-Tohami, M., V. Lovass-Nagy and R. Mukundan (1987). Design of observers for time-varying discrete-time descriptor systems. Int. J. Control, 46, 841-848. EI-Tohami, M., V. Lovass-Nagy and D. L. Powers (1984). On input function observers for generalized state-space systems. Int. J. Control, 40, 903-922. EI-Tohami, M., V. Lovass-Nagy and D. L. Powers (1985). On minimal-order inverses of discrete-time descriptor systems. Int. J. Control, 41, 991-1004. Fountain, D. W. and F. L. Lewis (1989). Generalized notions in geometry and duality. Proc. American Control Conf., pp. 2146-2151. Gantmacher, F. R. (1977). The Theory of Matrices. Chelsea, New York. Gray, S. (1989). Applications to the design of singular systems. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA. Grimm, J. (1988). Realization and canonicity for implicit systems. SIAM J. Control Optimiz. 26, 1331-1347. Kailath, T. (1980). Linear Systera~. Prentice-Hail, New Jersey. Karcanias, N. and G. E. Hayton (1981). Generalised autonomous dynamical systems, algebraic duality, and geometric theory. IFAC VIII Word Congress, Kyoto, Japan. Karcanias, N. and G. Kalogeropoulos (1987). A matrix pencil approach to the study of singular systems: algebraic and geometric aspects. Proc. Int. Symp. on Singular Systems. Atlanta, GA, pp. 29-33. Karcanias, N. (1987). Regular state-space realizations of singular system control problems. Proc. IEEE Conf. Decision and Control. Los Angeles, CA, pp. 1144-1146. Ku~era, V. and P. Zagalak (1988). Fundamental theorem of state feedback for singular systems. Automatica, 24, 653-658. Loiseau, J. J., K. Oz~aldiran, M. Malabre and N. Karcanias (1989). A feedback classification of singular systems. Proc. 1FAC Workshop on System Structure and Control. Prague, pp. 109-114. Lewis, F. L. (1983). Descriptor systems: expanded descriptor equation and Markov parameters. IEEE Trans. Aut. Control, AC-28, 623-627. Lewis, F. L. (1984). Descriptor systems: Decomposition into forward and backward subsystems. IEEE Trans. Aut. Control, AC-29, 167-170. Lewis, F. L. (1986). A survey of linear singular systems. Circuits, Syst., Signal Process. 5, 3-36. Lewis, F. L. (1990). Geometric design techniques for observers in singular systems. Automatica, 26, 411-415. Lewis, F. L., M. A. Christodoulou, B. G. Mertzios and K. ~zqaldiran (1989). Chained aggregation of singular systems. 1EEE Trans. Aut. Control, AC-24, 1007-1013. Lewis, F. L. and B. G. Mertzios (1987). Analysis of singular systems using orthogonal functions. IEEE Trans. Aut. Control, AC-32, 527-530. Lewis, F. L. and B. G. Mertzios (1990). On the analysis of discrete linear time-invariant singular systems. IEEE Trans. Aut. Control, AC-35, 506-511. Lewis, F. L. and K. Ozqaldiran (1989). Geometric structure and feedback in singular systems. IEEE Trans. Aut. Control, AC.34, 450-455. Lovass-Nagy, V. and D. L. Powers (1975). On rectangular systems of differential equations and their application to circuit theory. J. Franklin Inst., 299, 399-407. Lovass-Nagy, V., D. L. Powers and H.-C. Yah (1986). On

Geometric analysis of implicit systems controlling generalized state-space (descriptor) systems. Int. J. Control, 43, 1271-1281. Luenberger, D. G. (1989). Boundary recursion for descriptor variable systems. IEEE Trans. Aut. Control, AC-24, Malabre, M. (1987a). More geometry about singular systems. LAN ENSM Internal Report No. 1.87, Nantes, France. Malabre, M. (1987b). More geometry about singular systems. Proc. IEEE Conf. Decision and Control, pp. 1138-1139. Malabre, M. (1988). Geometric algorithms and structural invariants for linear singular systems. Proc. 12th IMACS World Congr. Paris, pp. 181-183. Malabre, M. (1989). On infinite zeros for generalized linear systems. Proc. MTNS. Amsterdam, 1, 271-278. Malabre, M. (1989). Generalized linear systems: geometric and structural approaches. Linear Algebra Applic., 122-124, 591-621. Mertzios, B. G. and F. L. Lewis (1989). Fundamental matrix of discrete singular systems. Circuits, Syst. Signal Process. I~,. Molinari, B. P. (1976). A strong controllability and observability in linear multivariable control. IEEE Trans. Aut. Control, AC-21, 761-764. Newcomb, R. W. (1966). Linear Multiport Synthesis. McGraw-Hill, New York. Newcomb, R. W. (1982). Semistate design theory---binary and swept hysteresis. Circuits Syst. Signal Process., 1, 203-216. Newcomb, R. W. and B. Dziurla (1989). Some circuits and systems applications of semistate theory. Circuits Syst. Signal Process., 8, 235-260. Nikoukhah, R. and A. S. Willsky (1987). Generalized Riccati equations for two-point boundary-value descriptor systems. Proc. 26th Conf. on Decision and Control, Los Angeles, CA, 1140-1142. Nikoukhah, R., A. S. Willsky and B. C. L6vy (1987). Boundary-value descriptor systems: Well-posedness, teachability and observability. Int. J. Control, 46, 17151737. Nikoukhah, R., A. S. Willsky and B. C. L6vy (1989). Reachability, observability, and minimality for shiftinvariant two-point boundary-value descriptor systems. Circuits Syst. Signal Process., 8, 313-340. Oz~aidiran, K. (1985). Control of descriptor systems. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA. Oz~aldiran, K. (1986). A geometric characterization of the reachable and the controllable subspaces of descriptor .. systems. Circuits Syst. Signal Process., 5, 37-48. Oz~aldiran, K. (1987). Geometric notes on descriptor systems. Proc. 26th Conf. on Decision and Control, Los .. Angeles, CA, pp. 1134-1137. Oz~aldiran, K. and F. L. Lewis (1989). Generalized reachability subspaces for singular systems. SIAM J. .. Control Optimiz., 26, 495-510. Ozqaldiran, K. and F. L. Lewis (1990). On the regularizability of singular systems. IEEE Trans. Aut. Control, AC-35, 1156-1160. Rassai, R. and R. W. Newcomb (1988). Van der Pol realization of torus knot oscillators. IEEE Trans. Circuits Syst., CS-35, 215-220. Rosenbrock, H. H. (1970). State-space and Multivariable Theory. Nelson, London.

137

Rosenbrock, H. H. (1974). Structural properties of linear dynamical systems. Int. J. Control, 20, 191-202. Schumacher, J. M. (1983). Algebraic characterization of almost invariance. Int. J. Control, 38, 107-124. Shafai, B. and F. L. Carroll (1987). Design of a minimal-order observer for singular systems. Int. J. Control, 45, 1075-1081. Syrmos, G., R. Rassai and R. W. Newcomb (1989). Semistate equations for solid-holed torus knots. Proc. Am. Control Conf., pp. 2156-2159. Syrmos, V. (1991). Feedback design techniques in linear system theory: Geometric and algebraic approaches. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA 30332. Tan, S. and J. Vandewalle (1990). The use of homogeneous form in the study of singular systems. Circuits Syst. Signal Process., 9, 301-317. Tan, S. and J. Vandewalle (1987). Irreducibility and joint controllability observability in singular systems. Proc. 26th Conf. on Decision and Control. Los Angeles, CA, pp. 1118-1123. Trazska, Z. (1987). An efficient algorithm for partial fraction expansion of the linear matrix pencil inverse. J. Franklin Inst. 324, 465-477. Van Dooren, P. (1979). The computation of Kronecker's canonical form of a singular pencil. Lin. Algebra Applic., 27, 103-140. Verghese, G. C. (1981). Further notes on singular systems. Proc. JACC. Charlottesville, VA. Verghese, G. C., B. C. L6vy and T. Kailath (1981). A generalized state-space for singular systems. IEEE Trans. Aut. Control, AC-26, 4, 811-831. Verhagen, M. H. and P. Van Dooren (1986). A reduced order observer for descriptor systems. Syst. Control Lett. 8, 29-37. Willems, J. C. (1981). Almost invariant subspaces: An approach to high gain feedback design. Part I: Almost controlled invariant subspaces. IEEE Trans. Aut. Control, AC-26, 235-252. Willems, J. C. (1982). Almost invariant subspaces: an approach to high gain feedback design. Part I: almost conditionally invariant subspaces. IEEE Trans. Aut. Control, AC-27, 1071-1085. Willems, J. C. (1989). Time reversibility. Proc. IFAC Workshop on System Structure and Control. Prague, pp. 9-11. Wong, K. T. (1974). The eigenvalue problem ;tTx + Sx. J. Diff. Equations 16, 270-280. Wonham, W. M. (1979). Linear Multivariable Control: A Geometric Approach. Springer, New York. Yamada, T. and D. G. Luenberger (1985). Algorithms to verify generic causality and controllability of descriptor systems. IEEE Trans. Aut. Control, AC-30, 874-880. Yamada, T. and D. G. Luenberger (1985). Generic controllability theorems for descriptor systems. IEEE Trans. Aut. Control, AC-38, 144-152. Yip, E. L. and E. F. Sincovec (1981). Solvability, controllability, and observability of continuous descriptor systems. IEEE Trans. Aut. Control, AC-26, 702-707. Zhang, S. Y. (1989). Generalized power inverse of polynomial matrices and the existence of infinite decoupling zeros. IEEE Trans. Aut. Control, AC-34, 743-745.