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On some structural properties of linear control systems--reachability from W in N and controllability to W in N

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ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS-REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N









By

FUMIO HAMANO


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILI4ENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA 1979










ACKNOWLEDGED NTS


I wish to express my sincere appreciation to all those who contributed in various degrees toward the fulfillment of this work.

I am particularly grateful to Professor R. E. KALMAN, the chairman of my supervisory committee, for his constant encouragement in seeking a concrete understanding of system theory and other areas in science. His guidance has been of a great help in establishing an organized view in this dissertation. Without the financial support which he arranged for me during the past four years and without the stimulating environment of the CENTER FOR MATHEMATICAL SYSTEM THEORY, this work may not have existed.

I am thankful to Professor C. V. SHAFFER, co-chairman of my supervisory committee, who has not only given me valuable comments concerning my dissertation, but also made favorable arrangements for me during my personal emergency.

The specific motivation for the research reported here was provided by the stimulating discussions with Professor G. BASILE relating to his earlier works concerning "geometric" views on various control and system problems. I appreciate his friendship and his deep interest in this work.

A dissertation is only a part of a doctorate. In the educational process during the past four years the influence of Professors E. EMREE. D. SONTAG, Y. YAMAMOTO, G. SONNEVEND, T. E. BULLOCK, M. E. WARREN, C. A. BURNAP, M. HEYMANN, M. L. J. HAUTUS, V. KUCERA and others was essential in the preparation for a doctoral degree.

Of course, no research would be made were it not for the long-term love and encouragement of a few close people. My parents and my wife, Shoko, have been constant sources of encouragement. To them I dedicate this work.

Needless to say, I am grateful to Ms. Karen Todd for her understanding and patience not only as a typist but as a friend.

This research was supported in part by US Army Grant DAAG 29-77-GO225 and US Air Force Grant AFOSR 76-3034 through the Center for Mathematical System Theory, University of Florida, Gainesville, FL 32611, USA. The













research was also supported in part by the Foundation for International Information Processing Education, 2-6-1 Marunouchi, Chiyoda-ku, Tokyo 100, JAPAN.












TABLE OF CONTENTS


ACKNOLEDGEMENTS . . ii

ABSTRACT . . v

CHAPTER

I. INTRODUCTION . . 1

II. THE r-STEP REACHABLE SUBSPACE FROM W IN N . . 7

1. F mod G Invariant Subspaces . . 7
2. Reachability From W in N . . 8
3. Nonrecursive Characterization of the r-Step
Reachable Subspace from W in N . . . 12
4. Concluding Remarks . .14

III. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N . . 16

5. Controllability to W in N . . .16
6. Nonrecursive Characterization of the r-Step
Controllable Subspace to W in N . . . 25
7. Concluding Remarks . . . 27

IV. UNKNOWN INPUT OBSERVABILITY . . 28

8. Unknown Input Final State Observability . . 28
9. Unknown Input Final State Observability-Part 2 (Special Cases) . . 37
10. Unknown Input Initial State Observability . . 39
11. Concluding Remarks . . .43

V. STABILIZABILITY, OUTPUT ZEROING AND DISTURBANCE
DECOUPLIN . . . . . 45

12. Stabilizability . . . 45
13. Output Zeroing . . 46
14. Disturbance Decoupling . . .48
15. Concluding Remarks . . . 51

VI. CONCLUSION . . .52

APPENDIX . 3

REFERENCES . . . . . 54 BIOGRAPHICAL SKETCH . . 57











Abstract of Dissertation Presented to the Graduate Council
of' the University of' Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy




ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS-
REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N






By
FUMIO HAMANO
August, 1979



Chairman: Dr. R. E. Kalman
Major Department: Electrical Engineering

In the "geometric approach" to the study of linear systems two important notions have been successfully used; namely, the maximal reachability subspace Xrac (N) contained in a given subspace N and the maximal F mod G invariant subspace contained in N. However, the definition of X ach (N) is not as natural in discrete time systems as in continuous time systems. It loses an important meaning when it is applied to discrete time systems, i.e., Xreach (N) is not the set of states reachable from 0 via trajectmax
onies in N. In this work similar notions which are suitable to treat discrete time systems are developed. In general the study is concerned with ttreachabilityf and "controllability" internal to the subspace N in discrete time systems. More specifically, the notions of the r-step reachable subspace from W in N and the r-step controllable subspace to W in N are introduced for given subspaces W and N (satisfying W C N). These are respectively defined to be the set of states reachable from W via trajectories in N in r steps and the set of states that can reach W in r steps via trajectories in N. Algebraic characterizations and sequential












and "feedback" properties of the above notions are first given, and then applications to the unknown-input observability and other problems in control systems are given to show the significance of the results.













CHAPTER I. INTRODUCTION


This dissertation discusses some structural properties of finitedimensional, discrete time, constant, linear dynamical systems. Great efforts have been made to study "geometric" properties in the continuous time systems as well as those which are common to both discrete and continuous time systems. (See for instance BASIIE and MARRO [1968a and b], WONHAM and MORSE [19701, WONHAM [1974, Chapters 1 through 51 and SILVERMAN [1976, Section III].) However, distinctive features of differentiating discrete and continuous time systems have not received much attention. This work intends to point out that there are important differences between the two kinds of systems and that discrete time systems, therefore, should be treated separately in such cases.

The following notation will be used in the sequel: "im", "ker" and
"dim" respectively stand for "the image of", "the kernel of" and "the dimension of". "E" and ":=" mean "is an element of" and "is defined to be", respectively.

Let us now turn to the definition of systems which is pertinent to the discussion of this dissertation. Let k be an arbitrary field, and let m, n and p be positive integers. A finite-dimensional, constant (coefficient), discrete time, linear dynamical system is a triple (F, G, H) whose dynamical interpretation is given by

(0.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1,

(0.2) y(t) = Hx(t), t = 0, 1, .


where x(t) E X := kn, u(t) G U := ki, y(t) E Y :- kp for t = 0, 1, . F: X -4X, G: U -*X and H: X -*Y are k-homomorphisms (or matrices) independent of time. The vector spaces X, U and Y are called the state,
the input (value) and the output (value) spaces, respectively. The elements of X, U and Y are called states, inputs and outputs, respectively.
We shall refer to the triple defined above as the system (F, G, H). However, when the output is of no interest, we shall simply say the pair (F, G) disregarding (0.2).












In this chapter we shall sometimes refer to continuous time interpretation for comparision. We choose the continuous time and set k to be complex numbers. Then we replace (0.1) by


a pair (F, G) with the This is defined as follows: either the set of real or


('0.1)' i*(t) = Fx(t) + Gu(t), t > 0,


in the definitions of (discrete time) pair we are interested in, and the pair (F, G)


the (discrete time) system (F, G, H) and the (F, G). Since continuous time systems are not what unless otherwise specified the system (F, G, H) will always be in discrete time.


Our main concerns are placed in the structural properties of the
trajectories governed by (0.1). Due to the algebraic nature of difference equations, these properties can be studied in a purely algebraic way without losing the intuition of the original dynamical nature of (0.1).

At this stage it seems appropriate to give a quick review of related concepts which have been treated in the literature:

Mi The reachable subspace (of the pair (F, G)), reach .is defined to be the set of states which can be reached from the zero state (via some trajectories) in a finite number of steps. It is known that Xreach is equal to

(0.3) im G +Fim G + . +F nl im G.

The pair (F, G) is said to be reachable if and only if xreac
(ii) The controllable subspace (of the pair (F, G)), Xcontr ,i the set of states from which the zero state can be reached (via some trajectories) in a finite number of steps. It is characterized by


(0.4) X nr = (F n)- (imG +Fim G+ .+ F'1 im G)


where (F n)lX = x E: X: F n x E: X 3 for a subspace X sof X. Note that we have Xr hC X cnrin general. The pair (F, G) is called













controllable if and only if Xcontr = X. (In the continuous time case Xreach and Xcntr can be defined similarly. The characterization of Xcntr , however, is different from the one in the discrete time case, cntr
namely, we have for the continuous time case Xc = im G + F im G +
.+ Fn-1 im G = Xreach.)
unob
(iii) The unobservable subspace, Xu , is the set of initial states
which can not be distinguished from the zero state by any input/output experiment. The subspace Xunmb is given by

H

(0.5) Xunob = ker . �
: n-1
L.

The system (F, G, H) (or simply the pair (H, F)) is said to be observable if and only if Xunob = 0. The condition Xub = 0 is necessary and sufficient for the initial state x(O) to be uniquely determined based on a sufficiently long interval of input/output measurement. For more details about reachability, controllability and observability the reader should refer to KAIMAN [1968], FURUTA [1973, Chapter 2, Section 7 through 101, MARRO [1975, Chapter 61 and WONHAM [1974, Chapters 1, 2 and 3].

(iv) A subspace V is called an F mod G invariant subspace if and only if

0.6) FVCV+imG.

It is well-known that (0.6) holds if and only if there exists a k-homomorphism ("state feedback") K: X -4U satisfying

(0.7) (F + GK)V C V.

The set of F mod G invariant subspaces is closed under subspace addition. Therefore, for a given subspace N of X there is an F mod G invariant subspace which contains any other F mod G invariant subspace. This is called the maximal F mod G invariant subspace contained in N, and we












shall denote it by V mx(N).

(v) Closely related to V a(N) is the subspace called the maximal reachability subspace contained in N which is denoted by X () It is defined as follows: Let K: X -+U be a k-homomorphism such that (F + GK)V a(N) C V(N) and let L: U -4 U be another k-homomorphism
satisfying im. GL = im G n Vm=(N). m1~ah(N), then, is defined to be the reachable subspace of the pair (F + GK, GL). It is evident that
reach Fe,
(F + GK)Xmax (N) MA ~ec(N) if one uses (0.3) and the Cayley-Hamilton theorem. Consequently, Xrec (N) is an F mod G invariant subspace
max
contained in N (or equivalently in V M(N). Also, since V Max
V V N) riillitfolwsta Fe~(N) = Xreach(V mx(.N)).
m~ax( max () trvalifoowtht max max max
The definition of -Zeach (N) does not depend on whether the pair
max
(F, G) represents a discrete time system or a continuous time system. However, there is an important difference in interpretation of Xreac (N).
each max
In the continuous time case )F (N) can be interpreted as the set of
max
states that can be reached from the zero state in a finite time via trajectories contained (at each time) in N. On the contrary this interpretation fails in the discrete time case. This point has not been clarified in the literature, not to mention its importance. The above observation then raises the following questions: What is the set of states reachable from the zero state via trajectories contained in N, XN ec
in the (discrete time) system (F, G, H)? How is it characterized? Are there any interesting properties? A natural question which comes next is: What is then the set of states controllable to the zero state via trajectories in N? What are its properties?

In Chapters II and III we shall attempt to answer the above questions in more general contexts.

We shall introduce the following new notion in Chapter II. Let r be a positive integer, and let W and N be subspaces of X satisfying W C N. The r-step reachable subspace from W in N, denoted by
rah(r), is the set of states which can be reached from some states
in W in r steps via trajectories contained (at each instant of time)










reach.
in N. We shall characterize re N (r) and study the properties of the
each X
sequence of subspaces 2w^eN (i), i = 1, 2, . with respect to W and
N. What appears to be intriguing is the fact that the properties of reach.
,N (i), i = 1, 2, . change drastically depending on W.
_reach(
In Chapter III we shall introduce the natural counterpart of XWN (r) which we shall call r-step controllable subspace to W in N,
contr,
,WN (r). This subspace is defined to be the set of states in N from which some state in W can be reached in r steps via trajectories contained in N. It should be noted that, except for the special cases of X ntr) (MARRO [1975, Chapter 4]) and �Ntr (r) (which we shall isu shor , contr
discuss shortly), the more general subspace XWN rr) has not been
contr,
studied as such. It is this generality that makes coN (r) an interesting object.
reach,
The significance of introducing aN (r) will become clear when we consider the unknown input observability at the final time r in Chapter IV (Sections 8 and 9). The problem is stated as follows: Given (F, G, H), a priori information about the initial state v(O) = Jx(O) and the output sequence y(l), y(2), ., y(r), find the state x(r) at the final time r. We shall see that the best we can do to identify x(r) is to determine the coset x(r) + keach (r) which in fact can be determined. Thus,
X'kerJ ,kerH
for an in-depth understanding of the unknown input observability at the final time r, it is essential to study the properties of reach r) \erJ,kerH~r
which depend on J, H and r. The results in Sections 8 and 9 appear to be new.
contrt contr,
A special case of Xo N (r), namely, XNN (r) has been discussed in the literature in relation to the unknown input initial state observability, and it is known that, if we are given (F, G, H) and the output c ont r~r a
sequence y(O), y(l), ., y(r), only the coset x(O) + XNN (r) can
be recovered based on the above data. (See, for example, BASILE and MARRO [1973, Theorem 1 and Corollary 11 and SILVERMAN [1976, Definition 2 in Section III],) We shall include this problem with more generality in Chapter IV, Section 10 to implement our knowledge of unknown input observability. The method used here to recover x(O) (or its coset) is less complex to understand than the ones which have been used in the












literature so far.

The subspace cWNt (r) is closely related to "state feedback"
K: X --)U. Those results in Chapter III which are related to state feedback will be applied in Chapter V in which we consider stabilizability, output zeroing and disturbance decoupling. Problems of these kinds have been treated in the literature in different fashions. (See WONHAM [1974, Theorem (2.3) for stabilizability, Theorem (4.4) for output stablization and Theorems (4.2) and (5.8) for disturbance decoupling].) The contents of Chapter V will also serve to exemplify the significance of oNtr (r) in the cases where W N. The theorem (12.1 (i), (iii)) concerning etabilility is of interest in the sense that it gives a new interpretation of stabilizability.












CHAPTER II. THE r-STEP REACHA.BIE SUBSPACE FROM W IN N


We study a finite-dimensional, constant (coefficient) discrete time, linear dynamical system (F, G, H) over an arbitrary field k; in this chapter we shall be interested in properties of the pair (F, G). We shall define a new notion of "the r-step reachable subspace from W in N", denoted by each (r), and study the properties of the sequence of
subspaces each (i5 i = 1, 2, . The subspace defined here will
find its application in Chapter IV.

We begin this chapter with a well-known notion of F mod G invariant. subspaces.


1.F mod G Invariant Subspaces

Consider a finite dimensional, constant, discrete time, linear dynamical system given by

(1.1) x(t + 1) Fx(t) + Gu(t), t = 0, 1,

where x(t) eX:=kn u(t)EU :=kI?) t =O0 1, .,and k is an
arbitrary field. We call X the state space and U the input (value) space. Since the output is of no interest at the moment, we shall refer to (1.1) by the pair (F, G).

(1.2) DEFINITION. A subspace V of X is an F mod G invariant subspace (or simply F mod G invariant) iff

(1.3) FV C V + i G.

The most important properties of F mod G invariant subspaces are expressed by the following

(1.4) LEM. Let V be a subspace of X. Then the following statements are equivalent:

(i) V satisfies (1.3).

(ii) For any x,, E V there exists an input u E U such that











Fx. + Gu E V.

(iii) There exists a k-homomorphism K: X -.U ("state feedback") such that

(1.5) (F + GK)V C V.

PROOF. The proof for the equivalence between (i) and (iii) can be found in BASILE and MARRO [1968a, Theorem 31, WONHAM and MORSE 11970. Le na (3.2)] and WONHAM [1974, Lemma (4.2)].

For convenience we shall give the proof here. Suppose that (1.5) holds. Let x. G V. Then (F + GK)x* = Fx. + GKx. C V. So Fx.
v + GKx. for some v E V, which implies (1.3).

Conversely, assume that (1.3) holds. Let (vl, ., Vq ) be a basis of V where q := dim V. Then (1.3) implies that for each i 1, ., q there exist wi C V and ui C U satisfying Fv = wi + Gui. Now define a k-homomorphism K: X -4U by Kvi = -ui, i =1, ., q. Then we have (F + GK)vi = wi C V, i = 1, ., q, which implies (1.5).

The equivalence of (i) and (ii) is easily proved and omitted here. 0

2. Reachability from W in N

We consider a finite dimensional, constant, discrete time, linear dynamical system (F, G) given by (1.1).

From here on, W and N will be subspaces of X satisfying
W C N C X, and r will be a positive integer.

(2.1) DEFINITION. A state x. C N is r-step reachable from W in N iff there exist another state xo* C W and an input sequence u(t) C U, t = 0, ., r - 1 such that x(O) = xo., x(t) C N, t = 0, ., r and x(r) = x.

The set of states x.'s satisfying the above requirement form a (linear) subspace.











reach,
(2.2) DEFINITION. The r-step reachable subspace ,N (r) from W in N is the set of states r-step reachable from W in N.

We show that the sequence eaNch(i), i = 1, 2, . can be
recursively computed. Let (2.3) XW,N(0) = W

(2.4) XW,N(i) = (FXW,N(i - 1) + im G) iN, i = 1, 2, .


Then we have

(2.5) THEOREM. each(r) = X, N(r).

PROOF. The proof is done by induction.

(i) X,N(1) = (EW + im G) N
= (x E N: x = Fw + Gu for some w E W and u E km = each .
= JW,N ()
(ii) Assume that XW,N(j) is equal to the j-step reachable subspace from W in N, J > 0.


Xw,N(J + 1) = (FXW,N(J) + im G) ON

= (Xj+1 E N: xJ+ = Fxj + GuJ for some xj E X N(j)' and uj E km].

By induction assumption any xj E XWIN(j) has at least one pair of sequences xt E N, t = O, ., j - 1, and ut. kn," t=. , ., j - 1
such that xo 0 W, xt =Fxt + Gt-, t = 1, ., j. Therefore


XW,N(j + 1) = (xj+l E N: There exist sequences xt E N, t = 0, ., j and ut E k, t = 0,., J satisfying xt+1 = Fxt + Gut, t = 0, ., j










and x E W)
o
Sreachj ). O

We now study properties of the sequence W, XWN (), each (2), . by looking at some properties of the sequence XW,N(i), i = 0, , .
The sequence XW,N(i), i = 0, 1, . (equivalently, W, reacWN ),
each(2), .) has conditional monotone properties. (2.6) LEMMA. (i) If X,N(i) CXWN(i + 1) for some integer i > 0, then X,N(j) C XW,N( + 1) for all integer j > i.
(ii) If XW,N(i) D W,N(i + 1) for some integer i > 0, then XW,N(J) D XW,N(j + 1) for all integer j > i.
(iii) If x,N(i) = XW,N(i + 1) for some integer i > 0, then X,N(i) = XW,N(i + j) for all j = 0, 1, .
PROOF. (i) Let XW,N(i) C X,N(i + 1) for some integer i > 0. Assume that XW,N(s) C XW,N(s + 1) holds for some integer s > i. Then

xW,N(s + 2) = (FXW,N(s + 1) + im G) N D (FXW,N(s) + im G) N

= XW,N(s + 1).

Therefore, by induction

XW,N(J) C W,N( + 1)

for all integer j > i.
(ii) Similar to the proof of (i).
(iii) Suppose X, N(i) = XW,N(i + 1) for some integer i > 0. If we assume that XW,N() = XW,N(S + 1) for some integer s > i, then
we have










XW,N(s + 2) = (FXWN(s + 1) + im G)C N = (FXW, N(s) + im G) nhN

= XW,N(s + 1).

Therefore, by induction

XW,N(j) = XW,-N(j + 1)

for all j =i i i + 1, . (2.7) REMARK. It is possible that none of three conditions in Lemma (2.6) may hold. For instance, let F :=[0 1], G :=[, N := X and W := span 1 , and let i be any positive integer.

However, if either XW,N(i) C XW,N(i + 1) or XW,N(j)D XW,N(j + 1) holds for some i, j = 0, 1, ., the sequence XW,N(), A = 0, 1, . will stop increasing or decreasing in a finite number of steps since X is finite dimensional. If it is the case, let v be the least integer i > 0 such that XWN(i) = xN(i + 1). (2.8) LEMMA. (i) If W C XW, N(1), then v < dim N - dim W < n.

(ii) If W D X~WN(1), then V < dim W < n.

PROOF. Immediate from Lemma (2.6) by using the finite dimensionality of X. Ol

Let v be as in the paragraph prior to Lemma (2.8). XWN(v) is not an F mod G invariant in general. However, we have (2.9) LEMMA. If FN C N + im G, then FXWN(v) C XWN(v) + im G holds.

PROOF. Since XWN(V) = (FXWN(v) + im G)iN, it follows that FXWN(V) C FN C N + im G. Therefore, we have












FXW,N((v) C (Fxw,N(v) + im G) n (N + im G)

C (FXWN(v) + im G) nTN + im G


= XN(v) + im G.


3. Nonrecursive Characterization of thy, r-Step Reachable Subspace from
W in N.

The sequence defined by (2.3) and (2.4) determines reach(r) recursively.
contr "
We now give a nonrecursive characterization of XWN (r).

Let B be a matrix having n rows. Let C and A be matrices with
n columns. Define


fo(B, A, C) := AB


rCB 0l
fl(B, A, C) := CAFB AG 1 [F G]

CB 0 0 f2(B, A, C) := CFB CG 0 ,
AF2B AFG At_


where 0 is the zero matrix of appropriate size. procedure, we define for r = 1, 2, .


f (B, A, C) :=
r


CB CFB CF2B
CF B


CFr--B
AFrB


0
CG CFG

CFr-2G
AFr-1G


Extending the above


. . . CG 0 . . . AFG AG


In this section we only use a special case where B = In, n X n identity matrix. (Another case where B j In will be used in Chapter III, Section 6.)

As before, W C N C X. Let N be a matrix with n columns satisfying


. . . O
. . . O
.












Denote by BW a basis matrix of W. Then we have


0


0~ 0 MFBW M G.



MFBw r-lG
InrF~ VrlG1


gr(Bw) := [IFBW;


Fr-1G . FG G],


r > 0.


(3.1) THEOREM. -reach(r) = gr(B) ker fr( , , ),

PROOF. Let x. E X.

x E gr(BW) ker fr (, M;, M;)


r > 0.


iff there exist E E kdiM



(3.2) uo E ker fr(Bw,

U
_ r-.



u
(3.3) x = gr(BW) o

r-1


and u , ., u E k such that
o r-1


M MN),


The conditions (3.2) and (3.3) are equivalent to


0 . 0 0 . O



. 0
. 0N


Define


ker * = N.











0


MqBW + 1A Guo = 0,


MFrBW + FG-u + + M GUr_1 0,



x. = Fr + F r-Gu + .+ Gu
o r-l'


which in turn is equivalent to the conditions




FBW + Gu 0C N


r r-1
x=FrBW + Fr 1Gu 0+ . + Gu 1 CN.


The last set of relations hold for some , C kdimW and uo, U, .,
Ckm reach,
ur1 c iff x reach (r) (by Definition (2.2)).

(3.4) REMARK. A set of vectors B, ut C km, t = 0, ., r -1
satisfying (3.2) and (3-3) are seen to be an initial state and a sequence of inputs satisfying the conditions of Definition (2.1) for the final state x*.


4. Concluding Remarks.

In our discussion of this chapter we have assumed W C N. This
assumption has been made since we are interested in structural properties of the system (F, G) inside the subspace N. Technically speaking, however, the above assumption is not essential. With slight modification all the statements still hold without assuming W C N. Theorem (3.1) can be easily modified to yield the corresponding result:

reachA
(4.1) WN (r) = gr(EW) ker fr(Bw, MN, M ), r > 0












where f r(B, A, C) is the matrix obtained by eliminating the first block row in fr(B, A, C) (hence, fr(B, A, C) consists of r block rows).

From Lemma (2.6) it follows that X,,N(i) CXWN(i + i), i = 0, , ., iff




and that XW, IN(i) D X,N(i + 1), i = 0, 1, ., iff

(4.3) W :XWN(l).


The natural question is then what is the significance of the conditions (4.2) and (4.3). The significance of (4.2) will be partially answered in Chapter III, Section 5. However, the implication of (4.3) is yet to be clarified.
_reach,
There are other questions to ask: when does the sequence XWN (i),
I reach , reach ,X ,
i = 1, 2, . "oscillate", i.e., a,N rJ) XcN hj + 1) for every

j > 0?; when does the sequence "oscillate" at the outset and stop its oscillation some time later?; etc. These are open problems.











CHAPTER III. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N.

In the previous chapter we have studied the r-step reachable subspace from W in N. In this chapter we introduce another new notion which we call "the r-step controllable subspace to W in N". Its applications will be found in Chapters IV and V.


5. Controllability to W in N.

Consider a finite dimensional, constant discrete time, linear dynamical pair (F, G) represented by

(5.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1,

where x(t) C X := kn, u(t) e U := km, t = 0, 1, ., and k is an
arbitrary field. We denote the system (5.1) by the pair (F, G).

As before W and N denote subspaces of X satisfying W C N C X, and r is a positive integer.

(5.2) DEFINITION. A state x. C N is r-step controllable to W in N iff there exists an input sequence u(t) C U, t = 0, ., r - 1 such that x(O) = x., x(r) ( W and x(t) C N, t = 0, ., r.

The set of states x.'s satisfying the above conditions form a (linear) subspace.
.contr,

(.53) DEFINITION. The r-step controllable subspace X CN (r) to W
in N is the set of states r-step controllable to W in N.
contrt
We show that XWN (r) can be computed recursively. Let

(5 WN (O) = w


(5.5) ,N() = F-1(WN(i - 1) + im G) rT)N, i = 1, 2,


where F-X := (x C X: Fx C X ) for a subspace X of X.











(5.6) THEOREM. X ontr(r) = RW,N(r), r > 0.

PROOF. The proof is by induction. For r = 1,


XW,N(1) = F-1(W + im G) n

= (xI E X: FX1 = - Gul for some wE W and ul km,

and xlE N)

= {xl E N: Fxl + Gul E W for some ul e km)

reach1)


contr
Now let WN(j) = XW,N (j), j > 0. Then we have


W,N(j+1) = (xj+1 E N: Fx+1 = xj - Guj+1 for some x E j,N(j) u+l E km. )


By assumption of ),N N (J) this is equal to

{Xj+1 G N: Fx1 + Guj+ = x, Fx + Gu = Xl, ., Fx1 + Gu1 = w

contr m
for some w E W, x, 2 XW,N (a), u k, a = , ., j and

uj+l 6 km1
Scontr,.
=),N (0 + ). O
contr contr
We now study properties of the sequence W, X,N (1), WN (2), by examining properties of the sequence (5.4), (5.5).
contr contr
The sequence (5.4), (5.5) (equivalently WW,N tr(1), (2),.)
has conditional monotone properties. (5.7) LEMMA. (i) If N(i) ,N(i + 1) for some integer i > 0,










then XW,(j) CXW,N(j + 1) for all integer j > i.
(ii) If (i) D X.,N(i + 1) for some integer i > 0, then
,N(i) D ,(J+ I for all integer j > i.
(ii) If ,N (i) = W,N(i + i1) for some integer i > 0, then

X,N(J) = X,N(i + j) for all integer j > 0.
PROOF. (i) The proof is by induction. Let XW,N(i) C X,N(i + 1). Assume XW,N() C XW,N( + 1) where 2 is a nonnegative integer. Then

kW,N(I + 2) = F-I (W,N( + 1) + im G) (IN D F-I(kW,N(I) + im G)(-hN = ,N( + i).

Therefore, by induction we get XW,N(j) C i,N(j + 1) for all integer j > i.
(ii) The proof is similar to that of (i).
(iii) This is again proved by induction. We assume w,N(i) =
XW,N(i+l) for some i > 0. If XW,N() = XW,N(i + 1) for some integer A > 0, then we have
WN(I + 2) = F-WN(4 + 1) + im G) AN

= F-( ,N(2) + im G) TnN

= XW,N(A + 1).

Therefore, XWN() = ,N(i + j) for all integer j > 0. 0 (5.8) REMARK. It is possible that neither one of three conditions in Lemma (5.7) may hold (e.g., let F, G, N, W be as in Remark (2.7), and consider any positive integer i).

However, if either XW,N(i) C WN(i + 1) or W,N(J)D W,N(j + 1)










happens for some i, j = O, 1, ., the sequence XWNW'
9 = O, 1, . will stop increasing or decreasing in a finite number of steps since X is finite dimensional. If it is the case, let P be the least integer i > 0 such that XWN(i) = WN(i + 1). (5.9) LEMMA. (i) If W C X (1), then 0 < dim N - dim W < n.

(ii) If W D X (1), then p < dim W < n.
~~,N -PROOF. Immediate from Lemma (5.7) and the above comment. O (5.10) LEMMA. Let 0 be as in the paragraph preceeding Lemma (5.9). Then

(5.11) FX~W, ' ) C X wYN() + im G. (5.12) WN() c N.

Scontr
So XWN (-) is an F mod G invariant subspace in N.

PROOF. By the definition of XWN(P) = XWN( + 1).
Therefore

WN WN( + 1)

- F-1(XW I() + im G) nN.


Hence, FXN(R) C XW(9) + im G, XWN(ii) C N. rn

(5.13) REMARK. Lemma (5.10) guarantees the existence of a feedback contr contr
K: X -~U such that (F + GK) XWN (0) C XWN i). (See Lemma (1.4).)

The set of F mod G invariant subspaces is closed under subspace addition. (See BASILE and MARRO [1968a, Section 2, Assertion 11 and WONHAM [1974, Lemma (5.3)].) Therefore, the following is well defined: (5.14) Vmax(N) := max[V C N: FV CV + im G).










The subspace V max(N) is called the maximal F mod G invariant subspace
max
in N.

It has been known (SILVERMAN [1976, Section III, A]) that Vmax(N)
is equal to the set of states in N for which there are input sequences such that the corresponding trajectories remain in N for v units of time. It should be noted that this statement is a special case of Theorem (5.6). It should also be noted that the algorithm for computing V max(N) is a max
special case of (5.4), (5.5) where W = N. (See BASILE and MARRO [1968a, Section 3, Corollary 1] and WONHAM [1974, Theorem (4.3)].) Summarizing, we have

(5.15) COROLLARY OF THEOREM (5.6). Let p be as in Lemma (5.9), and let Vmax(n) be as used above. Then

maax
X~ontr(, =, mxN 2,
Scontr

The sequence XWN (i), i = 1, 2, . has a conditional monotone
property. (See Theorem (5.6) and Lemma (5.7).) The natural question to ask is then when it is monotonically nondecreasing or nonincreasing.

( .J) LEMMA. (i) W C XWN(1) iff FW CW + im G.

(ii) If W D XN(1), then N(0) = Vmax(W) where 9 is as
in Lemma (2.9) and V max(W) is as in (5.14).
max

PROOF. (i) If W C XWN(1), then W CF-1 (w + im G) N. Hence
-1 CxN')
W CF (W +im G). So we have FW CW + im G.

Conversely, if FW CW + im G, then W C F (W + im G). Since
W CN by assumption, we get W C F-1(W + im G)(- )'N, i.e., W C XwN(1).

(ii) If W D XWN(1), by Lemma (5 .7 ii) we have XWN(i) CW
for all i = 0, 1, . . Then obviously XWN(i) = XW(i), i = 0, 1,. Therefore, by Corollary (5.15) ,N(p) = Vmax(W). 11

contr
For the nondecreasing sequence XN (i), i = 1, 2, ., we have











the following as the immediate consequence of Lemma (5.7 i), Lemma (5.16 i) and Theorem (5.6).

contr contr
(5.17) PROPOSITION. X (i) C X (i + 1) for all i = l, 2,
iff FW CW + im G.

contr contr
(5.18) COROLLARY. If FW C W + im G, then FeNr (r) CWN (r) + im G
for each r = 1, 2.

PROOF. Suppose FW CW + im G. Then by Proposition (5.17) we
contr contr
have XW,N (i - 1) C X,N (i), i = 1, 2, . By the definitions of the i and (i - l)-step controllable subspaces to W in N, for every
contr
v. c r i) there must exist u. EU such that Fv. + Gu. XWN(i - 1) C contr contr contr '
XWN (i). Therefore, FXN (i) C XN (i) + im G, i = 1, 2, .

As for the nonincreasing sequence we just note a special case of Lemma
contr contr
(5.7), i.e., XWN i) WN i + ), i = i, 2, . iff
1(i) : XW,N ( + 1)
W CF (W + im G) 'N.
contr
XWN (r) has the following properties in relation to state feedback.
c contr,
By the definition of contr (r) it is clear that the subspace is
state feedback invariant, i.e., the r-step controllable subspace to W
contr
in N of the system (F + GK, G) is equal to coN (r) of (F, G).

Under some conditions, the input sequence u(t) E U, t = O, ., r - 1 given in Definition (5.2) can be implemented by a suitably chosen state feedback u(t) = K rx(t), t = 0, ., r- 1.

contr contr
(5.19) THEOREM. (i) If XW,N (i) C WN + ), i = 1, 2, .,
i.e., if FW CW + im G, then for each r = , 2, . there is a feedback K : X -+V such that
r

(5.20) (F + GKr contr(r) C N, j = 0, ., r -,
r XN(r) ,

j contr
(5.21) (F + GKr) XWN (r) CW, j = r, r + 1,.











Scontr contrr
(ii) If YDN (i) DXWN (i + 1), i = 1, 2, ., then for
each r = 1, 2, ., there is a feedback K : X -4U such that
r

(5.22) (F + GKr) contr(r) CN, j = 0, ., r - 1
r contr
(5.23) (F + GKr) r cont (r) CW.
Scontr ontr

PROOF. (i) By Proposition (5.17) ,ontr () ontr(i + 1),
i = 1, 2, ., iff W is an F mod G invariant.
contr contr
Assume XWN i) C otr(i + ), i = 1, 2, . For each
r = 1, 2, . we:choose a basis of X as-follows. Let (eol, ., eolo
be a basis of W. Extend this basis to get the basis (eol, ., eoo;
ell, ., el) of coN tr(1). Repeat the extension until we get no more vectors to add to, say r' times where r' < r, and we obtain the basis e1, ., e~o e, ., er'- l,r ; er'l, ., e rA of
oN tr(r') = XcoN (r). We further extend this basis arbitrarily to get the basis of X, (e , ., e., e r. Here :=
contr Ontr r
dim Xk N nt) - dim X N (j - 1), j ., r' and A=
' contr ' contr
n - dim XW,N (r). Since ejsj E IN (J), J = O, ., r and s = 1 s = 1, ., AI , there exists an input ujsj such that


(5.24) Fe + Gu E ontr - 1)
jsy jsj XWN

where j = 1, ., r' and sj = 1, ., Aj. (See Definitions (5.2) and (5.3)). Define K : X -U so that it satisfies the following
r
conditions:

(a) Ujsj = Kes for j = , ., r and s = l, ., Aj,

(b) (F + GK )eos E W for s = 1, ., ao (such a Kr always exists since FW CW + im G. See Lemma (1.4) and Proposition (5.17).)


(c) Krej, j = 1, ., qr are arbitrary. Then











(5.25) (F + GKr)eos E W, s = 1, ., o'


(5.26) (F + GKr)ejsj C otr(j - 1), j = 1, ., r; s= , .


It is easy to check that the following relations hold:

(5.27) (F + GKr)ejsj C N


for j = O, ., r'; s. = i, ., e.; i = , .,r - 1 and


(5.28) (F + GK ) ejsj C W


for j = 0, ., r
contr contr
(ii) Suppose WN ,N i + ), i = i, 2, . For
each r = 1, 2, . we choose a basis of X in the following way.
contr . contr
Let (erl, ., erer) be a basis of XoWN (r) where := dim XcoNtr).
Extend the basis to get the new basis (e rl, ., err; erl,l, e ) contr
r-l,r-l of XW,N (r - 1). Repeat the procedure, and we obtain the basis ferl, ., erB; er-1,,., el�; 01, ., eo) of W where
S:= dim cont (j) - dim Xontr(j + ), j = , ., r - 1. Note that if �. = 0 we do not extend the basis at this step and go to the next step. By adding linearly independent vectors el' . qr we complete the basis
of X as erl, ., el l; o, ., eoAo; ell ., eqr where

r := n - dim W. Since ejsj c XWN tr), = 1, ., r and sj = , ., ,there must be an input u.jsj such that


contr
(5.29) Fe. + Gu. ntr( - 1)
asj jsj XWN

where j = 1, ., r and s =1,., . It is straightforward to
check

(5.31) (F + GK )re rs W, s = 1, ., r'
r rsr rr











(5.32) (F + GKr) rsre N, J = 0, .' r s 1 r


hold. 5


(5.33) REMARK. The choice of K : X -)U
r
This (limited) freedom in choosing Kr (r

application. (See Chapter V.)


(r > O) is by >0) is rather


no means unique. useful in


(5.34) REMARK. The inputs u. 's (s. = 1, ., ; .Jsj
or r) based on which K is defined, can be determined
r
(but not necessarily uniquely). See Remark (6.3).


= 0, ., r'
explicitly


Recall the definition of reach(r). Given x c w there may not " ,N Go
be any input sequence producing x(O)= X , x(t) C N, t = 1, ., r
reach,0
and x(r) XWN (r), r > 0. The condition that x must satisfy to
have such an input sequence is as follows. Let x C W. Then there is an
0
input sequence u(t) C U, t = 0, ., r - 1 such that the state
trajectory satisfies x(O) = X, x(t) C N, t = 1, ., r and x(r) C
reach,
rec (r) iff

ontr
x CW , (r)
o 7,


- reach
where X :=WN (r).

We now turn to the question of what is the implication of has been posed at the end of Chapter II.


(5.35) i > 0,


(4.2) which


contr
LEMMA. Let W(i) := X W ri), i > 0. Then for each integer
(4.2) holds iff


.reach
(5.36) W .rah()
Reach
PROOF. Suppose (4.2) holds. Trivially, W D (i),Wi). To show
the converse inclusion let w ( W. Then by (4.2) there exist x1 T W, u C U such that w = Fx + Gu N. Since w ( W we obtain w = Fx +
Gu 1 W. Repeating the above argument i times, we see that there also












exist xt W, ut C U, t - 2, ., i satisfying xt-1 = Fxt + Gut,
t 2 *., .contr,.+ U�
t = 2 ,., i. Then x contr i). Hence, noting that xt C W, t = 1,
;ach E 0 .,, i, we get w-6 XW1(i).W()

Conversely, if (5.36) holds, then there exists a subspace W C W
reach.
satisfying each (1) = W, which implies (4.2). 0

Consequently,
each (� reach~i+i ,2 f
(5.37) PROPOSITION. reach(i) 0~each WN(i + 1), i = 1, 2, .,
r each

(5.38) w = rmax ( )

where V is as in Lemma (2.8).


PROOF. Immediate by using the above lemma and Lemma (2.8).


6. Nonrecursive Characterization of the r-step Controllable Subspace to
W in N.

The sequence defined by (5.4) and (5.5) determines the r-step controlcontr
lable subspace to W in N, X N (r), recursively. We now give a
'' contr,
nonrecursive characterization of XWN (r).

As before, W CN CX and r = 1, 2, .

Recall the definition of fr(B, A, C) in Section 3 of Chapter II.
Denote by I the n X n identity matrix, and let 0 be the zero matrix
n
of suitable dimension. Let M be a matrix with n columns such that
ker = W, and define M similarly. Then


frIn, MMMi)=


0
M 0

.2



M r-1 M Fr


0

NFG


Mr-2 M FrlG
r- 1


0 . . . . 0.


. M 0

. G MG













r = 2, 3, ., and



M F M G

Write

P := [I 0. 0]
r n

where we have r blocks of n X m zero matrices.

contr (6.1) THEOREM. Xc,N (r) = Pr ker fr(In, Mw, N), r > 0.

PROOF. The condition

x
0
(6.2) Uo C ker fr(In ,


- r-1.

x 0 X, ut k , t = 0, ., r - 1, is equivalent to


Xo = 0O

xo + MGuo = 0,
r-i o r-2u
x F r 2Gu + . + MGUr-2 = 0,

S+ r-Guo + . + Gur-1 = 0,


which in turn is equivalent to the conditions

x E N,

Fx + Gu ( N,
o o


F x + F Gu +.+ Gu r-2 N,

Frx + Fr-1Gu + . + Gu E W.
o o r-1











c contr,
The last statement is equivalent to x G WN (r) (by Definitions (5.2) and (5.3)). 0


(6.3) REMARK. Vectors x� E X, ut C kc , t = 0, ., r - 1 satisfying (6.2) are viewed as an initial state and a sequence of inputs satisfying the conditions of Definition (5.2).

(6.4) REMARK. Methods of computing the maximal F mod G invariant subspace in N have received considerable attention. (See BASILE and MARRO [1968a, Section 3, Corollary 11, WONHAM [1974, Theorem (4.3)], SILVERMAN [1976, Lemma 6 in Section III, C1 and MOORE and LAUB[1978. Also recall Corollary (5.15).) Corollary (5.15) and Theorem (6.1) yield

(6.5) Vmax(N) = P ker f (I ,

This gives a new nonrecursive method of computing Vma(N).

7. Concluding Remarks

The remarks similar to those in Section 4 of Chapter II apply to the results of this chapter.
.r each,
As is the case with X ,N Cr) the assumption W C N is not
.' contr,
essential in the discussion of XW,N (r) from the technical point of view; the statements in this chapter can be modified straightforwardly to fit the case where W C N is not assumed. (See also Section 4 of the previous chapter.)
. contr contr
From Lemma (5.7) we know that XW,N (i) D XWN r' + 1), i = 0, 1, ., iff

(7.1) WDF' (W+im.G) nTN.

The significance of the condition (7.1), however, is not yet clear.










CHAPTER IV. UNKNOWN INPUT OBSERVABILITT


The notions of the r-step reachable subspace from W in N and
of the r-step controllable subspace to W in N as developed in Chapters II and III are now applied to the study of unknown input observability of a system (F, G, H).

Sections 8 and 9 discuss the unknown input final state observability. Section 10 treats the unknown input initial state observability.

Hereafter Y denotes the output (value) space and is defined by Y := kp. Also p := dim Y is used throughout Chapters IV and V.


8. Unknown Input Final State Observability

We consider a finite-dimensional, constant, discrete time linear dynamical system (F, G, H) given by

(8.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1,

(8.2) y(t) = Hx(t), t = 0, 1, .

where x(t) E X := kn, u(t)CU:=km, y(t)Y-kp (t = 0, 1, .)
and k is a field. (Recall the definition in the second paragraph of Chapter I.)

It is assumed that we have some degree of a priori information about the initial state given by

(8.3) W(x(O)) = x(O)

where X: X -kq is a k-homomorphism and q is a positive integer. In particular, if J is an isomorphism where q = n, then the initial state is a priori known. If J is zero, then the initial state is a priori unknown.

Depending on an initial state x(O) = x C X and an input sequence
0
u(t) = ut G U, t = 0, 1, ., the system produces the corresponding state and output sequences (or trajectories) x(t) C X, t = 0, 1, .











and y(t) e Y, t = 0, 1, ., respectively. As a result of (8.3) we also have v(xo) = Jx0 which we call v0
Considering the initial state, we have

v(x(o)) J
(8.4) - x(o).
y(x(o)) H

Let

v(O) j

y(x(0)) 1
For each r = 1, 2, . the sequence [v(0), y(l), ., y(r)) is called an initially modified output sequence (till time r). When we discuss two initially modified output sequences we prefer to denote them by (vol Yll' -' Yrl) and (vo2) Yl21 .' Yr2)" (8.5) PROBLEM. Given an initially modified output sequence (v(O), y(l), ., y(r)) of (F, G, H), find the corresponding final state x(r) C X for r = 1, 2l.

Note that the definition of v(O) and J and (8.4) clearly implies

(8.6) ker J C ker H.

(See Section 11 for the remark on the condition (8.6).)
We say that the system (F, G, H) is unknown-input observable at the final time r iff Problem (8.5) has a unique solution.
It is not always the case that this problem has a unique solution. In fact, it will be seen that rank HG = rank G is necessary for the final state x(r) to be uniquely determinable. (See Theorem (9.4).) So the important question is "to what extent can we recover x(r)?" (8.7) DEFINITION. x. E X is unknown input indistinguishable from 0 at the final time r iff thereexist an input sequence u(t) G U,












t = 0, ., r - 1, and an initial state x C ker J such that the
0
resulting x(t) C ker H, t = 1, ., r and x(r) = x.


The set of states unknown-input indistinguishable from 0 at the
final time r is a subspace of X.

(8.8) DEFINITION. The unknown-input unobservable subspace at the final UO
time r, denoted as XerJ (r), is the set of states that are unknown- ierJa
input indistinguishable from 0 at the final time r.


(8.9) DEFINITION. Two states x*1, x.2 guishable at the final time r iff there and ( tr-1, x ) of input sequences
ut2 t=o 02)
j = 1, 2 and initial ataees x(0) = x ,
conditions hold:
conditions hold:


C X are unknown-input indistinr-1
exist pairs ({utl)t=o, xol) u(t) = ut ; t = 0, ., r - 1; j = 1, 2, such that the following


(i) x . is the final state (at the time r) the pair (( tr-1, x2) where j = 1, 2,
utj t=o' 0o2
(ii) the initially modified output sequences the pairs coincide.


corresponding to corresponding to


The significance of X UO(r) is clarified by kerJ

(8,10) PROPOSITION. Two states x*1, x*2 7 X are unknown-input
indistinguishable at the final time r iff U0
(8.11) x.1 - x2 krJ(r).


PROOF] [Necessity.1 Let (v., Ylj, ., y .) be the 3 lj rj .
initially modified output sequence corresponding to the pair (fut t=o' tzro'
xoj) where j = 1, 2. Suppose v1 = v2 and ytl Yt2, t = 1, ., r. Let the new initial state of (F, G, H) be x(O) := xol - xo2. Since vl = v2, we have










Jx(0) = Jxol - Jxo2 = 1 - v2 = 0O, i.e., x(0) C ker J.


Apply the input u(t) = utl - ut2, t = O, ., r - 1 to the system (F, G, H). Then it is straightforward to show

x(r) = x.1 - x*2

y(t) = ytl - Yt2 = 0, t = 0, ., r. Therefore, x1 - x2 X O(r).
x*1 - *2 CXerJ


[Sufficiency.] Suppose (8.11) holds.
UO
Then clearly x E XkerJ(r). Therefore, there
r-1
fut)r- 1 and an initial state x C ker J of
t=o o
corresponding trajectory satisfies x(t) C ker
x(r) = x*.


Write x := X1 - x2. exist an input sequence (F, G, H) such that the H, t = O, ., r and


Define a new input sequence and a new initial state of (F, G, H) by u(t) := ut2 + ut, t = O, ., r - 1 and x(0) := xo2 + x where
r-1
xo2 and (ut2 t=o are the initial state and input sequence, respectively, giving rise to the final state x*2. It is easy to show that


v(0) = Jx(0) = Jxo2,

y(t) = Yt2 t = 0, ., r,

x(r) = x*2 + x. = xl'


where (yt2 rt=o is the output sequence produced by ut )r-t
t2 t=o N2 t=o


and xo2.


It follows from Proposition (8.10) that (8.11) defines equivalence
classes, each of which consists of the states unknown-input indistinguishable at the final time r and the collections of these classes are denoted by


X/XU0 (r).
kerJ











X/ rj(r) is a linear space over k where addition and sealar multiplication are defined in the obvious ways.

X rj(r) is characterized as follows: (8.12) THEOREM.
(8.13)

(8.13) X rJ(r) = XkerJ ker(r)
YlierJ~r Xker,kerH~r

where kerJkerH (r) is defined by (2.1) and-(2.2).
XrerJ,kerH-PROOF. Immediate by using Definitions (2.1) and (2.2) and Definitions (8.7) and (8.8). 0

Proposition (8.10) tells us that to recover x(r) we cannot do
better than identifying the equivalence class x(r) + X rJ(r). Then the kerJ
question is, "can we really find the equivalence class containing x(r)?"

(8.14) THEOREM. If x(r) is the final state at the time r, then
x(r) + X~ rj(r) can be uniquely determined based on the knowledge of the corresponding initially modified output sequence (v(O), y(1), ., y(r)).

PROOF. Given the initially modified output sequence (v(O), y(1), ., y(r)) of (F, G, H). Choose an x 0 X satisfying (8.15) v(O) = Jx ,

e.g., on a fixed coordinate basis, x may be taken to be

(8.16) o : tv(0)

where J is the pseudo-inverse of J. (See Appendix for the definition of the pseudo-inverse.) Then the initial state x(0) is written as

(8.17) x(O) = + x
0 0











for some xo E ker J. 9(t) := y(t) - HFtx , are given by (8.1) and Then it is easy to see


t
Define 2(t) := x(t) - F x0, t = 0, 1, ., and t = 0, 1, . where x(t), y(t), t = 0, 1, . (8.2) with the initial condition x(O) = Xo + Xo. that 2(t), 9(t), t = 0, 1, ., satisfy


(8.18) 2(t + 1) = F2(t) + Gu(t), t = 0, 1, .,

(8.19) 9(t) = HR(t), t = 0, 1, .

with the initial condition 2(0) = x0 E ker J. Since
o

rUO U
(8.20) x(r)+ r(r) = (r)+ FrX) X (r),
+XkeJr + o + erJ


it suffices to determine 2(r) + Xrj(r). "Notice that xo is known.
The next lemma provides a way to obtain 2(r) + XrJ(r).

We now consider the system (F, G, H) with the assumption that the initial state x(O) is in ker J.

Let x(0) C ker J. Then (8.21) x(O) = BerJ 5


for some t G kn where B,e n' := dim ker J. For each r = be an input sequence (till time t = 0, ., r corresponding to is given by


r(0)(8.22) y(1)

y(r)


is a basis matrix of ker J and 1, 2, ., let u(t) C U, t = 0, ., rr - 1). The output sequence y(t) G Y, the above initial state and input sequence


= f(BkerJ, H,


1)


where fr(BkerJ, H, H): kn'+rm -k(r+1l)p is the matrix given in Section 3











(in which BW is replaced by BkerJ and M by H). Note that y(O) = 0 since x(O) C ker J C ker H. The corresponding final state x(r) C X is obtained by


u(O)
(8.23) x(r) = gr(BkerJ : u(r - 1)

where gr(BkerJ) is defined in Section 3 (Chapter II).
Our problem is to determine x(r) + rJ(r) from the knowledge of
y(1), ., y(r). (v(O) = Jx(O) = 0 since x(O) C ker J. Also y(O) = 0.)
Define


u
o n' +rm: k" ,
r (BkerJ) := " i ,
Ur-l

where n' := dim ker J and BkerJ is a basis

r(ker J) := inm fr(BkerJ, H, H).

(8.24) LEMMa. There is a unique homomorphism X/r (BkerJ) ker fr(NerJ, H, H) such that the


f (BkerJ, H, H) Sr (BkerJ) -


gr(BkerJ)


u km, j = 0, .,r- 1) matrix of ker J. Write


0kerJ: P (ker J) 4 r r
following diagram co-mutes:


rr (ker J)
I kerJ
r


X/r (BkerJ) ker fr(BkerJ, H, H)


where f r(,', ) and gr(.) are as in Section 3 and p: X X/r(BkerJ) ker fr(BkerJ, H, H) is the canonical projection.


-.1p










Furthermore,

(8.25) gr(BkerJ) ker fr(BkerJ H, H) = X (r).


Therefore,


kerJ r (ker J) -X/ rJ(r).


PROOF. Let yr E r (ker J). such that


There is an element


Ur C 1r(BkerJ)


(8.26) fr(BkerJ, H, H) (Ur) = yr. Define -kerJ r(ker J) -X/gr(BkerJ) ker fr(BkerJ, H, H) (8.27) 'erJ r) = Pgr(BkerJ) (r)


kerJ
r


is well-defined.


satisfying (8.26).

(8.28) pgr(BkerJ) (-u)
r


= pgr(BkerJ) (ur + 7)


Indeed, let "u be another element in r0(BkerJ)
r r kerJ


(for some


7 E ker fr(BkerJ, H, H))


= P(gr(BterJ) (ur) + gr(BkerJ)()) = gr(BkerJ) ( + gr(BkerJ (Y) = gr kerJ) (ur


To show the linearity of kerJ, let yrl' Yr2 e r(ker J).
Then there exist Ur, Ur2 E nr(BkerJ) such that rj = f (B kerJ, H, H)(u rj), j = 1i, 2. Therefore, y1 + = f(Bk ,rJ H, H) (rl + r2) and
we have


kerJ + + U
r l +r2)= Pr(BkerJ) (u r + ur2











= Pgr(BkerJ) (url) + gr(BkerJ) (r2) kerJ - kerJ,=r Yrl) + r Jr2 kerJ,Similarly, for each a E k, 7r E r (ker J) we have rr (r) =
SerJ,r r
To show the uniqueness of the map suppose kerJ: r (ker J) -+ r r
X/gr(BkerJ) ker fr(BkerJ, H, H) is another map for which the diagram commutes. Then for each ur E Sr(ker J)


e r(BkerJ , H) r )= (BkerJ (r kerJ ,
r r(BkerJ H, H) r

Hence, noting that fr(BkerJ, H, H) is onto, we obtain kerJ = kerJ
(8.25) follows from Theorems (3.1) and (8.12). 0

(8.29) REMARK. ker r(ker J) - rX/X rj(r) does not depend on the
choice of BkerJ'

(8.30) REMARK. For a fixed coordinate basis 0kerJ Pr(ker J) -s X/Xr(r) may be defined by

(8.31) erJ r(BkerJ) kerJ H, H)


where fr(BkerJ, H, H) is the pseudo-inverse of fr(BkerJ, H, H). (See Appendix.)
As an immediate consequence of Proposition (8.10) and Theorems (8.12) and (8.14) we have

(8.32) THEOREM. A system (F, G, H) is unknown-input observable at the final time r iff

(8.33) erJch kr(r) = 0.
4erJ,kerHf










9. Unknown Input Final State Observability--Part 2 (Special cases).

Let r = 1, 2, .R. ecall Theorem (8.12). The unknown input unobservable subspace Xerj(r) at the final time r depends on the time r in general. However, in some cases the time dependence disappears in a finite time. (See Lemma (2.6) and the paragraph prior to Lemma (2.8).) And if

(9.1) ker J C (F ker J +im G) ker H holds, more can be said, namely, the sequence DOjM) i = 1, 2, .,

is monotonically nondecreasing (Lemma (2.6 i)) as well as it stops increasing in at most n steps, i.e.,

(9.2) Xerj(n) = Xerj(n + 1) =

(See Lemma (2.8 i).) Also, if

(9.3) ker J D (F ker J + im G) nker H


is true, then the sequence Xerj(i), i = 1, 2, ., is monotonically
nonincreasing (Lemma (2.6 1i)) and (9.2) holds. (See Lemma (2.8 ii).)

Note that if ker J = 0, i.e., the initial state is known a priori, then (9.1) holds, and that if ker J = ker H, i.e., if the initial state is not known a priori, then (9.3) holds. Therefore, for these cases the above statements are true.

Let us consider the first situation where ker J = 0 (i.e., the
initial state is a priori known). The condition under which Problem (8.5) has a unique solution becomes particularly-simple.

(9.4) THEOREM. If ker J = 0, then the system (F, G, H) is unknown
input observable at the final time r iff


(9.5) dim im HG = dim im G,









i.e.,

(9.6) rank HG = rank G.

PROOF. By Theorem (8.32) and Theorem (2.5) (F, G, H) is unknown input observable at the final time r iff (9.7) XokerH(r) = 0.
o,kr


Since Lemma


X ker(i) C X (i + ), i = 0, ., it follows from
O,kerH o,kerH
(2.6) that (9.7) holds iff


(9.8) o = Xo,kerH(1) = im Gn ker H.


It remains to show that (9.8) holds iff (9.5) holds.


Suppose im G fker H = 0. Let m' := rank G = dim im G. Denote
by (g g2' . gm, a basis of im G. m'e claim that Hg1, Hg2, ., Hgm, are linearly independent. In fact, let . , jHg. =0 for some a.j k, j = 1,
m j=l 3
2,., m, not all zero. Then H jI gj 0, which implies

j.Zajgj = 0 since im Gnker H = 0. Now since gl' ". gm, are linearly
independent, we get a. = 0 for all j = 1, ., m'. Contradiction. Hence, dim H im G = m'.


Conversely, suppose im G^) ker H / (0). Let Trivially el is a basis of Span el C im GE(ker H. a basis (el, e2' ., em,) of im G. Then


0 j el G im G()ker H. Extend el to form


(9.9) im HG = Span (Hel, ., Hem')

= Span (He2, ., Hem,).


Therefore, dim im HG < m'. OE (9.10) COROLLARY. Let m = p = 1 and define polynomials H[(z) and X(z) by I(z)/X(z) = H(zI - F)-IG ( 0). Suppose the initial state is known. Then the system (F, G, H) is unknown-input observable at the final








time r iff

(9.11) deg X(z) - deg 11(z) 1.

PROOF. Since m = p 1, it follows that rank G = 1 iff G / 0
and that rank HG = 1 iff HG / 0. Hence, (9.6) holds iff HG / 0 (since G 1 0 by assumption). El

(9.12) COROLLARY. Assume that the initial state is known, i.e., ker J 0.
Then there is a unique state trajectory x(t) G X, t =, 1, ., r corresponding to an initially modified output sequence (v(O), y(l), ., y(r)) iff (9.5) (i.e., (9.6)) is satisfied.

(9.13) REMARK. Assume that the initial state is known and that rank G m (i.e., dim im G = m). It can be shown that the input sequence corresponding to an initially modified output sequence (v(O), y(O), ., y(r)) is unique iff (9.5) (or (9.6)) is true.

10. Unknown Input Initial State Observability.

Consider the system (F, G, H) given by (8.1) and (8.2). As before r = l, 2,.

We wish to find x(O) based on the measurement of the output sequence y(O), ., y(r). Suppose that we are allowed to make an extra measurement at the final time r. (Suppose for example, that the "motion" of the system stops at the end (at the time r) and that we can examine the system more carefully to get an extra information at the time.) We will denote this extra information by

(10.1) Ve(r) = e x(r),

where v' (r) C ks and J : X -)k is a k-homomorphism (for some integer
e e
s > 0). Then


(10.2) [e x(r).
LY(r)- H








Denote

v (r) J
V (r) := e e :
e y(r) e H


Now we shall call the sequence (y(O), ., y(r - i), ve(r)) r-modified output sequence. Consequently, we have

(10.3) PROBLEM. Given an r-modified output sequence [y(O), ., y(r - 1),
ve(r)), find the corresponding initial state x(O).

By (10.2), clearly,

(10.4) ker J C ker H.
e

(See the remark on this condition in Section 11.)
We say that the system (F, G, H) is unknown-input initial state observable at the time r iff Problem (10.3) has a unique solution.
A state x. E ker H is said to be unknown-input initial state indistinguishable from 0 at the time r iff there is an input sequence u(t) E U, t = 0, 1, ., r - 1 such that the corresponding state sequence satisfies x(0) = x,, x(t) E ker H, t = 0, ., r - 1 and x(r) E ker Je. The set of states that are unknown-input initial state indistinguishable from
0 at the time r is a subspace of X.

(10.5) DEFINITION. The unknown-innut initial state unobservable subspace at the time r, denoted by XrJe(r), is the set of states that are unknowneinput initial state indistinguishable from 0 at the time r.

From Definitions (10.5) and (5.5) we obtain

(10.6) THEOREM. XrIU e(r) = Xkeontr (r).
TCerJe TierJe,kerH
Recall the results given in Chapter III. We know the characterizations
contr
and the properties of erjekerH(r).
The following result given by SILVERMAN [1976, Section III, A, therJekerH
The following result given by SILVERMAN (1976, Section III, A) the









paragraph after theorem 5] is a special case of the above theorem.

(10.7) COROLLARY. X rU (n) = V (ker H) where Vmax(ker H) is defined
XerH maxma
lr(5.14).

PROOF. Let J = H, and then use Corollary (5.15). O
e

Two states x.1, x*2 E X are unknown-input initial state indistinguishable at the time r iff there are two input sequences utj; t = 0, 1, ., r - 1; j = 1, 2, such that the r-modified output sequences corresponding � r-1r-1
to the pairs (x*1, (utl)r o) and (x*2, (ut2)t=O) coincide,

(10.8) PROPOSITION. Two states x.1, x42 E X are unknown-input initial state indistinguishable at the time r iff IU ()
(10.9) x.1 - x2 e irJe(r).

PROOF. Similar to the proof of Proposition (8.10). Therefore, we omit the proof here. 0

XI rJe(r) defines the set of the equivalence classes X/XU rJe(r). By Proposition (10.8) the best that we can recover of the initial state x(O) is the equivalence class x(O) + XUr(r). What remains to be IU + aere
shown is how to determine x(O) + XkerJe(r) from (y(O), ., y(r - 1),
ve(r)).


Suppose we have the initial state x(O) E X u(t) E U, t = 0, ., r - 1. The corresponding sequence y(t) 6 Y, t = 0, ., r - 1, ve(r) C


and input sequence r-modified output kq is given by


y(O) - x(O)
: = fr(I n, J e H) u(0) , y(r - 1) ev (r) u(i
-e


where In is the n X n identity matrix (or identity map kn -4 kn)









and f (., *, *) is as in Section 3 of Chapter II and Section 6 of Chapter III. Define (In) := kn+rm and r (ker J ) := im f (In, Je' H).
IU
(10.11) THEOREM. There is a unique homomorphism : P (r) such
that the following diagram commutes:

f (In, J e H)
Sr(I ) e - r(ker J )
r n r e

P
I


X X/XrJe(r)


wher- I is the n x n identity matrix (or the identity map kn - kn), fr(., ., ') and Pr are as in Sections 3 and 5, U: X -X/XUrJe(r) is the canonical projection.

PROOF. By Theorems (6.1) and (10.6) we have


(10.11) XIU (r) = P ker f (I , J , H).


Let y E r(ker Je). There is an Ur e r(I ) such that


(10.12) 7~ = f (I , J H)(- ).
r n e

Define Wr: Pr(ker J) - X/xrJe(r) = X/Pr ker fr(In, Je' H) by
kr e

(10.13) r r ) = O-P ur


We claim that rr is well-defined. In fact, let u' C Q (In) be another
r r r n
vector satisfying (10.12). Then

u - u ker f (In, J e H).
Sr n e


Therefore,










(io.14) fr(In, Je' H)(-) = fr (In, Je' H)(T + U) (for some

C ker f (I, Je , H))
-- fr(I n Jer H)(e



The linearity and the uniqueness can be shown similarly to Lemma (8.24).

By Proposition (10.8) (or the remark after the proposition) and
Theorem (10.12), we get the necessary and sufficient condition for (F, G, H) to be unknown input initial state observable at the time r.

(10.15) THEOREM. The system (F, G, H) is unknown input initial state observable at the time r iff

(10.16) count (r ) = 0. 0
(I016)XierJe,kerH


(10.17) REMARK. If r > n and if J = H, then (10.16) is equivalent to

(10.18) Vmax(ker H) = 0,


where Vmax(ker H) is the maximal F mod G invariant subspace in ker H.

11. Concluding Remarks

If one does not go through the arguments at the beginnings of Sections
8 and 10, i.e., if one does not want such conditions as (8.6) and (10.4), one can easily accomodate the discussion in this chapter to the new reach c r) (without
situations. The modified versions of XW,N r) and WNt (r) the assumption of W C N) should be used accordingly. (See Section
4 of Chapter II and Section 7 of Chapter III.)

Techniques similar to those used in this chapter can be applied to study unknown input observability problems in non-constant dynamical systems (F(t), G(t), H(t)). (The problem statements (8.5) and (10.3) should be modified in the obvious ways.) The unknown input unobservable subspace X UO(s, s + r) at the final time based on the observation over the








time interval [s, s + r] is defined similarly to (8.8). XUO(s, s + r) UO
can be characterized by X UO(s, s + r) = X(r) where X(O) = ker H(s), X(i) = (F(i - l)X(i - 1) + im G(i - l))Qf^ker H(s + i), i = 1, 2,. The unknown input initial state unobservable subspace XIU(s, s + r) based on the measurement over the time interval [s, s + r] is defined similarly to (10.5) and it is characterized by XIU(s, s + r) = X(r) where X(O) = ker H(s + r), x(i) = F-1(s + r - i)(X(i - 1) + im G(s + r - i + l))n ker H(s + r - i), i = 1, 2, . Non-recursive characterizations can also be given in analogous ways to those in constant dynamical systems.









CHAPTER V. STABILIZABILITY, OUTPUT ZEROING AND DISTURBANCE DECOUPLING reachh contr
In the previous chapter we have applied WeN (r) and XN. (r) to the problems of unknown input observability.
contr(
In this chapter we shall demonstrate the use of XWN tr(r) in the
discussions of stabilizability, output zeroing and disturbance decoupling.

12. Stabilizability

We consider the system (F, G) given by (8.1). In this section we assume that k := R or C where R is the field of real numbers and
C is the field of complex numbers.

The pair (F, G) or a map F (over k = R or C) is said to be asymptotically stable if Jim I Ftx.i = 0 (t = 0, 1, 2, .) for all x E X where 11*11: X -R+ is a norm defined on X and R is the set of nonnegative real numbers. It is known that the pair (F, G) is asymptotically stable iff jij < 1, i = 1, 2, ., n where ?i', i = 1, 2, ., n are the eigenvalues of F. (See, for example, FREEMAN [1965, Chapter 7, Section 9], MARRO [1975, Chapter 5, Section 4].)
If there exists a feedback K: X -)U such that %imJ (F + GK)tx*I = 0 (t 0, 1, 2, .) for all x* G X, then the pair (F, G) is said
to be asymptotically stabilizable. Let i(A) = ous(?)) s(N) be the minimal polynomial of X with respect to F where the roots of Ous(P)
have the magnitudes greater than or equal to unity and the roots of s(A) have the magnitude less than unity. Then we have

(12.1) THEOREM. The following statements are equivalent:

(i) The pair . (F, G) is asymptotically stabilizable.

(ii) ker $us(F) C Xcontr(n).
o,X
(iii) contr ,
Xii k(F) X(n) = X.

PROOF. (iii) = (i). Suppose (iii) holds. Since F ker ae(F) c ker $s(F), by applying Theorem (5.19) we see that there is a feedback K: X -4U satisfying










(12.2) (F + GK) X C ker i(F), i = n, n + 1, .


K can be chosen so that, in addition,


(12.3) (F + GK)Iker S(F) = FIker s(F). (See Remark (5.33).) Therefore, for any state x* C X we have (12.4) im(F + GK)txII = 1im llFt n(F + GK)nxl1 = 0 where t takes on the values n, n + 1, . (Note that ker s(F) is the stable mode.)

(i) = (ii). It can be shown that (i) implies ker US(F) C im G + F im G + ,,, + Fn-l im G. (See Theorem 2.2 in WONHAM [1974] .) n-1 contr
Since im G + F im G + . + F'l im G CoXtr (n), the implication follows.

(ii) (iii). Let xCE X. Write x = xs + x where
xs 6 ker $S(F) and xu E ker Us(F). Since xu E ker (F) C X ontr (n)
there exist u(O), u(1), ., U(n - 1) E U satisfying

(12.5) Fnxu + Fn-Gu(O) + . + FGu(n - 2) + Gu(n - 1) = 0. Therefore, for the initial state x(O) = x = xs + xu we have


(12.6) x(n) = Fn(xs + x ) + Fn-Gu(O) + . + Gu(n - 1) = Fnxs C ker OS(F).


13. Output Zeroing

We are interested in the system (F, G, H) given by (8.1) and (8.2) over an arbitrary field k.

(13.1) PROBLEM. (Output Zeroing by State Feedback) Given (F, G, H), find a k-homomorphism K: X -4U ("state feedback") for which there exists an integer i > 0 such that, for every x(O) E X, we have y(t) = O, t > i where y(t) is the output of the system (F + GK, 0, H) due to








the initial state x(O).

(13.2) THEOREM. Let x. e X. Then there exists a state feedback K: X ->U for which there is an integer i > 0 such that y(t) = 0, t > i for x(O) = x. where y(t) is the output of the system (F + GK, 0, H) due to the initial state x. iff

(13) contr
(1.5) x. E max(kerH),X(n)

where V (ker H) is defined by (5.14).

PROOF. ["if"] Since V max(ker H) is an F mod G invariant subspace in N, by Theorem (5.19) there is a feedback K : X -U such that
n contr
(F + GK) x(kerH),(n) C Vmax(ker H). This implies that, if x(O) = x., then

x(1) = (F + GK)x.

x(2) = (F + GK)x(1) = (F + GK)2x


x(n) = (F + GK)x(n - 1) = . = (F + GK)nx E V (ker H).


If K is so chosen as to satisfy (F + GK) V (ker H) C V (ker H) max
(see the proof of Theorem (5.19)), we have x(t) = (F + GK) -nx(n) C V (ker H) for all t > n. Thus, y(t) = Hx(t) = 0 for all t > n.

["only if"] Let x(t) C X, t = 0, 1, ., be the trajectory of the system (F + GK, 0, H) with the initial state x(O) = x. Then by assumption we have for some integer i > 0

x(i + j) e ker H, j = 0, ., n.

Therefore, by Corollary (5.15) we know x(i) C Vmax(ker H), which implies that x(O) = x. r Vi(V max(ker H), X) by Definition (5.2) and (5.3). Appealing to Proposition (5.17) we conclude








contr contr
x Vax(kerH),X XVmax(kerH),X


(15.4) COROLLARY. Let x EC X. Then there exists a feedback K : X -- U such that y(t) = 0, for all t = r, r + 1, . where y(t) is the output of the system (F + GK , 0, H) for x(O) = x, iff

contr
(13.5) x, e V (kerH),X(r).

PROOF. Similar to the proof of Theorem (13.2). O

(13.6) COROLLARY. The output zeroing problem by state feedback has a solution iff

contr
(13.7) XV (kerH),(n) = X.
max

PROOF. Immediate from Theorem (13.2). O]


14. Disturbance Decoupling

Consider a finite-dimensional, constant coefficient, discrete time, linear dynamical system with disturbance given by (14.1) x(t + 1) = Fx(t) + Gu(t) + Dv(t), t = 0, 1,

(14.2) y(t) = Hx(t), t = 0, 1, .


where X := kn, U := kim, Y := k, V:= ks (v(t) e V, t = 0, 1, .)
and k is a field. We may denote the system (14.1), (14.2) as (F, G, H, D). As before r denotes a positive integer. (14.3) PROBLEM. (r-step Disturbance Decoupling Problem) Given (F, G, H, D) with x(O) = 0, find (if possible) a feedback Krl: X -4U such that y(t) = O, t = 0, ., r for any v(t) C ks, t = 0, 1, ., where y(t) is the output of the system (F + GKr-l, O, H, D) with x(O) = 0.









(14.4) THEOREM. The r-step disturbance decoupling problem has a solution iff

(14.5) im D c: xontr r - 1).


PROOF. [Sufficiency] By assumption x(O) = 0. let Kl: X -4U r-1
be as in Theorem (5.19 ii) where we assume W = N = ker H. Replacing u(t) in (14.1) by Kr_1x(t), we get


(14.6) x(t + 1) = (F + GKr_ 1)x(t) +- Dv(t).


Consider the system (14.6), (14.2). Let v ( ks, j 0, 1, . . For
each j = 0, ., r - 1 suppose the disturbance v(t), t = 0, 1, ., be such that v(j) = vj(f 0 possibly) and that v(t) = 0 if t J j. Then we have

x(t) = O, t = 0, ., j,

x(t) = (F + GK r_)tiDv., t = j+ , j + 2,. .


Therefore, by Theorem (5.19 ii) with W and N both replaced by ker H,

y (t) := Hx(t) = 0, t = 0, ., r + j.

Now we superpose all the disturbance used above, i.e., we use v(t) such that v(t) = vt for t = 0, ., r - 1. Then the output y(t), t = 0, 1, ., satisfies

r-1
y(t) = jO j(t) = 0, t = 0, ., r.

[Necessity] Suppose that the problem has a solution but that im D keH kerH(r - 1). Since x(O) = 0, we have x(1) = Dv(O).
s contr
There must be v E k such that Dv ontr, - 1). Then by
So erH,kerH
Definitions (2.2), (2.3) there does not exist an input sequence u(t) C U, t = 1, 2, ., such that the corresponding state trajectory








(sequence) satisfies x(t) C ker H for t - 2, ., r. In particular there is no input sequence of the form Kx(t) satisfying the above requirement where K: X --U. Thus, we have contradiction. CI

The limit case of this problem ( r -)-) is the well-known Disturbance Decoupling Problem. It is known that the problem has a solution iff im D C V max(ker H) where V max(ker H) is the maximal F mod G invariant subspace in ker H. (Refer to WONTAM and MORSE [1970, Theorem (3.1) and WONHAM [1974, Theorem (4.2)].)

Let us now add a constraint to the above problem so that the state of the resulting system (F + GK, 0, D, H) will eventually reach zero when the disturbance becomes zero.

(14.7) PROBLEM. (Disturbance Decoupling Problem with Reset). Given (F, G, H, D) with x(O) = 0, find (if possible) a state feedback K: X -4U such that the resulting system (F + GK, 0, D, H) (with x(O) 0)
has zero output sequence and that (F + GK)i = 0 for some i > 0.

(14.8) THEOREM. Problem (14.7) has a solution iff the following conditions hold:

(14.9) im ID C X contrH(n),
contr
(14.) imD C o, kerHn,


(14.10) X contr) = X.
0 Z, X n=

PROOF. [Sufficiency] Assume (14.9) is true. By Theorem (5.19 i) there exists K: X -4U such that

(14.11) (F + GK)JXcntr (n) C ker H, j = 0, . n- 1,
o, kerH

(14.12) (F + GK)JXb, (n) = 0, j = n, n + 1.
b, kerh

(14.9), (14.11) and (14.12) then imply


(14.15) (F + GK)j im D C ker H, j = 0, 1, .








which means that the output y(t), t = 0, 1, . of the system (F + GK,
O, D, H) with x(0) = 0 is zero for all t = 0, 1, .

It remains to show that there is an integer i > 0 satisfying
i contr eontr
(F + GK) = 0. If (4.10) holds, then X = Xo X (n) C Xc X (n) for
contr 6ontr
any subspace W of X. So X =X X (n) where W := X kerH(n).
cotr WX o kerH
contr Noticing that 0 and X kerH(n) are F mod G invariant subspaces and
repeatedly using Theorem (5.19 i), we see that there is a state feedback
K: X ->U satisfying (14.11), (14.12) and n contr
(14.14) (F + GK)nX C Xcontr (n).
okerH

Hence,

(14.15) (F + GK)2nX = 0.

[Necessity] Let K: X ->U be a solution of the problem. Then n contrt .Nwdfn
clearly (F + GK) = 0. Hence Xco (n) = X. Now define
o,X(n

V := im D + (F + GK) im D + . + (F GK)imD.(FGK)n-1 im D.

Then clearly im DC V, (F + GK)V CV and V C ker H. Since (F + GK)n = O,
contr
we have (F + GK)nV = 0. Therefore, V C X .r Thus im D C V C
Xcontr (n). okerH
o,kerH
contrr contr.
(14.16) REMARK. Recall (0.4). We have Xcontr (n) = Xcontr oX

15. Concluding Remarks reach xreach(N
There is an important difference between X N each(n) and Xreach(N).
neach max
By choosing K: X -)U the spectrum of (F + GK)jXrax (N) can be assigned
max reach
arbitrarily, while to discuss the spectrum of (F +- GK)IX eN (n) may reach reach
not make sense since (F + GK) XoN (n) C Xo e (n) may not hold (unless
_reach,
N is an F mod G invariant subspace). However, though X each(r) loses contr
the arbitrary pole assignability, XcN (r) can treat stabilizability by state feedback (Theorem (12.1)).


The usefulness of Theorem (12.1) is yet to be clarified.










CHAPTER VI. CONCLUSION


We have introduced the r-step reachable subspace each (r) from contr),N
W in N and the r-step controllable subspace xi;,N (r) to W in N of the finite-dimensional, constant, discrete-time, linear dynamical system (F, G, H) over a field k. We have characterized these subspaces and discovered several interesting properties pertaining to them.
The notions eN ,r). (Ntr) are natural generalizations

of reachable and controllable subspaces when we are interested in structural properties of state trajectories contained in the subspace N and when initial and final states of the trajectories may not be zero.
reach ( a contr,
The significance of xWN (r) and XWN r) in control problems
has been demonstrated in Chapters IV and V. Among the applications presented in these chapters, the unknown-input observability at the final time r is the most important in the sense that it has motivated the author to study the subjects treated here.

As has been mentioned in the concluding remarks of each chapter, .reach, ic
there are several other interesting topics concerning WN (r) which are still open to further research. The relation between the transfer function M(zI - F)'IG and the subspaces each r), cont r(r) should also be studied.










APPENDIX


Al. Pseudo-inverse


Given an r X r nonsingular matrix M in a field k. There is
-1
a unique r X r matrix M satisfying


(Al.l) MM- = M- 1 = I.

-i
The matrix M is called the inverse of M.

Now let M be r1 X r2 matrix in k (which may be singular). The inverse of M does not exist in general. The idea of inverse, however, can be generalized as follows: Let s := rank M, and let M be an s X r2 matrix consisting of a collection of s linearly independent rows of M. Then the pseudo-inverse M of M is defined by


(Al.2) M:t M M (MT)

where A' denotes the transpose of a matrix A. The meaning of M is as
follows: Using trdom" to denote "the domain of", we have (Al.3) LEMMA. Let x ( dom M, and write x = x1 + x2 where x1 C im M" and x2 G ker M. Then M Mx =x.


PROOF. If xl r im M then xI M x1 for some s-vector x1.
Therefore


M Mx M f(MM( 1

Mx X1*


If x2 C ker M, then trivially MtMx2 -O. L


If M is of full column rank, Mt is called the left inverse of M; if M is of full row rank, M is called the right inverse of M; if M
i 7 n n s i n 7 u l a r , t h e n T.' t - 7A s ,e_.1S ~ i ] O <


1 2









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[1976] "A note on transmission zeros of a transfer function matrix",
IEEE Trans. Automatic Control, AC-21: 589-591. G. BASILE

[19691 "Some remarks on the pseudoinverse of a nonsquare matrix",
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Scienze dell'Istituto di Bologna.

G. BASILE and G. MARRO

[1968&] "Controlled and conditioned invariant subspaces in linear
system theory", Report No. AM-68-7, University of California,
Berkeley.

[1968b] "On the observability of linear time-invariant systems with
unknown inputs", Report No. AM-68-8, University of California,
Berkeley.

[1969] "L'invarianza rispetto ai disturbi studiata nello spazio
degli stati", Rendiconti Della LXX Riunione Annuale AEI.

[1973] "A new characterization of some structural properties of linear
systems: unknown input observability, invertibility and
functional controllability", Int. J. Control, 17: 951-945. S. P. BHATTACHARYYA, J. B. PEARSON and W. M. WONHAM

[19721 "On zeroing the output of a linear system", Information and
Control, 20: 135-142.

R. W. BROCKETT and M. D. MESAROVIC

[19651] "The reproducibility of multi-variable systems", J. Math.
Analysis and Applications, 11: 548-563. E. EMRE and M. L. J. HAUTUS

[1978] "A polynomial characterization of (A, B)-invariant and
reachability subspaces", Eindhoven Univ. of Technology,
Memorandum COSOR 78-19.

H. FREEMAN


I oali3 Son2. New -:










K. FURUTA

[1973] Senkei Shistem Seigyo Riron (Theory on Linear Systems and
Control), Shohkodoh, Tokyo. F. HAMANO and K. FURUTA

[1975] "Localization of disturbance and output decomposition in
decentralized linear multi-variable systems", Int. J. Control,
22: 551-562.

B. HARTLEY and T. 0. HAWKES

[1970] Rings, Modules and Linear Algebra, Chapman and Hall, London. D. G. LUENBERGER

[1966] "Observers for multivariable systems", IEEE Trans. Aut.
Control, AC-11: 190-197. R. E. KALMAN

[1963] "Mathematical description of linear dynamical systems", SIAM
J. Control, 1: 152-192.

[1968] "Lectures on controllability and observability", Proc. C.I.M.E.
Summer School, Edizioni, Cremonese, Roma, 1-149. R. E. KAIMAN, P. L. FALB and M. A. ARBIB

[1969] Topics in Mathematical System Theory, McGraw-Hill, New York. R. LASCHI and G. MARRO

[1969] "Alcune considerazioni sull'osservabilith dei sistemi dinamici
con ingressi inaccesibility", Rendiconti Della LXX Riunione
Annuale AEI.

G. MARRO

[19751 Fondamenti di Theoria dei Systemi, Patron, Bologna. B. C. MOORE and A. LAUB

[1978] "Computation of supremal (A, B)-invariant and controllability
subspaces", IEEE Trans. Auto. Control, AC-32: 783-792.









M. K. SAIN and J. L. MASSEY

(1969] "Invertibility of linear time-invariant dynamical systems",
IEEK Trans. Auto. Control, AC-14: 141-149. L. M. SILVERMAN

[1976] "Discrete Reccati equations: alternative algorithms, asymptotic
properties and system theoretic interpretations", in Control and Dynamical Systems: Advances in Theory and Applications,
Vol. 12 (edited by C. T. Leondes), Academic Press, pages
313-386.

L. M. SILVERMAN and H. J. PAYNE

[1971] "Input-output structure of linear systems with application to
the decoupling problem", SIAM J. Control, 2: 199-233. E. D. SONTAG

[1979] "On the observability of polynomial systems, I: finite-time
problems", SIAM J. Control and Optimization, 17: 139-151. L. WEISS and R. E. KALMAN

[1965) "Contributions to linear system theory", Int. J. Engineering
Science, 3: 141-171.

W. M. WONHAM

[1974] Linear Multivariable Control: A .Geometric Approach, Springer,
New York.

W. M. WONHAM and A. S. MORSE

[1970] "Decoupling and pole assignment in linear multivariable systems:
A geometric approach", MIAM J. Control, 8: ' 1-18.





















BIOGRAPHICAL SKETCH


Fumio HAMANO was born on August 28, 1949,. in Wikayama, JAPAN, to Zenichi HAMANO and Mitsuko HAMANO. He received his Bachelor of Engineering from Tokyo Institute of Technology in 1973 and his Master of Science and Engineering from the same institute in 1975.
















II certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Thil6sophy.




Rudolf E. Kalman, Chairman Graduate Research Professor



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Thilisophy.




Charles V. Shaffer, Co-cha Professor of Electrical Engineering



I certify that I have read this study and that in my opinion It conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.




Thomas E. Bullock
Professor of Electrical Engineering















I certify that I have read this study and that in my opinion it conforms to ceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.





Assistant Pr fe~sor of Electrical Engineering



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.




Charles ABmnap
Assistant Professor of Mathematics



This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

August, 1979




Dean, College of Engineering


Dean Graduate School




Full Text
CHAPTER I. INTRODUCTION
This dissertation discusses some structural properties of finite
dimensional, discrete time, constant, linear dynamical systems. Great efforts
have been made to study "geometric" properties in the continuous time systems
as well as those which are common to both discrete and continuous time
systems. (See for instance BASILE and MARRO [1968a and b], WONHAM and MORSE
[1970], WONHAM [1974, Chapters 1 through 5] and SILVERMAN [1976, Section
IH].) However, distinctive features of differentiating discrete and continuous
time systems have not received much attention. This work intends to point
out that there are important differences between the two kinds of systems
and that discrete time systems, therefore, should be treated separately
in such cases.
The following notation will be used in the sequel: "im", "ker" and
"dim" respectively stand for "the image of", "the kernel of" and "the
dimension of". "" and ":=" mean "is an element of" and "is defined
to be", respectively.
Let us now turn to the definition of systems which is pertinent to the
discussion of this dissertation. Let k be an arbitrary field, and let
m, n and p be positive integers. A finite-dimensional, constant
(coefficient), discrete time, linear dynamical system is a triple (F, G, H)
whose dynamical interpretation is given by
(0.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, ... ,
(0.2) y(t) = Hx(t), t 0, 1, ... ,
where x(t) X := kn, u(t) U := km, y(t) Y := kP for t = 0, 1, ... ;
F: X -X, G: U ->X and H: X -* Y are k-homomorphisms (or matrices)
independent of time. The vector spaces X, U and Y are called the state,
the input (value) and the output (value) spaces, respectively. The elements
of X, U and Y are called states, Inputs and outputs, respectively.
We shall refer to the triple defined above as the system (F, G, H). However,
when the output is of no interest, we shall simply say the pair (F, G)
disregarding (0.2).
1


APPENDIX
Al. Pseudo-inverse
Given an r X r nonsingular matrix M in a field k. There is
-1
a unique r X r matrix M satisfying
(Al.l) MM_1 = M_1M = I.
The matrix M ^ is called the inverse of M.
Now let M he r^ x r^ matrix in k (which may be singular). The
inverse of M does not exist in general. The idea of inverse, however,
can be generalized as follows: Let s := rank M, and let M be an
s X r matrix consisting of a collection of s linearly independent rows
t
of M. Then the pseudo-inverse M of M is defined by
(A1.2) M+ := m'{(Mm')'(Mm'))'1(MM/)'
' t
where A denotes the transpose of a matrix A. The meaning of M is as
follows
: Using
"dom" to denote "the domain
of", we have
(A1.3)
LEMMA.
Let x £ dom M, and write
/
x = x^ + x^ where x^ £ im M
and Xg
ker M.
Then M Mx = x .
PROOF.
If x^ £ im M then x^ =
: M x^ for some s-vector x .
Therefore
M+Mx1 = T{(Mm')'(MM'))"1(MM')'mM/x1
= M x
= xr
If x0 £ ker M, then trivially M^Mx^ =0.
t
If M is of full column rank, M is called the left inverse of M;
if M is of full row rank, M is called the right inverse of M; if M
is nonsingular, then M = M 1. (Also see G. BASILE [1969].)
53


time interval [s, s + r] is defined similarly to (8.8). X^(s, s + r)
can be characterized by X (s, s + r) = X(r) where X(0) = ker H(s),
X(i) = (F( i l)X(i l) + imG(i 1))P ker H(s + i), i = 1, 2, ... .
The unknown input initial state unobservable subspace X^(s, s + r)
based on the measurement over the time interval [s, s + r] is defined
similarly to (10.5) and it is characterized by X^(s3 s + r) = X(r)
where X(0) = ker H(s + r), x(i) = F ^(s + r i)(x(i l) +
; im G(s + r i + l))Oker H(s + r i), i = 1, 2, ... Non-recursive
characterizations can also be given in analogous ways to those in constant
dynamical systems.


CHAPTER II. THE r-STEP REACHABLE SUBSPACE FROM W IN N
We study a finite-dimensional, constant (coefficient) discrete time,
linear dynamical system (F, G, H) over an arbitrary field k; in this
chapter we shall be interested in properties of the pair (F, G). We
shall define a new notion of "the r-step reachable subspace from W in
N'f, denoted by XL. (r), and study the properties of the sequence of
.reach/ <
subspaces ^ (i), i = 1, 2, ... The subspace defined here will
find its application in Chapter IV.
We begin this chapter with a well-known notion of F mod G invariant
subspaces.
1. F mod G Invariant Subspaces
Consider a finite dimensional, constant, discrete time, linear dynamical
system given by
(1.1) x(t + l) = Fx(t) + Gu(t), t = 0, 1, ... ,
where x(t) X := kn, u(t) U := km, t = 0, 1, ..., and k is an
arbitrary field. We call X the state space and U the input (value)
space. Since the output is of no interest at the moment, we shall refer to
(1.1) by the pair (F, G).
(1.2) DEFINITION. A subspace V of X is an F mod G invariant
subspace (or simply F mod G invariant) iff
(1.3) FV C V + im G.
The most important properties of F mod G invariant subspaces are
expressed by the following
(1.4) LEMMA. Let V be a subspace of X. Then the following statements
are equivalent:
(i) V satisfies (1.3).
(ii) For any x* £ V there exists an input u£U such that
7


46
(12.2) (F + Gk/x C ker /(F), i = n, n + 1, ... .
K can be chosen so that, in addition,
(12.3) (F + GK) Jker /(F) = Fjker /(F).
(See Remark (5.33).) Therefore, for any state x^ G X we have
(12.4) £imJ|(F + GK/xJ = Jimj||Ft"n(F + GK)\\\ = 0
where t takes on the values n, n + 1, ... (Note that ker /(F) is
the stable mode.)
(i) = (ii). It can be shown that (i) implies ker /S(f) C
im G + F im G + ,,, + F*1"1 im G. (See Theorem 2.2 in WONHAM [1974].)
Since im G + F im G + ... + / ^ im G C XC0^r(n), the implication follows
O y A
(ii) =5- (iii). Let x# G X. Write x_^ = xg + x where
xg G ker /(F) and x^ ker /S(f). Since Xy^ ker ^S(f) C X;Cn^r (n)
there exist u(0), u(l), ..., U(n l) G U satisfying
(12.5) Fnxu + Fn_1Gu(0) + ... + FGu(n 2) + Gu(n l) = 0.
Therefore, for the initial state x(0) = x = x + x we have
(12.6) x(n) = Fn(x + x ) + Fn~1Gu(0) + ... + Gu(n l) = Fnx G ker /(F)
s u s
13. Output Zeroing
We are interested in the system (F, G, H) given by (8.l) and (8.2)
over an arbitrary field k.
(13.1) PROBLEM. (Output Zeroing by State Feedback) Given (F, G, H),
find a k-homomorphism K: X -U ("state feedback") for which there exists
an integer i > 0 such that, for every x(0) G X, we have y(t) = 0,
t > i where y(t) is the output of the system (F + GK, 0, H) due to


paragraph after theorem 5] is a special case of the above theorem.
(10.7)COROLLARY. X*Y._(n) V..___(ker H) where V (ker H) is defined
xCQITn THELX 1 luclX
by (5.14).
PROOF. Let Jg = H, and then use Corollary (5.I5).
Two states
able at the time
x#1, x#2 £ X are unknown-input initial state indistinguish
r iff there are two input sequences u
r 1; 3 = 1, 2,
to the pairs (x.
t y
t 0, 1,
such that the r-modified output sequences corresponding
(u+ilb and (x#2> coincide.
*1 ltl't=oJ
(10.8)PROPOSITION. Two states x^, x*g e X are unknown-input initial
state indistinguishable at the time r iff
(10.9)**! x*2e3We(r>-
PROOF. Similar to the proof of Proposition (8.10). Therefore,
we omit the proof here.
xj^erJe(r) defines the set of the equivalence classes X/X^^Je(r).
By Proposition (10.8) the best that we can recover of the initial state
x(0) is the equivalence class x(0) + xjj-grje(r)* 'What remains to be
shown is how to determine x(0) + Xj^rje(r) fro (y(0), y(r l),
ve(r)).
Suppose we have the initial state x(0) £ X and input sequence
u(t) £ U, t = 0, ..., r 1. The corresponding r-raodified output
sequence y(t) £ Y, t = 0, ..., r 1, v (r) £ k^ is given by
6
fy() 1


t (I J H)
0 0
1
y(r 1)
r n e


v (r)
L- ev -
u(i il
where 1^ is the n X n identity matrix (or identity map k11 kn)


33
for some xJ ker J. Define £(t) := x(t) F^X-, t = 0, 1, ..., and
$(t) := y(t) HF x t = 0, 1, .... where x(t), y(t), t = 0, 1, ...
~ j
are given by (8.1) and (8.2) with the initial condition x(0) = xq + xq.
Then it is easy to see that £(t), £(t), t = 0, 1, ..., satisfy
(8.18) £(t + 1) = Fifc(t) + Gu(t), t = 0, 1, ...,
(8.19) $(t) = Hfc(t), t = 0, 1, ...
J
with the initial condition £(0) = x ker J. Since
o
(8.20) *(r) + x£rJW = (S(r) + + 3¡2rJ>
TO
it suffices to determine £(r) + ^erj(r) 'Notice that xq is known.
uo
The next lemma provides a way to obtain &(r) + j(r).
We now consider the system (F, G, H) with the assumption that the
initial state x(0) is in ker J.
Let x(o) f ker J. Then
(8.21) x(0) t
for some f kn where B, T is a basis matrix of ker J and
ker-J
n" := dim ker J. For each r = 1, 2, ..., let u(t) U, t = 0, ..., r 1
be an input sequence (till time r l). The output sequence y(t) f. Y,
t = 0, ..., r corresponding to the above initial state and input sequence
is given by
y(o)
"l
(8.22)
y(i)


i'r^BkerJ>
H, H)
u(0)



y(r)
u(r 1).
where H, H): kn +rm the matrix given in Section 3


18
then X^ N(j) C ^N(d + 1) for all integer j > i.
(ii) If X^ (i) D X^. N(i + 1) for some integer i > 0, then
X^) D X^N(j + l5 for all integer j > i.
(iii) If X^. N(i) = X^. ^(i + l) for some integer i > 0, then
X^N(j) = N(i + J) for all integer > 0.
PROOF, (i) The proof is by induction. Let X^ N(i) C X^. N(i + l).
Assume X^ N(£) N( + l) where Z is a nonnegative integer. Then
yN(i + 2) = + i) + im a) On
3^ ,(0 + In S)Hn
=vu + 1)-
Therefore, by induction we get 3^. ^(j) CX^ + l) for all integer
i > i.
(ii) The proof is similar to that of (i).
(V .
(iii) This is again proved by induction. We assume X^ ^(i) =
X^ N(i + 1) for some i > 0. If X^ N() = X^ ^( + l) for some integer
Z > 0, then we have
3^n( + 2) = + l) + im G) PlN
= i,1(5^ N(J) + im G)flH
= V + 1)
Therefore, ^(i) = ^(i + j) for all integer j > 0.
(5.8) RIMARK. It is possible that neither one of three conditions in
Lemma (5.7) may hold (e.g., let F, G, N, W be as in Remark (2.7), and
consider any positive integer i).
However, if either X^N(i) c^jN(i + 1) or D X^N(A + 1)


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27
The last statement is equivalent to xq X^tr(r) (by Definitions (5.2)
and (50)).
(6.3) REMARK. Vectors x X, u, £ t = 0, ..., r 1 satisfying
O X
(6.2) are viewed as an initial state and a sequence of inputs satisfying
the conditions of Definition (5*2).
(6.4) REMARK. Methods of computing the maximal F mod G invariant
subspace in N have received considerable attention. (See BASILE and
MARRO [1968a, Section 3, Corollary!], WONHAM [1974, Theorem (4.3)],
SILVERMAN [1976, Lemma 6 in Section III, C] and MOORE and LAUB>.:[1978] Also
recall Corollary (5.15).) Corollary (5.15) and Theorem (6.1) yield
<6-5) W> = 3. W V V-
This gives a new nonrecursive method of computing VnnY(N).
7. Concluding Remarks
The remarks similar to those in Section 4 of Chapter II apply to the
results of this chapter.
As is the case with X^e^C^(r) the assumption WCI is not
contr
essential in the discussion of X^. ^ (r) from the technical point of
view; the statements in this chapter can be modified straightforwardly
to fit the case where W C N is not assumed. (See also Section 4 of the
previous chapter.)
iff
From Lemma (5.7) we know that X^^r(i) D X^^r(i + l), i = 0, 1,
(7.1) W D F_1(W + im G) PlE.
The significance of the condition (7.1), however, is not yet clear.



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the following as the immediate consequence of Lemma (5-7 i), Lemma (5l6 i)
and Theorem (5-6).
(5-17) PROPOSITION. X^JJtr(i) C X^JJtr(i + 1) for all i = 1, 2, ...
iff i¥ C W + im G.
(5.18)COROLLARY. If FW C W + im G, then FX^JJtr(r) C X^JJtr(r) + im G
for each r = 1, 2, ... .
PROOF. Suppose FW C W + im G. Then by Proposition (5.17) we
have X^. (x 1; C X^. (1), 1 = 1, 2, ... By the definitions of
the i and (i l)-step controllable subspaces to W in N, for every
f* OTVt" T*
v. G XL, (i) there must exist u. U such that Fv. + Gu 6 X,, (i l) C
^tr(i). Therefore, FX^Ri) C x£¡fr(i) + im G, i = 1, 2, .. . n
As for the nonincreasing sequence we just note a special case of Lemma
(5-7), i.e., ^tr(i) D X^JJtr(i + 1), i = 1, 2, ... iff
W C F_1(W + im G) f>N.
rcontr
X^. (r) has the following properties in relation to state
feedback.
By
rcontr
the definition of X^ ^ (r) it is clear that the subspace is
state feedback invariant, i.e., the r-step controllable subspace to W
in N of the system (F + GK, G) is equal to X^^r(r) of (F, G).
Under some conditions, the input sequence u(t) U, t = 0, ..., r 1
given in Definition (5*2) can be implemented by a suitably chosen state
feedback u(t) = K^x(t), t = 0, ..., r 1.
(5.19) THEOREM. (i) If X^^tr(i) C X^J¡tr(i + 1), i = 1, 2, ...,
i.e., if FW C W + im G, then for each r =1, 2, ... there is a feedback
K : X -> V such that
r
(5.20) (F + GKr)JX^JJtr(r) CN, j = 0, ..., r 1,
(5.21)(F + GKr)JX^JJtr(r) CW, j = r, r + 1,


CHAPTER V. STABILIZABILITY, OUTPUT ZEROING AND DISTURBANCE DECOUPLING
In the previous chapter we have applied X^.e^Cil(r) and X^^tr(r)
to the problems of unknown input observability.
In this chapter we shall demonstrate the use of X^^ r(r) in the
discussions of stabilizability, output zeroing and disturbance decoupling.
12. Stabilizability
We consider the system (F, G) given by (8.1). In this section we
assume that k := R or C where R is the field of real numbers and
C is the field of complex numbers.
The pair (F, G) or a map F (over k = R or C) is said to be
asymptotically stable if lim )| F^x !| =0 (t = 0, 1, 2, ...) for all
> 00 ^
x# G X where || |¡: X > R is a norm defined on X and R is the set
of nonnegative real numbers. It is known that the pair (F, G) is
asymptotically stable iff ¡A¡ <1, i = 1, 2, ..., n where A,
i = 1, 2, ..., n are the eigenvalues of F. (See, for example, FREEMAN
[1965, Chapter 7, Section 9l, MARRO [1975, Chapter 5, Section 4].)
If there exists a feedback K: X U such that £imj| (F + GK)^x*|| = 0
(t =0, 1, 2, ...) for all x# X, then the pair (F, G) is said
to be asymptotically stabilizable. Let $((A) = ^US(A)^S(A) be the
minimal polynomial of X with respect to F where the roots of ^US(A)
have the magnitudes greater than or equal to unity and the roots of
(A) have the magnitude less than unity. Then we have
(12.1) THEOREM. The following statements are equivalent:
(i) The pair (F, G) is asymptotically stabilizable.
(ii) ker jUS(F) C Xtr(n).
x-
PROOF, (iii) =* (i). Suppose (iii) holds. Since F ker tfS(F) C
ker $S(F), by applying Theorem (5.19) we see that there is a feedback
K: X -> U satisfying
45


35
Furthermore,
to
(8.25) g^j) ker f^B^, H, H) = ^(r).
Therefore. (^erJ: r.(ker J) X/X^rJ(r).
PROOF. Let yr G r^(ker j). There is an element G nr(BkerJ)
such that
(8.26) f^B^j, H, H) (i.) = yr.
Define rr(ker J) X/gr(BkerJ) ker f^B^,, H, H) by
(8.27) <6rJ(5r) (\).
5^er^ is well-defined. Indeed, let he another element in ^r(Bkerj)
satisfying (8.26).
(8.28) pg^j) (;)
= pgr(\erj) (\ + y) (for some 7 ker H, H))
= p(sr(\erJ) (ur) + gr(BkerJ^7^
= PSr(\erj) + ^r^erJ^ ^
PgpC^grj) (ur)*
To show the linearity of ^erJ, let yrl, yr2 G Tr(ker J).
Then there exist rl, rg G -Qr(BkerJ) such that = ^^kerJ^ H
3 =1, 2. Therefore, yrl + yy2 = fr(BkerJ> H, H) (url + u^) and
we have
^kerJ( + y ) = pg (B _) ( + c)
rr wrl r2 r kerJ rl r2'


13
ker = N. Denote by a basis matrix of W. Then we have
0 ... 0
^ ^G. 0 0
MjFCS
LV\ M?r'1(5
0
Define
gr(Bfrr) := [Y\; Fr-1G ... FG G], r >
0.
(3.1) THEOREM. xJ¡**Ch(r) = gr(y ker f^, Mj, Mj), r > 0.
PROOF. Let x* G X.
** gr(V ker fr(^, Kj, Hj)
iff there exist | G and u ..... u .. G km such that
o r-1
(3-2)
u
u i
L r-U
£ ker W k- *9'
(3-3) X = 8r(B)
u
u
L r-1
The conditions (3*2) and (3*3) are equivalent to


REFERENCES
B. 0. ANDERSON
[1976] "A note on transmission zeros of a transfer function matrix",
l Kb!hi Trans. Automatic Control, AC-21: 589-591*
G. BASILE
[1969] ,fSome remarks on the pseudoinverse of a nonsquare matrix",
Rendiconti Serie XII Tomo VT, Atti della Accademia delle
Scienze dell'1stituto di Bologna.
G. BAS HE and G. MARRO
[I968&] "Controlled and conditioned invariant subspaces in linear
system theory", Report No. AM-68-7, University of California,
Berkeley.
[1968b] "On the observability of linear time-invariant systems with
unknown inputs", Report No. AM-68-8, University of California,
Berkeley.
[1969] "L'invarianza rispetto ai disturb! studiata nello spazio
degli stati", Rendiconti Della LXX Riunione Annuale AEI.
[1973] "A new characterization of some structural properties of linear
systems: unknown input observability, invertibility and
functional controllability", Int. J. Control, 17: 931"9^3
S. P. BHATTACHARYYA, J. B. PEARSON and W. M. WONHAM
[1972] "On zeroing the output of a linear system", Information and
Control, 20: 135-lte.
R. W. BROCKET! and M. D. MESAROVIC
[1965] "The reproducibility of multi-variable systems", J. Math.
Analysis and Applications, 11: 5^8-563.
E. EMRE and M. L. J. HAUTU3
[1978] "A polynomial characterization of (A, B)-invariant and
reachability subspaees", Eindhoven Univ. of Technology,
Memorandum C0S0R 78-19
H. FREEMAN
[1965] Discrete-time Systems; An Introduction to the Theory, John
Wiley and Sons, New york.


TABLE OP CONTENTS
ACKNOWLEDGEMENTS
ABSTRACT v
CHAPTER
I. INTRODUCTION 1
II. THE r-STEP REACHABLE SUBSPACE FROM W IN N 7
1. F mod G Invariant Subspaces 7
2. Reachability From W in N 8
3. Nonrecursive Characterization of the r-Step
Reachable Subspace from W in N 12
4. Concluding Remarks 14
HI. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N l6
5. Controllability to W in N 16
6. Nonrecursive Characterization of the r-Step
Controllable Subspace to W in N 25
7. Concluding Remarks 27
IV. UNKNOWN INPUT OBSERVABILITY 28
8. Unknown Input Final State Observability 28
9. Unknown Input Final State Observability
Part 2 (Special. Cases) 37
10. Unknown Input Initial State Observability 39
11. Concluding Remarks 43
V.STABILTZABILITY, OUTPUT ZEROING AND DISTURBANCE
DECOUPLING 45
12. Stabilizability 45
13. Output Zeroing 46
14. Disturbance Decoupling 48
15. Concluding Remarks 51
VI. CONCLUSION 52
APPENDIX 53
REFERENCES 54
BIOGRAPHICAL SKETCH 57
iv


26
y = 2, 3, ,,,, and
fi Write
*1 o'
o
where we have r blocks of n X m zero matrices.
(6.1) THEOREM. x£jjtr(r) = Pr ker fr(ln, Mj), r > 0.
PROOF. The condition
(6.2)
u
u n
L r-1
ker fr(ln, M, Mj),
x X, u, £ km, t = 0, ..., r 1, is equivalent to
O tl
fro -
^ = 0,
+ + " + V2 =
VS + + ' + y Vl
which in turn is equivalent to the conditions
x N,
o *
Fx + Gu N,
o o
Fr_1x + Fr2Gu + ... + Gu N,
o o r-2
Frx + Fr-1Gu + ... + Gu ,
o o r-1

e w.


(5.25)(F + GK )e GW, s = 1, ..., Z ,
\ r OS o
(5.26)(F + GKr)e.s. G X^tr(j 1), g=l, r; s.. = 1, ...,
It is easy to check that the following relations hold:
(5.27)(F + GK )1e. G N
ry jsj
for 0, ..., r'; s = } ...} .i i = 0, ..., r 1 and
J J
(5.28)(F + GK )Xe. GW
r jsj
for j =0, ..., r
^contr/. ^ ,rcontr,.
(11) Suppose 3^ N (x) N (1 + 1), 1 = 1, 2, ... For
each r = 1, 2, ... we choose a basis of X in the following way.
Let ierl, e^ ) be a basis of X^JJtr(r) where : = dim X^jJtr(r),
Extend the basis to get the new basis {e .... e : e _ ,
rl rlr r-1,1,...
er-l,ir_i^ of X^^r(r l). Repeat the procedure, and we obtain the
basis ierl, e^; er-:L,:L,..., ei1 ''ol
^ := dim r(o) dim Xjjjtr(j + l), J = 0, r 1. Note that if
i = 0 we do not extend the basis at this step and go to the next step.
D
e we complete the basis
Or
ev,_n i e, ; ertl, ..., e^ ) of W where
By adding linearly independent vectors e..,
of X as {e .,..., e. .; ,, ..., e ; en, ..., ) where
rl 1^1 01 Contr £ 1 qr
q := n dim W. Since e. G X (j) j = 1, r and s. =
r jsj w,N j
1, ..., ., there must be an input u. such that
0 Jsj
(5.29) Fe.s. Gu.s. C 1)
where 3 = 1, ..., r and s. = 1, ..., £.. It is straightforward to
J d
check
(5.31) (F + GKr)rers GW, s^ = 1,
J


12
fxw,n(v) C (fxw,n(v) + im G) n (N + im Q)
C (FX^. n(v) + im G) pin + im G
= + im G.
3. Nonrecursive Characterization of the, r-Step Reachable Subspace from
W in N.
X*6&cll
The sequence defined by (2.3) and (2.4) determines X-. (r) recursively.
We now give a nonrecursive characterization of
Let B be a matrix having n rows. Let C and A be matrices with
n columns. Define
f (B, A, C) := AB
f (B A C) 4CB "
V > '~|afb AGJ
f2(B, A, C)
CB 0 0
CFB CG 0
af2b AFG AQ_
where 0 is the zero matrix of appropriate size. Extending the above
procedure, we define
for r = 1,
2, ...
CB
0
0 .

. 0
CFB
CG
0 .
. 0
fr(B, A, C)
: =
cf2b

CFG


CG



CFr-1B
CFr"2G
CG 0
AFrB
AFr_1G
. .
AFG AG
In this section
we
only use
a special case
where
B I n X n
rr
identity matrix. (Another case
where
B 4 I
T n
will be used in Chapter
IH, Section 6.)
As before, W C N C X. Let be a matrix with n columns satisfying


I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.
.^Wxv\
Rudolf E. Kalman, Chairman
Graduate Research Professor
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philisophy.
C&aJIqa 2/.
Charles V. Shaffer,
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.
Thomas E. Bullock
Professor of Electrical Engineering


The subspace V (N) is called the maximal F mod G invariant subspace
max
in N.
It has been known (SILVERMAN [1976, Section III, A]) that V (n)
max
is equal to the set of states in N for which there are input sequences
such that the corresponding trajectories remain in N for p units of time.
It should be noted that this statement is a special case of Theorem (5.6).
It should also be noted that the algorithm for computing V (n) is a
max
special case of (5.4), (5-5) where W = N. (See BASILE and MARRO [1968a,
Section 3, Corollary 1] and WONHAM [1974, Theorem (4.3)1*) Summarizing,
we have
(5.I5) COROLLARY OF THEOREM (5.6). Let p be as in Lemma (5.9)* and let
V (n) be as used above. Then
max
''max D
The sequence X^^r(i), i = 1, 2, ... has a conditional monotone
property. (See Theorem (5*6) and Lemma (5*7)*) The natural question to ask
is then when it is monotonically nondecreasing or nonincreasing.
( 5 .6) LEMMA, (i) W C X^ N(l) iff FW C W + im G.
(ii) If WDX^N(l), then X^ n(p) =vmax(w) where P is as
in Lemma (2.9) and V (w) is as in (3.14).
" 1 max ,
PROOF. (i) If W C X^ (1), then ¥ C F_1(W + im G)On. Hence
W C F ^(W + im G). So we have FW C W + im G.
Conversely, if FW C W + im G, then W C F ^(W + im G). Since
W C N by assumption, we get W C F ^(W + im G) ^ K, i.e., W CX^ .^(l).
(ii) If W D w(l), by Lemma (5.7 ii) we have ^(i) C W
for all i = 0, 1, ... Then obviously X^ N(i) = w(i), i = 0, 1, ... .
Therefore, by Corollary (5-15) ^ N(p) = V^W).
For the nondecreasing sequence X^^r(i), i = 1, 2, > we have


22
each r = 1, 2,
(ii) If X^tr(i) D Xjjjtr(i + 1), i = 1, 2,
then for
there is a feedback K : X
r
U such that
(5.22) (F + GKr)jxJJ}tr(r) CH, 3=0,
(5 .23) (F + GKr)rx£"tr(r) C W.
PROOF, (i) By Proposition (5.17) xJJ¡tr(i.) C X^j}tr(i + l),
i = 1, 2, ..., iff W is an F mod G invariant.
Assume x£jjtr(i) C x£JJtr(i + l), i = 1, 2, ... For each
r = 1, 2, ... we'choose a basis of X as follows. Let {e^, ...
o^o
)
'oV
be a basis of W. Extend this basis to get the basis {e ,, ...
contr ol
ell> *> eL?i^ N ^ePea^ the extension until we get no
more vectors to add to, say r' times where r' < r, and we obtain the
J of
basis (eQ1, > e^S e^, .,
X^ r(r') = X^0^ r(r). We further extend this basis arbitrarily to get
6 ^ ^ 5 6 /<.
r -l,r'_i* r V
/ 0
r
e ). Here l
and
3
\ : =
the basis of X, {e ..., ^ -
dim Xw N (j) dim X^ ^ (3-1), 3 = 2, ..., r
n dii x£*tr(r). Since e^ 3 = 0, ..., r' and 8^=1
s, =1, ..., there exists an input u^ such that
(5.24)
Fe. + Gu.
sj )sj
e
i)
where 3=1, ..., r' and s^ = 1, ..., y (See Definitions (5.2)
and (5*3)). Define Kp: X ->U so that it satisfies the following
conditions:
(a) u. = K e. for 3 =1, ..., r' and s. = 1,
Js3 r jsj ' 0 ' A'
(b) (F + GKr)eQs 6 W for s = 1, ..., Z (such a always
exists since FW C W + im G. See Lemma (1.4) and Proposition (5.17).)
(c) 3=1, ..., are arbitrary. Then


25
exist xtPM, ut e U, t 2, ..., i satisfying x^ = Fxt + Gut,
t = 2, i. Then x £ X^^tr(i). Hence, noting that x f. W, t = 1,
ria.i'h Z
i, we get w X^^i).
Conversely, if (5.36) holds, then there exists a subspace t Cl
satisfying X~e^Ch(l) = W, which implies (4.2).
Consequently,
(5.3?) PROiosmoN. X^h(i) c + 1), 1 = 1,2,..., Iff
<5-38> ff = C£(W),W where V is as in Lemma (2.8).
PROOF. Immediate by using the above lemma and Lemma (2.8).
6. Nonrecursive Characterization of the r-step Controllable Subspace to
W in N.
The sequence defined by (5.4) and (5.5) determines the r-step control
lable subspace to W in N, X^^tr(r), recursively. We now give a
nonrecursive characterization of X^^r(r).
As before, W C N C X and
N.c<
*N
r = 1, 2,
Recall the definition of fr(B, A, C) in Section 3 of Chapter II.
Denote by 1^ the n X n identity matrix, and let 0 be the zero matrix
of suitable dimension. Let be a matrix with n columns such that
ker W
, and define M similarly. Then
*4
0
0 . . 0
>4
0 . . 0

W V -

*
*4. :
0 0
r1
Vr'2s

\ 0
. . MJFG M^G


1. i
Vn(s + 2) = (fxh,n(s + 1} + im G)nH
= (F\ = Vn(s +
Therefore, by induction
Vn(J) = Vn< + 1)
for all j = i, i + 1, ...
(2.7) REMARK. It is possible that none of_ three conditions in
(2.6) may hold. For instance, let F :=
'0
W := span
0 1
1 0
G : =
N := X
and let i be any positive integer.
Lemma
and
However, if either X^N(i) C X^N(i + l) or => ^jjC + l)
holds for some i, 3 = 0, 1, ..., the sequence X^. ^(.0)> & = 0, 1, ...
will stop increasing or decreasing in a finite number of steps since X
is finite dimensional. If it is the case, let v be the least integer
i > 0 such that X^N(i) = X^N(i + 1).
(2.8) LEMMA. (i) If W C Xw N(l), then v < dim N dim W < n.
(ii) _If WDX^^(l), then V < dim W < n.
PROOF. Immediate from Lemma (2.6) by using the finite dimension
ality of X.
Let V be as in the paragraph prior to Lemma (2.8). X^ ^(v) is not
an F mod G invariant in general. However, we have
(2.9) LEMMA. If FN C N + im G, then FXW N(v) C N(v) + im G holds.
PROOF. Since X^. ^(v) (FX^. ^(v) + im G)flN, it follows that
FX^ N( v) C FN C N + im G. Therefore, we have


CHAPTER IV. UNKNOWN INPUT OBSERVABILITY
The notions of the r-step reachable subspace from W in N and
of the r-step controllable subspace to W in N as developed in Chapters
II and III are now applied to the study of unknown input observability of a
system (F, G, H).
Sections 8 and 9 discuss the unknown input final state observability.
Section 10 treats the unknown input initial state observability.
Hereafter Y denotes the output (value) space and is defined by
Y := k^. Also p := dim Y is used throughout Chapters IV and V.
8. Unknown Input Final State Observability
We consider a finite-dimensional, constant, discrete time linear
dynamical system (F, G, H) given by
(8.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, ....
(8.2) y(t) = Hx(t), t = 0, 1, ...
where x(t) F X : = kn, u(t) F U := k, y(t) Y = k^ (t = 0, 1, ...)
and k is a field. (Recall the definition in the second paragraph of
Chapter I.)
It is assumed that we have some degree of a priori information about
the initial state given by
(8.3) v(x(0)) = Jx(0)
where J: X -k^ is a k-homomorphism and q is a positive integer. In
particular, if J is an isomorphism where q = n, then the initial state
is a priori known. If J is zero, then the initial state is a priori
unknown.
Depending on an initial state x(0) = xq F X and an input sequence
u(t) = Uj_ F u, t 0, 1, ..., the system produces the corresponding
state and output sequences (or trajectories) x(t) F X, t = 0, 1, ...
28


(sequence) satisfies x(t) G ker H for t 2, .., r. In particular
there is no input sequence of the form Kx(t) satisfying the above
requirement where K: X ->U. Thus, we have contradiction.
The limit case of this problem ( r -<>) is the well-known Disturbance
Decoupling Problem. It is known that the problem has a solution iff
im D C V (ker H) where V (ker H) is the maximal F mod G invariant
max max
subspace in ker H. (Refer to WONHAM and MORSE [1970, Theorem (3.1) and
WONHAM [1974, Theorem (4.2)].)
Let us now add a constraint to the above problem so that the state of
the resulting system (F + GK, 0, D, H) will eventually reach zero when
the disturbance becomes zero.
(l4.7)PROBLEM. (Disturbance Decoupling Problem with Reset). Given
(F, G, H, D) with x(0) = 0, find (if possible) a state feedback
K: X -4 U such that the resulting system (F + GK, 0, D, H) (with x(0) = 0)
has zero output sequence and that (F + GK)1 = 0 for some i > 0.
(14.8) THEOREM. Problem (l4.7) has a solution iff the following condi
tions hold:
(14.9) 4.Cl*(n),
(14.10)XGOJtr(n) = X.
Oj A
PROOF. [Sufficiency] Assume (l4.9) is true. By Theorem (5.19 i)
there exists K: X -> U such that
(14.11) (F + GK)jXG£^H(n) C ker H, j = 0, ..., n 1,
(14.12) (F + GK)JXG¡^H(n) =0, j = n, n + 1, . .
(l4.9), (l4.ll) and (14.12) then imply
(14.13)(F + GK)J im D C ker H, j = 0, 1, ...,


research was also supported in part by the Foundation for International
Information Processing Education, 2-6-1 Marunouchi, Chiyoda-ku, Tokyo
100, JAPAN.
iii


in N.
^.reach/ N
V> (r)
and study the properties of the
1, 2,
with respect to W and
We shall characterize
sequence of subspaces X^e^|C^(i),
N. What appears to be intriguing is the fact that the properties of
X^. jy (i), i = 1, 2, ... change drastically depending on W.
In Chapter III we shall introduce the natural counterpart of X^ ^ (r)
which we shall call r-step controllable subspace to W in N,
X^^r(r). This subspace is defined to be the set of states in N from
which some state in W can be reached in r steps via trajectories
contained in N. It should be noted that, except for the special cases of
X^^r(r) (MARRO [1975> Chapter 4]) and X^^r(r) (which we shall
discuss shortly), the more general subspace X^^r(r) has not been
WjN contr
studied as such. It is this generality that makes X^ ^ (r) an inter
esting object.
The significance of introducing X^ (r) will become clear when we
consider the unknown input observability at the final time r in Chapter
TV (Sections 8 and 9)* The problem is stated as follows: Given (F, G, H),
a priori information about the initial state v(o) = Jx(0) and the output
sequence y(l), y(2), ..., y(r), find the state x(r) at the final time
r. We shall see that the best we can do to identify x(r) is to determine
(r) which in fact can be determined. Thus,
the coset x(r) + xfea^1.
v herJ,kerH
for an in-depth understanding of the unknown input observability at the
final time r, it is essential to study the properties of X^ j kerirr^
which depend on J, H and r. The results in Sections 8 and 9 appear
to be new.
A special case of X^oatr(r), namely, X^atr(r) has been discussed
in the literature in relation to the unknown input initial state observa
bility, and it is known that, if we are given (F, G, H) and the output
sequence y(0), y(l), ..., y(r), only the coset x(0) + Xaatr(r) can
be recovered based on the above data. (See, for example, RASILE and MARRO
[1973, Theorem 1 and Corollary 1] and SILVERMAN [1976, Definition 2 in
Section III].') We shall include this problem with more generality in
Chapter IV, Section 10 to implement our knowledge of unknown input
observability. The method used here to recover x(0) (or its coset)
is less complex to understand than the ones which have been used in the


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS
REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N
By
FUMIO HAMANO
August, 1979
Chairman: Dr. R. E. Kalman
Major Department: Electrical Engineering
In the "geometric approach" to the study of linear systems two important
notions have been successfully used; namely, the maximal reachability sub
space Xf*ck(N) contained in a given subspace N and the maximal F mod G
ulQJC .
invariant subspace contained in N. However, the definition of X (N)
max
is not as natural in discrete time systems as in continuous time systems.
It loses an important meaning when it is applied to discrete time systems,
i.e., Xch(N) is not the set of states reachable from 0 via traject-
max
ories in N. In this work similar notions which are suitable to treat
discrete time systems are developed. In general the study is concerned with
"reachability" and "controllability" internal to the subspace N in dis
crete time systems. More specifically, the notions of the r-step reachable
subspace from W in N and the r-step controllable subspace to W in N
are introduced for given subspaces W and N (satisfying WCN), These
are respectively defined to be the set of states reachable from W via
trajectories in N in r steps and the set of states that can reach W in
r steps via trajectories in N. Algebraic characterizations and sequential
v


ACKNOWLEDGEMENTS
I wish to express my sincere appreciation to all those who contributed
in various degrees toward the fulfillment of this work.
I am particularly grateful to Professor R. E. KALMAN, the chairman of
my supervisory committee, for his constant encouragement in seeking a
concrete understanding of system theory and other areas in science. His
guidance has been of a great help in establishing an organized view in
this dissertation. Without the financial support which he arranged for me
during the past four years and without the stimulating environment of the
CENTER FOR MATHEMATICAL SYSTEM THEORY, this work may not have existed.
I am thankful to Professor C. V. SHAFFER, co-chairman of my supervisory
committee, who has not only given me valuable comments concerning my
dissertation, but also made favorable arrangements for me during my personal
emergency.
The specific motivation for the research reported here was provided
by the stimulating discussions with Professor G. BASILE relating to his
earlier works concerning ''geometric" views on various control and system
problems. I appreciate his friendship and his deep interest in this work.
A dissertation is only a part of a doctorate. In the educational
process during the past four years the influence of Professors E. EMRE,
E. D. SONTAG, Y. YAMAMOTO, G. SONNEVEND, T. E. BULLOCK, M. E. WARREN,
C. A. BURNAP, M. HEYMANN, M. L. J. HAUTUS, V. KUCERA and others was
essential in the preparation for a doctoral degree.
Of course, no research would be made were it not for the long-term love
and encouragement of a few close people. My parents and my wife, Shoko,
have been constant sources of encouragement. To them I dedicate this work.
Needless to say, I am grateful to Ms. Karen Todd for her understanding
and patience not only as a typist but as a friend.
This research was supported in part by US Army Grant DAAG 29-77-G0225
and US Air Force Grant AFOSR 76-3034 through the Center for Mathematical
System Theory, University of Florida, Gainesville, FL 326II, USA. The
ii


9
(2.2) DEFINITION. The r-stex reachable subspace X^e^ch(r) from W
in N is the set of states r-step reachable from W in N.
We show that the sequence X^e^ch(i), i = 1, 2, ... can he
recursively computed. Let
(2.3) Xw>N(0)=W
(2.4) ^ (i) = (FX^jjd 1) + im G)Hn, i = 1, 2, ... .
Then we have
(2.5) THEOREM. x£**Ch(r) = X^r).
PROOF. The proof is done by induction.
(i) ^ N(l) = (IW + im G)ON
= {x, £ N: x. = Fw + Gu for some w W and u £ km)
1 l o o
= Ch(1)-
(ii) Assume that ^(d) is equal to the j-step reachable
subspace from W in N, d > 0.
YjjJ + 1) = (FX^J) + im G)Hn
= {xJ+l £ N: xi+1 = FXj + Gu. for some Xj ^^(d)'
and u. £ km).
d
By induction assumption any X. £ X^^(d) has at least one pair of
o > HI
sequences N, t = 0, ..., j 1, and u^. £ k, t.- 0, ..., d 1
such that Xq £ W, ]_ + ^ t = 1, ..., j. Therefore
y(j +1)
(x3+l 6 N:
ii 0,
There exist sequences x^ N,
d and u km, t = 0, ..., d
t
= FjCj. + Gu^., t = 0, d
satisfying x


6
literature so far.
The subspace X^^r(r) is closely related to "state feedback"
K: X -U. Those results in Chapter III which are related to state feedback
will be applied in Chapter V in which we consider stabilizability, output
zeroing and disturbance decoupling. Problems of these kinds have been
treated in the literature in different fashions. (See WONHAM [1974,
Theorem (2.3) for stabilizability, Theorem (4.4) for output stablization
and Theorems (4.2) and (5.8) for disturbance decoupling].) The contents
of Chapter V will also serve to exemplify the significance of X^^r(r)
in the cases where W j N. The theorem (12.1 (i), (iii)) concerning
etabilility is of interest in the sense that it gives a new interpretation
of stabilizability.


CHAPTER III. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N.
In the previous chapter we have studied the r-step reachable subspace
from W in N. In this chapter we introduce another new notion which we
call ,rthe r-step controllable subspace to W in N,r. Its applications
will be found in Chapters IV and V.
5. Controllability to W in N.
Consider a finite dimensional, constant discrete time, linear dynamical
pair (F, G) represented by
(5.1) x(t + l) = Fx(t) + Gu(t), t = 0, 1, ...
where x(t) 6 X := kn, u(t) £ U := km, t = 0, 1, ..., and k is an
arbitrary field. We denote the system (5.1) by the pair (F, G).
As before W and N denote subspaces of X satisfying W C N C X,
and r is a positive integer.
(5-2) DEFINITION. A state x^ £ N jLs r-step controllable to W in N
iff there exists an input sequence u(t) £ u, t = 0, ..., r 1 such
that x(0) = x^, x(r) £ W and x(t) £ N, t = 0, r.
The set of states x^'s satisfying the above conditions form a
(linear) subspace.
(5.3) DEFINITION. The r-step controllable subspace X^^r(r) to. W
in N is the set of states r-step controllable to W _in N.
We show that X^j^r(r) can be computed recursively. Let
(5*If) Vn(0)=w
(5.-5) 3^ N(i.) = F_1(^^N(i 1) + im G) On, i = 1, 2, ...,
where F "Si := (x £ X: Fx £ X ] for a subspace X of X.
s s s
16


I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.
Chtos d- ftuAna#
Charles A. Burnap
Assistant Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the
College of Engineering and to the Graduate Council, and was accepted
as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
August, 1979
Dean, College of Engineering
Dean Graduate School


BIOGRAPHICAL SKETCH
Fundo HAMANO was "born on August 28, 1949, in Wakayama, JAPAN, to
Zenichi HAMANO and Mitsuko HAMANO. He received his Bachelor of
Engineering from Tokyo Institute of Technology in 1973 and his Master of
Science and Engineering from the same institute in 1975.
57


6 + ^Guo = 0,
£+ Vr"lGuo+ '' + ^GVl = >
x* = F1^ i + Fr 1Guq + ... + Gur_1,
which in turn is equivalent to the conditions
% e G N
F% I + Gu0 G N
= F ^ + F Gu^ + + Gu^ ^ £ N.
d. imW
The last set of relations hold for some | £ k and u u^, ...,
ur_1 km iff x# G X^G^Ch(r) (by Definition (2.2)).
(3-4) REMARK. A set of vectors i^|, u^_ £ k, t = 0, ..., r 1
satisfying (3-2) and (3-3) are seen to be an initial state and a sequence
of inputs satisfying the conditions of Definition (2.1) for the final
state x#.
4. Concluding Remarks.
In our discussion of this chapter we have assumed ¥ C N. This
assumption has been made since we are interested in structural properties
of the system (F, G) inside the subspace N. Technically speaking,
however, the above assumption is not essential. With slight modification
all the statements still hold without assuming W C N. Theorem (3.1) can
be easily modified to yield the corresponding result:
(4.1) x^ch(r) = gr(y 4er ?r(^, Mj, Mj),
r > 0


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17
(5.6) THEOREM. Xjjjtr(r) = J^N(r), r > 0.
PROOF. The proof is by induction. For r = 1,
3^ N(l) = F (W + in G) O N
= {x^ X: Fx^ = w Gu^
for some w G W and
m
\ k ,
and xx G N)
(x^ G N: Fx^ + Gu^ G W for some G km)
each/
-cr-
Now
,contr
let ^>N(d) = N (j), d > 0. Then we have
Vn(3 + 1) ix3+l N: *Vi xj GVl for some *3 e Vh(j)'
,m >
Vie k
By assumption of ^(3) = X^^rf3) this is equal to
(*J+1 E B> F*3+1 + GUJ+1 *3, FXj + GU3 = FXj_ + GUX a
for some
_ vcontr/v .m t .
w G W, x^ G Xy jj \£), Ug G k £ ~ 2.f > J
and
VI e ^
-*£?'<* + l5-
We now study properties of the sequence W, X^^r(l), X^^r(2),
by examining properties of the sequence (5.4), (5.5).
The sequence (5.4), (5*5) (equivalently Wtr(l), X^jJtr(2), ...)
has conditional monotone properties.
(5.7) LEMMA, (i) If X^ N(i) C Xw N(i + l) for some Integer i > 0,


time r iff
')'-J
(9.11) deg X(z) deg ]l(z) = 1.
PROOF- Since m = p = 1, it follows that rank G = 1 iff G / 0
and that rank HG = 1 iff HG ^ 0. Hence, (9*6) holds iff HG 4 0 (since
G 4 0 by assumption).
(9*12) COROLLARY. Assume that the initial state is known, i.e., ker J = 0.
Then there is a unique state trajectory x(t) G X, t = 0, 1, ..., r
corresponding to an initially modified output sequence (v(0), y(l), ...,
y(r)} iff (9*5) (i.e., (9*6)) is satisfied.
(9.I3) REMARK. Assume that the initial state is known and that rank G = m
(i.e., dim im G = in). It can be shown that the input sequence correspond
ing to an initially modified output sequence (v(0), y(0), ..., y(r)) is
unique iff (9*5) (or (9*6)) is true.
10. Unknown Input Initial State Observability.
Consider the system (F, G, H) given by (8.l) and (8.2). As before
r = 1, 2, ... .
We wish to find x(0) based on the measurement of the output sequence
y(0), ..., y(r). Suppose that we are allowed to make an extra measurement
at the final time r. (Suppose for example, that the "motion" of the system
stops at the end (at the time r) and that we can examine the system
more carefully to get an extra information at the time.) We will denote
this extra information by
(10.1) vg(r) = Jgx(r),
where ve(r) £ ks and Jg: X >kS is a k-homomorphism (for some integer
s > 0). Then
(10.2)
v (r)
J
ev
r
e
y(r)_
H_


(13.4) COROLLARY. Let x £ X. Then there exists a feedback K : X ->U
such that y(t) = 0, for all t = r, r + 1, ... where y(t) is
the output of the system (F + GK^, 0, H) for x(o) = x# iff
(13-5)
x* G
_ contr / %
1 \aX PROOF. Similar to the proof of Theorem (13.2).
(13.6) COROLLARY. The output zeroing problem by state feedback has a
solution iff
(13-7)
contr
\ (kerH),X
max'
(n)
= X.
PROOF. Immediate from Theorem (13.2).
14. Disturbance Decoupling
Consider a finite-dimensional, constant coefficient, discrete time,
linear dynamical system with disturbance given by
(14.1) x(t + l) = Fx(t) + Gu(t) + Dv(t), t = 0, 1, ...
(14.2) y(t) = Hx(t), t = 0, 1, ...
where -X;= kn, U km, Y := kP, V : = kS (v(t) V, t = 0, 1, .. )
and k is a field. We may denote the system (l4.l), (l4.2) as
(F, G, H, D). As before r denotes a positive integer.
(14.3) PROBLEM. (r-step Disturbance Decoupling Problem) Given (F, G, H, D)
with x(0) = 0, find (if possible) a feedback K : X U such that
g
y(t) = 0, t = 0, ..., r for any v(t) G k t = 0, 1, ..., where
y(t) is the output of the system (F + GK^ 0, H, D) with x(0) = 0.


i.e.,
(9.6)rank HG = rank G.
PROOF. By Theorem (8.32) and Theorem (2.5) (F, G, H) is unknown
input observable at the final time r iff
(9.7) X (r) = 0.
w o,kerHv '
Sinoe Xo^kerH(i) C ^kerH^ + l), i = > ^ follows from
Lemma (2.6) that (9*7) holds iff
(9.8) 0 = X^kerH(l) = im GHker H.
It remains to show that (9-8) holds iff (9-5) holds.
Suppose im G O ker H = 0. Let m' := rank G = dim im G. Denote
by [g g0, g ,) a basis of im G. We claim that Hg.., Hg, ..., Hg ,
x m m" x m
are linearly independent. In fact, let ,Z,a.Hg. =0 for some a. £ k, j =
TT1 0JO J
2,,..., m, not all zero. Then H .Z,Qhg. = 0, which implies
Ill J X J J
g.. = 0 since im G O ker H = 0. Now since g^, ..., g^, are linearly
independent, we get a. = 0 for all j = 1, ..., m". Contradiction. Hence,
J
dim H im G = m'.
1,
Conversely, suppose im G Oker H 4 (0j. Let 0 / e^ £ im G ^ ker H.
Trivially e^ is a basis of Span e^ C im G O ker H. Extend e^ to form
a basis (e^, e^, ..., e^,) of im G. Then
(9-9) im HG = Span {He ..., He^,}
= Span {He, ..., He ,
2 m
Therefore, dim im HG < m'.
(9.10) COROLLARY. Let m = p = 1 and define polynomials Jl(z) and X(z)
by il(z)/x(z) = H(zl F) ~*"G (5/ 0). Suppose the initial state is known.
Then the system (F, G, H) is unknown-input observable at the final


ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS
REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N
By
FUMIO HAMANO
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979


10
and
reach
x W)
o
N
(J + 1).

Vie now study properties of the sequence W, X^e^ch(l), X^e^|ch(2), ...
hy looking at some properties of the sequence X^. ^(i), i = 0, 1, ... .
The sequence 7L. N(i), i = 0, 1, ... (equivalently, W, X^^ch(l),
N (2), ...) has conditional monotone properties.
(2.6) IEMMA. (i) If X^ N(i) C XW w(i + 1) for some integer i > 0,
then X^ N(j) C X^. N(j + l) for all integer j > i.
(ii) If X^ N(i) D X^j. ^(i + 1) for some integer i > 0, then
Xw^N(j) D Xtf N(d + 1) for all integer > i.
(iii) If Xw (i) = X^j N(i + 1) for some integer i > 0, then
Xw>N(i) = X^N(i + ) for all = 0, 1, ... .
PROOF, (i) Let
X^N(i) CX^N(i + 1) for some integer
Assume that X^^ N(s) C X^^ N(s + l) holds for some integer s > i.
i > 0.
Then
X^N(s + 2) = (fx^jH(b + 1) + im G) On
D (FXw>n(s) + im G)PlN
= + 1).
Therefore, hy induction
V(3)cxw,n<3 + 1)
for all integer > i.
(ii) Similar to the proof of (i).
(iii) Suppose X^ N(i) = X^ N(i + 1) for some integer i
If we assume that X^ ^(s) = X^ ^(s + l) for some integer s > i,
we have
> 0.
then


CHAPTER VI. CONCLUSION
We have introduced the r-etep reachable subspace X^e^C^(r) from
W in N and the r-step controllable subspace X^.^r(r) to W in N
of the finite-dimensional, constant, discrete-time, linear dynamical
system (F, G, H) over a field k. We have characterized these subspaces
and discovered several interesting properties pertaining to them.
The notions X^e^ch(r). and X^^r(r) are natural generalizations
of reachable and controllable subspaces when we are interested in structural
properties of state trajectories contained in the subspace N and when
initial and final states of the trajectories may not be zero.
The significance of X^e^|ch(r) and 3^^tr(r) in control problems
has been demonstrated in Chapters IV and V. Among the applications pre
sented in these chapters, the unknown-input observability at the final time
r is the most important in the sense that it has motivated the author to
study the subjects treated here.
As has been mentioned in the concluding remarks of each chapter,
there are several other interesting topics concerning X^e^C^(r) which
are still open to further research. The relation between the transfer
function M^(zl F) and the subspaces X^e^ch(r), X^^r(r) should
also be studied.
52


2
In this chapter we shall sometimes refer to a pair (F, G) with the
continuous time interpretation for comparision. This is defined as follows:
We choose the continuous time and set k to he either the set of real or
complex numbers. Then we replace (0.1) by
( 0.1)' x(t) = Fx(t) + Gu(t), t > 0,
in the definitions of the (discrete time) system (F, G, H) and the
(discrete time) pair (F, G). Since continuous time systems are not what ,
we are interested in, unless otherwise specified the system (F, G, H)
and the pair (F, G) will always be in discrete time.
Our main concerns are placed in the structural properties of the
trajectories governed by (O.l). Due to the algebraic nature of difference
equations, these properties can be studied in a purely algebraic way without
losing the intuition of the original dynamical nature of (O.l).
At this stage it seems appropriate to give a quick review of related
concepts which have been treated in the literature:
(i) The reachable subspace (of the pair (F, G)), is defined
to be the set of states which can be reached from the zero state (via some
trajectories) in a finite number of steps. It is known that X**60,011
is equal to
(0.3) im G + F im G + ... + Fn_1 im G.
The pair (F, G) is said to be reachable if and only if xreac^ x.
(ii) The controllable subspace (of the pair (F, G)), xcorrtr, is
the set of states from which the zero state can be reached (via some
trajectories) in a finite number of steps. It is characterized by
(0.4) XCOntr = (Fn)_1 (im G + F im G + ... + Fn_1 im G)
where (Fn) := {x X: Fn x Xg} for a subspace Xg of X. Note
that we have a60,0*1 c XCOn^r in general. The pair (F, G) is called


k
shall denote it by V (N).
J max' '
(v) Closely related to V (N) is the subspace called the maximal
reachability subspace contained in N which is denoted by X~ (N).
It is defined as follows: Let K: X -+U be a k-homomorphism such that
(F + GK)V (n) CV (N) and let L: U -U be another k-homomorphism
max max ,
satisfying im GL = im GOV (N). xXeacn(N), then, is defined to be
max max
the reachable subspace of the pair (F + GK, GL). It is evident that
(F + GK)Xreach(N) c XreaC^1(N) if one uses (0.3) and the Cayley-Hamilton
max max
theorem. Consequently, X^fty (T) is an F mod G invariant subspace
contained in N (or equivalently in V^y(lQ). Also, since =
V (V (N)) trivially, it follows that X^^fa) = xrach(v (U)).
max' max' 7' max ' max max'
The definition of ]Coh(N) does not depend on whether the pair
(F, G) represents a discrete time system or a continuous time system.
However, there is an important difference in interpretation of X^ach(lQ.
In the continuous time case can be interpreted as the set of
states that can be reached from the zero state in a finite time via
trajectories contained (at each time) in N. On the contrary this inter
pretation fails in the discrete time case. This point has not been
clarified in the literature, not to mention its importance. The above
observation then raises the following questions: What is the set of states
reach
reachable from the zero state via trajectories contained in N, XI ,
in the (discrete time) system (F, G, H)? How is it characterized? Are
there any interesting properties? A natural question which comes next
is: What is then the set of states controllable to the zero state via
trajectories in N? What are its properties?
In Chapters II and III we shall attempt to answer the above questions
in more general contexts.
We shall introduce the following new notion in Chapter II. Let r be
a positive integer, and let W and N be subspaces of X satisfying
W C N. The r-step reachable subspace from W in N, denoted by
resell
^ (r), is the set of states which can be reached from some states
in W in r steps via trajectories contained (at each instant of time)


30
t = 0, r 1, and an initial state x G ker J such that the
resulting x(t) G ker H, t = 1, . ., r and x(r) = x^.
The set of states unknown-input indistinguishable from 0 at the
final time r is a subspace of X.
The unknown-innut unobservable subsoace at the final
U0
(8.8) DEFINITION.
time r, denoted as X^rj(r), is the set of states that are unknown-
input indistinguishable from 0 at the final time r
*1
x0 G X are unknown-input indistin-
(8.9) DEFINITION. Two states x
guishable at the final time r iff there exist pairs ((u+1)^_^, x^n)
((ut2 ^t-o* ^o2/ '~'J~ / '(^0
j = 1, 2 and initial ataees x(0) = x
({u+0)J~^, x 0) of input sequences u(t) = u, t = 0, ..., r 1;
tl t=o ol
9
oj-
1, 2, such that the following
conditions hold:
(i)
x^ is the final state (at the time r) corresponding to
the Palr (fVw V5
where j =1? 2,
(ii) the initially modified output sequences corresponding to
the pairs coincide.
The significance of ^grj(r)
(8,10) PROPOSITION. Two states
is clarified by
xn, xv G X are unknown-input
indistinguishable at the final time r iff
(8.11) x^ x^g G ^erj(r)
PROOF] [Necessity.] Let {v., y .,
J -*-J
' yr.i!
be the
r-1
initially modified output sequence corresponding to the pair ((u ,
CJ oO
x .) where
03
j = 1, 2. Suppose v1 = v2 and ytl = y
t2J
1,
r.
Let the new initial state of (F, G, H) be x(0)
x T x O*
ol o2
Since
vl
v.
we have


15
where ?r(B, A, C) is the matrix obtained by eliminating the first block row
in fr(B, A, C) (hence, fr(B, A, C) consists of r block rows).
From Lemma (2.6) it follows that X^. ^(i) C! X^ ^(i + l), i = 0, 1, ...,
iff
(4.2) wcx^N(i)
and that X^N(i) d ^ jjC1 + 1)* 1 = > 1> > iff
(4.3) WDX^n(1).
The natural question is then what is the significance of the conditions
(4.2) and (4.3). The significance of (4.2) will be partially answered
in Chapter III, Section 5. However, the implication of (4.3) is yet to
be clarified.
There are other questions to ask: when does the sequence X^ ^ (i),
1, 2, ... "oscillate", i.e., X^^ch(j) 4 X^JJCh(j + l) fr every
j > 0?; when does the sequence "oscillate" at the outset and stop its
oscillation some time later?; etc. These are open problems.
*


29
and y(t) £ Y, t = 0, 1, respectively. As a result of (8.5)
we also have v(x ) = Jx_ which we call v .
o o o
Considering the initial state, we have
(8.10
Let
*(x(0))
7n
J
y(x(o))
H
x(0),
V(x(0))
T .
J
y(x(o))
> 0
H_
v(O) ::
For each r = 1, 2, ... the sequence {v(0), y(l), ..., y(r)) is called
an initially modified output sequence (till time r). When we discuss two
initially modified output sequences we prefer to denote them by
(v.
y ,} and {v
oi> JU) *riJ 1 02> J,12 * Jr2
(8.5)PROBLEM. Given an initially modified output sequence (v(o), y(l),
..., y(r)) of (F, G, H), find the corresponding final state x(r) £ X
for r = 1, 2, ... .
Note that the definition of v(0) and J and (8.4) clearly implies
(8.6) ker J C ker H.
(See Section 11 for the remark on the condition (8.6).)
We say that the system (F, G, H) is unknown-input observable at the
final time r iff Problem (8.5) has a unique solution.
It is not always the case that this problem has a unique solution. In
fact, it will be seen that rank HG = rank G is necessary for the final
state x(r) to be uniquely determinable. (See Theorem (9-4).) So the
important question is "to what extent can we recover x(r)?"
(8.7) DEFINITION. x# £ X is unknown input indistinguishable from 0
at the final time r iff there exist an input sequence u(t) £ U,


43
(10.14) fr(ln, Je, H)(u^) = fr(ln, Je, H)(r + ) (for some
u C ker JqI H))
f (I J H)(u ).
r' n e' r
The linearity and the uniqueness can he shown similarly to Lemma (8.24).
By Proposition (10.8) (or the remark after the proposition) and
Theorem (10.12), we get the necessary and sufficient condition for (F, G, H)
to he unknown input initial state observable at the time r.
(10.15) THEOREM. The system (F, G, H) is unknown input initial state
observable at the time r iff
<*> D
(10.17) REMARK. If r > n and if Jg = H, then (10.16) is equivalent to
(10.18) V (ker H) = 0,
' max' *
where vmax(ker H) the raaxiraal F mod G invariant subspace in ker H.
11. Concluding Remarks
If one does not go through the arguments at the beginnings of Sections
8 and 10, i.e., if one does not want such conditions as (8.6) and (10.4),
one can easily accomodate the discussion in this chapter to the new
situations. The modified versions of X^e^ch(r) and X^^r(r) (without
the assumption of WCl) should be used accordingly. (See Section
4 of Chapter II and Section 7 of Chapter III.)
Techniques similar to those used in this chapter can be applied to
study unknown input observability problems in non-constant dynamical
systems (F(t), G(t), H(t)). (The problem statements (8.5) and (IO.5)
should be modified in the obvious ways.) The unknown input unobservable
uo
subspace X (s, s + r) at the final time based on the observation over the


32
is a linear space over k where addition and scalar multi
plication are defined in the obvious ways.
X? T(r) is characterized as follows:
kerJv '
(8.12) THEOREM.
(8.) *SrJM -
where xJerifkerH^ is defined by (2.1) and-(2.2).
PROOF. Immediate by using Definitions (2.1) and (2.2) and
Definitions (8.7) and (8.8).
Proposition (8.10) tells us that to recover x(r) we cannot do
IK)
better than identifying the equivalence class x(r) + ^erj(r)* Then the
question is, "can we really find the equivalence class containing x(r)?"
(8.14)
THEOREM.
JJO
If x(r) is the final state at the time r, then
x(r) + Xjcerj(r) can be Uniquely determined based on the knowledge of the
corresponding initially modified output sequence (v(0), y(l), ..., y(r)}.
PROOF. Given the initially modified output sequence (v(0), y(l),
..., y(r)} of (F, G, H). Choose an xq X satisfying
(8.15) v(0) = Jxq,
e.g., on a fixed coordinate basis, xq may be taken to be
(8.16) Xo := J+v(0)
where J* is the pseudo-inverse of J. (See Appendix for the definition
of the pseudo-inverse.) Then the initial state x(0) is written as
(8.17) x(0) = xq + x^


56
- PSr(Bkerj) (uri) + pgr^BkerJ^ ^Ur2^
= ^rJ(yrl)+ Similarly, for each a k, y r (ker J) we have ^er<^(oy ) =
cer J\ r r r r
r (yr)
To show the uniqueness of the map suppose $^erJ: r^ker J) -
x/gr(Bkerj) ker fr(B^erj^ H) is another map for which the diagram
commutes. Then for each u £ (ker j)
r r
$cerJf (B H h) (u ) = pg (B. _) (u )
rr, r' kerJ v r r kerJ r
,kerJ
= erJfr(W H Hence, noting that *r(Bkerj> H, H) is onto, we obtain = $^er<^".
(8.25) follows from Theorems (3.1) and (8.12).
(8.29) REMARK. ^erJ: ^(ker J) ->X/X¡^rJ(r) does not depend on the
choice of B^erj.
(8.30) REMARK. For a fixed coordinate basis ^erJ; Tr(ker J) ->
X/X^rj(r) &y defined by
(8.31) iff" := P^OW,) H> K>
j*
where **r(Bkerj> H> H) is the pseudo-inverse of fr(Bkerj> H). (See
Appendix.)
As an immediate consequence of Proposition (8.10) and Theorems (8.12)
and (8.lU) we have
(8.32)THEOREM. A system (F, G, H) is unknown-input observable at the
final time r iff
(8-35) rH=0-


the initial state x(0).
(13.2) THEOREM. Let x^ G X. Then there exists a state feedback
K: X -U for which there is an integer i > 0 such that y(t) = 0,
t > i for x(0) = x* where y(t) is the output of the system (F + GK,
0, H) due to the initial state x^ iff
(13.3)
rcontr
(n)
where V (ker H) is defined by (5.14).
max
PROOF. ["if"] Since V (ker H) is an F mod G invariant sub
max
space in N, by Theorem (5.19) there is a feedback K^: X -4 U such that
(F + x(n) C Vmax(ker H). This implies that, if x(0) = x#,
then
x(l) = (F + GK)x#
x(2) = (F + GK)x(l) = (F + GK)2x^
x(n) = (F + GK)x(n l) = ... = (F + GK)nx G V (ker H).
^ max
If K is so chosen as to satisfy (F + GK) V (ker H) C V (ker H)
max
(see the proof of Theorem (5.19))* we have x(t) = (F + GK) nx(n) G V (ker H)
max
for all t > n. Thus, y(t) = Hx(t) = 0 for all t > n.
["only if"] Let x(t) 6 X, t = 0, 1, ..., be the trajectory
of the system (F + GK, 0, H) with the initial state x(0) = x#. Then
by assumption we have for some integer i > 0
x(i + j) G ker H, j = 0, ..., n.
Therefore, by Corollary (5.I5) we know x(i) G V (ker H), which implies
max
that x(0) = x^ G v^(vmax(ker x) "by Definition (5.2) and (5.3).
Appealing to Proposition (5.17) we conclude


3
(in the continuous time case
control1able if and only if xCon^r = X.
and Xcontr can he defined similarly. The characterization of
yContr
, however, is different from the one in the discrete time case,
corrtr
namely, we have for the continuous time case X = im G + F im G +
+ F^1 im G = Xreach.)
unob
(iii) The unobservahle subspace, xww, is the set of initial states
which can not be distinguished from the zero state by any input/output
experiment. The subspace X^0^ is given by
/ vunob .
(0.5) X = ker
H
HF
The system (F, G, H) (or simply the pair (H, F)) is said to be
observable if and only if X^0*5 = 0. The condition X^0*5 = 0 is necessary
and sufficient for the initial state x(0) to be uniquely determined based
on a sufficiently long interval of input/output measurement. For more
details about reachability, controllability and observability the reader
should refer to KAIMAN [1968], FURUTA [1973, Chapter 2, Section 7 through
10], MARRO [1975, Chapter 6] and WONHAM [1974, Chapters 1, 2 and 3].
(iv) A subspace V is called an F mod G invariant subspace if and
only if
(0.6) FVCV + imG.
It is well-known that (0.6) holds if and only if there exists a
k-homomorphism ("state feedback") K: X -U satisfying
(0.7) (F + gk)v C V.
The set of F mod G invariant subspaces is closed under subspace addition.
Therefore, for a given subspace N of X there is an F mod G invariant
subspace which contains any other F mod G invariant subspace. This is
called the maximal F mod G invariant subspace contained in R, and we


24
(502) (F + GK )Je <2 N, J = 0,
I Sj*
hold
(533) REMARK. The choice of K^: X ->U (r > 0) is by no means unique.
This (limited) freedom in choosing K^ (r >0) is rather useful in
application. (See Chapter V.)
r 1;
= 1,
(5.34)REMARK. The inputs
or r)
(but not necessarily uniquely). See Remark (6.3).
j = 0,
u. s (s. = 1.
0sj I
based on which K^ is defined, can be determined explicitly
Recall the definition of X^e^C^(r).
Given x GW,
o
there may not
, r
be any input sequence producing x(0) = x x(t) £ I, t = 1,
reach ^
and x(r) £ ^ ^ (r), r > 0. The condition that x^ must satisfy to
have such an input sequence is as follows. Let xq £ W. Then there is an
input sequence u(t) £ U, t = 0, ..., r 1 such that the state
xq, x(t) G N, t = 1, ..
iff
trajectory satisfies x(0)
r and x(r) G
1, N
, 0 ,rreach/ \
Yhere X := (r).
We now turn to the question of what is the implication of (4.2) which
has been posed at the end of Chapter II.
co nt Y* /
(5.35)LEMMA. Let W(i) (i), i > 0. Then for each integer
i > 0, (4.2) holds iff
(5.36)W
vreach ,. N
PROOF. Suppose (b.2) holds. Trivially, W !) ^(i). To show
the converse inclusion let w G W. Then by (4.2) there exist x^ G W,
ux G U such that w = Fx^ + Gu^ G N. Since w G W we.obtain w = Fx^ +
Gu^ G W. Repeating the above argument i times, we see that there also


and "feedback properties of the above notions are first given, and then
applications to the unknown-input observability and other problems in
control systems are given to show the significance of the results.
vi


19
happens for some i, j = 0, 1, ..., the sequence (i),
£=0,1, ... will stop increasing or decreasing in a finite number
of steps since X is finite dimensional. If it is the case, let |i be
the least integer i > 0 such that X^ i.) = X^. ^(i + l).
(5.9) LEMMA, (i) Lf WCX^jj(l), then p < dim N dim W < n.
(ii) _If ¥DX^^(l), then M < dim W < n.
PROOF. Immediate from Lemma (5-7) and the above comment.
(5*10) LEMMA. Let p be as in the paragraph preceeding Lemma (7.9)*
Then
(5-H) c y N(n) + im G.
(5.12) yH(p)cN.
So X^fr(p) is an F mod G invariant subspace in N.
PROOF. By the definition of p, X^. ^(p) = ^(p + l).
Therefore
+
= F_1(XW (M) + im G)ON.
Hence, Ry(n) + im G, yN(p) Cl. n
(5.13) REMARK. Lemma (5.IO) guarantees the existence of a feedback
K: X -U such that (F + GK) X^JJtr(p) CX^JJtr(p). (See Lemma (1.4).)
The set of F mod G invariant subspaces is closed under subspace
addition. (See BASILE and MARRO [1968a, Section 2, Assertion 1] and
WONHAM [1974, Lemma (5-3)]*) Therefore, the following is well defined:
(5.14) V (N) : = max[V C N: FV C V + im G].
max


56
M. K. SAIN and J. L. MASSEY
[1969] "invertibility of linear time-invariant dynamical systems",
TWER Trans. Auto. Control, AC-14: 141-149.
L. M. SILVERMAN
[1976] "Discrete Reccati equations: alternative algorithms, asymptotic
properties and system theoretic interpretations", in Control
and Dynamical Systems: Advances in Theory and Applications,
Vol. 12 (edited by C. T. Leondes), Academic Press, pages
513-586.
L. M. SILVERMAN and H. J. PAYNE
[1971] "Input-output structure of linear systems with application to
the decoupling problem", SIAM J. Control, 9: 199-233.
E. D. SONTAG
[1979] "On the observability of polynomial systems, I: finite-time
problems", SIAM J. Control and Optimization, 17: 139-151.
L. WEISS and R. E. KAIMAN
[1965] "Contributions to linear system theory", Int. J. Engineering
Science, J: 141-171.
W. M. WONHAM
[1974] Linear Multivariable Control: A.Geometric Approach,. Springer,
New York. .. -
W. M. WONHAM and A. S. MORSE
[1970] "Decoupling and pole assignment in linear multivariable systems:
A geometric approach", AM J. Control,"8f1-18.


and fr(*, ,
Chapter III.
) is as in Section 3 of Chapter II
Define ft (I ) := kn+rni and p (ker
r n r
and Section 6 of
J ) := im f (I J H).
e r n e
(lO.ll) THEOREM. There is a unique homomorphism
ithat the following diagram commutes:
>We(r)
ft (I )
r n
f (I J H)
rv n e
T (ker J )
rx e
P
r
X
cr
I
*r
-ku
erJe
(r)
where I is the n x n identity matrix (or the identity map kn ->kn),
ft IU
f (, , ) and P are as in Sections 3 and 5, cr: X X/X. (r)
r r kerJe
is the canonical projection.
PROOF. By Theorems (6.1) and (10.6) we have
i10-11 Cje(r) Pr ker fr(l Je H>'
Let y G T (ker J ). There is an u G ft (I ) such that
r rk e r rs n
(10.12) yr = fr(ln, Je, H)(ur).
Define ^(ker Jg) ->X/X^rJe(r) = X/Pp ker fr(ln, J,, H) by
(10.13) *r(yr) = aPrr.
We claim that is well-defined. In fact, let G ^r^n^ be anot':ier
vector satisfying (10.12). Then
' G ker f (I J H).
r r r n e
Therefore,


55
K. FURUTA
f1975] Senkei Shistem Seigyo Riron (Theory on Linear Systems and
Control), Shohkodoh, Tokyo.
F. HAMANO and K. FTJRUTA
[1975] "Localization of disturbance and output decomposition in
decentralized linear multi-variable systems", Int. J. Control,
22: 551-562*
B. HARTLEY and T. 0. HAWKES
[1970] Rings, Modules and Linear Algebra, Chapman and Hall, London.
D. G. LUENBERGER
[1966] "Observers for multivariable systems", IEEE Trans. Aut.
Control, AC-11: 190-197*
R. E. KALMAN
[1963] "Mathematical description of linear dynamical systems", SIAM
J. Control, 1: 152-192.
[1968] "Lectures on controllability and observability", Proc. C.I.M.E.
Summer School, Edlzioni, Cremonese, Roma, 1-149*
R. E. KALMAN, P. L. FALB and M. A. ARBIB
[1969] Topics in Mathematical System Theory, McGraw-Hill, New York.
R. LASCHI and G. MARRO
[1969) "Alcune considerazioni sull'osservabilita del sistemi dinamici
con ingressi inaccesibility", Rendiconti Della LXX Riunione
Annuale AEI.
G. MARRO
[1975] Fondamenti di Theoria dei System!, Patron, Bologna.
B. C. MOORE and A. LAtJB
[1978] "Computation of supremal (A, B)-invariant and controllability
subspaces", IEEE Trans. Auto. Control, AC-23: 783-792.


(l4.4) THEOREM. The r-step disturbance decoupling problem has a solu
tion iff
(111.5) m D C \erH>kerH(r !)
PROOF. [Sufficiency] By assumption x(0) = 0. Let K X -i
be as in Theorem (5.19 ii) where we assume W = N = ker H. Replacing
u(t) in (l4.l) by K ^(t), we get
(14.6) x(t + 1) = (F + GKr_1)x(t) + Dv(t).
g
Consider the system (14.6), (l4.2). Let v C k j = 0, 1, ... For
ti
each j = 0, ..., r 1 suppose the disturbance v(t), t = 0, 1, ...,
be such that v(j) = v.(^ 0 possibly) and that v(t) =0 if t / j. Then
we have
*t = Oj y J }
x(t) = (F + GK )t"1Dv., t = j + 1, j + 2, ... .
Therefore, by Theorem (5*19 ii) with W and N both replaced by ker H,
y_j(t) := Hx(t) =0, t = 0, ..., r + j.
Now we superpose all the disturbance used above, i.e., we use v(t) such
that v(t) = for t = 0, .... r 1. Then the output y(t),
t = 0, 1, ..., satisfies
r-1
y(t) = X0 y^(t) = o, t = o, ..., r.
[Necessity] Suppose that the problem has a solution but that
im D ^kgyHrker{j(r i)* Since x(0) = 0, we have x(l) Dv(0).
There must be v kS such that Dv X^n^.r. (r l). Then by
o o K6rii) Kern
Definitions (2.2), (2.3) there does not exist an input sequence
u(t) f U, t = 1, 2, ..., such that the corresponding state trajectory


37
9. Unknown Input Final State ObservabilityPart 2 (Special Cases).
Let r = 1, 2, ... Recall Theorem (8.12). The unknown input un-
Tin
observable subspace X^rJ(r) at the final time r depends on the time
r in general. However, in some cases the time dependence disappears in
a finite time. (See Lemma (2.6) and the paragraph prior to Lemma (2.8).)
And if
(9.1) ker J C (F ker J + im G) Oker H
holds, more can be said, namely, the sequence X^rJ(i), i 1, 2, ...,
is monotonically nondecreasing (Lemma (2.6 i)) as well as it stops
increasing in at most n steps, i.e.,
(9'2) O1') X2rJ(n + 1) =
(See Lemma (2.8 i).) Also, if
(9.3) ker J 3 (F ker J + im G) Oker H
uo
is true, then the sequence xkerj(i) i = 1, 2, ..., is monotonically
nonincreasing (Lemma (2.6 ii)) and (9-2) holds. (See Lemma (2.8 ii).)
Note that if ker J = 0, i.e., the initial state is known a priori,
then (9.1) holds, and that if ker J = ker H, i.e., if the initial state
is not known a priori, then (9.3) holds. Therefore, for these cases the
above statements are true.
Let us consider the first situation where ker J = 0 (i.e., the
initial state is a priori known). The condition under which Problem (8.5)
has a unique solution becomes particularly..simple.
(9-4) THEOREM. If ker J 0, then the system (F, G, H) is unknown
input observable at the final time r iff
(9*5) dim im HG = dim im G,


40
Denote
v (r)~
e
, J :
J
e
y(r)
e
H
How we shall call the sequence (y(0), y(r l), v (r)) r-modified
e
output sequence. Consequently, we have
(10.3) PROBLEM. Given an r-modified output sequence (y(0), ..., y(r l),
vg(r)), find the corresponding initial state x(0).
By (10.2), clearly,
(10.4) ker J C ker H.
e
(See the remark on this condition in Section 11.)
We say that the system (F, G, H) is unknown-input initial state
observable at the time r iff Problem (IO.3) has a unique solution.
A state x# 6 ker H is said to be unknown-input initial state indis
tinguishable from ,0 at the time r iff there is an input sequence u(t) U,
t = 0, 1, ..., r 1 such that the corresponding state sequence satisfies
x(0). x#, x(t) ker H, t = 0, ..., r 1 and x(r) ker Je. The
set of states that are unknown-input initial state indistinguishable from
0 at the time r is a subspace of X.
(10.5)DEFINITION. The unknown-input initial state unobservable subsuace
at the time r, denoted by ^erje(r) j is the set of states that are
unknown jnput initial state indistinguishable' from 0 at the time r.
i From Definitions (10.5) and (5.3) we obtain
0.0.6) IHEOEM. CeAerHM-
Recall the results given in Chapter HI. We know the characterizations
and the properties of ^^)!terH(r).
The following result given by SILVERMAN [1976, Section III, A, the


Jx(0) = Jx
i.e., x(0) £ ker J.
01 JXo2 T1 V2 = 0
Apply the input u(t) = u u
t2
0,
(F, G, H). Then it is straightforward to show
1 to the system
x(r) x*p x*2
y(-t) = ytl yt2 = o, t = o, ..., r.
Therefore, x^ x^2 £ X^rJ(r).
[Sufficiency.] Suppose (8.11) holds. Write x^ := x1 x0.
Then clearly x^ £ j(r). Therefore, there exist an input sequence
(u,), and an initial state x £ ker J of (F, G, H) such that the
t t=o o 7
corresponding trajectory satisfies x(t) £ ker H, t = 0, r and
x(r) = x#.
Define a new input sequence and a new initial state of (F, G, H)
by u(t) := ut2 + u t = 0, ..., r 1 and x(0) := Xq2 + xq where
I* 1
xq2 and ^ut2^t-o are state and input sequence, respectively,
giving rise to the final state x*2. It is easy to show that
v(0) = Jx(0) = Jxq2,
y(t) = yt2> t = 0, ..., r,
x(r) = x^2 + x* = x#1,
where fy^^t-o 0U^PU^ sequence produced by {u^2}^_q and xq2.
It follows from Proposition (8.10) that (8.1l) defines equivalence
classes, each of which consists of the states unknown-input indistinguishable
at the final time r and the collections of these classes are denoted by
x/x
UO
kerJ
(r).


(in which is replaced by BkgrJ and by H). Note that y(0) = 0
since x(0) G ker J C ker H. The corresponding final state x(r) G X
is obtained by
(8.25) x(r) = gr(\erJ)
u(0)
u(r 1)
where
r) is defined in Section 3 (Chapter II).
,U0
M^erJ'
Our problem is to determine x(r) 4 X^rj(r) from the knowledge of
y(l), ..., y(r). (v(0) = Jx(0) = 0 since x(0) G ker J. Also y(0) 0.)
Define
i i
u
u
r-1
_ n 4rm t n m A ,, n >
G k : | G k u. fe k j = 0, ...,,r lj
0
where n' := dim ker J and B, is a basis matrix of ker J. Write
kerJ
T (ker J) := im f (B, t, H, H).
r' rv kerJ* 1
(8.24) LEMMA. [There is
ker VW
a unique homomorphism ^er^: r^ker J) ->
H, H) such that the following diagram commutes:
(B, t)
rv kerJ
kerJ*
H, H)
X
P
ry(ker J)
! /erj
ker VW
H, H)
where ?T('> '> *) and g^() are as in Section 3 and p: X -
X/gr(Bkerj) ker fr(BkerJ> H, H) is the canonical projection.


which means that the output y(t), t = 0, 1, ... of the system (F + GK,
0, D, H) with x(o) =0 is zero for all t = 0, 1, ... .
It remains to show that there is an integer i > 0 satisfying
(F + GK)1 = 0. If (4.10) holds, then X = X^Jtr(n) C X^Jtr(n) for
any subspace W of X. So X = XTS^r(n) where W := X^.n^rTT(n).
, W,X o.kerH'
Noticing that 0 and X w(n) are F mod G invariant subspaces and
O j K6!Tri
repeatedly using Theorem (5*19 i)> we see that there is a state feedback
K: X ~>U satisfying (l4.ll), (14.12) and
CL.,.11,) (f + gk)"xcx£>).
Hence,
(14.15) (F + GK)2nX = 0.
[Necessity] Let K: X -> U be a solution of the problem. Then
clearly (F + GK)n = 0. Hence XC^r(n) = X. Now define
O ) .
*1
V := im D + (F + GK) im D + ... + (F + GK) im D.
Then clearly im D C V, (F + GK)V C V and V C ker H. Since (F + GK)n
we have (F + GK)^ = 0. Therefore, V C xcon"^r
. O KG^Tii
rcontr
= 0,
Thus im D C V C
x;;k¡;H(n)- D
(l4.l6) REMARK. Recall (0.4). We have X^Jtr(n) = xCOntr.
15 Concluding Remarks
There is an important difference between Xre^C^(n) and Xreac'1(N).
o,N v max v
By choosing K: X ->U the spectrum of (F + GK)|xreac (n) can be assigned
T13.X !TG3C]l
arbitrarily, while to discuss the spectrum of (F + GK)¡X^ w (n) may
not make sense since (F + GK) Xq (n) C XQ (n) may not hold (unless
) y X*G9/0l^l
N is an F mod G invariant subspace). However, though X. (r) loses
contr w,N
the arbitrary pole assignability, X^ (r) can treat stabilizability
by state feedback (Theorem (l2.l)).
The usefulness of Theorem (12.1) is yet to be clarified.


Fx* + Gu V.
(iii) There exists a k-homomorphism K: X -+U ("state feedback")
such that
(1.5) (f + gk)vcv.
PROOF. The proof for the equivalence between (i) and (iii) can
be found in BASILE and MARRO [1968a, Theorem 3], WONHAM and MORSE [1970.
Lemma (3.2)] and WONHAM [1974, Lemma (4.2)].
For convenience we shall give the proof here. Suppose that (1.5)
holds. Let x* V. Then (F + GK)x# = Fx* + GKx* V. So Fx* =
v + GKx* for some v V, which implies (1.3).
Conversely, assume that (1.3) holds. Let {v., ..., v ) be
x Si
a basis of V where q := dim V. Then (1.3) implies that for each
i 1, ..., q there exist w^ V and u^ £ U satisfying Fv^ =
w^ + Gu^. Now define a k-homomorphism K: X -U by Kv^ = -u^,
i = 1, ..., q. Then we have (F + GK)v^ = w^ £ V, i = 1, ..., q, which
implies (1.5).
The equivalence of (i) and (ii) is easily proved and omitted
here.
2. Reachability from W in N
We consider a finite dimensional, constant, discrete time, linear
dynamical system (F, G) given by (l.l).
From here on, W and N will be subspaces of X satisfying
W C N C X, and r will be a positive integer.
(2.1) DEFINITION. A state x# N is r-step reachable from W in N
iff there exist another state xq^ W and an input sequence u(t) U,
t = 0, ..., r 1 such that x(0) = xq^, x(t) N, t = 0, ..., r
and x(r) = x#.
The set of states x^'s satisfying the above requirement form a
(linear) subspace.