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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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Hamano, Fumio, 1949-
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Integers ( jstor )
Linear systems ( jstor )
System observability ( jstor )
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Dissertations, Academic -- Electrical Engineering -- UF
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Thesis--University of Florida.
Bibliography:
Bibliography: leaves 54-56.
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Typescript.
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Vita.
Statement of Responsibility:
by Fumio Hamano.

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Full Text

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

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 52611, 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.

ACKNOWLEDGEMENTS . . . .... . 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
DECOUPLIIN . . . . ... 45

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

VI. CONCLUSION . . . ... . .. 52

APPENDIX . . . . . . . .

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 sub-
space reach (N) contained in a given subspace N and the maximal F mod G
maxreach
invariant subspace contained in N. However, the definition of reach (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., mreac (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 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 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 53 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) U := ki, y(t) Y := k 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 comparison.
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)' x(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:

(i) The reachable subspace (of the pair (F, G)), Xreach, 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 + F im G + ... + Fn1l im G.
reach
The pair (F, G) is said to be reachable if and only if x = X.
(ii) The controllable subspace (of the pair (F, G)), contr, 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 = (F-1 (im G + F im G + ... + F-1 im G)

where (Fn)"-X := x E X: Fn x E X ) for a subspace X of X. Note
that we have Xreach C ontr in general. The pair (F, G) is called

controllable if and only if Xcontr = X. (In the continuous time case
Xreach and Xcontr can be defined similarly. The characterization of
Xcontr, however, is different from the one in the discrete time case,
contr
namely, we have for the continuous time case X = im G + F im G +
n-1 each
... + F1 im G = rech.)

(iii) The unobservable subspace, Xunob, is the set of initial states
which can not be distinguished from the zero state by any input/output
unob
experiment. The subspace X is given by

H

(0.5) Xunb = ker

The system (F, G, H) (or simply the pair (H, F)) is said to be
observable if and only if Xunob = 0. The condition Xunb = 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
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) FV CV + im G.

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 CV.

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 (N).
max
(v) Closely related to V (N) is the subspace called the maximal
maxeach
reachability subspace contained in N which is denoted by Xmch(N).
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 U be another k-homomorphism
max max Feach
satisfying im GL = im G( fmV (N). reach(N), then, is defined to be
max max
the reachable subspace of the pair (F + GK, GL). It is evident that
reach :reach
(F + GK)Xeac (N) Ceach(N) if one uses (0.3) and the Cayley-Hamilton
max max
theorem. Consequently, Xreach (N) is an F mod G invariant subspace
max
contained in N (or equivalently in V (N)). Also, since V (N)
maxreah max
V (V ax(N)) trivially, it follows that Xa( = reach(V ()).
max max max max max
The definition of xreach(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 reach (N).
each max
In the continuous time case Xeach(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 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
reachable from the zero state via trajectories contained in N, Xeach
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
reach
X aN (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 X h N (r) and study the properties of the
reach '
sequence of subspaces X-e (i), i = 1, 2, ... with respect to W and
N. What appears to be intriguing is the fact that the properties of
reach
XN (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,
Pcontr
Xcor (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 Xotr (r) (which we shall
dis o '. contr
discuss shortly), the more general subspace XWN (r) has not been
contr
studied as such. It is this generality that makes XN (r) an inter-
esting object.
reach
The significance of introducing XW, (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) + Xreach (r) which in fact can be determined. Thus,
XkerJ,kerH
for an in-depth understanding of the unknown input observability at the
reach (r)
final time r, it is essential to study the properties of XerJkerH)
\erJ,kerH
which depend on J, H and r. The results in Sections 8 and 9 appear
to be new.
contr contr
A special case of XW,N (r), namely, XNN (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
contr
sequence y(O), y(l), ..., y(r), only the coset x(O) + XN,N (r) can
be recovered based on the above data. (See, for example, BASILE and MARRO
[1975, 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(0) (or its coset)
is less complex to understand than the ones which have been used in the

literature so far.

The subspace Wontr 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 stabilization
and Theorems (4.2) and (5.8) for disturbance decoupling].) The contents
of Chapter V will also serve to exemplify the significance of Xo tr(r)
in the cases where W N. The theorem (12.1 (i), (iii)) concerning
stabilility is of interest in the sense that it gives a new interpretation
of stabilizability.

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", denoted by Xeac (r), and study the properties of the sequence of
subspaces each(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 + 1) = Fx(t) + Gu(t), t = 0, 1, ... ,

where x(t) X := kn, u(t) U := km, t = 0, ..., 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 CV + 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, E V there exists an input u E U such that

Fx. + Gu V.

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

(1.5) (F + GK)V CV.

PROOF. The proof for the equivalence between (i) and (iii) can
be found in BASILE and MARRO [1968a, Theorem 31, 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 E V, which implies (1.3).

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

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

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 CX, and r will be a positive integer.

(2.1) DEFINITION. A state x, E N is r-step reachable from W in N
iff there exist another state xo. W and an input sequence u(t) 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 X (r) from W
in N is the set of states r-step reachable from W in N.
We show that the sequence ch(i), i = 1, 2, ... can be
recursively computed. Let

(2.5) XW,N(0) = W

(2.4) XWN(i) = (FX,N(i 1) + im G) nN, i = 1, 2, ...

Then we have

(2.5) THEOREM. Xeachr) = XN(r).

PROOF. The proof is done by induction.

(i) X,N(1) = (FW + im G) N
= (xE N: x = Fw + Gu for some w W and u E km)
1 1 o o
Each,
= XN (1).
(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) N

= (X1 E N: x+1 = Fx. + Guj for some xj E XN(j)
and uj E k ].

By induction assumption any xj E X 'N(j) has at least one pair of
m
sequences xt E N, t = ..., j 1, and ut. kn," t.= 0, ..., j 1
such that xo W, xt = Fxt- + Gut, t = 1, ..., j. Therefore

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

and x E W)
o
=reach ).
reach,- _reach,_
We now study properties of the sequence W, XWN l), X ch (2) ...
by looking at some properties of the sequence XW,N(i), i = 0, ....
The sequence X,N(i), i = 0, 1, ... (equivalently, W, rXWN ),
each2, ...) has conditional monotone properties.

(2.6) IEMMA. (i) If W,N(i) CXWN(i + 1) for some intege i > 0,
then W,N(j) C X,(j + 1) for all integer j > i.

(ii) If X,N(i) D ,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
WN(i) = XW,N(i + ) for all j = 0, 1,....
PROOF. (i) Let X,N(i) C X,N(i + 1) for some integer i > 0.
Assume that XWN(s) C X,N(s + 1) holds for some integer s > i. Then

',N(s + 2) = (FXW,N(s + 1) + im G)nN

D (FXW,N(s) + im G)nN

= ,N(s + 1).

Therefore, by induction

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

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

XW,N(S + 2) = (FXWN(s + 1) + im G) ~N

= (FXWN(s) + im G) hN

= XW,N(s + 1).

Therefore, by induction

X,N) = XWN(j + 1)

for all j = i, i + 1, .... O

(2.7) REMARK. It is possible that none of three conditions in Lemma
(2.6) may hold. For instance, let F := 0], G := N := X and
W:= span i and let i be any positive integer.
However, if either XW,N(i) C XW,N(i + 1) or XW,N(j)D X,(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 XWN(1), then v < dim N dim W < n.

(ii) If W D XWN(1), then v < dim W < n.

PROOF. Immediate from Lemma (2.6) by using the finite dimension-
ality of X. O

Let v be as in the paragraph prior to Lemma (2.8). XW,N(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 XW,N(v) + im G holds.

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

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

C (FXW,N(v) + im G) nN + im G

= XN(v) + im G.

5. Nonrecursive Characterization of thg r-Step Reachable Subspace from
W in N.

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

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

fo(B, A, C) := AB

fl(B, A, C):= FB

CB 0 0

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

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

Extending the above

f (B, A, C) :=
r

0
CG
CFG

CFr-2G
AFr-1G

. 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 M be a matrix with n columns satisfying

CB
CFB
CF2B

CFr--B
AFrB

. 0
. O
0
o

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

N0

G 0
MFBw M4G.

SF2B MW FG

MF\r Vr-lG

gr(Bw) := [FrB;

Fr-1G ... FG G],

r > 0.

(3.1) THEOREM. ieach(r) = gr(B) ker frl(, M, MN),

PROOF. Let x, E X.

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

r > 0.

iff there exist E kdiM

(3.2) uo ker fr w,

(3.3) x = gr(BW) o

-r

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

M, M;),

The conditions (3.2) and (33.) are equivalent to

0 ... 0

0 ... 0

0

... M (

Define

ker M* = N.

00

MFBW + MNGuo = O,

MFrB + MN'r-lGuo + + MGUr1 = 0,

x = Fr + Fr-Gu + ... + Gur-l'

which in turn is equivalent to the conditions

BW N

FBW ( + Guo C N

o r-1
x =FrB + Fr Guo + ... + Gur-1 C N.

dimW
The last set of relations hold for some E kdW and uo, l, ...,
km reach
ur1 c km iff x aN (r) (by Definition (2.2)).

(3.4) REMARK. A set of vectors B, ut C km, t = 0, ..., r 1
satisfying (3.2) and (33.5) 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:

(.1) reach) er M) r
(4.1) rWN (r) = gr(W) ker fr(Bw, M, M(B, r > 0

where f (B, A, C) is the matrix obtained by eliminating the first block row
in f (B, A, C) (hence, f (B, A, C) consists of r block rows).
From Lemma (2.6) it follows that XN(i) C X,N(i + ), i = 0, 1, ...,
iff

(4.2) W C X,N(l)

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

(4.3) WDX ,N(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 = 1, 2, ... "oscillate", i.e., each() r Xeach( + 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 = O, 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 EC N is r-step controllable to W in N
iff there exists an input sequence u(t) E U, t = 0, ..., r 1 such
that x(O) = x., x(r) ( W and x(t) C N, t = O, ..., r.

The set of states x.'s satisfying the above conditions form a
(linear) subspace.
contr
(5.5) DEFINITION. The r.-step controllable subspace X, N (r) to W
in N is the set of states r-step controllable to W in N.

contr
We show that XW,N (r) can be computed recursively. Let

(5 XWN() = W

(5.5) W,N(i) = F- WN( i 1) + im G) O N, i = 2, ...

where F-1X := (x X: Fx C X ) for a subspace X of X.
s s s

(5.6) THEOREM. Xontr(r) = W,N(r), r > 0.

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

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

= (x E X: FX = w Gul for some w W and u k ,

and xl N)

= (x E N: Fxl + Gul W for some ul km)
reachh()

contr
Now let ,N(j) = X jW,N (), j > 0. Then we have

W,N(j+1) = xj+1 N: Fxj+ = xj Guj+l for some x j,N'j)

uj+l E km. )

contr
By assumption of X N() = XN () this is equal to

{Xj+1 N: Fx1 + Guj = x, Fx + Gu = xl, ..., Fx1 + Gu = w
contr m
for some w E W, XEW,C N (a), u k, a = 1, ..., j and

uj+l 6 km1
Scontr,.
,N (0 + ). o
contr contr
We now study properties of the sequence W, XN (), XN (2), ...
by examining properties of the sequence (5.4), (5.5).
contr contr
The sequence (5.4), (5.5) (equivalently WW,N (1), o r(2),...)
has conditional monotone properties.

(5.7) LEMMA. (i) If ,N(i) C ,N(i + 1) for some integer i > 0,

then XW (j) CXWN(j + 1) for all integer j > i.
(ii) If N(i) D XN(i + 1) for some integer i > 0, then
XN() D ,N + 1 for all integer j > i.
(iii) If ,N(i) = j,N(i + 1) for some integer i > 0, then
XN(J) = XW,N(i + j) for all integer j > 0.
PROOF. (i) The proof is by induction. Let W,N(i) C XN(i + 1).
Assume W,N(M) C XW,N( + 1) where 2 is a nonnegative integer. Then

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

= X ( + 1).

Therefore, by induction we get XW,N () C j,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 W, N() = XW,N( + 1) for some integer
i > 0, then we have
WN( + 2) = F N(4 + 1) + im G) rN

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

= XW,N(A + 1).

Therefore, XN(i) = ,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 W,N(i) C ,N(i + 1) or ,N() D W,N( + 1)

happens for some i, j = 0, 1, ..., the sequence XW,N(),
9 = 0, 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 ,N(i) = W,N(i + 1).

(5.9) LEMMA. (i) If W C XWN(1), then < < dim N dim W < n.
(ii) If W D X (l1), then p < dim W < n.
-,W N -- N

PROOF. Immediate from Lemma (5.7) and the above comment. C

(5.10) LEMMA. Let t be as in the paragraph proceeding Lemma (5.9).
Then

(5.11) FXRN(0) C XWY(V ) + im G.

(5.12) ,N(u) c N.

contr
So W,N (i) is an F mod G invariant subspace in N.

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

WN' = XWN + 1)

= F-1(XW N() + im G) nN.

Hence, FXwN(k ) C XN() + im G, W,N(i) C N. r7

(5.13) REMARK. Lemma (5.10) guarantees the existence of a feedback
contr contr
K: X -)U such that (F + GK) XW,N (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) V (N) := max(V C N: FV C V + im GI.
max

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

It has been known (SILVERMAN [1976, Section III, Al) that Vmax(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 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 (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
V m(n) be as used above. Then

maxmax

contr
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.

(c.16) LEMMA. (i) W C (N(1) iff FW CW + im G.

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

PROOF. (i) If W C XN(1), then W CF -(W + im G) N. Hence
-1 c(), 1
W CF (W + im G). So we have FW CW + im G.
-1
Conversely, if FW CW + im G, then W C F (W + im G). Since
W C N by assumption, we get W C F-(W + im G) ('N, i.e., W C XWN(1).

(ii) If W D XWN(1), by Lemma (5.7 ii) we have XW N(i) CW
for all i = 0, 1, ... Then obviously XW N(i) = X (i), i = 0, 1,...
Therefore, by Corollary (5.15) ,N(P) = Vmax(W).

contr
For the nondecreasing sequence XVN (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. Xw (i) C (i + 1) for all i = 2, ...
iff FW CW + im G.

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

PROOF. Suppose FW CW + im G. Then by Proposition (5.17) we
contr contr
have XWN (i 1) C XN (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 N (i) there must exist u. EU such that Fv. + Gu. XN (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., XN i) WN i +), i = 2, ... iff
1(i) : XW,N ( + 1)
W CF (W + im G) 'N.
contr
,N (r) has the following properties in relation to state
feedback.
c contr,
By the definition of conr (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 x(t), t = 0, ..., r- 1.

contr contr
(5.19) THEOREM. (i) If XWN (i) N +), 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 cotr(r) C N, j = 0, ..., r ,
(5.21) (F + GK N r) CW j = r r + 1 .. .
j contr
(5.21) (F + GKr) XWN (r) CW, j = r, r + 1,....

Scontr contrr
(ii) If WN (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)X c (r) C N, j = 0, ..., r 1
r contr
(5.23) (F + GKr) ontr(r) CW.
Scontr ontr

PROOF. (i) By Proposition (5.17) ontr i) tr(i + 1),
i = 1, 2, ..., iff W is an F mod G invariant.
contr. contr
Assume XWN (i) C n(i + 1), i = 1, 2, ... For each
r = 1, 2, ... we:choose a basis of X as follows. Let (eol, ..., eol)
be a basis of W. Extend this basis to get the basis (eol, ..., eoo;
ell, ..., el1) of Xontr (1). Repeat the extension until we get no
more vectors to add to, say r' times where r' < r, and we obtain the
basis (e0, ..., e e ..., e r'-l,r_; er', ., e r' of
N otr(r') = XcoN (r). We further extend this basis arbitrarily to get
the basis of X, (e ,; eI, .., e ). Here :=
contr Ontr r
dim Xk N (j) dim X N (j 1), = 2, ..., r and qr
I contr contr
n dim X,N (r). Since ejs .,N (j), j = 0, ..., r' and s = 1
s = 1, ..., A there exists an input u s such that

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

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

(a) ujs = Kes for j = ..., r and s = ..., A ,

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

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

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

(5.26) (F + GKr)ejsj tr(j ), j = 1, ..., r; s= 1, ...
r) J( + W,N J j

It is easy to check that the following relations hold:

(5.27) (F + GK ) e.j N
r asj

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

(5.28) (F + GK )e.jsj W
r asj

for j = 0, ..., r
Scontr. contr
(ii) Suppose XN i) ( + 1),( 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 XWoN (r) where ~ := dim Xct(r).
Extend the basis to get the new basis (e ..., err ; er l ,
e ) contr
r-l,rr-l of WN (r 1). Repeat the procedure, and we obtain the
basis erl, .., erar; er-l,l,..., el~l; e0l, ..., eoo of W where
S:= dm ontr (j) dim ontr(j + ), j = 0, ..., 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', .., we complete the basis
1 Gqr
of X as erl, ..., el ; ol', ..., eoo; el' .."" eq where
q := n dim W. Since ejsj XWN (), j = 1, ..., r and s =
, ..., there must be an input u.jsj such that

contr
asj jsj WN

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

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

(5.52) (F + GKr )'e N, J = 0, ..., r 1; s = ...,r
hold. rs
hold. 5

(5.53) 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, ..., ; j
.Jj
or r) based on which K is defined, can be determined
r
(but not necessarily uniquely). See Remark (6.5).

= 0, ..., r
explicitly

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

x W ontr
x W ,N (r)
o0 7,

reach
where X := XN 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 > O,

(4.2) which

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

reach
(5.36) w reach(i).

PROOF. Suppose (4.2) holds. Trivially, W D (i),Wr i). To show
the converse inclusion let w C W. Then by (4.2) there exist xl C W,
u C U such that w = Fx + Gu C N. Since w ( W we obtain w = Fx +
Gul 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,
2 ,contr t
t = 2, ..., i. Then x contr i). Hence, noting that xt C W, t = 1,
;ach "
..., i, we get w- XW(i)W(i).

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

Consequently,

(5.57) PROPOSITION. each ) CXeaC h(i + 1), i= 1, 2, ...,

__each
(5.38) w = X max(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-
contr
lable subspace to W in N, X N (r), recursively. We now give a
nonrecursive characterization of XWN (r).

As before, W CN C X and r = 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 M = W, and define M similarly. Then

fr(In M ) =

M 2
MF

Mr-1

M r

0G
NFG

r-2G
M F'lG
M7 G

0 .
O..

.. .

. MFG MG

r = 2, 5, ..., and

1n VF MG

Write

P := [I 0 .. ]
r n

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

contr
(6.1) THEOREM. XN (r) = Pr ker fr(n, Mw, M-), r > 0.

PROOF. The condition

(6.2) o C ker fn' M;, ME),

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

M~Xo = 0,

x0o + MGuo = 0,
r-i o r-2Gu 0
xF o + Mr2Guo + ... + MGUr-2 = 0,

K Xo + Fr- Guo + ... + MGu- = 0,

which in turn is equivalent to the conditions

x E N,

Fx + Gu ( N,
o o

r-1 r-2
F x + F Gu + ... + Gu r_2 N,

o o r-x

contr
The last statement is equivalent to xo XWtN (r) (by Definitions (5.2)
and (5.3)). o

(6.3) REMARK. Vectors xo E X, ut c k"i, 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 11974, 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) Vax(N) = P ker f (In, )

This gives a new nonrecursive method of computing V (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 XrN (r) the assumption W C N is not .
.'N 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.)

From Lemma (5.7) we know that XW, ti) D ,N (i + 1), i = 0, ...,
iff

(7.1) W D F'(W + im G) n N.

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

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) X := kn, u(t) U := km 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

(8.5) v(x(O)) = Jx(O)

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(O) = x C X and an input sequence
u(t) = ut 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 (8.5)
we also have v(x ) = Jx0 which we call v

Considering the initial state, we have

v(x(0)) J
(8.4) ().
y(x(0)) H
Let

Wv(x(0) J
v(O) := J :=
y(x(0)) H_
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' Y11' "'" rl) and (vo2 2 2 "' 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) E 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 thereexist an input sequence u(t) E U,

t = 0, ..., r 1, and an initial state x C ker J such that the
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 Xk (r), is the set of states that are unknown-
- erJr
input indistinguishable from 0 at the final time r.

(8.9) DEFINITION. Two states x*1, x2
guishable at the final time r iff there
and ((utr-t= x ) of input sequences
ut2 t=o 02)
j = 1, 2 and initial ataees x(0) = x .,
conditions hold:

X are unknown-input indistin-
exist pairs (utlt=o, xl)
u(t) = u ; t = 0, ..., r 1;
j = 1, 2, such that the following

(i) x is the final state (at the time r)
the pair ((urtjt=, x2) where j = 1, 2,
u-tj t=o --2 -
(ii) the initially modified output sequences
the pairs coincide.

corresponding to

corresponding to

UO
The significance of Xk (r) is clarified by

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

UO
(8.11) x. x2 Xkrj(r).

PROOF] [Necessity.1 Let (v., Ylj ..., y .) be the
initially modified output sequence corresponding to the pair ([u tj=o'
x oj where j = 1, 2. Suppose v = v2 and yt Yt2' t = 1, ..., r.
Let the new initial state of (F, G, H) be x(O) := ol Xo2. Since
vl = v2, we have

Jx(0) = Jxo Jx2 =v v2= 0, i.e., x(0) E ker J.

Apply the input u(t) = ut ut2, t = 0, ..., r 1 to the system
(F, G, H). Then it is straightforward to show

x(r) = xw1 x*2

y(t) = ytl t2 = 0, t = 0, ..., r.

Therefore, x.1 x.2 C OJ(r).
*1 *2 CXerJ

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

Write x. := xl x2.
exist an input sequence
(F, G, H) such that the
H, t = 0, ..., 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) = t2 t = 0, ..., r,

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

where (ytrt=o is the output sequence produced by (u, )tr-
t2t =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/X er(r) is a linear space over k where addition and solar multi-
plication are defined in the obvious ways.

X rJ(r) is characterized as follows:

(8.12) THEOREM.

(8.13) XerJ()= kerJ erH(r)

where erJker (r) is defined by (2.1) and-(2.2).
X"kerJkerH ---

PROOF. Immediate by using Definitions (2.1) and (2.2) and
Definitions (8.7) and (8.8). O

Proposition (8.10) tells us that to recover x(r) we cannot do
better than identifying the equivalence class x(r) + X r(r). Then the
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
UO
x(r) + XUer(r) can be uniquely determined based on the knowledge of the
corresponding initially modified output sequence (v(O), y(l), ..., y(r)).

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

(8.15) v(O) = Jx ,

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

(8.16) o :=Jtv(0)

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

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

for some x E ker J.
9(t) := y(t) HFt 0,
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(0) = x + 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

U(8.20) x(r) + (r) = ((r) + F(),
(8.20) x(r)+ Oer)= (2(r) Fr%=) + C(r),
+XkeJ~r + o + erJ

it suffices to determine 2(r) + X r(r). Notice that x 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(O) C ker J. Then

(8.21) x(0) = BerJ

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

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

y(o)7
(8.22) y(1)

y(r)

= BkerJ' H,

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

1)

(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(0)
(8.25) x(r) = gr(BkerJ :
u(r 1)

where gr(BerJ) is defined in Section 3 (Chapter II).

Our problem is to determine x(r) + X e(r) from the knowledge of
y(l), ..., y(r). (v(O) = Jx(0) = 0 since x(O) C ker J. Also y(0) = 0.)
Define

S erJ kn +rm E kn" ,
Ur-1

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

rr(ker J) := imn fr(BkerJ, H, H).

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

f (BkerJ H, H)
Gr(BkerJ) "r P

gr(BkerJ)

u. E km, j = 0, ...,r,- 1)

matrix of ker J. Write

0kerJ: r (ker J) -
r r
following diagram commutes:

rr(ker J)

S kerJ
r

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

where f (-, *, -) and gr(.) are as in Section 3 and p: X ->
Xr (BkerJ) ker fr(BkerJ, H, H) is the canonical projection.

I I -

Furthermore,

(8.25) gr(BkerJ) ker f(BkerJ H, H) = (r).

Therefore,

rkerJ r(ker J) -X/ rJ(r).
r r I* er)

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

There is an element

r r r(BkerJ)

(8.26) fr(BkerJ, H, H) (ur) = y.-

Define kerJ: r(ker J) -X/g(BkerJ) ker fr(BkerJ, H, H)

(8.27) erJr) = Pgr(Bke^rJ) (r)'

4kerJ
r

is well-defined.

satisfying (8.26).

(8.28) pgr(BkerJ) (u)
r

= pgr(BkerJ (ur + 7)

Indeed, let u' be another element in 0 (BkerJ)
r r kerJ

(for some

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

= P(gr(BerJ) (ur) + gr(BkerJ)(7))

= gr(BkerJ) (r + (kerJ (

= Pg 0r kerJ) (ur

kerJ
To show the linearity of er let yrl' Yr2 rr(ker J).
Then there exist u, Ur2 C r(BkerJ) such that rj = f (BkerJ' H, H)(urj)
j = 1, 2. Therefore, yr + y2 = f(BkrJ, H, H) (ur + u2) and
we have

kerJ ++
Yrl + r2) Pgr(BkerJ) (url + Ur2)

= pgr(BkerJ) (url) + P(BkerJ) (r2)
IkerJ kerJ,-
= r rl) + r2
kerJ -
Similarly, for each a E k, r Ec r (ker J) we have r (oy ) =
arerJ,-
r r.
To show the uniqueness of the map suppose kerJ: P (ker J) -*
r r
X/gr(BkerJ) ker fr(BkerJ, H, H) is another map for which the diagram
commutes. Then for each u E S(ker J)

kerJf (B H_ ) r (
r r(BkerJ' H, H) ( )= kerJ (r
kerJ ,
-r r(BkerJ H, H) r

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

(8.29) REMARK. er: r(ker J) ->X/X rJ(r) does not depend on the
choice of BkerJ

(8.30) REMARK. For a fixed coordinate basis kerJ: r (ker J) -)
X/XIer(r) may be defined by

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

where fr(Bker, 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) erJcke(r) = 0.
4erJkerHf

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

Let r = 1, 2, .... Recall Theorem (8.12). The unknown input un-
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) nker H

holds, more can be said, namely, the sequence r(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) Xer(n) = r(n + 1) ...

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

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

is true, then the sequence Xerj(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,

i.e.,

(9.6) rank HG = rank G.

PROOF. By Theorem (8.52) 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 keri) C X k (i + 1), i = 0, ..., it follows from
O,kerH o,kerH
(2.6) that (9.7) holds iff

(9.8) 0 = X o,kerH(1) = im G-n ker H.
It remains to show that (9.8) holds iff (95) holds.
It remains to show that (9.8) holds iff (9.5) holds.

Suppose im G f ker H = 0. Let m' := rank G = dim im G. Denote
by (g g ..2' gm,) a basis of im G. We claim that Hg1, Hg2, ..., Hgm,
are linearly independent. In fact, let .Z a.Hg = 0 for some a. k, j = 1,
m j=1l 3 J
2 ,... m not all zero. Then H j =a 0, which implies

jIaj gj = 0 since im G nker 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 G(Tker H.
a basis (el, e2' ..., em,) of im G. Then

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

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

= Span (He2, ..., Hem).

Therefore, dim im HG < m'. E

(9.10) COROLLARY. Let m = p = 1 and define polynomials Jl(z) and X(z)
by l1(z)/X(z) = H(zI F)-IG (j 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 1(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 4 0 by assumption). n

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

(9.15) 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 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 (8.1) and (8.2). As before
r = 1, 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) e(r) = J x(r),
ks ss

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).
y(r)_ [H

Denote

~ ..
v (r) J
v (r) := e : = e
y(r) e H

Now we shall call the sequence (y(O), ..., y(r 1), 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(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 (10.3) has a unique solution.

A state x, E 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) 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-input initial state unobservable subspace
at the time r, denoted by XrJe(r), is the set of states that are
unknown.jinput initial state indistinguishable from 0 at the time r.

From Definitions (10.5) and (5.5) we obtain

(10.6) THEOREM. XtrJe(r) Xeor,kerH(r).

Recall the results given in Chapter III. We know the characterizations
contr
and the properties of erkeH(r).

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) COROLARY. X rH(n) = V (ker H) where V (ker H) is defined
KerH max max
by (5.14).

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

Two states x,1, x*2 X are unknown-input initial state indistinguish-
able 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
to the pairs (x*1, (utl)=) and (x*2, [Ut2)t=O) coincide,

(10.8) PROPOSITION. Two states x,1, x*2 E X are unknown-input initial
state indistinguishable at the time r iff

(10.9) x1 x2 E XrJe(r).

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

X rJe(r) defines the set of the equivalence classes X/X rJe(r).
By Proposition (10.8) the best that we can recover of the initial state
x(O) is the equivalence class x(O) + X (r). What remains to be
shown is how to determine x(O) + XerJe (r) from (y(O), ..., y(r 1),
e (r)).

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

and input sequence
r-modified output
kq is given by

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

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

and f (., *, *) is as in Section 5 of Chapter II and Section 6 of
Chapter III. Define 0 (In) := kn and r (ker J ) := im f (I J e H).
IU
(10.11) THEOREM. There is a unique homomorphism : P (r) such
-r erjer
that the following diagram commutes:

f (I J H)
fr(n e r(ker J )
r (In) P(ker J
r n r e

P '
r I r

X X/ rJe(r)

wher- I is the n x n identity matrix (or the identity map kn kn),
fr (, ., *) and P are as in Sections 3 and 5, c: X -X/XUJ(r)
is the canonical projection.

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

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

Let y E r(ker J ). There is an ur ( (I ) such that

(10.12) r = f (I J H)(d ).
r re

Define r: Pr(ker Je) -X/Xre(r) = X/Pr ker fr(In, J H) by

(10.13) *r( r) = -P ur

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

u u ker f (I J H).
Srr n e

Therefore,

(10.14) fr(I n Je H)(I) = f(In, J, H)(r( + -) (for some

1 C ker f (I Je, H))
= fr(In Je H)(' ).

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) contr (r) = 0. O
(10.16) ^XerJe,kerH

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

(10.18) V (ker H) = 0,

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

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 accommodate the discussion in this chapter to the new
reach contr withoutt
situations. The modified versions of XeN chr) and r(r) (without
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 O(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 (s, s + r) = X(r) where X(O) = ker H(s),
X(i) = (F(i 1)X(i 1) + im G(i 1))('Qker 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 X U(s, s + r) = (r)
where X(0) = ker H(s + r), x(i) = F (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

In the previous chapter we have applied rech (r) and XcoN tr)
to the problems of unknown input observability.

In this chapter we shall demonstrate the use of XWtr (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 im (I Ftx.l| = 0 (t = 0, 1, 2, ...) for all
x X where 1'|j1: 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 hi l < 1, i = 1, 2, ..., n where hi'.
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 limJI (F + GK)t x* = 0
(t = 0, 1, 2, ...) for all x. C X, then the pair (F, G) is said
to be asymptotically stabilizable. Let i(w) = ous(X) s(?) be the
minimal polynomial of X with respect to F where the roots of (us(P)
have the magnitudes greater than or equal to unity and the roots of
8s(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 Xontr(n).
oX
(iii) contr
(i-ii) xerk (F),X(n) = X.

PROOF. (iii) = (i). Suppose (iii) holds. Since F ker os(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)iX C ker s(F), i = n, n + 1, ....

K can be chosen so that, in addition,

(12.3) (F + GK)jker S(F) = FIker \$S(F).

(See Remark (5.33).) Therefore, for any state x* C X we have

(12.4) imllJ(F + GK)txII = imlFt-n(F + GK)nxl = 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- im G. (See Theorem 2.2 in WONHAM [1974].)
n-1 contr
Since im G + F im G + ... F im GC XoX (n), the implication follows.
(ii) = (iii). Let x X. Write x = xs + x where
xs E ker \$s(F) and xu E ker \$US(F). Since xu E ker S(F) C X on (n)
there exist u(O), u(l), ..., U(n 1) E U satisfying

(12.5) Fnx + Fn- Gu(O) + ... + FGu(n 2) + Gu(n 1) = 0.

Therefore, for the initial state x(O) = x. = xs + x we have

(12.6) x(n) = Fn(x + x ) + Fn-lGu(O) + ... + Gu(n 1) = Fnx r ker \$(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.

(15.1) 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) = 0,
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

(1-) contr
(1.5) x e max(kerH),X(n)

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

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

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

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

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

If K is so chosen as to satisfy (F + GK) V (ker H) C V (ker H)
maxax
(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) E Vmax(ker H), which implies
that x(O) = x. ( V.(V a(ker H), X) by Definition (5.2) and (5.3).
Appealing to Proposition (5.17) we conclude

contr contr
SX x(kerH),X(i) C XVmax(kerH),X(n). [

(15.4) COROLLARY. Let x. E 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
(15.5) x, XVa (kerH),X().

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

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

S contr
(137) (kerH),X(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 := km, Y : kp, V := ks (v(t) 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.

(14.5) PROBLEM. (r-step Disturbance Decoupling Problem) Given (F, G, H, D)
with x(O) = 0, find (if possible) a feedback Krl: X U such that
y(t) = O, t = 0, ..., r for any v(t) E 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 solu-
tion iff

(14.5) im D C XentrkerHr 1).

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

(14.6) x(t + 1) = (F + GKr_1)x(t) + Dv(t).

Consider the system (14.6), (14.2). Let v ( k 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. Then
we have

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

x(t) = (F + GKrl)t- vj, t = j + 1, 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) = 30 y(t) = 0, t = 0, ..., r.

[Necessity] Suppose that the problem has a solution but that
contrs contr
im D t kerH,kerHr I) Since x(O) = O, we have x(1) = DV(0).
There must be v0 G k such that Dv Xker, (k r 1). Then by
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. O

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 CV (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.

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

(14.8) THEOREM. Problem (14.7) has a solution iff the following condi-
tions hold:

(14.9) im DC Xco (n),
o,kerH

(14.10) Xcontr(n) = X.
o, X

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

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

(14.12) (F + GK) Xcontr (n) = 0, j = n, n + ....
(4.), (b,) a ( t ikerH

(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(O) = 0 is zero for all t = 0, 1, ....

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

(14.14) (F + GK)nX C Xcontr (n).
o,kerH

Hence,

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

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

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

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

15. Concluding Remarks

reach Xreach(N)
There is an important difference between X N e (n) and Xrea(N).
oN r max
By choosing K: X U the spectrum of (F + GK)(Xreac (N) can be assigned
max
ax reach
arbitrarily, while to discuss the spectrum of (F + GK)IXW eN (n) may
reach reach
not make sense since (F + GK) XocN (n) C X h(n) may not hold (unless
N is an F mod G invariant subspace). However, though Xeach (r) loses
contr
the arbitrary pole assignability, XwN (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 ech (r) from
contr,
W in N and the r-step controllable subspace Xon tr(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 ~ (r). and Xontr(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 XW c(r) and Xconr(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,
.reach ich
there are several other interesting topics concerning XWr (r) which
are still open to further research. The relation between the transfer
function M (zI F)-G and the subspaces each, r), cont (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) MM1 = M-1 = I.

-1
The matrix M is called the inverse of M.

Now let M be rl 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 r2 matrix consisting of a collection of s linearly independent rows
c-1-
of M. Then the pseudo-inverse M of M is defined by

(Al.2) Mt := M(MM' )(MM~)}l(M')

where A denotes the transpose of a matrix A. The meaning of Mt is as
follows: Using "dom" to denote "the domain of", we have

(Al.3) LEMMA. Let x C dom M, and write x = x + x where x C im M
and x2 C ker M. Then M Mx = x.
21

PROOF. If x C im M then xl = M xl for some s-vector xl.
Therefore

= Mx

X1*
= xl.

If x2 C ker M, then trivially M Mx2 = 0. [

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
t -no th n l
" nns-i~rsid~lar. then M MV

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BIOGRAPHICAL SKETCH

Fumio 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.

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.

Rudolf E. Kalman, Chairman

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.

Charles V. Shaffer, Co-chaain
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 acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.

Michael E. Wa ren
Assistant 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.

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

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