ZEROS, OUTPUT-NULLING SUBSPACES AND ZERO

DYNAMICS IN MIMO LTI SYSTEMS

Signal Processing, Systems Modelling and Control

Jerzy Tokarzewski

Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland

Lech Sokalski

Automotive Industry Institute, Jagiellonska 55, 03-301 Warsaw, Poland

Keywords: Linear systems, Output-zeroing problem, Zeros, Zero dynamics, Markov parameters

Abstract: In standard MIMO LTI continuous-time systems S(A,B,C) the classical notion of the Smith zeros does not

characterize fully the output-zeroing problem nor the zero dynamics. The question how this notion can be

extended and related to the state-space methods is discussed. Nothing is assumed about the relationship of

the number of inputs to the number of outputs nor about the normal rank of the underlying system matrix.

The proposed extension treats zeros (called further the invariant zeros) as the triples (complex number,

nonzero state-zero direction, input-zero direction). Such treatment is strictly connected with the output

zeroing problem and in that spirit the zeros can be easily interpreted even in the degenerate case (i.e., when

any complex number is such zero). A simple sufficient and necessary condition of degeneracy is presented.

The condition decomposes the class of all systems S(A,B,C) such that 0B

≠

and into two disjoint

subclasses: of nondegenerate and degenerate systems. In nondegenerate systems the Smith zeros and the

invariant zeros are exactly the same objects which are determined as the roots of the so-called zero

polynomial. The degree of this polynomial equals the dimension of the maximal (A,B)-invariant subspace

contained in Ker C, while the zero dynamics are independent upon control vector. In degenerate systems the

zero polynomial determines merely the Smith zeros, while the set of the invariant zeros equals the whole

complex plane. The dimension of the maximal (A,B)-invariant subspace contained in Ker C is strictly larger

than the degree of the zero polynomial, whereas the zero dynamics essentially depend upon control vector.

0C ≠

1 INTRODUCTION

During the past three decades considerable attention

has been paid to the determination and computation

of zeros of a LTI MIMO system S(A,B,C). A large

number of types of zeros has been defined

(MacFarlane,Karcanias, 1976; Schrader, Sain,

1989). The commonly used definitions employ the

Smith form (Callier, Desoer, 1982; Chen, 1984;

Gantmacher, 1988) of the system matrix

. Recall (Callier, Desoer,

1982) that for P(s) there exist unimodular matrices

⎥

⎦

⎤

⎢

⎣

⎡

−−

BAsI

)s(P

U(s) and V(s) and a polynomial matrix such

that and has the form

)s(Ψ

)s(V)s()s(U)s(P Ψ=

)s(Ψ

⎥

⎦

⎤

⎢

⎣

⎡

=Ψ

0000

0)s(

0.0

00)s(

)s(

Here

)s(

is called the Smith form of P(s) when

polynomials

)s(

are monic and divides )s(

)s(

1i+

for

1,...,1i

−

, and is the normal rank

of P(s). The polynomials are known as

invariant factors of P(s) and their product

)s(

)s(...)s()s(

1 ν

is called the zero polynomial

of P(s) (and of S(A,B,C)). The roots of are the

Smith zeros of S(A,B,C) (they are commonly known

rather as invariant zeros (Basile, Marro, 1992; Marro

)s(ψ

114

Tokarzewski J. and Sokalski L. (2004).

ZEROS, OUTPUT-NULLING SUBSPACES AND ZERO DYNAMICS IN MIMO LTI SYSTEMS - Signal Processing, Systems Modelling and Control.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 114-121

DOI: 10.5220/0001136001140121

 SciTePress

et al., 2002)). The transmission zeros of S(A,B,C)

are the Smith zeros of its minimal (controllable and

observable) subsystem. The zeros of a transfer-

function matrix G(s) can be defined (Misra et al.,

1994) as the Smith zeros of the system matrix

obtained from any given minimal state-space

realization of G(s). The Smith zeros of the pencil

(i.e., uncontrollable (

[

BAsI −−

]

) modes of A)

are the input decoupling (i.d.) zeros and the Smith

zeros of

(i.e., unobservable (

⎥

⎦

⎤

⎢

⎣

⎡

−

AsI

o ) modes of

A) are the output-decoupling (o.d.) zeros of

S(A,B,C). The input-output decoupling (i.o.d.) zeros

of S(A,B,C) are those o.d. zeros which disappear

when the i.d. zeros are eliminated (Rosenbrock,

1970; 1973). Defined above multivariable zeros are

involved in several problems of control theory such

as zeroing the system output, tracking the reference

output, disturbance decoupling, noninteracting

control, model matching and output regulation

(Basile, Marro, 1992; Isidori, 1995; Marro, 1996;

Sontag, 1990; Wonham, 1979). The Smith zeros

were discussed, at various simplifying assumptions

concerning the systems considered, by many authors

(Schrader, Sain, 1989). As is known (MacFarlane,

Karcanias, 1976) the Smith zeros are related

(through the corresponding zero directions) with

zeroing of the system output. Simple examples (see

5) show however that they do not characterize fully

the output-zeroing problem (in particular, the zero

dynamics nor the maximal (A,B)-invariant subspace

contained in Ker C). In order to remove this

disadvantage we consider a set (denoted as

) of

complex numbers such that for each its element

there exists a zero direction with nonzero state-zero

direction. The set

of the Smith zeros, where

, is

contained in

. To any element of there

corresponds an output-zeroing input which produces

nontrivial solution of the state equation. Under

typical tansfomations,

has the same invariance

propeties as

. For the reasons mentioned above,

is treated as an extension of and to the

elements of

we do not assign a new name; we

call

)}s(Pranknormal)(Prank:{: <λ∈λ= C

them simply the invariant zeros.

The paper is organized as follows. In section 2 we

give an overview concerning the basic properties

and the algebraic characterization of

(based on

singular value decomposition (SVD) of the first

nonzero Markov parameter) and explicit formulas

for the maximal (A,B)-invariant subspace contained

in Ker C. Main results are given in sections 3 and 4.

we denote the fields of real and complex

numbers;

CR,

}0)AIdet(:{)A( =−λ

∈

C stands for

the spectrum of matrix A and the Moore-Penrose

pseudoinverse of a matrix M is denoted by

2 PRELIMINARY RESULTS

2.1 Definition and Basic Properties of

Invariant Zeros

Consider system S(A,B,C) of the form

)t(Cx)t(y),t(Bu)t(Ax)t(x

, (1)

0t ≥ ,

, , , where A

and

()∈R

ut()∈ R

yt()∈R

≠

0, C

≠

0 are real matrices of appropriate

dimensions. The set

of admissible inputs is

assumed to consist of all piecewise continuous

functions

. The first nonzero

Markov parameter of (1) is denoted by

where

u(.):[ , )0 ∞→R

≤

−

n , i.e.,

and

CA B

== =

−

...

CA B

≠

The four-fold canonical decomposition (Kalman,

1982) (2) of (1)

⎥

⎦

⎤

⎢

⎣

⎡

34oc

24co

141312oc

A000

AA00

A0A0

AAAA

⎥

⎦

⎤

⎢

⎣

⎡

, (2)

[

]

occo

C0C0C

⎥

⎦

⎤

⎢

⎣

⎡

is not unique, however in any such form the orders

nnn

co co co

and

of the corresponding

matrices on the diagonal of the A-matrix are

uniquely determined by the order of A (n), the

degree of G(s) (

) and rank defects of the

controllability and observability matrices (

and

) as

nnnn

co co c

−

nnnn

co co c o

=++ n

−

nn-nn

co co o

−

. The characteristic polynomials

(up to a constant) of these matrices also remain

ZEROS, OUTPUT-NULLING SUBSPACES AND ZERO DYNAMICS IN MIMO LTI SYSTEMS

115

unchanged and the elements of

σ(A

)

are known

as controllable and unobservable (

co) modes of (1);

analogously, co-modes are eigenvalues of

(these are poles of G(s)),

-modes are eigenvalues

and

-modes are eigenvalues of

Definition 1 (Tokarzewski, 2000; 2002a,b)

(i) A number

λ∈

is an invariant zero of (1) if and

only if (iff) there exist vectors

(state-

zero direction) and

(input-zero direction)

such that the triple

satisfies

≠∈x

∈C

,,xg

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

−−λ

BAI

. (3)

The set of the invariant zeros is denoted by

System (1) is called degenerate iff

is infinite

(otherwise, (1) is called nondegenerate).

(ii) The transmission zeros of (1) are the invariant

zeros of its minimal subsystem.

(iii) The o.d. zeros are the unobservable (

) modes

of (1). The i.d. zeros are the uncontrollable (

)

modes of (1). The i.o.d. zeros are the uncontrollable

and unobservable

()co

modes of (1).

The set

is invariant under nonsingular coordinate

transformations in the state-space, nonsingular

transformations of the inputs or outputs, constant

state or output feedback to the inputs, constant

output feedback to the integrator input. Any o.d.

zero of (1) is in

as well as any transmission zero

of (1) is in

The Kalman form (2) determines individual kinds of

decoupling zeros (including multiplicities) of (1) via

the polynomials

iod co co

ssIA

...

( ) det( )=− and

od co co co co

s sI A sI A

( ) det( )det( )=− − and

id co co co co

s sI A sI A

( ) det( )det( )=− −.

Definition 2 (Tokarzewski, 2000; 2002a) For a

transfer-function matrix G(s) a number

∈

λ is a

transmission zero of G(s) iff it is an invariant zero of

any given minimal realization of G(s).

2.2 Invariant Zeros and Output-

Zeroing Problem

The dynamical interpretation of the elements of

in (1) is based on the following formulation of

the output-zeroing problem (Isidori, 1995). Find all

pairs

consisting of an initial state

(,()xut

)

∈

and a such that the

corresponding system response satisfies

U∈(.)u

()

for

all

≥ 0. Any nontrivial pair of this kind (i.e., such

that

≠

0(.)u

≠

) is called an output-

zeroing input. The internal dynamics of (1)

consistent with the constraint

for all

()= 0

≥ 0

are called the zero dynamics of the system.

The same symbol

is used to denote state-zero

direction (Definition 1(i)) and initial state in output-

zeroing inputs. The state-zero direction must be a

nonzero vector (real or complex), whereas the initial

state must be a real vector (not necessarily nonzero).

If state-zero direction

x is a complex vector, then it

gives two initial states

and

(see

(Tokarzewski, 2000; 2002a) for an explicit form of

output-zeroing inputs and the corresponding

solutions of the state equation generated by the

elements of

Z ).

The set of all output-zeroing inputs completed by the

trivial pair

forms a linear space

over

. In this space we can distinguish a subspace

of all pairs

(, such that and

for all

()xut

=0 )≡0

))

(xut

= 0 U∈(.)u

u t Ker B

()∈

≥ 0. Any such pair affects (1)

in the same way as the trivial pair, i.e., it gives

identically zero solution and

for all

()= 0

≥ 0.

We do not relate these pairs with invariant zeros (we

associate them with the trivial pair).

Recall (Wonham, 1979) that a subspace

X is

(A,B)-invariant if there exists a

mxn real matrix F

such that

()

⊆ R

()ABFX X

⊆

(in (Basile, Marro,

1992) X is called an (A,B)-controlled invariant). The

maximal (A,B)-invariant subspace contained in

(denoted as

KerC

∗

(,,)

) is an unique (A,B)-

invariant subspace contained in

Ker with the

property that any (A,B)-invariant subspace X

contained in

Ker must satisfy

. If

is an output-zeroing input and

the corresponding solution, then

for all . Moreover, for

any

there exists an output-

zeroing input such that the corresponding solution

passes through

(Tokarzewski, 2002a).

ABC⊆

∗

(,,)

(,()xut

) t

()

)C,B,A(X)t(x

o ∗

∈ 0t ≥

)C,B,A(Xx

o ∗

∈

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116

2.3 Relationship between

and

S I

Proposition 1 (Tokarzewski, 2002b) In system (1)

the sets

and

are interrelated as follows.

S I

(i)

⊆

(ii) System (1) is nondegenerate iff

(iii) System (1) is degenerate iff

Thus in (1) the set

may be empty, finite or equal

to C, and when (1) is nondegenerate, then

is its

invariant zero iff

λ is a root of the zero polynomial.

If in (1) there exists at least one invariant zero which

is not a Smith zero, then (1) is degenerate. On the

other hand, if (1) is degenerate, we can have

(see section 5). Moreover, if

≠∅

=∅

is a Smith zero of (1), then there exist

and

such that

satisfy (3) (i.e., to

any Smith zero there corresponds zero direction with

nonzero state-zero direction).

≠∈x

g ∈C

,,xg

2.4 Invariant Zeros and (A,B)-

Invariant Subspaces

Lemma 1 (Tokarzewski, 2002a) If in (1) is

(i.e., all Markov parameters are zero), then

Gs()≡ 0

and (i.e.,

is the unobservable subspace for (1)).

CAKer)C,B,A(X

−

∗

XABC

∗

(,,)

Suppose now that in (1) not all Markov parameters

are zero and let the first nonzero Markov parameter

, CA B

0≤≤−1

n , have rank .

Define the projective matrix (Tokarzewski, 2000)

},{min0 rmp ≤<

KIBCABCA

:()=−

(4)

The approach presented below (based on SVD of

) enables us to decide the question of

degeneracy/nondegeneracy and to characterize the

invariant zeros as well as the subspace

In general, the invariant zeros of (1) will be

characterized as invariant zeros (in particular, when

(1) is nondegenerate, as output-decoupling zeros) of

certain closed-loop system (obtained from (1) via

introducing appropriate pre- and postcompensator

and state feedback matrix), while

will

be characterized as the unobservable subspace for

that system.

CA B

∗

(,,)

∗

(,,)

Let us write SVD (Callier, Desoer, 1982) of

VUBCA Λ=

, where (5)

⎥

⎦

⎤

⎢

⎣

⎡

=Λ

r x m-dimensional, is p x p nonsingular and

diagonal (with positive singular values of

)

and

rxr U and mxm V are orthogonal. Introducing

into (1)

V and U as pre- and postcompensator we

associate with (1) a new system

SABC(,,)

)t(xC)t(y),t(uBAx)t(x =+=

, (6)

where

BBVCUC

==,

and

uVuyUy

==,

are decomposed as follows

[

]

pmp −

= BBB ,

⎥

⎦

⎤

⎢

⎣

⎡

− pr

⎥

⎦

⎤

⎢

⎣

⎡

− pm

⎥

⎦

⎤

⎢

⎣

⎡

− pr

(7)

and

B consists of first p columns of B whereas

consists of first p rows of C. Moreover,

CA B

is the first Markov parameter for (6) and

ppp

BACM

= . Since (6) is obtained from (1) by

nonsingular transformations of inputs and outputs,

the sets of the invariant zeros for S(A,B,C) and

SABC(,,) coincide. For the associated with (1)

system (6) we form the projective matrix

KIBCABCA

:()=−

(8)

which, in view of (5) and (7), may be expressed as

ACMBIK

ppp

−

−= (9)

Lemma 2 The matrix

in (9) satisfies

(i)

KK =

(ii)

ACKer}xxK:x{:

===Σ

pn −=Σ

dim

(iii)

BIm}0xK:x{:

===Ω ,

p=Ω

dim

(iv)

)( Ω⊕Σ=

RC ,

ZEROS, OUTPUT-NULLING SUBSPACES AND ZERO DYNAMICS IN MIMO LTI SYSTEMS

117

(v) 0BK =

pmpm −−

= BBK

0KAC

(vi)

⎪

⎩

⎪

⎨

⎧

+≥

≤≤

1kfor0

k0forAC

)AK(C

(vii)

k0forAC)AK(C

≤≤= l

Remark 1 For (4) and (8),(9) is KK

= .

Proposition 2 In (1) let and

mp <=BCArank

SABC(,,) in (6) let 0B ≠

− pm

. Then the

sequence of transformations

)C,B,AK(S)C,B,A(S)C,B,A(S

k pm−

→→

has the following properties:

(i) it preserves the set of the invariant zeros, i.e.,

)C,BA,KS(

)C,BS(A,

C)B,S(A,

-k

ZZZ

(ii) it preserves the maximal (A,B)-invariant

subspace contained in

, i.e.,

CKer

)C,B,AK(X)C,B,A(X)C,B,A(X

k pm−

∗∗∗

(iii) it preserves the zero polynomial for S(A,B,C),

i.e.,

)s()s()s(

)C,B,A

)C,B,A(S

pm−

ψ=ψ=ψ

KS(

and consequently, the set of the Smith zeros, i.e.,

)C,BA,KS(

)C,BS(A,

C)B,S(A,

-k

ZZZ

Proposition 3 (Tokarzewski, 2002a) In (1) let

. Then (1) is nondegenerate and

rank CA B

= m

λ∈

is an invariant zero of (1) iff

is an o.d. zero

)C,B,AK(S

. Moreover,

equals the

unobservable subspace for

∗

(,,)

)C,B,AK(S

, i.e.,

)AK(CKer)C,B,A(X

−

∗

Proposition 4 (Tokarzewski, 2002a) In (1) let

m > and let

have full row rank r. Then:

(i) S(A,B,C) is degenerate iff in

SABC(,,) in (6) is

0B ≠

−rm

. Moreover,

∈

is an invariant zero of

(1) iff

is an invariant zero of the system

)C,B

,AK(S

k r−

whose transfer-function matrix

equals zero identically. Furthermore,

equals the unobservable subspace for

∗

(,,)

)C,B,AK(S

k rm−

, i.e.,

)AK(CKer)C,B,A(X

−

∗

(ii) S(A,B,C) is nondegenerate iff

0B =

−rm

Moreover,

∈

is an invariant zero of (1) iff

an o.d. zero of the system

)C,B,AK(

k r

Furthermore,

XABC

∗

(,,)

equals the unobservable

subspace for

)C,B,AK(S

k r

, i.e.,

)AK(CKer)C,B,A(X

−

∗

Proposition 5 (Tokarzewski, 2002a) In (1) let

have rank

CA B

},min{ rmp

and in SABC(,,)

in (6) let

0C =

− pr

. Then:

(i) S(A,B,C) is degenerate iff

0B ≠

− pm

Moreover,

∈

is an invariant zero of (1) iff

an invariant zero of the system

)C,B,AK(

k pm−

whose transfer-function matrix equals zero

identically. Furthermore,

equals the

unobservable subspace for

∗

(,,)

)C,B,AK(S

k pm−

, i.e.,

)AK(CKer)C,B,A(X

−

∗

(ii) S(A,B,C) is nondegenerate iff

0B =

− pm

Moreover,

∈

is an invariant zero of (1) iff

an o.d. zero of the system

)C,

B,AK(S

Furthermore,

XABC

∗

(,,)

equals the unobservable

subspace for

)C,B,AK(S

k p

, i.e.,

)AK(CKer)C,B,A(X

−

∗

Proposition 6 (Tokarzewski, 2002a) In (1) let

and in },min{BCArank

rmp <= SA BC(,,) in

(6) let

0C ≠

− pr

and let 0B =

− pm

. Then S(A,B,C)

is nondegenerate; moreover

λ∈

is its invariant

zero iff

is an o.d. zero of the system

)C,

B,AK(S

. Furthermore,

equals

the unobservable subspace for

∗

(,,)

)C,B,AK(S

k p

, i.e.,

)AK(CKer)C,B,A(X

−

∗

ICINCO 2004 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

118

Propositions 2-6 and Lemma 1 yield a recursive

procedure for the computation of invariant zeros and

for system S(A,B,C) in (1).

XABC

∗

(,,)

Procedure 1 (Tokarzewski, 2002a)

has full column rank.

Invariant zeros of (1) are o.d. zeros of

)C,B,AK(S

and

)AK(CKer)C,B,A(X

−

∗

has full row rank r and

m >

2a. 0B =

−rm

. Invariant zeros of (1) are o.d. zeros

)C,B,AK(S

k r

and

)AK(CKer)C,B,A(X

−

∗

2b.

0B ≠

−rm

. S(A,B,C) is degenerate and

)AK(CKer)C,B,A(X

−

∗

3. . },min{BCArank

rmp <=

3a.

0C =

− pr

3a1.

0C =

− pr

and 0B =

− pm

. Invariant zeros of

(1) are o.d. zeros of

)C,B,AK(S

k p

and

)AK(CKer)C,B,A(X

−

∗

3a2.

0C =

− pr

and 0B ≠

− pm

. S(A,B,C) is

degenerate and

)AK(CKer)C,B,A(X

−

∗

3b.

0C ≠

− pr

3b1.

0C ≠

− pr

and 0B =

− pm

. Invariant zeros of

(1) are o.d. zeros of

)C,B,AK(S

k p

and

)AK(CKer)C,B,A(X

−

∗

3b2.

0C ≠

− pr

and 0B ≠

− pm

. The question is

not decided at this step. Start the next step applying

Procedure 1 to system

)C,B,AK(S

k pm−

4. In (1) all Markov parameters are zero. S(A,B,C)

is degenerate and

CAKer)C,B,A(X

−

∗

In the case 3b2 we begin the second step applying

Procedure 1 to system

with the

matrices

B C(',',')

AK'A

= ,

pm−

= B'B , C'C = and

pmm

−

inputs (i.e., we find the first nonzero

Markov parameter for

and its SVD and

then we form the associated system

)'C,'B,'A(S

; in

the cases 2a and 2b are not possible

since its Markov parameters have no full row rank).

The process ends after at most n steps. At the last

step we can meet only two possible situations: we

get a system whose transfer-function matrix is

identically zero or a system with the first nonzero

Markov parameter of full column rank.

SA B C(',',')

Corollary 1

(i) The question of seeking invariant zeros and

XABC

∗

(,,)

for (1) can be decided at the first step

(the cases 1, 2a, 2b, 3a1, 3a2, 3b1 or 4 in Procedure

1) or, in case 3b2, applying successively Procedure

1, after at most

n steps.

(ii) The recursive process generated by point 3b2

preserves

C)B,S(A,

)s(

)C,B,A(S

and

(comp. Proposition 2). Thus,

can be

found out as the set of the invariant zeros of the

system obtained at the last step (similarly for

∗

(,,)

C)B,S(A,

)s(

)C,B,A(S

and

XABC

∗

(,,)

(iii) The process ends when we approach a

nondegenerate system (the case 1, 3a1 or 3b1) or a

degenerate system (the case 3a2 or 4).

3 SMITH ZEROS, INVARIANT

ZEROS AND ZERO DYNAMICS

Proposition 7 If (1) is nondegenerate, then its

Smith zeros and the invariant zeros are the same

objects (including multiplicities). Moreover, the

degree of the zero polynomial for (1) equals

dim ( , , )XABC

∗

, while the zero dynamics, in

appropriate coordinates, have the form

, where the characteristic polynomial of

matrix N equals the zero polynomial for (1) and

)t(N)t( ξ=ξ

belongs to the subspace

(when taken in

the same coordinates).

)C,B,A(X

∗

Proposition 8 If (1) is degenerate, then the

dimension of

is larger than the degree

of the zero polynomial for (1), i.e.,

)C,B,A(X

∗

ZEROS, OUTPUT-NULLING SUBSPACES AND ZERO DYNAMICS IN MIMO LTI SYSTEMS

119

)s(deg)C,B,A(Xdim

)C,B,A(S

ψ>

∗

. Moreover,

the Smith zeros of (1) are i.o.d. zeros of certain

system whose transfer-function matrix equals zero

identically and the zero dynamics for (1) depend

esentially upon control vector.

4 SUFFICIENT AND NECESSARY

CONDITION OF DEGENERACY

Proposition 9 System (1) is degenerate iff

Brank)s(Pranknormal +< n . (10)

Proposition 10 System (1) is nondegenerate iff

Brank)s(Pranknormal += n . (11)

From Proposition 9 and from the relation

we get )s(Granknormal)s(Pranknormal += n

Proposition 11 Let be a )s(G

transfer-

function matrix and let S(A,B,C) (1) stand for its

minimal

n-dimensional state-space realization. Then

is degenerate (comp. Definition 2) iff )s(G

Brank)s(Granknormal < . (12)

Proposition 12 G(s) is nondegenerate iff

Brank)s(Granknormal = . (13)

5 EXAMPLES

Example 1 In (1) let

⎥

⎦

⎤

⎢

⎣

⎡

−−−

121

100

010

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

−−

010

012

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

The system is minimal, asymptotically stable and has

no Smith zeros. In SVD of

CB we take

⎥

⎦

⎤

⎢

⎣

⎡

−

2/12/1

⎥

⎦

⎤

⎢

⎣

⎡

=Λ

⎥

⎦

⎤

⎢

⎣

⎡

Via Procedure 1 (3b2) we consider system

)C,B,AK(S

k pm−

, where

⎥

⎦

⎤

⎢

⎣

⎡

−−−

−=

121

010

⎥

⎦

⎤

⎢

⎣

⎡

−

⎥

⎦

⎤

⎢

⎣

⎡

−

Since all Markov parameters of

)C,B,AK(S

k pm−

are zero,

)C,B,AK(S

k pm−

and consequently,

system (1) are degenerate. The zero dynamics for (1)

are

133

uxx

−

and . CKer)C,B,A(X =

∗

Example 2 (Emami-Naeini, Van Dooren, 1982) In

(1) let

⎥

⎦

⎤

⎢

⎣

⎡

−

001

000

012

⎥

⎦

⎤

⎢

⎣

⎡

[]

010C −=

The system has one single Smith zero at 2. Since

Gs()

≡

, we have

; moreover,

X A B C Ker C

∗

=(,,)

Example 3 In S(A,B,C) in (1) let and

let

0)s(G ≡

Hx'x

denote a change of coordinates which

transforms (1) to its Kalman form

)'C,'B,'A(S

⎥

⎦

⎤

⎢

⎣

⎡

34oc

1413oc

A00

AA0

AAA

⎥

⎦

⎤

⎢

⎣

⎡

, (14)

[

]

C00'C

⎥

⎦

⎤

⎢

⎣

⎡

Since in (1) we have assumed

and , in 0B ≠ 0C ≠

(14) is always

>n and 0

>n , while 0

≥n

(i.e., i.o.d. zeros for S(A,B,C) may not exist). The

normal rank of the system matrix for (14) is

nnnn

ocococ

and the zero polynomial for

(14) (and consequently, for S(A,B,C)) is

)AsIdet()s(

ococ)C,B,A(S

−

. Thus, the Smith

ICINCO 2004 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

120

zeros of (1) are i.o.d. zeros of (1). Of course,

ococ

)'C,'B,'A(Xdim)C,B,A(Xdim nn +==

∗∗

The zero dynamics in

are governed by

the equations

)'C,'B,'A(S

)t(xA)t(x

)t(uB)t(xA)t(xA)t(x

ocococ

ococ13ocococ

++=

where

≠ , and their solutions remain in

)'C,'B,'A(X}0x:'x{X

∗

==∈=

R . If we

constrain initial conditions to the subspace

}0x,0x:'x{X

ococ

==∈=

R , then in this part

X the zero dynamics are governed by

)t(uB)t(xA)t(x

ococococ

The source of degeneracy of (14) (and consequently,

of (1)) lies in this part (i.e., controllable and

unobservable) of the system, since for any

)A(

σ∉λ the triple

⎥

⎦

⎤

⎢

⎣

⎡

=λ

with

gB)AI(x

ococ

−

−λ= and

BKerg∉ ,

satisfies Definition 1(i) for

. )'C,'B,'A(S

6 CONCLUSION

The purpose of this paper was to discuss certain

geometric aspects of multivariable zeros that are not

commonly known from the relevant literature. The

presented approach can be extended on non strictly

proper systems.

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