Lyapunov stability of 2D finite-dimensional behaviors

20 de Maio de 2010

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International Journal of Control

Na_Ra_Ro(Lyapunov_stability_2D)_submitted_IJC


Vol. 00, No. 00, Month 200x, 114

RESEARCH ARTICLE Lyapunov stability of

2D

nite-dimensional behaviors

Napp Avelli, D.a∗ Rapisarda, P.b and Rocha, P.c a

RD Unit Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal

b c

ISIS group, University of Southampton, UK, e-mail:

;

[email protected];

Department of Electrical and Computer Engineering, Faculty of Engineering, University of Oporto, Portugal, e-mail:

[email protected]

(Received 00 Month 200x; nal version received 00 Month 200x) In this paper we investigate a Lyapunov approach to the stability of nite-dimensional 2D systems. We use the behavioral framework and consider a notion of stability following the ideas in (15, 18, 19). We characterize stability in terms of the existence of a (quadratic) Lyapunov function and provide a constructive algorithm for the computation of all such Lyapunov functions. Keywords:

1

2-D system; Lyapunov function; quadratic dierence form; behavioral approach;

Introduction

The stability of two dimensional (2D ) systems has been the subject of extensive investigation in the past decades; among these research eorts, some have also been focused on the computation of Lyapunov functions. Past research has predominantly been concerned with systems whose set of trajectories is innite-dimensional, and almost exclusively has concerned specic class of models, for example Fornasini-Marchesini or Roesser models (see (3, 10)). Moreover, in those investigations a specic (usually nonnegative quarter-plane) notion of causality has been assumed. In this paper we follow the behavioral approach: we study the stability of

2D

systems described

by higher-order dierence equations without reference to special representations; the central object of interest in our investigation is the set of all admissible trajectories of the system, the

behavior, rather than any of its specic representations. Following the pioneering approach of (19), stability is accordingly dened at the level of trajectories, although we will be using a dierent but

ultimately equivalent denition to that proposed in (19). We also adopt the eminently reasonable position proposed in (19) to let the system dynamics themselves dictate what notion of causality is most appropriate for the case at hand. A Lyapunov analysis of stability of

innite-dimensional square 2D behaviors has been presented 2D discrete Rw . We dene

in (8); in this paper we concentrate our attention on the case of nite-dimensional behaviors, i.e. nite-dimensional subspaces of the set of trajectories from a nite-dimensional

2D

Z2

to

system stable if all trajectories of the behavior go to zero along every

discrete line in a cone, which without loss of generality we take to be the rst orthant of the lattice

Z2 ;

this is a dierent but equivalent denition from that adopted in section 3, Def. 3.1 of

(19), and follows the approach of (12, 15, 18). The main result of the paper is a characterization of stability for nite-dimensional

2D

behaviors in term of the existence of a

dened as a quadratic function of the system variables and their

∗

Corresponding author.Email: [email protected]

ISSN: 0020-7179 print/ISSN 1366-5820 online c 200x Taylor & Francis

DOI: 10.1080/0020717YYxxxxxxxx http://www.informaworld.com

2D

Lyapunov function,

shifts which is positive along

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2 each discrete line in the rst orthant, and whose increment along each such line is negative. In this paper we also give necessary and sucient conditions for a quadratic function of the system variables and their

2D

shifts to be a Lyapunov function; and we illustrate an algorithm to

compute a Lyapunov function for a given nite-dimensional

2D

behavior.

In the following we make extensive use of the concepts and calculus of

forms

2D

quadratic dierence

(see (9)), and their association with four-variable polynomial matrices. We will also use

extensively the concepts and terminology of the behavioral approach to

2D

systems. In order

to make the paper self-contained we have included some background material in section 2; the

2D

reader interested in a more thorough introduction to

behavioral system theory is referred to

(16, 17, 19). The main result of the paper is illustrated in section 3. In section 4 we outline an algorithm for the construction of Lyapunov functions.

2

Preliminaries

We consider sets

B

of trajectories dened over

Z2

that can be described by a set of linear partial

dierence equations, i.e., 2

B = ker R(σ1 , σ1−1 , σ2 , σ2−1 ) ⊆ (Rw )Z , where

σi 's

are the

2D

(1)

shift operators dened by

σi w(k1 , k2 ) = w((k1 , k2 ) + ei ) , −1 (k1 , k2 ) ∈ Z2 and ei the ith element of the canonical basis of R2 , i = 1, 2; and R(s1 , s−1 1 , s2 , s2 ) is a 2D (p × w)-dimensional Laurent-polynomial matrix. We call (1) a kernel representation of the behavior B .

for

We now introduce nite-dimensional behaviors, and briey discuss their representation by means of state equations. Given a full column rank Laurent-polynomial matrix

Laurent variety

−1 R ∈ Rp×w [s1 , s−1 1 , s2 , s2 ],

we dene its

(or simply variety) as

V(R) := {(λ1 , λ2 ) ∈ C2 | rank(R(λ1 , λ2 )) < rank(R)}, where

rank(R(λ1 , λ2 ))

is the rank of the complex matrix

the Laurent-polynomial matrix of

B

R.

R(λ1 , λ2 ),

while

rank(R)

is the rank of

It can be shown (21) that any two dierent representations

share the same Laurent variety; consequently in the following we refer to the

denoted by

V(B),

(see (2, 17)) that a behavior space over

R

variety of B,

as the Laurent variety of any of its kernel representations. It is well known

B

is nite dimensional (when considered as a subspace of the vector

consisting of all trajectories from

number of points, or equivalently if

B

Z2

Rw )

to

if and only if

V(B)

consists of a nite

admits a left factor prime representation (see (5) for a

denition).

B there exist a hybrid represenA2 ∈ Rn×n and C ∈ Rw×n such that B 2 n trajectory x : Z → R such that

It was shown in (4) that for every nite dimensional behavior

tation

of rst order, i.e., there exist matrices

consists of all trajectories

w

A1 ∈

Rn×n ,

for which there exists a

σ1 x = A1 x σ2 x = A2 x

(2)

w = Cx, holds, where also the matrices

A1

amd

A2

commute:

A1 A2 = A2 A1 .

In particular,

A1 , A2

and

C

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Na_Ra_Ro(Lyapunov_stability_2D)_submitted_IJC 3

can be chosen so that the

state variable x is observable [(w, x)

satisfy (2) and

from

w,

i.e.,

w = 0] =⇒ [x = 0] .

There are several dierent characterization of observability in terms of the algebraic properties of the representation; the rst one is



 σ 1 I − A1 ker  σ2 I − A2  = {0}. C This condition is equivalent with the

extended observability matrix, dened as the column block

matrix



 C  CA1     CA2     CA2  1   O(A1 , A2 , C) :=  CA1 A2  ,    CA22      . .   . n−1 CA2

(3)

n. This implies that there exists a matrix E such that E ·O(A1 , A2 , C) = In , n × n identity matrix. Thus we obtain that x = X(σ1 , σ1−1 , σ2 , σ2−1 )w where

having rank equal to where

In

is the



 C  Cs1     Cs2     Cs1 s2    −1 2  ∈ R[s1 , s−1 , s2 , s−1 ]n×w . X(s1 , s−1 Cs 2 1 1 , s2 , s2 ) := E  1    Cs22     ..   .  Csn−1 2 In this case the

x

(4)

state map X(σ1 , σ1−1 , σ2 , σ2−1 ) is minimal, i.e., the dimension of the state variable

is minimal among all possible representations (2). In the following, discrete lines in the lattice

Z2

will play an important role; we now introduce

the basic notation and discuss behaviors restricted to lines. The set of lines in the rst orthant of

Z2

is dened by

L := {` ⊂ N2 | ` = {(a, b) ∈ N2 | α ∈ N}, a, b ∈ N

are coprime}.

The lines in the rst orthant corresponding to the vertical, respectively horizontal, axes, will be denoted in the following with dene the

`1

restriction of B to `

and

`2

B |` := {w` : Z → Rw | where

w |`

respectively. Given a

there exists

w∈B

denotes the restriction of the trajectory

trajectory, while

w |`

2D

behavior

B

and a line

` ∈ L,

we

as

w

such that

w | ` = w` } ,

to the domain

`.

Note that

w`

is a

1D

is a trajectory depending on two indices. It has been shown in (11, Th.6

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4

2D behavior restricted to a line is a 1D behavior; that is, B is the kernel of some 1D shift dened by a Laurent-polynomial matrix R(s) ∈ R[s, s−1 ]. It that if B is described by (2) and if B1 and B2 denote the restrictions of B to the

and Th.7) that a

polynomial operator in the is easy to see axes, then

Bi

is described in the state-space form as

σi x = Ai x w = Cx, i = 1, 2.

(5)

Note that these state representations may be non-minimal even if (2) is minimal.

In many modeling and control problems it is necessary to study quadratic functionals of the system variables and their derivatives; for example, in linear quadratic optimal control,

H∞ -

control, dissipativity theory, etc. Following the seminal work of (20), successively extended to the

2D

4 variables as a tool to express 2 independent variables and their shifts. We next review the

case in (7, 8, 16), we will use polynomial matrices in

quadratic functionals of functions of

denitions regarding these functionals which are most relevant to the problems treated in this paper. In the following, we use the multi-indices k = (k1 , k2 ) and l = (l1 , l2 ), and indeterminates by ζ = (ζ1 , ζ2 ) and η = (η1 , η2 ). We also denote ζ k = ζ1k1 ζ2k2 and η k = η1k1 η2k2 . We denote with Rw×w [ζ, η] the set of real polynomial w × w matrices in the 4 indeterminates ζ and η ; that is, an w×w [ζ, η] is of the form element of R

Φ(ζ, η) =

X

Φk,l ζ k η l

(6)

k,l

Φk,l ∈ Rw×w ; the sum ranges over a nite set of multi-indices k, l ∈ N2 . The 4-variable > > polynomial matrix Φ(ζ, η) is called symmetric if Φ(ζ, η) = Φ(η, ζ) , equivalently if Φk,l = Φl,k w×w [ζ, η], for all l and k. In this paper we restrict our attention to the symmetric elements in R w×w [ζ, η]. Any symmetric Φ induces a quadratic functional and denote this subset by Rs where

QΦ : (Rw )Z×Z × (Rw )Z×Z −→ (R)Z×Z X QΦ (w) = (σ k w)> Φk,l σ l (w) k,l

where the

k -th

shift operator

quadratic dierence form QDFs

QΦ1 , QΦ2

σk

is dened as

σ k = σ k1 σ k2

(similarly for

σ l ).

we say that

QΦ1

B is equivalent to QΦ on B, denoted by QΦ = QΦ 2

QΦ1 (w) = QΦ2 (w)

1

for all

QΦ the Φ. Given two

We call

(in the following abbreviated with QDF) associated with

2

, if

w∈B.

B

We call a QDF

QΦ

QΦ

nonnegative along B, denoted QΦ ≥ 0, if QΦ (w) ≥ 0 for all w ∈ B. We call B

B

positive along B, denoted QΦ > 0, if QΦ ≥ 0, and moreover ∀w ∈ B [QΦ (w) = 0] =⇒ [w = 0].

In the following we will be also using operations on QDFs. Given a QDF

` = {α(a, b) | α ∈ N} ∈ L,

we dene the

QΦ

and a line

increment of QΦ along the line `, denoted ∇` (QΦ ), as

∇` QΦ (w)(α(a, b)) := QΦ (w)((α + 1)(a, b)) − QΦ (w)(α(a, b)). The increment along the vertical, respectively horizontal, line will be denoted by respectively.

∇`1 , ∇`2

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3 2D stability and Lyapunov functions In this paper we will consider stability as dened in (18), i.e. with respect to a cone

S ⊂ R × R is ball in R × R,

S.

A set

cone if αS ⊂ S for all α ≥ 0. A cone S is solid if it contains an open and pointed if S ∩ (−S) = {(0, 0)}. A cone is proper if it is closed, pointed, solid, called a

and convex. Since an appropriate change of independent variables transforms any proper cone into the rst orthant, in the following we assume, without loss of generality, that orthant in

S

S

is the rst

Z2 . For the sake of brevity, in the following we will use the expression stable

instead

of stable with respect to the rst orthant. The denition of asymptotic stability that we shall use in the rest of this paper is the following; note that it is the discrete counterpart of that considered in the continuous-time case in (15) for

nD

behaviors.

Definition 3.1 :

Let

B

be a

2D

behavior.

B

is

asymptotically stable

if

[w ∈ B] =⇒ [∀ (a, b) ∈ N2 lim w(α(a, b)) = 0] . α→∞

It is straightforward to see that that

w` ,

the

1D

B

is asymptotically stable if and only if

trajectory associated with the restriction

w |` ,

∀ ` ∈ L

it holds

goes to zero as the independent

variable goes to innity. This denition of stability is equivalent to the denition considered in (19) for (nite dimensional)

2D behaviors. It was shown in (18) that in the discrete case all stable

behaviors according to Denition 3.1 are nite dimensional; note that in the continuous case this is not necessarily true. Having dened stability as in Denition 3.1, we now dene Lyapunov functions as follows.

Definition 3.2 :

if for all

`∈L

A functional

it holds that for

2

F (w)|` > 0 If

F

is a quadratic functional of

(QLF) for

2

F : (Rw )Z → (R)Z all w ∈ B ,

w∈B

is a

Lyapunov function

for a

2D

behavior

B

∇` F (w) < 0.

and

and its shifts, we call it a

quadratic Lyapunov function

B.

In order to state the main result of this section, a characterization of asymptotic stability in terms of Lyapunov functions, we need some preliminary concepts and results. The rst one is the notion of quadratic functionals of the state. Let let (2) be a minimal state representation of

w∈B matrix

and its shifts is a

P

B.

B

be a nite-dimensional

2D

behavior, and

We say that a quadratic functional

QΦ (w)

of

quadratic function of the state of B if there exists a symmetric constant

such that for all trajectories

the following result shows,

any

(w, x)

satisfying (2) it holds that

QΦ (w) = x> P x.

As

quadratic functional of the system variables and their shifts is a

quadratic functional of the state.

Let B be a 2D behavior, and let (2) be a state representation of B. Let 2 2 QΦ : (Rw )Z → (R)Z be a QDF. Then there exists a symmetric matrix P ∈ Rn×n such that for all w ∈ B with associated state trajectory x it holds

Proposition 3.3 :

QΦ (w) = x> P x .

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6

Proof: Write QΦ (w) =

N X

(σ1i1 σ2i2 w)> Φi1 ,i2 ,k1 ,k2 (σ1k1 σ2k2 w)

i1 ,i2 ,k1 ,k2 =0



  Φ0000 Φ0010 Φ0001 . . . w     Φ1000 Φ1010 Φ0011 . . .   σ1 w  = w> σ1 w> σ2 w> . . .  Φ0100 Φ0110 Φ0101 . . .   σ2 w  .    . . .

Now observe that

σ1i1 σ2i2 w = CAi11 Ai22 x;

. . .

. . .

..

.

. . .

consequently the last expression can be rewritten as



  Φ0000 Φ0010 Φ0001 . . . Cx     Φ1000 Φ1010 Φ0011 . . .   CA1 x  (Cx)> (CA1 x)> (CA2 x)> . . .  Φ0100 Φ0110 Φ0101 . . .   CA2 x  .    . . .

. . .

. . .

..

.

. . .

Now dene



 Φ0000 Φ0010 Φ0001 . . .  Φ1000 Φ1010 Φ0011 . . .    P := O(A1 , A2 , C)>  Φ0100 Φ0110 Φ0101 . . .  O(A1 , A2 , C) ;   . . .

. . .

. . .

..

.

the claim follows.

QΦ can be written as QΦ (w) = w ∈ B , where X(σ1 , σ1−1 , σ2 , σ2−1 ) variable x when acting on the tra-

It follows from the proof of Proposition 3.3 that every QDF

(X(σ1 , σ1−1 , σ2 , σ2−1 )w)> P (X(σ1 , σ1−1 , σ2 , σ2−1 )w) = x> P x

for

is a polynomial operator in the shift that induces the state jectory

w,

and whose expression is derived in a straightforward manner from the extended ob-

QΦ is equivalent on B to a QDF QΦ0 induced Φ0 (ζ1 , ζ2 , η1 , η2 ) = X(ζ1 , ζ2 )> P X(η1 , η2 ), with P symmetric and X inducing a state variable. We call such a QDF QΦ0 a canonical representative of QΦ . Note that if xb is another minimal state variable for B, then it is easy to see that xb = T x. Consequently, QΦ (w) = x b> Pbx b where Pb = (T > )−1 P T −1 . If a well-ordering (see (1)) has been w×w [ζ , ζ , η , η ], then a unique canonical representative can xed in the space of polynomials R 1 2 1 2

servability matrix (3). Consequently, every QDF by a 4-variable polynomial matrix of the form

be dened, see (7). We now give an example of the computation of a canonical representative of a QDF. Example 3.4 Let

B = ker R(σ1 , σ1−1 , σ2 , σ2−1 )

be a

2D

behavior where

 (s2 − 21 )(s1 − 14 ) 0  (s1 − 1 )(s2 − 1 ) 0  −1 2 4  , R(s1 , s−1 1 , s2 , s2 ) =  0 s1 − 13  0 s2 − 15 

and

Φ(η1 , η2 , ζ1 , ζ2 )

a

4-variable

polynomial matrix given by

Φ(η1 , η2 , ζ1 , ζ2 ) =

1 + η1 ζ1 ζ1 η1 η 2 ζ2

.

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It is a matter of straightforward verication to check that the variety of

B

consists of the points

1 1 1 1 1 1 V(B) = {( , ), ( , ), ( , )} ; 2 2 4 4 3 5 since

V(B)

is nite, it follows that

B

is nite-dimensional.

Following the procedure illustrated in (4), a minimal state-space realization of

B

as in (2) is

given by the matrices

1

1

 00 A1 =  0 13 0  0 0 14

 00 A2 =  0 15 0  0 0 14

2

It is easy to compute that matrices

A1 , A2

and

C,

2

x = X(σ1 , σ1−1 , σ2 , σ2−1 )w

C=

101 010

.

is the state variable corresponding to the

where

 −4(s1 − 12 ) 0 −1  0 1. X(s1 , s−1 1 , s2 , s2 ) = 4(s1 − 41 ) 0 

We now compute a canonical representative of

QΦ .

It easy to check that

QΦ (w) = x> P x,

where



 Φ0000 Φ0010 Φ0001 . . .  Φ1000 Φ1010 Φ0011 . . .    P = O(A1 , A2 , C)>  Φ0100 Φ0110 Φ0101 . . .  O(A1 , A2 , C)   . . .

. . .

. . .

..

.



1 0 1    2 100100   01      0 0 0 0 0 0  4 1 0 21 0 14 0 14 0 41 0 14 0     1 1 0 0 1 0 0 0 0 =  0 1 0 13 0 91 0 19 0 25 0 15 1   1 0 0 0 0 0 1 1 1 4 1 0 14 0 16 0 14 0 16 0 16 0  0 0 0 0 0 0 0 1 000001  4 0  1 4 0 Hence, a canonical representative for

QΦ

 0 1 1 0  0 41   1  5 0  1  0 6  5 1 4  25 0  =  1 1  3 0 4  9 1  8 9 0  1  0 16  1  25 0  1  0 16 1 15 0

1 3 1 25 1 3

9 8 1 3 9 8

 .

is

Φ0 (ζ1 , ζ2 , η1 , η2 ) = X(ζ1 , ζ2 )> P X(η1 , η2 ) . The notion of canonical representative of a QDF is important for the proof of the following theorem, that constitutes the main result of this section. It relates the behavior

B

2D

stability of the

with the 1D stability of the 1D behaviors resulting from the restriction of

B

2D

to the

axes and with the existence of a quadratic Lyapunov functional.

Let B be a nite dimensional 2D behavior and denote with B1 , B2 the restrictions of B to the vertical, respectively horizontal, axis. The following statements are equivalent: (1) B is stable; Theorem 3.5 :

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8

(2) B1 and B2 are stable 1D behaviors; (3) There exist three 4 variable polynomial matrices Φ, ∆1 and ∆2 such that Bi

Bi

and

Q∆i > 0

QΦ > 0,

(4) There exists a QLF for B. Proof: We begin with some general B

i = 1, 2.

considerations about nite-dimensional behaviors which

will make the proof of the result easier. Since trajectory of

B

∇ì QΦ =i −Q∆i ,

B

is nite dimensional,

V(B)

is nite and every

is a linear combination of polynomial exponentials of the form

wλ1 ,λ2 (k1 , k2 ) = pλ1 ,λ2 (k1 , k2 )λk11 λk22 , for some suitable nonzero

w-vector

polynomial function

pλ1 ,λ2 (k1 , k2 ) =

X

pλ1 ,λ2 ,

(7)

i.e.

αij k1i k2j ,

ij∈I where

I ⊂ N2

is a nite bi-index set and

αij ∈ R2 .

This implies that the trajectories of

Bi

are

linear combinations of trajectories of the form

pλ1 ,λ2 (ki ei )λki i , i = 1, 2. Furthermore, it follows from (18, Th.8) that

(λ1 , λ2 ) ∈ V(B), 1) ⇒ 2):

Let

it holds that

b1 ∈ V(B1 ). λ

|λα1 1 λα2 2 | < 1

B

is stable if and only if

for all

By assumption

B

(8)

V(B)

is nite and for all

(α1 , α2 ) ∈ N2 . w∈B B1 are of the form described in (8) with b k1 λ bk2 pbλ1 ,bλ2 (k1 , k2 )λ 1 2 ∈ B . Since B is stable it

is nite dimensional and the trajectories of

are of the form described in (7). Also, the trajectories of

i = 1.

This implies that there exists

b2 λ

such that

2 b α1 λ b α2 |λ 1 2 | < 1, for all (α1 , α2 ) ∈ N . In particular, if α2 = 0 we < 1 for all α1 ∈ N, which amounts to saying that B1 is stable. that B2 is stable.

follows that

obtain that we have

b α1 | that |λ 1

The same argument

shows

2) ⇒ 1):

(λ1 , λ2 ) ∈ V(B). Since Bi is stable, i = 1, 2 and λ1 ∈ V(B1 ) and λ2 ∈ V(B2 ) |λα1 1 | < 1 and |λα2 2 | < 1 for all (α1 , α2 ) ∈ N2 . Therefore |λα1 1 λα2 2 | < 1 for all (α1 , α2 ) ∈ N2 , i.e., B is stable. Let

we have that

2) ⇒ 3):

Consider a representation of

we have that

A1 , A2

B

as in (2). Then

Bi

is described by (5); by assumption

are Schur matrices, i.e. all their eigenvalues have modulus less than one.

Since these matrices commute, there exists (see (13)) a matrix

e1 > 0 P > 0, ∆

and

e2 > 0 ∆

of

suitable sizes such that

e A> i P Ai − P = −∆i ,

for

i = 1, 2.

Dene

QΦ (w) := (X(σ1 , σ1−1 , σ2 , σ2−1 )w)> P X(σ1 , σ1−1 , σ2 , σ2−1 )w = x> P x, e i X(σ1 , σ −1 , σ2 , σ −1 )w = x> ∆ e ix , Q∆i (w) := (X(σ1 , σ1−1 , σ2 , σ2−1 )w)> ∆ 1 2 i = 1, 2, where X is the polynomial operator in the shift inducing the state variable x. Now > > since ∇ì QΦ (w) = x (Ai P Ai − P )x, ∀ w ∈ B, i = 1, 2, it is easy to verify that QΦ , Q∆1 and

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Q∆2

satisfy the conditions of statement 3).

3) ⇒ 4): We show that QΦ

B . It follows from the Proposition 3.3 that there exists QΦ (w) = x> P x with x = X(σ1 , σ1−1 , σ2 , σ2−1 )w. Observe that the 1D dynamics along a line ` = {α(i, j) | α ∈ N} ∈ L are described by σ` x = Ai1 Aj2 x, w = Cx, for some i, j xed but otherwise arbitrary. Thus, it is enough to prove that P i j > i j i j satises a matrix Lyapunov equation for A1 A2 , i.e. (A1 A2 ) P (A1 A2 ) − P < 0. Observe rst that is a QLF for

a symmetric polynomial matrix

P

such that

(Ai1 Aj2 )> P (Ai1 Aj2 ) − P = (Aj2 )> (Ai1 )> P (Ai1 )(Aj2 ) − P < (Aj2 )> P (Aj2 ) − P, where we have used the fact that

i > i [A> 1 P A1 − P < 0] ⇒ [(A1 ) P (A1 ) − P < 0] as

can

be

readily

(Aj2 )> P (Aj2 ) − P < 0

proved

by

induction.

From

the

same

argument

it

follows

that

(Ai1 Aj2 )> P (Ai1 Aj2 ) − P < 0. Now dene QΦ (w) := x> P x, holds that ∇` QΦ < 0, as was to be proved.

and consequently

and conclude that along the line

`

it

4) ⇒ 2) It is a matter of straightforward verication to check that x|ì is a state trajectory for 1D behavior Bi , i = 1, 2. Moreover, if P is a matrix corresponding to a canonical representa> tive of the Lyapunov function QΦ , then (x|ì ) P x|ì is a Lyapunov function for Bi . This implies that for i = 1, 2 it holds that wi goes to zero along the line ì , i = 1, 2, i.e., Bi is stable. the

We are now in the position of stating a characterization of Lyapunov functions in terms of canonical representatives. Definition 3.6 :

(CLS) for

A1

and

Let

A2

A1 , A2 ∈ Rn×n . A matrix P > 0 is said to be a common

Lyapunov solution

if

A> 1 P A1 − P < 0 A> 2 P A2 − P < 0.

(9)

Using this denition, we can give yet another equivalent statement to those of Theorem 3.5. Proposition 3.7 : Let B be a 2D behavior and Φ be a QDF. Φ is a QLF for B if and only if given any minimal state-space representation (A1 , A2 , C) of B together with a state map X ∈ Rn×w [ξ1 , ξ2 ] inducing the state variable x for the representation, the following two conditions hold: (1) There exists a symmetric matrix P > 0 which is a CLS for A1 and A2 ; (2) the canonical representative of QΦ is equal to X > P X . Proof: The claim follows easily using the same arguments of the proof 4) ⇒ 2), 2) ⇒ 3) and

3) ⇒ 4)

of Theorem 3.5.

An interesting question is the construction of all Lyapunov functions for a given

B;

4

2D

behavior

this is addressed in the next section.

Construction of Lyapunov functions

The result of Proposition 3.7 shows that the problem of nding a Lyapunov function can be reduced to that of nding a common solution to a pair of discrete-time

1D

Lyapunov equations.

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10 In this section we provide a constructive algorithm for the construction for Lyapunov functions for a given stable

2D

behavior.

We present some preliminary results which will be instrumental to this end. The rst one introduces two useful maps. In the following we denote with Definition 4.1 :

Let

A1 , A2 ∈ Rn×n .

The

Rn×n s

Lyapunov map

the

n × n symmetric matrices.

associated with

Ai , i = 1, 2

are

dened as

Li : Rn×n → Rn×n s s P 7→ A> i P Ai − P.

(10)

The name Lyapunov map is adopted following (14).

If the matrices A1 , A2 ∈ Rn×n in (10) commute then the Lyapunov maps L1 and L2 associated with A1 , A2 commute.

Lemma 4.2 :

Proof: (L1 L2 )(P ) = A>1 [A>2 P A2 − P ]A1 − [A>2 P A2 − P ] > > > = A> 1 A2 P A2 A1 − A1 P A1 − A2 P A2 + P > > > = A> 2 A1 P A1 A2 − A2 P A2 − A1 P A1 + P > > = A> 2 [A1 P A1 − P ]A2 − [A1 P A1 − P ]

= (L2 L1 )(P ).

The next result states a necessary and sucient condition for the existence of a common Lyapunov solution for

A1

and

A2 ,

given in terms of the Lyapunov maps

Li .

Let the matrices A1 , A2 ∈ Rn×n in (10) be Schur commuting matrices. Then the associated Lyapunov maps Li are invertible. Moreover, a matrix P > 0 is a CLS for A1 and A2 if and only if there exists a matrix S > 0 such that

Theorem 4.3 :

−1 P = (L−1 2 L1 )(S).

Proof:

The fact that the maps

Li

(11)

are invertible follows from standard knowledge regarding

stability of 1D systems. We prove the second part of the claim. (⇐): This part of the claim can be proved using the same argument of the proof of statement

L2 (P ) = L−1 1 (S) < 0 because of the > 0. Thus, P is a Lyapunov solution for A2 . In the same way we prove that P is a Lyapunov solution for A1 . Thus P is a CLS for A1 and A2 . (⇒): Let P be a CLS for A1 and A2 , and dene Qi = Li (P ), i = 1, 2. Note that Qi < 0, i = 1, 2. −1 −1 −1 Dene S = L1 (Q2 ) = L1 (L2 (P )) = L2 (L1 (P )) = L2 (Q1 ). Then, (L2 L1 )(S) = L2 (Q2 ) = P . Note that S > 0.

ii)

of Theorem 1 of (13). By assumption we have that

−1 linearity of maps Li , as L1 (−S)

The result of Theorem 4.3 characterizes the common Lyapunov solutions for a pair of Schur commuting matrices; it also constitutes a generalization of Theorem 1 of (13), since it shows that the condition (11) is not only sucient, but also necessary for the existence of a common Lyapunov function. Moreover, Theorem 4.3 also suggests an algorithm to compute a CLS by inversion of the maps

Li .

We now proceed to investigate further the properties of these maps; a similar approach has been adopted for one polynomial Lyapunov equation in (14). The assumption that the matrices

Ai , i = 1, 2,

have a basis of common eigenvectors is crucial in our approach; consequently, we

concentrate on the case in which both case.

A1

and

A2

are diagonalizable. Note that this is the generic

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Lemma 4.4 : Let A1 and A2 be two n × n diagonalizable commuting matrices and Li the Lyapunov maps associated with A1 , A2 . Then the set

n

V = {(v, λ, µ) ∈ R × R × R |

Iλ − A1 v = 0} Iµ − A2

has cardinality n; denote its i-th element with (vi , λi , µi ), i = 1, . . . , n. Moreover, the set Vb = {b vij = vi vj> + vj vi> | 1 ≤ i ≤ j ≤ n}

forms a basis of common eigenvectors of L1 and L2 of dimension the eigenvalues

n(n+1) 2

, each associated with

λi λj − 1 µi µj − 1 ,

for 1 ≤ i ≤ j ≤ n. Proof: That the

set

V

has cardinality

n

follows from the well-known fact that commuting

matrices are diagonalizable if and only if they have a basis of common eigenvectors. The claim

vbij are eigenvectors of Li follows easily after verifying that L1 (b vij ) = (λi λj −1)b vij and L2 (b vij ) = (µi µj − 1)b vij ; note that these formulas also show what the eigenvalues associated with each eigenvector are. In order to prove that these matrices form a basis for the set of n × n symmetric matrices, observe that any linear combination of the v bij can be written down as that the matrices

 v1>  v>   2   ..  K v1 v2 · · · vn  .  vn> 

with

K

a nonsingular symmetric matrix. The linear independence of the

the linear independence of the vectors

vi , i = 1, . . . , n.

vbij

then follows from

Ai , i = 1, 2 are diagonalizable, a basis Li , i = 1, 2, can be computed in a straightforward way from a eigenvectors of A1 and A2 . Consequently, the inversion necessary to compute

The result of Lemma 4.4 shows that if the matrices of common eigenvectors of basis of common the matrix

P

as in Theorem 4.3 is a straightforward matter. These considerations lead us to

stating the following algorithm to compute Lyapunov functions for a given stable behavior Lyapunov function for

B,

a

B. Algorithm

Input: A stable, nite-dimensional behavior Output:

Φ ∈ Rw×w [ζ1 , ζ2 , η1 , η2 ]

inducing a Lyapunov function for

Step 1: Compute a representation of

inducing the state variable

x

Step 2: Using the matrices

4.4.

B;

B

B.

as in (2), together with a state map

X ∈ Rn×w [ξ1 , ξ2 ]

for the representation.

A1

and

A2

from Step 1 construct

V

and

Vb

as described in Lemma

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12 Step 3: Select

αij , 1 ≤ i ≤ j ≤ n

such that

X

S=

αij vbij > 0.

1≤i≤j≤n

Step 4: Output

Φ(ζ1 , ζ2 , η1 , η2 ) := (X(ζ1 , ζ2 ))>

X 1≤i≤j≤n

αij vbij X(η1 , η2 ). (λi λj − 1)(µi µj − 1)

Some remarks are in order.

{b vij }1≤i≤j≤n forms a basis for the space of n×n symmetric matrices, S in the step 3 generates all negative denite matrices; consequently the algorithm produces all possible Lyapunov functions for a given stable 2D behavior. P > In particular, a selection of S can be performed for example by taking S = − 1≤i≤j≤n vi vi .

Remark 1 :

Note that since

the construction of the matrix

Remark 2 :

The increments

∇ì ; i = 1, 2

of the Lyapunov function

QΦ

computed in Step 4 are

easily seen to be equal respectively to

 ∆1 (ζ1 , ζ2 , η1 , η2 ) = X(ζ1 , ζ2 )> 

 X

1≤i≤j≤n

1 αij vbij  X(η1 , η2 ), λi λj − 1 

 ∆2 (ζ1 , ζ2 , η1 , η2 ) = X(ζ1 , ζ2 )> 

X

1≤i≤j≤n

Remark 3 :

1 µi µj − 1

αij vbij  X(η1 , η2 ).

It follows from Theorem 9.1.1 of (6) that there exist non-diagonalizable matrices

which do not have a basis of common

generalized

eigenvectors. The problem of nding ecient

algorithms to compute a common Lyapunov solution in this non-generic case is a matter of further investigation.

Consider (A1 .A2 , X) as in the example 3.4. In step 3, take, for instance S = I3 , the 3 × 3 identity matrix. Step 4 produces

Example 4.5

Φ(ζ1 , ζ2 , η1 , η2 ) =

5

1 1 9(ζ1 − 12 )(η1 − 12 ) + 225 16 (ζ1 − 4 )(η1 − 4 ) 0 64 0 75

.

Conclusions

In this paper we have illustrated a Lyapunov approach to the stability of nite-dimensional

2D

systems. We have adopted as denition of stability the one given in Def. 3.1, namely the

asymptotic stability along all lines in the rst orthant. The main results are Theorem 3.5, which characterizes stability in terms of the existence of a Lyapunov function, dened as a quadratic functional of the system variables which is positive along all lines, and whose increments are negative along all lines; and the algorithm given in section 4 for the computation of Lyapunov functions for a stable

2D

behavior

B.

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Acknowledgement The second author gratefully acknowledges the nancial support of

Engineering and Physical Sciences Research Council

The Royal Society

and of the

for nancially supporting the visit to the

Faculty of Electrical Engineering and Computer Science of the University of Oporto (Portugal) during which the results presented in this paper were obtained. The research of the other two authors has been nanced by the Fundação para a Ciência e a Tecnologia, through the R&D Unit Centro de Investigação e Desenvolvimento em Matématica e Aplicações.

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