BOUNDS FOR THE SOLUTION OF DISCRETE COUPLED

LYAPUNOV EQUATION

Adam Czornik

Silesian Technical University, Department of Automatic Control and Robotics

Akademicka Street 16, 44-100 Gliwice, Poland

Aleksander Nawrat

Silesian Technical University, Department of Automatic Control and Robotics

Akademicka Street 16, 44-100 Gliwice, Poland

Keywords:

Coupled Lyapunov equation, jump linear systems, eigenvalues bounds.

Abstract:

Upper and lower matrix bounds for the solution of the discrete time coupled algebraic Lyapunov equation for

linear discrete-time system with Markovian jumps in parameters are developed. The bounds of the maximal,

minmal eigenvalues, the summation of eigenvalues, trace and determinant are also given.

1 INTRODUCTION

It is well known that algebraic Lyapunov and Riccati

equations are widely applied to various engineering

areas including different problems in signal process-

ing and, especially, control theory. In the area of

control system analysis and design, these equations

play crucial role in system stability and boundedness

analysis, optimal and robust controllers and ﬁlters de-

sign, the transient behavior estimates, etc. During the

past two decades many bounds for the solution of var-

ious types of algebraic Lyapunov and Riccati equation

have been reported. The surveys of such results can be

found in (Mori and Derese, 1984), (Komaroff, 1996),

(Kwon et al., 1996), (Czornik and Nawrat, 2000). The

reasons that the problem to estimate upper and lower

bounds of these equations has become an attractive

topic are that the bounds are also applied to solve

many control problems such as stability analysis (Lee

et al., 1995), (Patel and Toda, 1980), time-delay sys-

tem controller design (Mori et al., 1983), estimation

of the minimal cost and the suboptimal controller de-

sign (Langholz, 1979), convergence of numerical al-

gorithms (Allwright,1980), robust stabilization prob-

lem (Boukas et al., 1997). Eigenvalue bounds can be

also used to determine whether or not the system un-

der consideration possesses the singularly perturbed

structure (Gajic and Qureshi, 1995). An excellent mo-

tivation to study the bounds for Lyapunov equation is

given in (Gajic and Qureshi, 1995) (Section 2.2). The

authors advocated the results in this area by saying

that sometimes we are just interested in the general

behavior of the underlying system and then the behav-

ior can be determined by examining certain bounds on

the parameters of the solution instead of the full solu-

tion.

Considering the linear dynamical systems with

Markovian jumps in parameter values, which have re-

cently attracted a great deal of interest, instead of one

equation a set of coupled algebraic equations arises.

They are called coupled algebraic Riccati and cou-

pled Lyapunov equation. All the reasons mentioned

above could be repeated to show how the bounds for

coupled algebraic Lyapunov equations can be used.

Bounds for the coupled Riccati equation have been al-

ready obtained in (Czornik and Swierniak,2001a) and

(Czornik and Swierniak, 2001b). To our knowledge

this paper is the ﬁrst where the bounds for coupled

algebraic Lyapunov equations are established.

The eigenvalues λ

i

(X), where i = 1, ..., n, of

a symmetric matrix X ∈ R

n×n

are assumed to be

arranged such that

λ

1

(X) ≥ λ

2

(X) ≥ ... ≥ λ

n

(X) .

When we consider the discrete time jump linear sys-

tem the following discrete coupled algebraic Lya-

punov equation (DCALE) arises (Chizeck et al.,

1986):

P

i

= Q

i

+ A

′

i

F

i

A

i

(1)

where

F

i

=

X

j∈S

p

ij

P

j

(2)

and A

i

, Q

i

, P

i

∈ R

n×n

, p

ij

∈ [0, 1] ,

P

j∈S

p

ij

=

1, i ∈ S , S is a ﬁnite set. The numbers p

ij

are the

transitions probabilities of a Markov chain.

We need the following lemma.

11

Czornik A. and Nawrat A. (2006).

BOUNDS FOR THE SOLUTION OF DISCRETE COUPLED LYAPUNOV EQUATION.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 11-15

DOI: 10.5220/0001201900110015

Copyright

c

SciTePress

Lemma 1 (Marshall and Olkin, 1979)Let X, Y ∈

R

n×n

with X = X

′

, Y = Y

′

, X, Y ≥ 0. Then the

following inequalities hold

λ

i+j−1

(XY ) ≤ λ

i

(X)λ

j

(Y ) , if i+j ≤ n +1 (3)

λ

i+j−n

(XY ) ≥ λ

i

(X)λ

j

(Y ) , if i+j ≥ n+1 (4)

l

X

k=1

λ

k

(X + Y ) ≤

l

X

k=1

λ

k

(X) +

l

X

k=1

λ

k

(Y ) (5)

l

X

k=1

λ

n−k+1

(X + Y ) ≥

l

X

k=1

λ

n−k+1

(X) +

l

X

k=1

λ

n−k+1

(Y ) . (6)

2 MAIN RESULTS

The next theorem contains the main result of the pa-

per.

Theorem 2 For the eigenvalues λ

k

(P

i

) , k =

1, ..., n, i ∈ S of positive deﬁnite solution P

i

, i ∈ S of

DCALE (1), the following inequalities hold

l

X

k=1

λ

k

(P

i

) ≤

l

X

k=1

λ

k

(Q

i

) +

max

j∈S

p

ij

λ

1

(A

i

A

′

i

) ·

·

P

i∈S

P

l

k=1

λ

k

(Q

i

)

1− max

j∈S

λ

1

A

j

A

′

j

max

j∈S

P

i∈S

p

ij

=

= α (l, i) , (7)

for l = 1, ..., n, if max

j∈S

λ

1

A

j

A

′

j

max

j∈S

P

i∈S

p

ij

<

1, and

l

X

k=1

λ

n−k+1

(P

i

) ≥

l

X

k=1

λ

n−k+1

(Q

i

) +

min

j∈S

p

ij

min

j∈S

λ

n

A

j

A

′

j

P

i∈S

P

l

k=1

λ

n−k+1

(Q

i

)

1− min

j∈S

λ

n

A

j

A

′

j

min

j∈S

P

i∈S

p

ij

=

= β (l, i) , (8)

for l = 1, ..., n, if min

j∈S

λ

n

A

j

A

′

j

min

j∈S

P

i∈S

p

ij

<

1.

Proof. From (1) it follows, by using (5) and (3), that

l

X

k=1

λ

k

(P

i

) ≤

l

X

k=1

λ

k

(Q

i

) +

l

X

k=1

λ

k

(A

′

i

F

i

A

i

) =

=

l

X

k=1

λ

k

(Q

i

) +

l

X

k=1

λ

k

(F

i

A

i

A

′

i

)

≤

l

X

k=1

λ

k

(Q

i

) + λ

1

(A

i

A

′

i

)

l

X

k=1

λ

k

(F ) . (9)

Applying (5) to (2) leads to

l

X

k=1

λ

k

(F ) ≤

X

j∈S

p

ij

l

X

k=1

λ

k

(P

j

)

!

. (10)

Combining (9) with (10) yields to

l

X

k=1

λ

k

(P

i

) ≤

l

X

k=1

λ

k

(Q

i

) +

+ max

j∈S

λ

1

A

j

A

′

j

X

j∈S

p

ij

l

X

k=1

λ

k

(P

j

)

!

. (11)

Summing the above inequality over i ∈ S we have

X

i∈S

l

X

k=1

λ

k

(P

i

) ≤

X

i∈S

l

X

k=1

λ

k

(Q

i

)

+ max

j∈S

λ

1

A

j

A

′

j

X

i,j∈S

p

ij

l

X

k=1

λ

k

(P

j

)

!

=

X

i∈S

l

X

k=1

λ

k

(Q

i

) + max

j∈S

λ

1

A

j

A

′

j

·

·

X

j∈S

X

i∈S

p

ij

!

l

X

k=1

λ

k

(P

j

)

!

≤

X

i∈S

l

X

k=1

λ

k

(Q

i

) + max

j∈S

λ

1

A

j

A

′

j

·

·

max

j∈S

X

i∈S

p

ij

!

X

i∈S

l

X

k=1

λ

k

(P

i

) .

Solving this inequality respect to

P

i∈S

P

l

k=1

λ

k

(P

i

) and taking into account

that

max

j∈S

λ

1

A

j

A

′

j

max

j∈S

X

i∈S

p

ij

< 1

we obtain

X

i∈S

l

X

k=1

λ

k

(P

i

) ≤

ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

12

P

i∈S

P

l

k=1

λ

k

(Q

i

)

1− max

j∈S

λ

1

A

j

A

′

j

max

j∈S

P

i∈S

p

ij

. (12)

(9) implies also that

l

X

k=1

λ

k

(P

i

) ≤

l

X

k=1

λ

k

(Q

i

) +

λ

1

(A

i

A

′

i

)

X

j∈S

p

ij

l

X

k=1

λ

k

(P

j

)

!

≤

l

X

k=1

λ

k

(Q

i

) +

max

j∈S

p

ij

·

·λ

1

(A

i

A

′

i

)

X

j∈S

l

X

k=1

λ

k

(P

j

) . (13)

Applying (12) on the right hand side of (13) we have

(7).

To proof (8) let’s observe that the use of (6) and (4)

to (1) to gives

l

X

k=1

λ

n−k+1

(P

i

) ≥

l

X

k=1

λ

n−k+1

(Q

i

) +

l

X

k=1

λ

n−k+1

(A

′

i

F

i

A

i

) =

l

X

k=1

λ

n−k+1

(Q

i

) +

l

X

k=1

λ

n−k+1

(F

i

A

i

A

′

i

) ≥

l

X

k=1

λ

n−k+1

(Q

i

) + λ

n

(A

i

A

′

i

) ·

l

X

k=1

λ

n−k+1

(F

i

) ≥

l

X

k=1

λ

n−k+1

(Q

i

) +

min

j∈S

λ

n

A

j

A

′

j

l

X

k=1

λ

n−k+1

(F

i

) . (14)

Summing (14) over i ∈ S we have

X

i∈S

l

X

k=1

λ

n−k+1

(P

i

) ≥

X

i∈S

l

X

k=1

λ

n−k+1

(Q

i

) +

min

j∈S

λ

n

A

j

A

′

j

X

i∈S

l

X

k=1

λ

n−k+1

(F

i

) . (15)

Applying (6) to (2) leads to

l

X

k=1

λ

n−k+1

(F

i

) ≥

X

j∈S

p

ij

l

X

k=1

λ

n−k+1

(P

j

) . (16)

Combining (15) with (16) yields to

X

i∈S

l

X

k=1

λ

n−k+1

(P

i

) ≥

X

i∈S

l

X

k=1

λ

n−k+1

(Q

i

) +

min

j∈S

λ

n

A

j

A

′

j

·

·

X

i∈S

X

j∈S

p

ij

l

X

k=1

λ

n−k+1

(P

j

)

=

X

i∈S

l

X

k=1

λ

n−k+1

(Q

i

) + min

j∈S

λ

n

A

j

A

′

j

·

·

X

j∈S

X

i∈S

p

ij

!

l

X

k=1

λ

n−k+1

(P

j

)

!

≥

X

i∈S

l

X

k=1

λ

n−k+1

(Q

i

) + min

j∈S

λ

n

A

j

A

′

j

·

· min

j∈S

X

i∈S

p

ij

X

i∈S

l

X

k=1

λ

n−k+1

(P

i

) .

Solving this inequality with respect to

P

i∈S

P

l

k=1

λ

n−k+1

(P

i

) and taking into account

that

min

j∈S

λ

n

A

j

A

′

j

min

j∈S

X

i∈S

p

ij

< 1

we obtain

X

i∈S

l

X

k=1

λ

n−k+1

(P

i

) ≥

P

i∈S

P

l

k=1

λ

n−k+1

(Q

i

)

1− min

j∈S

λ

n

A

j

A

′

j

min

j∈S

P

i∈S

p

ij

. (17)

Combining (14) and (16) we conclude that

l

X

k=1

λ

n−k+1

(P

i

) ≥

l

X

k=1

λ

n−k+1

(Q

i

) +

min

j∈S

p

ij

·

· min

j∈S

λ

n

A

j

A

′

j

X

i∈S

l

X

k=1

λ

n−k+1

(P

i

) .

Applying (17) to the right hand side of the above in-

equality we obtain (8).

Using the Theorem 2 we can establish the follow-

ing general matrix bound for the solution of DCALE

(1).

Theorem 3 For the positive deﬁnite solution P

i

, i ∈

S of DCALE (1) we have

P

i

≤

X

j∈S

p

ij

α (1, j)

A

′

i

A

i

+ Q

i

, (18)

BOUNDS FOR THE SOLUTION OF DISCRETE COUPLED LYAPUNOV EQUATION

13

if max

j∈S

λ

1

A

j

A

′

j

max

j∈S

P

i∈S

p

ij

< 1 and

P

i

≥

X

j∈S

p

ij

β (1, j)

A

′

i

A

i

+ Q

i

(19)

if min

j∈S

λ

n

A

j

A

′

j

min

P

i∈S

p

ij

j∈S

< 1, where α (1, j)

and β (1, j) are given in Theorem 2.

Proof. In (Rugh, 1993) it has been shown that for any

symmetric matrix T ∈ R

n×n

and x ∈ R

n

λ

n

(T )x

′

x ≤ x

′

T x ≤ λ

1

(T )x

′

x.

Using this inequality to (1) we have

X

j∈S

p

ij

λ

n

(P

j

)

A

′

i

A

i

+ Q

i

≤ P

i

≤

≤

X

j∈S

p

ij

λ

1

(P

j

)

A

′

i

A

i

+ Q

i

.

Combining this inequality with (7) and (8) for l = 1

we get the conclusions of the theorem.

From Theorem 3 on the obvious way the bounds

for det (P

i

), t r (P

i

), λ

i

(P ) can be obtained and they

are collected in the next Remark.

Remark 1 For the positive deﬁnite solution P

i

, i ∈ S

of DCALE (1) we have

tr (P

i

) ≤

X

j∈S

p

ij

α (1, j)

tr (A

′

i

A

i

) + trQ

i

,

det (P

i

) ≤ det

X

j∈S

p

ij

α (1, j)

(A

′

i

A

i

) + Q

i

,

λ

k

(P

i

) ≤ λ

k

X

j∈S

p

ij

α (1, j)

(A

′

i

A

i

) + Q

i

,

if max

j∈S

λ

1

A

j

A

′

j

max

j∈S

P

i∈S

p

ij

< 1 and

tr (P

i

) ≥

X

j∈S

p

ij

β (1, j)

tr (A

′

i

A

i

) + tr (Q

i

)

det (P

i

) ≥ det

X

j∈S

p

ij

β (1, j)

(A

′

i

A

i

) + Q

i

λ

k

(P

i

) ≥ λ

k

X

j∈S

p

ij

β (1, j)

tr (A

′

i

A

i

) + Q

i

,

if min

j∈S

λ

n

A

j

A

′

j

min

P

i∈S

p

ij

j∈S

< 1. Where α (1, j)

and β (1, j) are given in Theorem 2.

Now we have bounds of λ

1

(P

i

) and tr (P

i

) in The-

orem 2 and in Corollary 1 similar for the lower bounds

of λ

n

(P

i

) and tr (P

i

) , but in general is difﬁcult to say

which one are better, however the example presented

in the next section suggests that the bounds from The-

orem 3 can be better.

3 NUMERICAL EXAMPLE

Consider the following fourth-order jump linear sys-

tem with three switching modes (Gajic and Qureshi,

1995): S = {1, 2, 3}

[p

ij

]

i,j∈S

=

"

0.1 0.3 0.6

0.5 0.25 0.25

0 0.3 0.7

#

A

1

=

0.0667 0.0665 0.0844 −0.2257

0.1383 −0.1309 0.0797 0.1162

0.0658 0.0298 0.0645 −0.1018

−0.2283 0.2438 −0.1990 0.2997

A

2

=

0.1885 −0.3930 −0.0894 −0.1919

−0.4230 0.3598 −0.1224 −0.1548

0.0350 −0.1950 −0.1967 −0.1017

−0.2648 −0.2440 −0.0542 0.0484

A

3

=

0.2746 0.0634 0.3414 −0.0692

0.0796 0.4167 0.0283 −0.1207

−0.1607 0.0344 −0.2227 0.1617

0.1175 −0.2969 0.4149 0.3314

Q

1

= Q

2

= Q

3

= I

4

.

For the solution P

1

, P

2

, P

3

we have

λ

1

(P

1

) = 1.3533, λ

2

(P

1

) = 1.1182, λ

3

(P

1

) = 1.0124,

λ

4

(P

1

) = 1.0000,

λ

1

(P

2

) = 1.7003, λ

2

(P

2

) = 1.2309, λ

3

(P

2

) = 1.0979,

λ

4

(P

2

) = 1.0104,

λ

1

(P

3

) = 1.6385, λ

2

(P

3

) = 1.3763, λ

3

(P

3

) = 1.0665,

λ

4

(P

3

) = 1.0019,

(7) and (8) give the following bounds

λ

1

(P

1

) ≤ 3.5806, λ

4

(P

1

) ≥ 1

λ

1

(P

2

) ≤ 4.7421, λ

4

(P

2

) ≥ 1

λ

1

(P

3

) ≤ 6.2760, λ

4

(P

3

) ≥ 1

which are not satisfying. However (18) and gives

1.0000 ≤ λ

4

(P

1

) ≤ 1.0000, 1.0101 ≤ λ

3

(P

1

) ≤ 1.0562,

1.0809 ≤ λ

2

(P

1

) ≤ 1.4486, 1.2928 ≤ λ

1

(P

1

) ≤ 2.6237,

1.0098 ≤ λ

4

(P

2

) ≤ 1.0445, 1.0807 ≤ λ

3

(P

2

) ≤ 1.3667,

1.2054 ≤ λ

2

(P

2

) ≤ 1.9334, 1.5095 ≤ λ

1

(P

2

) ≤ 3.3156,

1.0014 ≤ λ

4

(P

3

) ≤ 1.0082, 1.0550 ≤ λ

3

(P

3

) ≤ 1.3196,

1.3130 ≤ λ

2

(P

3

) ≤ 2.8202, 1.5134 ≤ λ

1

(P

3

) ≤ 3.9861.

ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

14

4 CONCLUSION

Upper and lower matrix bounds for the solution of

DCALE have been developed. By these bounds, the

corresponding eigenvalue bounds (i.e. for each eigen-

values including the extreme ones, the trace and the

determinant) have been deﬁned in turn.

ACKNOWLEDGEMENTS

The work has been supported by KBN grant No 0

T00B 029 29 and 3 T11A 029 028.

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