Further results on the measurement of solution bounds of the generalized Lyapunov equations

Journal of the Franklin Institute 340 (2003) 461–480

Further results on the measurement of solutionbounds of the generalized Lyapunov equations

C.H. Lee*

Department of Electrical Engineering, Cheng Shiu University, Kaohsiung 833, Taiwan ROC

Abstract

This paper discusses further results for the bounds of the solutions of the algebraic matrix

Generalized Lyapunov Equations (GLE). Several iterative procedures for more precise

estimations are proposed. Furthermore, some new matrix and eigenvalue bounds for the

solutions of the GLE are measured by making use of linear algebraic techniques. It is also

shown the majority of existing matrix bounds of the continuous and discrete Lyapunov

equations are the special cases of ours.

r 2003 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Generalized Lyapunov Equations; Matrix bound; Eigenvalue bound; Linear algebraic

technique

Notation

AT complex conjugate transpose of AACn�n

A > ðXÞ0 the matrix AACn�n is a positive (semi)definite Hermitian matrixliðAÞ the ith eigenvalue of a positive definite Hermitian matrix A and liðAÞ

is arranged in non-increasing order; i.e., l1ðAÞXl2ðAÞX?XlnðAÞjaj the absolute value of a complex number a%a the complex conjugate of a complex number asiðAÞ the ith singular value of a matrix AACn�n and siðAÞ is arranged in

non-increasing order; i.e., s1ðAÞXs2ðAÞX?XsnðAÞmðAÞ the matrix measure of AACn�n; mðAÞ ¼ l1ððA þ ATÞ=2Þ

ARTICLE IN PRESS

*Tel./fax: +886-7-554-6722.

E-mail address: [email protected] (C.H. Lee).

0016-0032/$30.00 r 2003 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfranklin.2003.10.001

1. Introduction

Stability is an important characteristic and the most fundamental requirement forcontrolled systems analysis and design. Therefore, various approaches have beenproposed to solve the stability analysis problem for linear system during the pastseveral decades. Among those approaches, the Lyapunov Theory possibly is themost used one. Continuous and discrete Lyapunov type equations are used to dealwith the above problem. Furthermore, the bounds of the solutions for the Lyapunovequations can also be utilized to treat many control problems such as robust stabilityanalysis, perturbation bound estimation, and so on. Therefore, a number of workshave also been presented to estimate the solution bounds of the continuous anddiscrete Lyapunov equations [1–11]. However, it is known that all the continuousand discrete Lyapunov equations are particular cases of the Generalized LyapunovEquations (GLE) [12]. In literature, the GLE can be applied to solve the rootclustering problem for linear system with/without parametric perturbations [13–16].However, it might be troublesome to solve the GLE especially when the dimensionsof a system become large. Recently, by extending the methods developed by Lee[5–7] and making use of linear algebraic techniques, the estimation problem of thesolution bounds of the GLE was first treated in [17]. Some matrix bounds andseveral eigenvalue bounds for the solutions of the GLE were developed. It is shownthat the majority of existing solution bounds of the continuous and discreteLyapunov equations are only the special cases of those of [17]. Furthermore, thosepresented solution bounds are also applied to treat the problem of robust rootclustering in subregions of the complex plane for linear perturbed systems. Thetolerance perturbation bounds in terms of the mentioned solution bounds wereestimated. The feature of those perturbation bounds is that it is not necessary tosolve any GLE and hence the computational burden can be reduced. It is also shownthat the tighter those solution bounds are, the better the tolerance perturbationbounds are. Therefore, the research objective of the estimation problem for thesolutions of the GLE is to obtain sharper solution bounds. To improve andsupplement existing work [17], this paper develops further results for themeasurement of solution bounds of the GLE. By using iterative procedures, severalmore precise estimations for matrix bounds presented in [17] can be obtained.Furthermore, some new matrix and eigenvalue bounds for the solutions of the GLEare also derived. These new bounds can supplement those given in [17]. It is alsoshown again that the majority of existing matrix bounds for the solutions of thecontinuous and discrete Lyapunov equations are the special cases of what ispresented in this paper. Therefore, the paper can also be considered as ageneralization work for the estimation problem of solution bounds of the classicalLyapunov equations.This paper is organized as follows. In Section 2, we simply introduce the GLE by

reviewing the nominal matrix root clustering theory. Furthermore, several usefullinear algebraic matrix inequalities and previous results are also reviewed. Section 3offers several iterative procedures for more precise estimations and measures somenew upper matrix and eigenvalue bounds of the solutions for the GLE. Furthermore,

ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480462

comparisons between these obtained results and existing bounds of the continuousand discrete Lyapunov equations are also made. An illustrative example is givenin Section 4 to demonstrate the feasibility of these obtained results. Finally,in Section 5, the conclusions are offered.

2. Preliminaries

In this section, we introduce the GLE by reviewing the root clustering theory.Define the regions O1; and O2 of the complex plane, respectively, as

O1 ¼ fðx; yÞjb0 þ b1x þ b2yo0g; ð1Þ

O2 ¼ fðx; yÞjj0 þ j1y2 þ j2x þ j3x

2o0g; ð2Þ

where b0;b1;b2;j0;j1;j2;j3AR; b21 þ b22a0 and j1X0:Furthermore, define a disk region Dðx; rÞ of the complex plane as

Dðx; rÞ ¼ fzjðz xÞð%z %xÞorg; ð3Þ

where xAC denotes the center and r is the radius.For these regions, some root clustering results are presented as follows.

Theorem 1 (Gutman and Jury [12]). If and only if for any given positive definite

Hermitian matrix Q there exists a unique matrix P > 0 such that

c0Pþ c1ATP þ c2PA ¼

12

Q; ð4Þ

where c0 ¼ b0; c1 ¼ 1=2ðb1 þ ib2Þ; and c2 ¼ 1=2ðb1 ib2Þ; then all eigenvalues of

AARn�n are located inside region O1:

Theorem 2 (Gutman and Jury [12]). The necessary and sufficient condition that

assure all the eigenvalues of AARn�n lie in the region O2 is for any given positive

definite Hermitian matrix Q there exists a unique matrix P > 0 which satisfies

d0P þ d1ðATP þ PAÞ þ d2ATPA þ d3½ðA2ÞTP þ PA2� ¼ Q; ð5Þ

where d0 ¼ j0; d1 ¼ j2=2; d2 ¼ ðj1 þ j3Þ=2; and d3 ¼ ðj3 j1Þ=4:

According to Theorems 1 and 2 and from [12,13], some useful regions and thecorresponding parameters ci and di are summarized in Table 1. Without loss ofgenerality, we assume that ðj1 þ j3ÞX0; (i.e., d2X0) in this paper.

Theorem 3 (Gutman and Jury [12]). All eigenvalues of A are located within Dðx; rÞ if

and only if for any given positive definite Hermitian matrix Q there exists a unique

matrix P > 0 which satisfies the following Lyapunov equation:

1

r2ðA xIÞPðA xIÞT P ¼ Q: ð6Þ

ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 463

It is seen that Eqs. (4), (5), and (6) are of the formXi;j

eijAiPðATÞj ¼ Q

The above equation is the so-called ’’Generalized Lyapunov Equation’’ (GLE).By setting c1 ¼ 0; c1 ¼ c2 ¼ 1=2; d0 ¼ d2 ¼ d3 ¼ 0; and d1 ¼ 1; Eqs. (4) and (5)

become the standard continuous Lyapunov equation

ATP þ PA ¼ Q; ð7Þ

where A is now assumed to be a stable matrix. Furthermore, letting d0 ¼ 1; d1 ¼d3 ¼ 0; d2 ¼ 1; x ¼ 0; and r ¼ 1; Eqs. (5) and (6) then become the discrete Lyapunovequation

ATPA P ¼ Q: ð8Þ

We assume that all eigenvalues of A are now located inside the unit circle centered atthe origin. Obviously the continuous and discrete Lyapunov equations (7) and (8)are special cases of the GLE (4), (5), and (6).Besides the GLE, some useful inequalities and previous results are given below.

Lemma 1 (Zhou and Khargonekar [18]). For any n � n complex matrices A and B

and any positive constant a;

ATB þ BTApa2ATA þ1

a2BTB ð9Þ

ARTICLE IN PRESS

Table 1

Some useful regions and the corresponding parameters

Region O1 c0 c1 c2

Sector

hðx þ aÞ yo0 ha ðh þ iÞ=2 ðh iÞ=2Left half plane

aþ pxo0; ap > 0 a p=2 p=2

Region O2 d0 d1 d2 d3

Vertical strip

x2 2ax þ a2 w20o0 a2 w2

0 a 1=2 1=4Horizontal strip

y2 w20o0 w2

0 0 1=2 1=4Parabola

y2 þ 4px þ 4pao0 4pa 2p 1=2 1=4Hyperbola

ðx aÞ2

a2þ

y2

b2þ 1o0

1 ða=aÞ2 a=a21

2b2

1

2a214a2

1

4b2

Ellipse

ðx aÞ2

a2þ

y2

b2 1o0

ða=aÞ2 1 a=a21

2b2þ

1

2a21

4a2

1

4b2

C.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480464

Lemma 2 (Komaroff [3]). Consider the continuous Lyapunov equation (7). Then for

any k ¼ 1; 2;y; n;Xk

1

liðATP þ PAÞXlnðAT þ AÞXk

1

liðPÞ: ð10Þ

For a symmetric matrix ðAT þ AÞ; it is obvious that

lnðAT þ AÞ ¼ l1ðAT AÞ ¼ 2mðAÞ: ð11Þ

Lemma 3 (Amir-Moez [19]). For any symmetric matrices A;BACn�n; the following

inequalities are true.

liþj1ðA þ BÞpljðAÞ þ liðBÞ; i þ jpn þ 1; ð12Þ

liþjnðA þ BÞXljðAÞ þ liðBÞ; i þ jXn þ 1 ð13Þ

Lemma 4 (Marshell and Olkin [20]). For any n � n matrices A and B,

liðABÞ ¼ liðBAÞ; i ¼ 1; 2;y; n: ð14Þ

Lemma 5 (Amir-Moez [19]). For any symmetric matrices A, BACn�n; the following

inequalities are held.

liþj1ðABÞpljðAÞliðBÞ; i þ jpn þ 1; ð15Þ

liþjnðABÞXljðAÞliðBÞ; i þ jXn þ 1: ð16Þ

Theorem 4 (Lee and Lee [17]). Define a matrix U as

U � 2c2A þ c0I : ð17Þ

Then the solution P of the GLE (4) can be measured as

PXP1 � a½Q a2UTU �1=2; ð18Þ

where the positive constant a is chosen such that

Q > a2UTU : ð19Þ

Theorem 5 (Lee and Lee [17]). If a positive constant a is selected so that

Q > a2I ; ð20Þ

then the GLE (4) satisfies

PXP2 �a

s1ðUÞðQ a2IÞ1=2: ð21Þ

Theorem 6 (Lee and Lee [17]). Define the matrix V as

V � d3A2 d1A

1

2d0I : ð22Þ


Then the lower bound of the solution P of the GLE (5) can be estimated as

PXP4 � afQ þ ad2AT½Q a2VTV �1=2A a2VTVg1=2; ð23Þ

where the positive constant a is chosen such that

Q > a2VTV : ð24Þ

Theorem 7 (Lee and Lee [17]). Selecting the positive constant a such that Eq. (20) is

met, then the solution P of the GLE (5) has the lower bound

PXP5 �a

s1ðV ÞQ þ

as1ðV Þ

d2AT½Q a2I �1=2A a2I

� �1=2

: ð25Þ

Theorem 8 (Lee and Lee [17]). Define the matrix W as

W � ðA xIÞ=r: ð26Þ

For the GLE (6), we have

PXW ðQ þ WQWTÞWT þ Q � P8 ð27Þ

and if 1 s21ðW Þ > 0; then

Ppl1ðQÞ

1 s21ðW ÞWWT þ Q � P10: ð28Þ

3. Further results for solution bounds of the GLE

Before deriving new parallel lower matrix and eigenvalue bounds for the solutionof the GLE, we declare that the following procedures can make more preciseestimations for the presented lower matrix bounds (23), (25), and (27).

Procedure 1 (for Eq. (23)). Step 1: Set k ¼ 0 and define P01 � a½Q a2VTV �1=2:Step 2: Define Pðkþ1Þ1 � afQ þ d2A

TPk1A a2VTVg1=2 and calculate Pðkþ1Þ1:If Pðkþ1Þ1 ¼ Pk1; stop this procedure. The more precise estimation is found as

PXPk1: Otherwise, set k ¼ k þ 1 and go to Step 2.

Procedure 2 (for Eg. (25)). Step 1: Set k ¼ 0 and define P02 � a=s1ðV Þ½Q a2I �1=2:Step 2: Define Pðkþ1Þ2 � a=s1ðV ÞfQ þ d2A

TPk2A a2Ig1=2 and calculate Pðkþ1Þ2: IfPðkþ1Þ2 ¼ Pk2; stop this procedure. The more precise estimation is found as PXPk2:Otherwise, set k ¼ k þ 1 and go to Step 2.

Procedure 3 (for Eq. (27)). Step 1: Set k ¼ 0 and define #P01 ¼ P10 ¼ W ðQ þWQWTÞWT þ Q:

Step 2: Define #Pðkþ1Þ1 ¼ W #Pk1WT þ Q and calculate #Pðkþ1Þ1:


Step 3: If #Pðkþ1Þ1 ¼ #Pk1 then stop this procedure, and the more precisemeasurement of P is found as PX #Pk1: Otherwise, set k ¼ k þ 1 and go to Step 2.

Consider Procedures 1–3. Matrices Pk1; Pk2; and #Pk1 posses the convergence

Pk1XPðk1Þ1X?XP11XP01; Pk2XPðk1Þ2X?XP12XP02;

and #Pk1X#Pðk1Þ1X?X #P11X

#P01:

These inequalities are proved in PP1 of the appendix.If the condition 1 s21ðW Þ > 0 is met, then we also have the following procedure

for obtaining a more precise estimate of the upper solution matrix bound (28).

Procedure 4. Step 1: Set i ¼ 0 and define *P0 ¼ P12 ¼ ðl1ðQÞ=ð1 s21ðW ÞÞÞWWT þ Q:Step 2: Define *Piþ1 ¼ W *PiW

T þ Q and calculate *Piþ1:Step 3: If *Piþ1 ¼ *Pi then stop this procedure and the more precise measurement of

P is found as Pp *Pi: Otherwise, set i ¼ i þ 1 and go to Step 2.

The convergence of Procedure 4 is given in PP2 of the appendix.

Theorem 9. Define

D � ðUQUTÞ1=2: ð29Þ

Then the solution P of the GLE (4) has the lower bound

PXP3 �1

aD1½DðaQ Q1ÞD�1=2D1 ð30Þ

where a > 0 is determined by

aQ2 > I : ð31Þ

Proof. In light of definition (17), the GLE (4) can be rewritten as

2ðc0P þ c1ATP þ c2PAÞ ¼ PU þ UTP ¼ Q: ð32Þ

This means that all eigenvalues of the matrix U lie in the left side of the complexplane and hence U is invertable. Therefore, one can conclude

ðUQUTÞ1=2ðaP þ Q1U1ÞUQUTðaP þ UTQ1ÞðUQUTÞ1=2

¼ Dða2PD2P þ aPUQUTUTQ1 þ aQ1U1UQUTP

þ Q1U1UQUTUTQ1ÞD

¼ D½a2PD2P þ aðPU þ UTPÞ þ Q1�D

¼ a2DPD2PD DðaQ þ Q1ÞDX0; ð33Þ

where definition (29) is used. Eq. (33) implies

ðDPDÞ2X1

a2DðaQ Q1ÞD: ð34Þ

Selecting a > 0 such that Eq. (30) holds, then Eq. (34) leads to bound (29). &


Remark 1. As mentioned in [21], it is hard or even impossible to compare thetightness between parallel bounds. We also found that the sharpness of the proposedresults (18), (21), and the new bound (30) cannot be compared by mathematicalmethods. However, they may give a supplement to each other for the measurementof the lower matrix bound of P for the GLE (4).

Theorem 10. The GLE (4) satisfies

l1ðPÞXmaxðl1ðP1Þ; a; bÞ; ð35Þ

lnðPÞpl1ðQÞ

2ReðlnðUÞÞ; ð36Þ

where P1 is defined as Eq. (18) and a and b, respectively, is defined by

a � maxi

liðQÞ2siðUÞ

; i ¼ 1; 2;y; n; ð37Þ

b � maxi

½li½DðaiQ Q1ÞD��1=2

ailiðD2Þ; i ¼ 1; 2;y; n; ð38Þ

where ai; i ¼ 1; 2;y; n are determined by

aiQ2 > I ; i ¼ 1; 2;y; n: ð39Þ

Proof. By bound (18), it is obvious that l1ðPÞXl1ðP1Þ: We rewrite Eq. (31) as

PðUÞ þ ðUTÞP ¼ Q: ð40Þ

Applying Lemma 4 and Eq. (15) of Lemma 5 to Eqs. (40) and (34), respectively, gives

liðQ a2IÞp1

a2liðP2UUTÞp

1

a2s2i ðUÞl1ðPÞ

2; i ¼ 1; 2;y; n;

l1ðPÞ2l2i ðD

2ÞXliðDPDÞ2X1

a2li½DðaQ Q1ÞD�; i ¼ 1; 2;y; n:

In fact, the above inequalities hold for any a that satisfies conditions (20) and (31),respectively. Therefore, the above inequalities, respectively, can be rewritten as

liðQ a2i IÞp1

a2is2i ðUÞl1ðPÞ

2; i ¼ 1; 2;y; n with Q > a2i I ; ð41Þ

l1ðPÞ2l2i ðD

2ÞX1

a2ili½DðaiQ Q1ÞD�; i ¼ 1; 2;y; n with a2i Q > I : ð42Þ

Choosing ai ¼liðQÞ2and solving Eq. (41) leads to

l1ðPÞXmaxi ai

½liðQ a2i IÞ�1=2

siðUÞ¼ maxi ai

½liðQÞ a2i �1=2

siðUÞ¼ maxi

liðQÞ2siðUÞ

¼ a

ð43Þ


Solving Eq. (42) with respect to l1ðPÞ yields l1ðPÞXb: Then bound (35) is directlyobtained.Since the real parts of all eigenvalues of U are negative (i.e., ReðliðUÞÞo0 for

i ¼ 1; 2;y; nÞ; the symbol FðUÞ now is defined to represent the set of the eigenvectorsof the matrix U : That is, FðUÞ � fziACnjUzi ¼ liðUÞzi; i ¼ 1; 2;y; ng: Then, pre-and post-multiplying zTi and zi; respectively, to Eq. (32) results in

zTi PUzi þ zTi UTPzi ¼ zTi Qzi

which implies

2ReðliðUÞÞXzTi Qzi

zTi Pzi

; i ¼ 1; 2;y; n:

According to Rayleigh’s principle [22], we have

2ReðlnðUÞÞXl1ðQÞlnðPÞ

:

Therefore, bound (36) is obtained. &

Theorem 11. If the matrix V is invertable, then the solution P of the GLE (5) can be

measured as

PXP6 �1

aaQ Q1

l1ðVQVTÞþ

d2

al1ðVQVTÞAT aQ Q1

l1ðVQVTÞ

� �1=2

A

" #1=2; ð44Þ

where the positive constant a is determined by Eq. (31).

Proof. If the matrix V is invertable, it is obvious that

ðaP Q1V1ÞVQVTðaP VTQ1Þ

¼ a2PVQVTP aðPV þ VTPÞ þ Q1

¼ a2PVQVTP aðQ þ d2ATPAÞ þ Q1

X0;

which infers

a2l1ðVQVTÞP2Xa2PVQVTPXaðQ þ d2A

TPAÞ Q1XaQ Q1: ð45Þ

Solving Eq. (45) for P; one gets

PX1

aaQ Q1

l1ðVQVTÞ

� 1=2:

Now, we can derive

a2PVQVTPX aðQ þ d2ATPAÞ Q1

X a Q þd2

aAT aQ Q1

l1ðVQVTÞ

� �A

� Q1: ð46Þ

Using the fact that l1ðVQVTÞP2XPVQVTP and solving this inequality with respect

to P; we obtain bound (44). &


In light of Theorem 11, we propose the following procedure for a preciseestimation of P:

Procedure 5. Step 1: Set k ¼ 0 and define

P03 �1

aaQ Q1

l1ðVQVTÞ

� 1=2:

Step 2: Define

Pðkþ1Þ3 �1

aaQ Q1

l1ðVQVTÞþ

d2

l1ðVQVTÞATPk3A

� 1=2and calculate Pðkþ1Þ2: If Pðkþ1Þ2 ¼ Pk3 then stop this procedure, and the more preciseestimation is found as PXPk3: Otherwise, set k ¼ k þ 1 and go to Step 2.Matrix Pk3 posses the convergence Pk3XPðk1Þ3X?XP13XP03 and the proof is

omitted.

Remark 2. Notice that the tightness of the proposed parallel results (23), (25), and(44) also cannot be compared by mathematical methods. However, they may alsogive a supplement to each other.

Theorem 12. The GLE (5) satisfies

l1ðPÞXmaxðl1ðP4Þ; c; dÞ; ð47Þ

where the matrix P4 is defined by (23) and c and d, respectively, is defined as

c � maxi

ai

siðV Þli Q þ

ai

s1ðV Þd2A

T½Q a2i I �1=2A a2i I

� �1=2

with Q > a2i I ; ð48Þ

d �maxi1

ai

li ai Q þd2

ai

AT aiQ Q1

l1ðVQVTÞ

� �A

� Q1

� �1=2

� liðVQVTÞ1=2 with aiQ > I : ð49Þ

Furthermore, if 2mðV Þ þ d2s21ðAÞo0; then

l1ðPÞpl1ðQÞ

2mðV Þ þ d2s21ðAÞ� gðQÞ; ð50Þ

lnðPÞpl1ðQ þ d2gðQÞATAÞ

2ReðlnðV ÞÞ; ð51Þ

where V is defined by (22).

Proof. Bound l1ðPÞXl1ðP4Þ is directly derived from Eq. (23). According todefinition (22), we can rewrite the GLE (5) as

PV þ VTP ¼ Q þ d2ATPAXQ: ð52Þ


Applying Lemma 1 to Eq. (52), we have

QpQ þ d2ATPAp

1

a22PVVTP þ a22I : ð53Þ

This implies

PXa2

s1ðV Þ½Q a22I �

1=2: ð54Þ

Substituting Eq. (54) into Eq. (53) yields

Q þ d2a2

s1ðV ÞAT½Q a22I �Ap

1

a22PVVTP þ a22Ip

1

a22s21ðV ÞP2 þ a22I : ð55Þ

Applying Lemmas 4 and 5 to Eqs. (55) and (46), respectively, and then solving themwith respect to l1ðPÞ gives l1ðPÞXc and l1ðPÞXd: Therefore, the lower eigenvaluebound (47) is obtained.From Eq. (52), the application of Lemma 2 givesXk

1

liðQ þ d2ATPAÞXlnðV þ VTÞ

Xk

1

liðPÞ: ð56Þ

Taking k ¼ 1 and utilizing Eq. (11), inequality (56) becomes

2mðV Þl1ðPÞpl1ðQ þ d2ATPAÞpl1½Q þ d2l1ðPÞATA�: ð57Þ

Using the fact ATPApl1ðPÞATA and applying Eq. (12) with i ¼ j ¼ 1 to Eq. (57)leads to

2mðV Þl1ðPÞpl1ðQÞ þ d2l1ðPÞs21ðAÞ: ð58Þ

Therefore, it is seen that if 2mðV Þ þ d2s21ðAÞo0; then bound (50) is obtained.As the similar ways of the proof of Theorem 10, we use the symbol CðV Þ to represent

the set of the eigenvectors of the matrix V ; i.e., FðV Þ � fziACnjVzi ¼ liðV Þzi; i ¼1; 2;y; ng: Then, pre- and post-multiplyingzTi and zi; respectively, to Eq. (52) results in

zTi PðV Þzi þ zTi ðV ÞTPzi ¼ zTi ðQ þ d2ATPAÞziX zTi ðQ þ d2l1ðPÞATAÞzi:

If 2mðV Þ þ d2s21ðAÞo0; then Eq. (59) is satisfied and we obtain

2ReðliðV ÞÞXzTi ðQ þ d2gðQÞATAÞzi

zTi Pzi

; i ¼ 1; 2;y; n:

It is well known that ReðliðV ÞÞpmðV Þ: Therefore, the condition 2mðV Þ þd2s21ðAÞo0 implies ReðliðV ÞÞo0 for i ¼ 1; 2;y; n: In light of Rayleigh’s principle,we have

2ReðlnðV ÞÞXl1ðQ þ d2gðQÞATAÞ

lnðPÞ

which infers bound (51). &

Remark 3. Setting c0 ¼ 0; c1 ¼ c2 ¼ 0:5; d0 ¼ d2 ¼ d3 ¼ 0; and d1 ¼ 1; the GLE (4)and (5) become the continuous Lyapunov equation (7). The bounds P1 in Eq. (18)


and P4 in Eq. (23) become

PXP1 ¼ P4 ¼ a½Q a2ATA�1=2 � P7; ð59Þ

bounds (21) and (25) become

PXP2 ¼ P5 ¼a

s1ðAÞðQ a2IÞ1=2; ð60Þ

and bounds (30) and (44), respectively, become

PXP3 ¼1

aA1½AðaQ Q1ÞA�1=2A1; ð61Þ

PXP6 ¼1

aaQ Q1

l1ðAQATÞ

� 1=2: ð62Þ

Bounds (59) and (60), respectively, are the same as those presented in [5,6] and thematrix bounds (61) and (62) are new results. Furthermore, the eigenvalue bounds(35) and (47) become

l1ðPÞXmaxðl1ðP7Þ; e; f Þ; ð63Þ

where e and f are defined, respectively, by

e � maxi

1

2siðAÞliðQÞ;

f � maxi

1

ai

lifaiQ Q1g1=2liðAQATÞ1=2 with a2i Q > I :

Note that the constants ai in Eq. (48) are chosen as a2i ¼ liðQÞ=2 for this case.Furthermore, for this case, Eqs. (36) and (51) become

lnðPÞpl1ðQÞ

2ReðlnðAÞÞð64Þ

and the upper bound (50) then becomes

l1ðPÞpl1ðQÞ2mðAÞ

: ð65Þ

In the literature, [5] proposed

l1ðPÞXl1ðP7Þ: ð66Þ

[4] suggested that

l1ðPÞXmaxi

1

2siðAÞliðQÞ: ð67Þ

[8] presented the bound

l1ðPÞXl1ðQÞ2s1ðAÞ

: ð68Þ


[10] derived

l1ðPÞXtrðQÞ2trðAÞ

ð69Þ

and [2,11] obtained

l1ðPÞXlnðQÞ

2maxi ReðliðAÞÞ: ð70Þ

Since l1ðPÞXmaxðl1ðP7Þ; e; f ÞXl1ðP7Þ; bound (63) is better than bound (66). In [6],it is shown that Eq. (67) is sharper than Eq. (68) and is tighter than Eq. (69) ifmini jliðAÞj ¼ maxi Re½liðAÞ�: Furthermore, besides mini jliðAÞj ¼ maxi Re½liðAÞ�;if Q is chosen as Q ¼ cI ; then Eq. (67) is also better than Eq. (70). However, sincel1ðPÞXmaxðl1ðP7Þ; e; f ÞXe; it is obvious the obtained result (63) is also sharperthan bound (67). Furthermore, surveying the literature, bounds (64) and (65),respectively, coincide with the previous results proposed in [2,11].

Theorem 13. For the GLE (6), we have the lower matrix bound

PXlnðQÞ

1 s2nðW ÞWWT þ Q � P11 ð71Þ

where the matrix W is defined by (26).

Proof. In light of Eq. (26), we rewrite the GLE (6) as

P ¼ WPWT þ Q: ð72Þ

Applying Lemma 3 to Eq. (72) gives

lnðPÞ ¼ lnðWPWT þ QÞXlnðWPWTÞ þ lnðQÞ:

The application of Lemmas 4 and 5 results in

lnðPÞXlnðPWWTÞ þ lnðQÞXlnðPÞs2nðW Þ þ lnðQÞ:

Solving this inequality with respect to lnðPÞ yields

lnðPÞXlnðQÞ

1 s2nðW Þ: ð73Þ

Note that the satisfaction of Eq. (72) means that the absolute values of alleigenvalues of W are less than one (i.e. jliðW Þjo1). Besides, from the factsnðW Þomini jliðW Þj; it is shown that s2nðW Þo1: Substituting the relationWPWT

XlnðPÞWWT and Eq. (73) into Eq. (72) leads to bound (71). &

In fact, we can use the following procedure to obtain more precise bound for P9:

Procedure 6 (for (71)). Step 1: Set k ¼ 0 and define

#P02 ¼ P9 ¼lnðQÞ

1 s2nðW ÞWWT þ Q:

Step 2: Define #Pðkþ1Þ2 ¼ W #Pk2WT þ Q and calculate #Pðkþ1Þ2:


Step 3: If #Pðkþ1Þ2 ¼ #Pk2 then stop this procedure, and the more precisemeasurement of P is found as PX #Pk2: Otherwise, set k ¼ k þ 1 and go to Step 2.

Consider the convergence of Procedure 6, matrix #Pk2 has to satisfy

#Pk2X#Pðk1Þ2X?X #P12X

#P02:

The proof of #Pk2X#Pðk1Þ2X?X #P12X

#P02 is similar to that of #Pk1X#Pðk1Þ1X?X

#P11X#P01 and hence is omitted.

According to Theorems 8 and 13, the following eigenvalue bounds are directlyobtained.

Corollary 1. For the GLE (6), the following eigenvalues bounds are satisfied:

liðPÞXmaxðliðP8Þ; liðP9ÞÞ; ð74Þ

trðPÞXmaxðtrðP8Þ; trðP9ÞÞ ð75Þ

and if 1 s21ðW Þ > 0; then

liðPÞpliðP10Þ; ð76Þ

trðPÞXtrðP10Þ; ð77Þ

where P8; P9; and P10 are defined, respectively, by Eq. (27), (71), and (28).

Remark 4. Letting x ¼ 0 and r ¼ 1; the GLE (6) becomes the discrete Lyapunovequation (8). Then bounds (27), (71), and (28), respectively, become

PXAðQ þ AQATÞAT þ Q � P11; ð78Þ

PXlnðQÞ

1 s2nðAÞAAT þ Q � P12; ð79Þ

Ppl1ðQÞ

1 s21ðAÞAAT þ Q � P13: ð80Þ

Note that bounds (79) and (80) are consistent with those presented by Montemayorand Womack [8] and the tightness between (78) and (79) cannot be compared.Furthermore, for this case, bounds (74)–(77) become respectively

liðPÞXmaxðliðP11Þ; liðP12ÞÞ; ð81Þ

trðPÞXmaxðtrðP11Þ; trðP12ÞÞ; ð82Þ

liðPÞpliðP13Þ if 1 s21ðAÞ > 0; ð83Þ

trðPÞptrðP13Þ if 1 s21ðAÞ > 0; ð84Þ

where P11; P12; and P13 are defined, respectively, by (78), (79), and (80). Bounds (83)and (84), respectively, are the same as those presented in [7].


In the literature, [7] also proposed

liðPÞXliðP12Þ; ð85Þ

trðPÞXtrðP12Þ: ð86Þ

[11] gave the following bounds:

lnðPÞXlnðQÞ

1 s2nðAÞ; ð87Þ

l1ðPÞpl1ðQÞ

1 s21ðAÞif 1 s21ðAÞ > 0 ð88Þ

[9] derived the trace bounds

trðPÞXtrðQÞ

1 s2nðAÞ; ð89Þ

trðPÞptrðQÞ

1 s21ðAÞif 1 s21ðAÞ > 0: ð90Þ

Furthermore, the following bounds were given in [1]

liðPÞXliðQÞ; ð91Þ

Xk

i¼1

liðPÞpXk

i¼1

liðQÞ1 s21ðAÞ

if 1 s21ðAÞ > 0: ð92Þ

Obviously bound (81) is tighter than Eqs. (85) and (82) is better than Eq. (86).In [7], it is proved that bound (85) with i ¼ n is sharper than Eq. (87) and bound

(85) is tighter than Eq. (88) when i ¼ 1: Furthermore, for the case Q ¼ cI ; [7] alsoshown that Eq. (84) is more precise than Eqs. (90) and (86) is sharper than Eq. (89).Since

liðP13ÞXli½AðQ þ AQATÞAT þ Q�XliðQÞ;

it is obvious that bound (81) is better than Eq. (91). From Eq. (80), we haveXk

i¼1

liðPÞpXk

i¼1

lil1ðQÞ

1 s21ðAÞAAT þ Q

� : ð93Þ

For the case Q ¼ cI ; the application of Lemma 3 givesXk

i¼1

li

l1ðQÞ1 s21ðAÞ

AAT þ Q

� p

Xk

i¼1

l1ðQÞ1 s21ðAÞ

s21ðAÞ þ liðQÞ�

¼Xk

i¼1

liðQÞ1 s21ðAÞ

:

Therefore, bound (93) is tighter than bound (92) for the case Q ¼ cI :

Remark 5. So far several parallel measures for the solutions of the GLE have beendeveloped. They are all new. We found those solution bounds of the continuous anddiscrete Lyapunov equations proposed in the literature are only the special cases ofthis work. As mentioned in [21], it is hard or even impossible to compare the


tightness between those parallel bounds by mathematical methods. In this paper, thesharpness of presented parallel matrix solution bounds cannot be compared.However, they may give a supplement to each other. Furthermore, in the followingsection, it will be demonstrated the feasibility of these obtained results.

4. An numerical example

Consider the matrix

A ¼

3:8 0 0

0 1:6 1

0 0:5 2:4

264

375:

It is seen that all eigenvalues of A are located within the intersection region ofDð3; 2Þ and the following regions

O1 ¼ fðx; yÞj 0:8x 2y þ 0:5o0g ðSectorÞ;

O21 ¼ fðx; yÞj y2 22o0g ðHorizontal stripÞ;

O22 ¼ fðx; yÞj y2 þ 2x þ 2o0g ðParabolaÞ:

In light of Table 1 and from Theorems 1–3, the corresponding GLE are

0:5P þ ð0:4 iÞATP þ ð0:4þ iÞPA ¼ Q for O1; ð94Þ

4P þ 12 ATPA 1

4 ½ðA2ÞTP þ PA2� ¼ Q for O21; ð95Þ

2P þ ðATP þ PAÞ þ 12

ATPA 14½ðA2ÞTP þ PA2� ¼ Q for O22; ð96Þ

14ðA þ 3IÞPðA þ 3IÞT P ¼ Q for Dð3; 2Þ: ð97Þ

Select Q ¼2 0 00 4 10 1 4

24

35 for the above Lyapunov equations (94)–(97). Then, from

Theorems 1, 2, 6–9, and 11, one can obtain the following bounds of P for theLyapunov equations (94)–(97).For Eq. (94):

PX

0:1053 0 0

0 0:2862 0:0388

0 0:0388 0:2695

264

375 ¼ P1 with a ¼ 0:15 ðTheorem 4; ½17�Þ;

PX

0:1222 0 0

0 0:2086 0:0358

0 0:0358 0:2086

264



PX

0:1222 0 0

0 0:3084 0:0307

0 0:0307 0:2755

264

375

¼P3 with a ¼ 0:5 ðTheorem 9; this paperÞ:

For Eq. (95):

PX

0:2383 0 0

0 0:3060 0:0394

0 0:0394 0:3261

264


PX

0:2696 0 0

0 0:3236 0:0534

0 0:0534 0:3593

264


PX

0:3370 0 0

0 0:3650 0:0578

0 0:0578 0:4307

264

375


For Eq. (96):

PX

0:2224 0 0

0 0:3096 0:0403

0 0:0403 0:3266

264


PX

0:2275 0 0

0 0:2812 0:0466

0 0:0466 0:3089

264


PX

0:2814 0 0

0 0:3155 0:0502

0 0:0502 0:3669

264

375


For Eq. (97):

PX

2:3712 0 0

0 7:4279 1:4822

0 1:4822 4:9974

264

375 ¼ P8 ðTheorem 8; ½17�Þ;


PX

2:3711 0 0

0 5:7441 1:0589

0 1:0589 4:3594

264

375 ¼ P9 ðTheorem 13; this paperÞ:

For this case, it is seen that P3XP2; liðP3Þ > liðP1Þ for all i; P61 > P51 > P41; P62 >P42; and P62 > P52: One cannot compare the tightness between bounds P8 and P9:However, by utilizing Procedure 6, one can obtain

PX

2:3809 0 0

0 7:7286 1:6395

0 1:6395 5:1058

264

375 ¼ #P12;

which infers #P12 > P8: This shows that all proposed bounds are the best for this case.

5. Conclusions

Some iterative procedures are proposed to make more precise estimations forprevious existing lower matrix bounds of the solutions of the GLE. Besides, severalnew matrix and eigenvalue bounds of the solutions for the GLE are presented. It isseen that the majority of the existing solution bounds for the continuous and discreteLyapunov equations are the special cases of these results. Therefore, this paper canbe considered as a generalization work for existing results. From the given numericalexample, it is also shown that these obtained parallel bounds are better for somecase(s).

Acknowledgements

The author would like to thank the National Science Council of the Republic ofChina for the financial support of this research under the grant NSC90-2213-E230-004.

Appendix

PP1: (A) The proofs of Pk1XPðk1Þ1X?XP11XP01 and Pk2XPðk1Þ2X?X

P12XP02:Set k ¼ 0; we obtain

P11 � afQ þ d2ATP01A a2VTVg1=2;

where P01 � a½Q a2VTV �1=2: Due to the fact ATP01AX0; one has

P11 ¼ afQ þ d2ATP01A a2VTVg1=2Xa½Q a2VTV �1=2 ¼ P01:


Now, we assume Pðk1Þ1XPðk2Þ1 then

Pk1 ¼ afQ þ d2ATPðk1Þ1A a2VTVg1=2

X afQ þ d2ATPðk2Þ1A a2VTVg1=2

¼Pðk1Þ1:

Therefore, by the inductive method, one can conclude that Pk1XPðk1Þ1X?X

P11XP01:Following similarly as the above proof, one can easily prove inequalities

Pk2XPðk1Þ2X?XP12XP02: We omit the proof.(B) The proof of #Pk1X

#Pðk1Þ1X?X #P11X#P01:

From the definition of #Pk1; we have

#P11 ¼ W #P01WT þ Q ¼ W ½W ðQ þ WQWTÞWT þ Q�WT þ Q: ðA:1Þ

Since Q þ WQWTXQ; Eq. (A.1) becomes

#P11XW ðQ þ WQWTÞWT þ Q ¼ #P01:

Assume that #Pðk1Þ1X #Pðk2Þ1: Then

#Pk1 ¼ W #Pðk1Þ1WT þ QXW #Pðk2Þ1W

T ¼ #Pðk1Þ1:

By the inductive method, one can conclude that #Pk1X#Pðk1Þ1X?X #P11X

#P01:PP2: The proof the convergence of Procedure 4.From the definition of *Pk and the fact *Pkpl1ð *PkÞI ; we have

*P1 ¼ W *P0WT þ Qpl1ð *P0ÞWWT þ Q: ðA:2Þ

Application of Eq. (12) leads to

*P0 ¼ l1l1ðQÞ

1 s21ðW ÞWWT þ Q

� �p

l1ðQÞ1 s21ðW Þ

s21ðW Þ þ l1ðQÞ

¼l1ðQÞ

1 s21ðW Þ: ðA:3Þ

Substituting Eq. (A.3) into Eq. (A.2) results in

*P1 ¼l1ðQÞ

1 s21ðW ÞWWT þ Q ¼ *P0:

Suppose that *Pk1X *Pk2: Then

*Pk ¼ W *Pk1WT þ QpW *Pk2W

T ¼ *Pk1:

By the inductive method, we can say

*Pkp *Pk1p?p *P1p *P0

which infers the convergence of Procedure 4.


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Further results on the measurement of solution bounds of the generalized Lyapunov equations

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