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
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 465
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:
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480466
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). &
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 467
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
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480468
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). &
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 469
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
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480470
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)
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 471
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
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480472
[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:
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 473
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].
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480474
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
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 475
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
375 ¼ P2 with a ¼ 1:0 ðTheorem 5; ½17�Þ;
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480476
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
375 ¼ P41 with a ¼ 0:15 ðTheorem 6; ½17�Þ;
PX
0:2696 0 0
0 0:3236 0:0534
0 0:0534 0:3593
264
375 ¼ P51 with a ¼ 0:15 ðTheorem 7; ½17�Þ;
PX
0:3370 0 0
0 0:3650 0:0578
0 0:0578 0:4307
264
375
¼P61 with a ¼ 0:5 ðTheorem 11; this paperÞ:
For Eq. (96):
PX
0:2224 0 0
0 0:3096 0:0403
0 0:0403 0:3266
264
375 ¼ P42 with a ¼ 0:15 ðTheorem 6; ½17�Þ;
PX
0:2275 0 0
0 0:2812 0:0466
0 0:0466 0:3089
264
375 ¼ P52 with a ¼ 0:15 ðTheorem 7; ½17�Þ;
PX
0:2814 0 0
0 0:3155 0:0502
0 0:0502 0:3669
264
375
¼P62 with a ¼ 0:5 ðTheorem 11; this paperÞ:
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;
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 477
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:
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480478
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.
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480 479
References
[1] J. Garloff, Bounds for the eigenvalues of the solution of discrete Riccati and Lyapunov equations and
the continuous Lyapunov equation, Int. J. Control 43 (1986) 423–431.
[2] N. Komaroff, Simultaneous eigenvalue lower bounds for the Lyapunov matrix equation, IEEE
Trans. Automat. Control 33 (1988) 126–128.
[3] N. Komaroff, Diverse bounds for the eigenvalues of the continuous algebraic Riccati, IEEE Trans.
Automat. Control 39 (1994) 532–534.
[4] C.H. Lee, Eigenvalue upper and lower bounds of the solution for the continuous algebraic matrix
Riccati equation, IEEE Trans. Circuits Systems, Part—I 43 (1996) 683–686.
[5] C.H. Lee, New results for the bounds of the solution for the continuous Riccati and Lyapunov
equations, IEEE Trans. Automat. Control 42 (1997) 118–123.
[6] C.H. Lee, On the upper and lower bounds of the solution for the continuous Riccati matrix equation,
Int. J. Control 66 (1997) 105–118.
[7] C.H. Lee, Upper and lower matrix bounds of the solution for the discrete Lyapunov equation, IEEE
Trans. Automat. Control 41 (1996) 1338–1341.
[8] J.J. Montemayor, B.F. Womack, Comments on ‘On the Lyapunov matrix equation’, IEEE Trans.
Automat. Control 20 (1975) 814–815.
[9] T. Mori, N. Fukuma, M. Kuwahara, On the discrete Lyapunov equation, IEEE Trans. Automat.
Control 27 (1982) 463–464.
[10] R.V. Patel, M. Toda, On norm bounds for algebraic Riccati and Lyapunov equations, IEEE Trans.
Automat. Control 23 (1978) 87–88.
[11] K. Yasuda, H. Hirai, Upper and lower bounds on the solution of the algebraic Riccati equation,
IEEE Trans. Automat. Control 24 (1979) 483–487.
[12] S. Gutman, E.I. Jury, A general theory for matrix root-clustering in subregions of the complex plane,
IEEE Trans. Automat. Control 26 (1981) 853–863.
[13] A.-A.A. Abdul-Wahab, Lyapunov-type equations for matrix root-clustering in subregions of the
complex plane, Int. J. Syst. Sci. 21 (1990) 1819–1830.
[14] S. Gutman, H. Taub, Linear matrix equations and root clustering, Int. J. Control 50 (1989)
1635–1643.
[15] H.Y. Horng, J.H. Chou, I.R. Horng, Robustness of eigenvalue clustering in various regions of the
complex plane for perturbed systems, Int. J. Control 57 (1993) 1469–1484.
[16] R.K. Yadavalli, Robust root clustering for linear uncertain systems using Generalized Lyapunov
theory, Automatica 29 (1993) 237–240.
[17] C.H. Lee, S.T. Lee, On the estimation of solution bounds of the generalized Lyapunov equations and
the robust root clustering for the linear perturbed systems, Int. J. Control 74 (2001) 996–1008.
[18] K. Zhou, P.P. Khargonekar, Robust stabilization of linear systems with norm-bounded time-varying
uncertainty, Systems Control Lett. 10 (1988) 17–20.
[19] A.R. Amir-Moez, Extreme properties of eigenvalues of a Hermitian transformation and singular
values of the sum and product of linear transformation, Duke Math. J. 23 (1956) 463–467.
[20] A.N. Marshell, I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic,
New York, 1979.
[21] T. Mori, I.A. Derese, A brief summary of the bounds on the solution of the algebraic matrix equation
in control theory, Int. J. Control 39 (1984) 247–256.
[22] B. Noble, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1969.
ARTICLE IN PRESSC.H. Lee / Journal of the Franklin Institute 340 (2003) 461–480480
Top Related