Solution Bounds of the Continuous and Discrete

Lyapunov Matrix Equations1

C. H. LEE2

Communicated by M. J. Balas

Abstract. A unified approach is proposed to solve the estimation prob-

lem for the solution of continuous and discrete Lyapunov equations.

Upper and lower matrix bounds and corresponding eigenvalue bounds

of the solution of the so-called unified algebraic Lyapunov equation

are presented in this paper. From the obtained results, the bounds for

the solutions of continuous and discrete Lyapunov equations can be

obtained as limiting cases. It is shown that the eigenvalue bounds of the

unified Lyapunov equation are tighter than some parallel results and

that the lower matrix bounds of the continuous Lyapunov equation are

more general than the majority of those which have appeared in the

literature.

Key Words. Matrix bounds, eigenvalue bounds, Lyapunov equation,

unified algebraic Lyapunov equation.

1. Introduction

Consider the so-called unified algebraic Lyapunov equation (UALE,

Ref. 1)

PA +ATP + DATPA +Q = 0, (1)

where A˛<n · n represents a stable constant matrix, the superscript T denotes

the transpose, Q˛<n · n is a positive-definite matrix, the matrix P˛<n · n is

1This work was supported by the National Science Council, Republic of China, Grant NSC

90-2213-E230-004.2Professor, Department of Electrical Engineering, Cheng-Shiu University, Kaohsiung, Taiwan,

ROC.

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 120, No. 3, pp. 559–578, March 2004 (g2004)

559

0022-3239=04=0300-0559=0 g 2004 Plenum Publishing Corporation

the positive-definite solution of (1), and D denotes the sampling period. In

Ref. 2, it is shown that the positive-definite solution P is unique.

By setting D= 0 in the UALE (1), we obtain the following continuous

algebraic Lyapunov equation (CALE):

PA +ATP+Q = 0: (2)

Furthermore, since

ATP +PA + DATPA = (DA+ I)TP=D(DA + I) – P=D, (3)

the UALE (1) can be rewritten as

(DA+ I)TP(DA+ I) + DQ =P: (4)

Let D= 1 and let A + I be replaced by A. Then, the UALE (1) becomes the

discrete Riccati equation (DALE)

ATPA +Q =P: (5)

From the above descriptions, it is seen that the continuous and discrete

Lyapunov equations are the limiting cases of the UALE (1). In other words,

the UALE (1) can unify the continuous and discrete cases. For control sys-

tems, the CALE (2) and the DALE (5) are usually utilized to solve the sta-

bility analysis problem. Sometimes, we need only bounds of the exact

solutions of the mentioned equations to reduce the computational burden.

Furthermore, the solution bounds of these equations can be applied also to

solve many control problems such as stability analysis for systems in the

presence of perturbations and=or time delay (Refs. 3, 4), robust root clus-

tering (Refs. 5, 6), determination of the size of the estimation error for mul-

tiplicative systems (Ref. 7), and so on (Ref. 8). Therefore, the estimation

problem for the solutions of these equations has become an attractive topic

of research. A number of bounds including extreme eigenvalues, trace,

determinant, summation and product of eigenvalues, and matrix bounds

have been proposed (Refs. 2, 9–26). Of these measurements, the matrix

bounds can determine directly all the corresponding eigenvalue bounds;

hence, they are the most general findings. The goal of this paper is to derive

new upper and lower matrix bounds of the solutions of the CALE (2) and the

DALE (5). As mentioned in the above descriptions, the UALE can unify the

CALE and the DALE. Therefore, we develop matrix bounds for the solution

of the UALE (1) and then the solution bounds of the CALE and the DALE

are obtained directly. This work can be considered also as a generalization of

the solution bounds for the continuous and discrete Lyapunov equations.

Comparisons show that the obtained bounds for the eigenvalues of the

solution of the UALE are tighter than some existing results and that the

lower matrix bound of the CALE is more general than the majority of those

560 JOTA: VOL. 120, NO. 3, MARCH 2004

reported in the literature. Numerical examples are given to show that these

obtained matrix bounds and the corresponding eigenvalue bounds have

good performances for some case(s).

The following conventions are used in this paper. A >B(A$B) means

that matrix A–B is positive definite (semidefinite); li(A) and si(A) denote the

ith eigenvalue and the ith singular value of a matrix A for i = 1, 2, . . . , n,

whereas li(A) and si(A) are arranged in nonincreasing order [i.e.,

l1(A)$l2(A)$ � � �$ln(A) and s1(A)$s2(A)$ � � �$sn(A)]. The identity

matrix with appropriate dimension is represented by I.

2. Main Results

We review first the following useful results.

Lemma 2.1. See Ref. 27. For any symmetric matrices A, B˛< n · n and

1# i, j#n, the following inequalities hold:

li+j – n(A +B)$lj(A) + li(B), i + j#n + 1, (6)

li+j – 1(A +B)#lj(A) + li(B), i + j#n + 1: (7)

Lemma 2.2. See Ref. 27. Let A and B be n· n real positive-

semidefinite matrices. Then, for 1# i, j#n,

li + j – 1(AB)#lj(A)li(B), i + j#n + 1: (8)

Lemma 2.3. See Ref. 28. For any n·n real matrices A and B,

li(AB) = li(BA), i = 1, 2, . . . , n: (9)

Utilizing the above useful results, the main results of this paper are

derived as follows.

Theorem 2.1. Let the positive constant matrix M be chosen such that

Q>M: (10)

Then, the solution P of the UALE (1) is such that

P$S–1(S(Q –M + DhATA)S)1=2S–1 ”Pll , (11)

where the positive-definite matrix S and the constant h are defined by

S ” (AM–1AT )1=2, (12)

JOTA: VOL. 120, NO. 3, MARCH 2004 561

h ”Ds 2

n(AS)+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2s 4

n(AS) + 4l2

1(AM–1AT )ln[S(Q –M)S]

q

2l2

1(AM–1AT ): (13)

Proof. For any positive constant matrix M, the following inequality

holds:

(M – 1=2ATP +M1=2)T (M – 1=2ATP +M1=2)

= PAM–1ATP+ PA +ATP+M$0: (14)

This implies that

–ATP – PA#PAM–1ATP+M: (15)

The UARE (1) is then rewritten as

DATPA +Q = –ATP – PA#PAM–1ATP +M, (16)

which implies that

PAM–1ATP$Q –M + ln(P)DATA, (17)

where the fact that P$ln(P)I is used. By the definition (12) and by pre-

multiplication postmultiplication of (17) by S, we obtain

(SPS)2 = SPSSPS$S(Q –M)S + ln(P)DSATAS: (18)

The application of (6), (8), (9) to (18) yields

l2

1(S2)l2

n(P)$l2

n(SPS) = l2

n(S2P)$ln[S(Q –M)S + ln(P)DSATAS]

$ln(S(Q –M)S)+ Ds 2n(AS)ln(P): (19)

Selecting M >0 such that (10) is satisfied gives

ln(P)$Ds 2

n(AS) +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2s 4

n(AS) + 4l2

1(AM–1AT )ln[S(Q –M)S]

q

2l2

1(AM–1AT )” h: (20)

Substituting (20) into (18) results in

(SPS)2$S(Q –M + hDATA)S: (21)

Solving the above inequality with respect to P leads to the lower bound (11).

u

Theorem 2.2. Assume that P is the solution of the UALE (1). If

a ” l1(A +AT + DATA)<0, (22)

562 JOTA: VOL. 120, NO. 3, MARCH 2004

then

P# [l1(Q)= – a ](DA + I)T (DA + I) + DQ ”Plu: (23)

Proof. From (4) and in light of the fact that P#l1(P)I, we have

P#l1(P)(DA + I)T (DA+ I) + DQ: (24)

Applying (7) of Lemma 2.1 to (24) leads to

l1(P)#l1(P)l1[(DA + I)T (DA + I)] + Dl1(Q): (25)

From (25) and using the relation

l1[(DA+ I)T (DA + I)]= Dl1(A +AT + DATA) + 1,

if condition (22) is met, then

l1(P)#l1(Q)=[– l1(A+AT + DATA)]= [l1(Q)= – a ]: (26)

Substituting this inequality into (24) results in the bound (23). u

We can use the bounds (11) and (23) to derive another upper matrix

bound for P.

Theorem 2.3. If the condition (22) holds, then the solution of the

UALE (1) is such that

P# [Q + (A + I)TPlu(A + I) – (1 – D)ATPllA], (27)

where the positive matrices Pll and Plu are defined by (11) and (23).

Proof. By the following matrix identity:

ATP+ PA = (A + I)TP(A+ I) –ATPA – P, (28)

the UALE (1) is rewritten as

Q + (A+ I)TP(A + I) – (1 – D)ATPA =P: (29)

From the introduction, it is known that D = 0 for the continuous case and

D= 1 for the discrete case. Hence, in this paper, we restrict the analysis to

1 – D$0. Therefore, if the condition (22) is met, then the application of (11)

and (23) to (29) yields the upper matrix bound (27). u

Remark 2.1. Surveying the literature, we find that only Refs. 23, 24

proposed several eigenvalue lower bounds and one eigenvalue upper bound

JOTA: VOL. 120, NO. 3, MARCH 2004 563

for the UALE (1); the results of the UALE (1) proposed in Ref. 24 coin-

cides with those presented in Ref. 23. As mentioned in the introduction, the

matrix bounds are the most general findings. Therefore, the obtained

matrix bounds are more general than the results in Refs. 23, 24. Further-

more, Refs. 23–24 proposed the following bound of the solution P of the

UALE (1):

�k

1li(P)# �

k

1li(Q)=(– a), if a<0, (30)

where a is defined as (18). Letting k = 1 in (30) gives the following eigenvalue

bound:

l1(P)#l1(Q)=(– a): (31)

From (23), we have the eigenvalue bounds

l1(P)#l1[[l1(Q)=(– a)](DA + I)T (DA + I) + DQ], (32)

�k

1li(P)# �

k

1li[[l1(Q)= – a ](DA + I)T (DA + I) + DQ]: (33)

From (7) and using the relation

l1[(DA + I)T (DA+ I)] = Da + 1,

we have

l1(P)# [l1(Q)=– a ]l1[(DA + I)T (DA + I)]+ Dl1(Q)

= [l1(Q)=– a ](Da + 1) + Dl1(Q)

= [l1(Q)=– a ], (34)

which means that the bound (32) is sharper than the bound (31).

When Q = cI and from (7), we have

�k

1li(P)# �

k

1li{[li(Q)=(– a)](DA + I)T (DA+ I) + DQ} (35)

# �k

1[l1(Q)=(– a)]l1[(DA + I)T (DA + I)]+ Dli(Q)

� �

= �k

1[li(Q)=(– a)](Da + 1)+ Dli(Q)f g

= �k

1li(Q)=(– a), (36)

which means that the bound (33) is tighter than (30) for this case.

564 JOTA: VOL. 120, NO. 3, MARCH 2004

Setting D = 0 in (11), (23), and (27) gives the following matrix bounds for

the CALE (2).

Theorem 2.4. For the CALE (2), the positive solution P has the fol-

lowing lower bound:

P$S–1(S(Q –M)S)1=2S–1 ” Pllc, (37)

where the positive matrix S is defined by (12). Furthermore, if

a ” l1(AT +A)<0, the solution P has also the upper bounds

P# [l1(Q)=(– a)]I ”Pluc, (38)

P# [Q + (A + I)TPluc(A + I) –ATPllcA] ” Pluc1: (39)

From (37)–(39), we have

li(P)$li[S–1(S(Q –Mi)S)1=2S–1], i = 1, 2, . . . , n, (40)

li(P)# min[li(Pluc), li(Pluc1)], i = 1, 2, . . . , n, (41)

where the positive-definite matrix Mi is determined by Q>Mi.

Remark 2.2. Notice that the bounds (37)–(39) are new. The bound

(38) may be somewhat conservative. However, this bound can be utilized to

derive the bound (39).

We do not find a systematical method to determine the tuning matrix M

such that the obtained bound is the best measurement. However, simple

selections of M, that are easy to compute and yield better results for some

cases, are made as follows. If the positive matrix M is selected as M = 0.5Q,

then we have the following corollary.

Corollary 2.1. The positive solution P of the CALE (2) has the fol-

lowing bound:

P$ (1=2)U–1(UQU)1=2U–1 ”Pllc1, (42)

where the positive-definite matrix U is defined by U ” (AQ–1AT)1=2.

In addition, using Q$ln(Q)I, we have another lower matrix bound.

Corollary 2.2. The positive solution P of the CALE (2) satisfies

P$ (1=2)ffiffiffiffiffiffiffiffiffiffiffiffiln(Q)

p(A–1QA–T )1=2 ”Pllc2: (43)

JOTA: VOL. 120, NO. 3, MARCH 2004 565

If Q = cI, where c is a positive constant, then (42) becomes (43). Note that the

bounds (42) and (43) coincide with those proposed in Ref. 9.

Setting

M =ATA=b

in (37) leads to the following result.

Corollary 2.3. For the CALE (2), the positive solution P satisfies

P$ (1=ffiffiffiffib

p)(Q –ATA=b)1=2 ” Pllc3, (44)

where the positive constant b is chosen such that bQ >ATA.

The bound (44) results in the eigenvalue bounds

li(P)$ (1=ffiffiffiffiffib i

p)li(Q –ATA=b i)

1=2, i = 1, 2, . . . , n, (45)

where the positive constant bi is chosen such that biQ>ATA.

Reference 18 presented the same bounds (44) and (45).

Furthermore, letting M = I=x results in the following lower matrix

bound.

Corollary 2.4. The positive solution P of the CALE (2) satisfies

P$ (Q – I=x)1=2=ffiffiffix

ps 1(A) ” Pllc4, (46)

where x>0 is determined by Q>xI.From (46), we have

li(P)$li(Q – I=x i)1=2=

ffiffiffix

pis 1(A), i = 1, 2, . . . , n, (47)

where xi >0 is determined by Q >xiI.

The bounds (46) and (47) are the same as those derived in Ref. 19.

If ln(Q) >1, by letting M be (1 – e)Q +Q–1=e, we have a new lower

bound for P.

Corollary 2.5. For the CALE (2), the positive solution P is such that

P$V –1[V (eQ –Q–1=e)V ]1=2V –1 ”Pllc5, (48)

where the positive constant e satisfies

l–1

n (Q)<e<1

and the positive matrix V is defined as

V = {A[(1 – e)Q +Q–1=e]–1AT}1=2:

566 JOTA: VOL. 120, NO. 3, MARCH 2004

If M is chosen as

M = e(Q +ATQ–1A),

with

0<e< [1 + l1(ATQ–1AQ–1)]–1,

then the following lower bound is obtained.

Corollary 2.6. The positive solution P of the CALE (2) is such that

P$E–1{Ee[(1 – e)Q – eATQ–1A]E}1=2E–1 ” Pllc, (49)

where the positive matrix E is defined by

E = {A[Q +ATQ–1A]–1AT}1=2:

Remark 2.3. In light of Corollaries 2.1–2.4, it is obvious that the

bounds (42)–(47) reported in Refs. 9, 18, 19 are only special cases of the

obtained bound (37). Moreover, Ref. 9 showed that the bound (42) is tight-

er than the bound (44). In Refs. 18–19, it is proved that the bounds (45)

and (47) are sharper than the majority of those reported in the literature.

Setting D = 1 and using A to replace A + I, the bounds (11), (23), (27)

become the following matrix bounds for the solution of the DALE (5).

Theorem 2.5. Let the positive-definite matrix P be the solution of the

DALE (5). Then,

P$R–1{R[Q –M + h(A – I)T (A – I)]R}1=2R–1 ” F(M, A, Q), (50)

where R is defined as

R ” [(A – I)M–1(A – I)T ]1=2: (51)

Furthermore, if

1 – s 21(A)>0, (52)

then

P#{l1(Q)=[1 – s 21(A)]}ATA +Q ”Plud (53)

and

P#Q +ATPludA ” Plud1, (54)

JOTA: VOL. 120, NO. 3, MARCH 2004 567

where the positive constant matrix M is determined by Q >M and h is

defined by

h ”s 2

n[(A – I)R] +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis 4

n[(A – I)R] + 4l2

1[(A – I)M–1(A – I)T ]ln[R(Q –M)R]

q

2l2

1[(A – I)M–1(A – I)T ]:

(55)

Remark 2.4. A simple choice of M for (50) is

M =M1 = 0:5Q:

For this case, (50) becomes

P$R–1{R[0:5Q + h(A – I)T (A – I)]R}1=2R–1 =F(M1, A, Q), (56)

with

R ”ffiffiffi2

p[(A – I)Q–1(A – I)T ]1=2, (57)

h ”0:5s 2

n[(A – I)R] +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:25s 4

n[(A – I)R] + l2

1[(A – I)Q–1(A – I)T ]ln(RQR)

q

2l2

1[(A – I)Q–1(A – I)T ]:

(58)

Other simple choices for the matrix M, which are easy to compute, can be

made as

(i) M =M2 = e[Q + (A – I)TQ–1(A – I)],

with

0<e<{1 + l1[(A – I)TQ–1(A – I)Q–1]}–1; (59)

(ii) M =M3 = (1 – e)Q +Q–1=e,

with

l–1

n (Q)<e<1; (60)

(iii) M =M4 = e(A – I)T (A – I),

with

e<ln[(A – I)–TQ(A – I)–1]; (61)

(iv) M =M5 = eI ,

with

e<ln(Q): (62)

568 JOTA: VOL. 120, NO. 3, MARCH 2004

Remark 2.5. The bound (54) is the same as that presented in Ref. 15.

In Ref. 15, it is shown that the bound (54) is tighter than (53) and that the

corresponding eigenvalue bounds are better than some parallel results.

Reference 15 gave also a numerical example to show the good performance

of (54). Furthermore, (50) is a new lower matrix bound for the DALE (5).

It is found that the sharpness between the matrix bound (50) and existing

results cannot be compared by mathematical methods. They may comple-

ment each other for estimating the solution bounds of the DALE (5).

Remark 2.6. From Theorems 2.4 and 2.5, it is seen that the results

presented for the UALE (1) can indeed unify the results for the continuous

and discrete cases.

Using the following notations, existing solution matrix bounds of the

CALE (2) and the DALE (5) are summarized in Tables 1 and 2:

MA(Q) ” max [– xT (ATQ +QA)x=2xTQx], x˛<n=[0], (63)

Table 1. Matrix bounds of the CALE (2).

P$S–1(S(Q – M)S)1=2S–1 ” Pllc This paper

P#l1(Q)– a I ” Pluc This paper

P# [Q+ (A+ I )TPluc(A+ I ) – ATPllcA] ” Pluc1 This paper

P$ (1=2)S–1(SQS)1=2S–1 ” Pllc1 Ref. 9, this paper

P$ (ffiffiffiffiffiffiffiffiffiffiffiffiln(Q)

p=2)(A–1QA–T )1=2 ” Pllc2 Ref. 9, this paper

P$ (Q – ATA=b)1=2=ffiffiffiffib

p” Pllc3, b >0, bQ$ATA Ref. 18, this paper

P$ (Q – I=x)1=2=ffiffiffix

ps 1(A) ” Pllc4, x >0, xQ$ I Ref. 19, this paper

Pllc7 ” Q=2MA(Q)#P#Q=2mA(Q) ” Pluc2 Ref. 11

Pllc8 ” ln(NTQN)N–TR1N

–1#P#l1(NTQN)N–TR1N

–1 ” Pluc3 Ref. 20

Pllc9 ” ln(Gcn)MnMnT#P#l1(Gcn)MnMn

T ” Pluc4 Ref. 22

Pllc10 ” ln(Gcm)MmMmT #P#l1(Gcm)MmMm

T ” Pluc5 Ref. 25

Table 2. Matrix bounds of the DALE (5).

P$R–1(R[Q – M + h(A – I )T(A – I )]R)1=2R–1 ” F(M, A, Q) This paper

P${l1(Q)=[1 – s12(A)]}ATA + Q ” Plld1 Ref. 17

P$Q + ATPlld1A ” Plld2 Ref. 15

P#{l1(Q)=[1 – s12(A)]}ATA + Q ” Plud Ref. 17, this paper

P#Q + ATPludA ” Plud1 Ref. 15, this paper

Plld3 ” ln(Mn M¢n)P1#P#l1(Mn M¢n)P1 ” Plud2, P1 – AnP1A¢n = I Ref. 21

Plld4 ” ln(Mn M¢n)[I – (AA¢)n]–1#P#l1(Mn M¢n)[I – (AA¢)n]–1

” Plud3, with AA¢ = A¢ARef. 21

Plld5 ” ln(Gdm)Mm MmT #P#l1(Gdm)Mm Mm

T ” Plud4 Ref. 25

Plld6 ” ln(NTQN)N–TR2N

–1#P#l1(NTQN)N–TR2N

–1 ” Plud5 Ref. 20

JOTA: VOL. 120, NO. 3, MARCH 2004 569

mA(Q) ” min [– xT (ATQ +QA)x=2xTQx], x˛<n=[0], (64)

R1 ” diag{1=[– 2Re(li(A))]}, (65)

R2 ” diag{1=[1 – jli(A)j2]}, (66)

A =NLN–1, L ” diag{li(A)}, (67)

Gcn ” {gij}˛<n · n, (68)

with

gij ”ðO

0

ai(t)aj(t)dt, (69)

eAT t = a1(t)I + a2(t)AT + � � � + an(t)(A

T )n–1, (70)

Mn ” [D, ATD, (AT )2D, . . . , (AT )n–1D], (71)

where Q =DDT,

Gcm ” {gij}˛<m · m,

with

gij =

ðO0

ai(t)aj(t)dt,

eAT t = a1(t)I + a2(t)AT + � � � + am(t)(AT )m–1, (72)

Gdm ” {gij}˛Rm · m, (73)

with

gij ” �O

k=0ai(k)aj(k), Ak = �

m–1

i=0ai(k)Ai, (74)

Mm ” [D, ATD, (AT )2D, . . . , (AT )m–1D], (75)

where Q =DDT,

m ” degree of the minimal polynomial of A: (76)

3. Numerical Examples

Example 3.1. This example is given in Ref. 25. Consider the CALE

(2) with

A =

– 1 1 0

0 – 1 0

0 0 – 1

264

375, Q =

5 0 1

0 8 1:4

1 1:4 5:5

264

375:

570 JOTA: VOL. 120, NO. 3, MARCH 2004

Then, the positive-definite solution P to the CALE (2) is

P =

2:5000 1:2500 0:5000

1:2500 5:2500 0:9500

0:5000 0:9500 2:7500

264

375,

with

l1(P) = 6:0756, l2(P) = 2:4408, l3(P) = 1:9836:

According to Theorem 2.4 and Corollaries 2.5–2.6, the upper and lower

matrix bounds of P can be estimated as

P$

2:2976 1:1936 0:4924

1:1936 4:9188 0:9216

0:4924 0:9216 2:7436

264

375 =Pllc5,

with

M = 0:45Q +Q–1=0:55

=

2:6280 0:0126 0:3781

0:0126 3:8383 0:5671

0:3781 0:5671 2:8347

2664

3775,

P$

2:3379 1:1731 0:4701

1:1731 4:8365 0:9249

0:4701 0:9249 2:7417

264

375 =Pllc6,

with

M = 0:5(Q +ATQ–1A)

=

2:6040 – 0:1005 0:4802

– 0:1005 4:1626 0:7025

0:4802 0:7025 2:8489

2664

3775,

JOTA: VOL. 120, NO. 3, MARCH 2004 571

P#

8:6752 0 0

0 8:6752 0

0 0 8:6752

2664

3775 =Pluc,

P#

2:7024 1:1040 0:5076

1:1040 11:8459 0:9707

0:5076 0:9707 2:7564

2664

3775 =Pluc1:

From Table 1, the bounds Pllc1 to Pllc4, Pllc7 to Pllc10, and Pluc2 to Pluc5 are

Pllc1 =

2:3174 1:1587 0:4635

1:1587 4:8800 0:9317

0:4635 0:9317 2:7427

264

375, Pllc2 =

3:3141 1:3731 0:3826

1:3731 2:4679 0:1835

0:3826 0:1835 2:3079

264

375,

Pllc3 =

2:2057 0:3982 0:2797

0:3982 2:6546 0:3762

0:2797 0:3762 2:3657

264

375, b = 0:7,

Pllc4 =

1:4930 – 0:0200 0:2497

– 0:0200 2:1212 0:2895

0:2497 0:2895 1:5898

264

375, x = 0:5,

Pllc7 =

1:0607 0 0:2121

0 1:6972 0:2970

0:2121 0:2970 1:1668

2664

3775,

Pllc9 =

0:0330 – 0:0330 0:0066

– 0:0330 0:1078 0:0026

0:0066 0:0026 0:0363

2664

3775,

Pllc10 =

0:4289 – 0:2145 0:0858

– 0:2145 0:9008 0:0772

0:0858 0:0772 0:4718

2664

3775,

Pluc2 =

7:4429 0 1:4886

0 11:9087 2:0840

1:4886 2:0840 8:1872

2664

3775,

572 JOTA: VOL. 120, NO. 3, MARCH 2004

Pluc4 =

56:3700 – 56:3700 11:2740

– 56:3700 184:1420 4:5096

11:2740 4:5096 62:0070

2664

3775,

Pluc5 =

14:5711 – 7:2855 2:9142

– 7:2855 30:5992 2:6228

2:9142 2:6228 16:0282

2664

3775:

Note that the bounds Pllc8 and Pluc3 cannot be calculated because the matrix

A is not diagonable. For this case, it is seen that

Pllc5>Pllc4>Pllc7>Pllc10>Pllc9,

Pllc6>Pllc4>Pllc7>Pllc10>Pllc9,

Pllc3>Pllc4>Pllc7>Pllc10>Pllc9,

Pllc1>Pllc4>Pllc7>Pllc10>Pllc9,

li(Pllc5)>li(Pllc2), for all i,

li(Pllc6)>li(Pllc2), for all i,

Pluc#Pluc4#Pluc5,

Pluc1#Pluc4#Pluc5,

li(Pluc1)<li(Pluc3), for all i:

The corresponding eigenvalue bounds are summarized in Table 3. From

Table 3, it should be noted that all corresponding eigenvalue bounds of Pllc5

are the best ones, except the minimum eigenvalue bound. Furthermore,

according to (41) and (42), we can conclude that

5:7403#l1(P)#8:6752,

2:4094#l2(P)#2:9962,

1:8573#ln(P)#2:2187:

This means that our results on the eigenvalue bounds are tighter than the

previous ones for this case. In fact, if the matrix M is selected as

M =

1:3786 – 1:4286 0

– 1:4286 2:8571 0

0 0 1:4286

264

375,

then we have

ln(P)$1:9607:

JOTA: VOL. 120, NO. 3, MARCH 2004 573

That is, with appropriate selections for M, we can obtain more accurate

estimations for the eigenvalue bounds.

Example 3.2. Since Ref. 15 has given a numerical example to show

the good performance of the bound (54), this paper compares only the

matrix bound (50) with existing results. Consider the DALE (5) with

A =

0:72 0:05 0:05

0 0:75 0

0 0 0:88

264

375, Q =

3 0 – 1:5

0 3 0

– 1:5 0 3

264

375:

For this case, the solution P of the DALE (5) is

P=

6:2292 0:4875 – 3:4818

0:4875 6:9763 – 0:3510

– 3:4818 – 0:3510 12:0087

264

375

and the corresponding eigenvalues are

l1(P)= 13:6849, l2(P) = 6:9709, l3(P) = 4:5584:

In light of (50), by choosing

M =M1 = 0:5Q,

Table 3. Eigenvalue bounds for the solution of the CALE (2).

l1(P) l2(P) l3(P)

Pllc5 5.7403 2.4094 1.8102

Pllc6 5.6661 2.3929 1.8573

Pllc1 5.6906 2.3947 1.8547

Pllc2 4.4105 2.2331 1.4463

Pllc3 3.9314 (b1 = 0.4) 2.2902 (b2 = 0.4) 1.9577 (b3 = 0.7)

Pllc4 2.6726 (x1 = 0.25) 1.7982 (x2 = 0.35) 1.2375 (x3 = 0.5)

Pllc7 1.8404 1.2347 0.8496

Pllc8 failed failed failed

Pllc9 0.1203 0.0390 0.0178

Pllc10 0.9871 0.5310 0.2835

Pluc 8.6752 8.6752 8.6752

Pluc1 12.0899 2.9962 2.2187

Pluc2 12.9137 8.6638 5.9613

Pluc3 failed failed failed

Pluc4 205.4561 66.6689 30.3940

Pluc5 33.5312 18.0380 9.6293

574 JOTA: VOL. 120, NO. 3, MARCH 2004

we estimate the lower matrix bound of P as

P$

5:5524 0:4759 – 3:0765

0:4759 6:3054 – 0:3188

– 3:0765 – 0:3188 11:2888

264

375=F(M1, A, Q),

which yields

l1(P)$12:6634, l2(P)$6:3141, l3(P)$4:1692:

If we select

M =M2 = (1=2)[Q + (A – I)TQ–1(A – I)]

=

1:5174 – 0:0031 – 0:7494

– 0:0031 1:5110 – 0:0001

– 0:7494 – 0:0001 1:5024

2664

3775,

then

P$

5:5622 0:4763 – 3:0795

0:4763 6:3086 – 0:3169

– 3:0795 – 0:3169 11:2877

264

375=F(M2, A, Q),

leading to

l1(P)$12:6659, l2(P)$6:3178, l3(P)$4:1747:

According to Table 3, the bounds Plld1 to Plld6 and the corresponding

eigenvalue bounds are

Plld1 =

4:5413 0:1070 – 1:3930

0:1070 4:6798 0:0074

– 1:3930 0:0074 5:3098

264

375,

l1(P)$6:3726, l2(P)$4:6844, l3(P)$3:4739,

Plld2 =

5:3542 0:2213 – 2:2191

0:2213 5:6518 – 0:0410

– 2:2191 – 0:0410 7:0007

264

375,

l1(P)$8:5532, l2(P)$5:6565, l3(P)$3:7970,

JOTA: VOL. 120, NO. 3, MARCH 2004 575

Plld3 =

3:6924 0:1326 0:1780

0:1326 3:9066 0:0618

0:1780 0:0618 6:0412

264

375,

l1(P)$6:0570, l2(P)$3:9617, l3(P)$3:6215,

Plld5 =

5:4471 0:2259 – 2:8652

0:2259 5:7509 – 0:1283

– 2:8652 – 0:1283 6:9403

264

375,

l1(P)$9:1714, l2(P)$5:7381, l3(P)$3:2288,

Plld6 =

0:8840 – 1:4733 – 0:2762

– 1:4733 6:1315 0:4604

– 0:2762 0:4604 2:1577

264

375,

l1(P)$6:5770, l2(P)$2:1115, l3(P)$0:4847:

Since the matrix A is not normal, the bound Plld4 cannot work. From the

above results, it is seen that

F(M1, A, Q)>Plld1,

F(M2, A, Q)>Plld1,

li[F(M2, A, Q)]>li[F(M1, A, Q)]>li(Plldj), i = 1, 2, 3 and j = 2, 3, 5, 6:

Therefore, it seems that the proposed lower bound F(M2, A, Q) is the best

one for this case.

4. Conclusions

A generalization work for the estimation bounds of the continuous and

discrete algebraic Lyapunov equations was developed. Upper and lower

solution bounds of the UALE including matrix bounds and eigenvalue

bounds were proposed. The results presented are new and become bounds of

the solutions of the continuous and discrete Lyapunov equations via limiting

cases. Comparisons showed that the eigenvalue bounds of the UALE are

tighter than some parallel results and that, for the CALE, the majority of the

lower matrix bounds which have appeared in the literature only are special

cases of the obtained result. Numerical examples are given to demonstrate

576 JOTA: VOL. 120, NO. 3, MARCH 2004

the good performances of the obtained bounds. It is seen also that different

choices of M lead to different bounds. Although simple selections for the

tuning matrix M are given, it is expected to find a systematic method for

determining M that can yields better bounds for the solutions of the CALE

and the DALE.

References

1. MIDDLETON, R. H., and GOODWIN, G. C., Digital Control and Estimation: A Uni-

fied Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1990.

2. KWON, B. H., YOUN, M. J., and BIEN, Z., On Bounds of the Riccati and Lyapunov

Equations, IEEE Transactions on Automatic Control, Vol. 30, pp. 1134–1135,

1985.

3. LEE, C. H., LI, T. H. S., and KUNG, F. C., A New Approach for the Robust

Stability of Perturbed Systems with a Class of Noncommensurate Time Delays,

IEEE Transactions on Circuits and Systems, Part 1, Vol. 40, pp. 605–608, 1993.

4. WANG, S. S., and LIN, T. P., Robust Stability of Uncertain Time-Delay Systems,

International Journal of Control, Vol. 46, pp. 963–976, 1987.

5. LEE, C. H., and LEE, S. T., On the Estimation of Solution Bounds of the

Generalized Lyapunov Equations and the Robust Root Clustering for the Linear

Perturbed Systems, International Journal of Control, Vol. 74, pp. 996–1008,

2001.

6. YEDAVALLII, R. K., Robust Root Clustering for Linear Uncertain Systems Using

Generalized Lyapunov Theory, Automatica, Vol. 29, pp. 237–240, 1993.

7. KOUIKOGLOU, V. S., and PHILLIS, Y. A., Trace Bounds on the Covariances of

Continuous-Time Systems with Multiplicative Noise, IEEE Transactions on

Automatic Control, Vol. 38, pp. 138–142, 1993.

8. MORI, T., and DERESE, I. A., A Brief Summary of the Bounds on the Solution of the

Algebraic Equations in Control Theory, International Journal of Control, Vol. 39,

pp. 247–256, 1984.

9. CHOI, H. H., and KUC, T. Y., Lower Matrix Bounds for the Continuous Algebraic

Riccati and Lyapunov Matrix Equations, Automatica, Vol. 38, pp. 1147–1152,

2002.

10. GARLOFF, J., Bounds for the Eigenvalues of the Solution of Discrete Riccati and

Lyapunov Equations and the Continuous Lyapunov Equation, International Jour-

nal of Control, Vol. 43, pp. 423–431, 1986.

11. GERMEL, J. C., and BERNUSSOU, J., On Bounds of Lyapunov’s Matrix Equation,

IEEE Transactions on Automatic Control, Vol. 24, pp. 482–483, 1979.

12. HMAMED, A., Discrete Lyapunov Equation: Simultaneous Eigenvalue Lower

Bounds, International Journal of Systems Science, Vol. 22, pp. 1121–1126, 1991.

13. KOMAROFF, N., Lower Bounds for the Solution of the Discrete Algebraic Lyapunov

Equation, IEEE Transactions on Automatic Control, Vol. 37, pp. 1017–1018,

1992.

JOTA: VOL. 120, NO. 3, MARCH 2004 577

14. KOMAROFF, N., and SHAHIAN, B., Lower Summation Bounds for the Discrete Ric-

cati and Lyapunov Equations, IEEE Transactions on Automatic Control, Vol. 37,

pp. 1078–1080, 1992.

15. LEE, C. H., Upper and Lower Bounds of the Solutions of the Discrete Algebraic

Riccati and Lyapunov Matrix Equations, International Journal of Control, Vol.

68, pp. 579–598, 1997.

16. LEE, C. H., Eigenvalue Upper and Lower Bounds of the Solution for the Continuous

Algebraic Matrix Riccati Equation, IEEE Transactions on Circuits and Systems,

Part 1, Vol. 43, pp. 683–686, 1996.

17. LEE, C. H., Upper and Lower Bounds of the Solution for the Discrete Lyapunov

Equation, IEEE Transactions on Automatic Control, Vol. 41, pp. 1338–1341,

1996.

18. LEE, C. H., New Results for the Bounds of the Solution for the Continuous Riccati

and Lyapunov Equations, IEEE Transactions on Automatic Control, Vol. 42, pp.

118–123, 1997.

19. LEE, C. H., On the Upper and Lower Bounds of the Solution for the Continuous

Riccati and Matrix Equation, International Journal of Control, Vol. 66, pp.

105–118, 1997.

20. LEE, C. H., Upper and Lower Matrix Bounds of the Solutions for the Continuous

and Discrete Lyapunov Equations, Journal of the Franklin Institute, Vol. 334B,

pp. 539–546, 1997.

21. MORI, T., FUKUMA, N., and KUWAHARA, M., Eigenvalue Bounds for the Discrete

Lyapunov Matrix Equation, IEEE Transactions on Automatic Control, Vol. 30,

pp. 925–926, 1985.

22. MORI, T., FUKUMA, N., and KUWAHARA, M., Explicit Solution and Eigenvalue

Bounds in the Lyapunov Matrix Equation, IEEE Transactions on Automatic

Control, Vol. 31, pp. 656–658, 1986.

23. MRABTI, M., and BENSEDDIK, M., Unified Type Nonstationary Lyapunov Matrix

Equation: Simultaneous Eigenvalue Bounds, Systems and Control Letters, Vol. 24,

pp. 53–59, 1995.

24. MRABTI, M., and HMAMED, A., Bounds for the Solution of the Lyapunov Matrix

Equation: A Unified Approach, Systems and Control Letters, Vol. 18, pp. 73–81,

1992.

25. TROCH, I., Improved Bounds for the Eigenvalues of solutions of the Lyapunov

Equation, IEEE Transactions on Automatic Control, Vol. 32, pp. 744–747, 1987.

26. WANG, S. D., KUO, T. S., and HSU, C. F., Trace Bounds on the Solution of the

Algebraic Matrix Riccati and Lyapunov Equations, IEEE Transactions on Auto-

matic Control, Vol. 31, pp. 654–656, 1986.

27. AMIR-MOEZ, R., Extreme Properties of Eigenvalues of a Hermitian Transformation

and Singular Values of the Sum and Product of Linear Transformations, Duke

Mathematical Journal, Vol. 23, pp. 463–467, 1956.

28. MARSHALL, A. W., and OLKIN, I., Inequalities: Theory of Majorization and Its

Applications, Academic Press, New York, NY, 1979.

578 JOTA: VOL. 120, NO. 3, MARCH 2004