# Some Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation

Authors: Zübeyde Ulukök; Ramazan Türkmen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We propose diverse upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE) by building the equivalent form of the CARE and using some matrix inequalities and linear algebraic techniques. Finally, numerical example is given to demonstrate the effectiveness of the obtained results in this work as compared with some existing results in the literature. These new bounds are less restrictive and provide more efficient results in some cases.


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## Body

## 1. Introduction

In many areas of optimal control [1–3], robust control [1], robust stability [4], filter design [5], stability theory and analysis [6–8], control design [9] in control theory [10, 11] including optimization stability theory, and transient performance performance nonlinear systems [12], the algebraic Riccati and Lyapunov matrix equations play an important role.For example, consider the following linear system such thatA∈ℝn×n,B∈ℝn×m,x0∈ℝn[13]:
(1)x.(t)=Ax(t)+Bu(t),x(0)=x0,
with the state feedback control
(2)u(t)=-Kx(t),K=BTP,
and the performance index
(3)J=∫0∞(xTQx+uTu)dt,
whereR=BBT∈ℝn×n,Q∈ℝn×nis positive semidefinite matrix, andPis the positive semidefinite solution to the continuous algebraic Riccati matrix equation (CARE)
(4)AP+PAT-PRP+Q=0.WhenB=0andAis stable matrix, the CARE (4) becomes the continuous algebraic Lyapunov matrix equation (CALE)
(5)ATP+PA=-Q.It is assumed that the pair(A,R1/2)is stabilizable. Then the CARE (4) has a unique symmetric positive semidefinite stabilizing solution if the pair(A,Q1/2)is observable.The problem of estimating solution bounds for the algebraic Riccati and Lyapunov matrix equations has widely been considered in the recent years, since these equations are widely used in many fields of control system analysis and design. A number of works have reported numerical algorithms to get the exact solution of the mentioned equations [7]. However, we should note that the analytical solution of these equations has some complications and computational burdens, specially, when the dimensions of the system matrices increase. Thus, for some applications such as stability analysis [8], it is the only preferred solution matrix bounds for the exact solution that can be obtained without hard and complicated burdens. Moreover, as mentioned in [12], in practice, the solution matrix bounds can also be used as approximations of the exact solution or initial guesses in the numerical algorithms for the exact solution [10].The existing results obtained during 1974–1994 have been summarized by Kwon et al. [14] only including all eigenvalue bounds such as the extreme eigenvalues, the summation, the trace, majorization inequalities, the product, and the determinant. Unfortunately, by this time, the upper matrix bounds for the solution of the CARE (4) have not been proposed in the literature. However, Lee in [15] has proposed upper and lower matrix bounds for the CARE (4) and henceforth many reports have been presented for the upper [16–20] and lower [18, 19, 21] bounds for the solution of the CARE (4). As matrix bounds include all eigenvalue bounds [14, 22, 23] particularly the minimum and maximum eigenvalues, trace [10, 24, 25], determinant [14], and norm [26] bounds, it is seen that they are the most general and useful. Therefore, this paper presents upper matrix bounds for the solution of the CARE (4) by utilizing various matrix identities and matrix inequalities.Letℝn×mbe the set ofn×mreal matrices. In this paper, we denote the eigenvalues of ann×nreal matrix byλi(X); ifX∈ℝn×nis a symmetric matrix, then its eigenvalues are arranged in the nonincreasing orderλ1(X)≥λ2(X)≥⋯≥λn(X). ForX∈ℝn×m, suppose that the singular values ofXare ordered in nonincreasing form; that is,s1(X)≥s2(X)≥⋯≥sn(X). Also, lettr(X),XT,X-1, anddet(X)denote the trace, transpose, inverse, determinant, respectively. Additionally, the spectral condition number of any matrixXis defined byκ2(X)=s1(X)s1(X-1). WriteX≥(>)0, ifXis a positive semidefinite (positive definite) matrix. For the symmetric matrices of the same sizeXandY, ifX-Yis positive semidefinite, we writeX≥YorY≤X. Then, Weyl’s monotonicity principle means thatY≤Xleads toλi(Y)≤λi(X),i=1,2,…,n. The identity matrix inℝn×nis shown byI.The following lemmas are used to prove the main result of this paper.Lemma 1 (see [27, 28]).
LetX∈ℝn×nbe symmetric matrix. Then the following inequality holds:
(6)λn(X)I≤X≤λ1(X)I.Lemma 2 (see [27, 28]).
For any matrixA∈ℝn×mand any positive semidefinite matricesX,Y∈ℝn×nsuch thatX≥Y>(≥)0, it holds thatATXA≥ATYA, with strict inequality ifXandYare positive definite andAis of full rank.Lemma 3 (see [27, 28]).
For any symmetric matricesX,Y∈ℝn×n, the following inequality holds:
(7)λi+j-1(X+Y)≤λj(X)+λi(Y),lhhhhi+j≤n+1,1≤i,j≤n.Lemma 4 (see [28]).
LetX,Y∈ℝn×n, fori=1,2,…,n, one has
(8)si(X+Y)≤si(X)+s1(Y).Lemma 5 (see [28]).
LetX∈ℝm×n,Y∈ℝn×m, fori=1,2,…,n, one has
(9)si(XY)≤si(X)s1(Y).Lemma 6 (see [29]).
LetX∈ℝn×n, fori=1,2,…,n, then
(10)|λi(A+AT2)|≤si(A).Lemma 7 (see [30]).
The following matrix inequality:(11)(WSSTV)>0,
whereW=WTandV=VT, is equivalent to either
(12)V>0,W-SV-1ST>0,
or
(13)W>0,V-STW-1S>0.Lemma 8 (see [17]).
The positive semidefinite solutionPof the CARE (4) has the following upper bound on its maximal eigenvalue:
(14)λ1(P)≤λ1(DTD)λ1[(Q+KTK)DTD]λn(ZDTD)≡η,
whereKis any matrix stabilizingA+BK(i.e.,Re(λi(A+BK))<0for alli) and the nonsingular matrixDand positive definite matrixZare chosen to yield the LMI
(15)(A+BK)TDTD+DTD(A+BK)≤-Z.
This eigenvalue upper bound (14) is always calculated if there exists a unique positive semidefinite solution of the CARE (4).


## 2. Main Results

Zhang and Liu in [19] obtained the lower and upper bounds for the solution of the CARE (4) which improve the results in [21]. Also, Lee in [18] proposed upper and lower bounds for the solution of the CARE (4) by considering a different approach. In this section, we will present diverse upper matrix bounds for the solution matrix of the CARE (4) in the light of the reported results in [18, 19], by utilizing the above lemmas and generating some matrix identities.Theorem 9.
Assume thatQis symmetric positive definite and there exists a unique symmetric positive semidefinite solutionPto the CARE (4). ThenPsatisfies the following inequality:
(16)P≤{11-λ1(I-εR)×{M+2φ1ελ1(I-εR)s1[A(I-εR)-1]I}11-λ1(I-εR)}1/2≡Pu1(ε,A,Q,R),
where the positive semidefinite matrixM1and the positive constantδare defined by
(17)M=εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT,δ=ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR),
whereεis any positive constant such that
(18)0<ε<∥R+ATQ-1A∥-1,
and positive constantφ1is defined by
(19)φ1≡11-λ1(I-εR)≡×{{[1-λ1(I-εR)]λ1(M)+δ2}1/2+δ}.Proof.
By adding and subtracting(1/ε)PP+A((1/ε)(I-R))-1ATfrom (4), one gets
(20)[P+A(1εI-R)-1](1εI-R)[P+A(1εI-R)-1]T-1εPP-A(1εI-R)-1AT+Q=0,
therefore,
(21)[P+A(1εI-R)-1](1εI-R)[P+A(1εI-R)-1]T=1εPP+A(1εI-R)-1AT-Q.
Applying Lemmas 1 and 2 to (21) gives
(22)1εPP+A(1εI-R)-1AT-Q=[P+A(1εI-R)-1](1εI-R)×[P+A(1εI-R)-1]T≤λ1(1εI-R)[P+A(1εI-R)-1]×[P+A(1εI-R)-1]T≤λ1(1εI-R)[P2+A(1εI-R)-2AThhhhhhhhhhhhhhl+P(1εI-R)-1AT+A(1εI-R)-1P].
For the partP((1/ε)I-R)-1AT+A((1/ε)I-R)-1Pof (22), applying Lemmas 1, 6, and 5, respectively, shows that
(23)P(1εI-R)-1AT+A(1εI-R)-1P≤λ1(P(1εI-R)-1AT+A(1εI-R)-1P)I≤|λ1(P(1εI-R)-1AT+A(1εI-R)-1P)I|≤2s1(P(1εI-R)-1AT)I≤2λ1(P)s1(A(1εI-R)-1)I.
Thus, in light of the fact (23), (22) becomes
(24)1εPP+A(1εI-R)-1AT-Q≤λ1(1εI-R)[(A(1εI-R)-1)P2+A(1εI-R)-2AThhhhhhhhhhhhhhl+2λ1(P)s1(A(1εI-R)-1)I].
IfQ>0andεsatisfies (18), then
(25)I-εR-εATQ-1A>0,εQ>0.
By the application of the Schur complement formula of Lemma 7 to (25), we can say that the above inequalities are satisfied if and only if
(26)(εQεAεATI-εR)>0,
which means that
(27)εQ-ε2A(I-εR)AT>0,I-εR>0.
Therefore, we say that (24) is equivalent to
(28)P2≤λ1(I-εR)[P2+A(1εI-R)-2AT≤+2λ1(P)s1(A(1εI-R)-1)I]≤+εQ-ε2A(I-εR)-1AT.
Since1-λ1(I-εR)>0, (28) can be rewritten as
(29)P2≤11-λ1(I-εR)≤×{[A(I-εR)-1]εQ-ε2A(I-εR)-1AT≤+ε2λ1(I-εR)A(I-εR)-2AT≤+2λ1(P)ελ1(I-εR)s1[A(I-εR)-1]I}.
Utilizing the relations in Lemmas 1 and 3, (29) becomes
(30)λ1(P2)≤11-λ1(I-εR)λ1×{[εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT]+2λ1(P)ελ1(I-εR)s1[A(I-εR)-1]}≤11-λ1(I-εR)×{λ1[εQ-ε2A(I-εR)-1AT+ε2λ1(I-εR)A(I-εR)-2AT]+2λ1(P)ελ1(I-εR)s1[A(I-εR)-1]}.
Solving (30) according toλ1(P)gives
(31)λ1(P)≤{[ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR)]211-λ1(I-εR)λ1≤×[εQ-ε2A(I-εR)-1AT≤+ε2λ1(I-εR)A(I-εR)-2AT]≤+[ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR)]2}1/2≤+ελ1(I-εR)s1[A(I-εR)-1]1-λ1(I-εR)≡φ1.
Substituting (31) into (29) results in the upper bound
(32)P≤{11-λ1(I-εR)≤×{εQ-ε2A(I-εR)-1AT≤+ε2λ1(I-εR)A(I-εR)-2AT≤+2φ1ελ1(I-εR)s1[A(I-εR)-1]I}11-λ1(I-εR)}1/2.
This completes the proof.Remark 10.
The inequality (3.5) in [19] is clearly as follows:
(33)P2≤λ1(I-εR){λ12(P)+2λ1(P)εs1[A(I-εR)-1]≤+ε2s12[A(I-εR)-1]}≤+εQ-ε2A(I-εR)-1AT.
Thus, when the inequality (28) is considered, from the facts
(34)P2≤λ12(P),[A(I-εR)-1][A(I-εR)-1]T≤s12[A(I-εR)-1],
it is seen that the upper bound in Theorem 9 is always sharper than the result given by Theorem3.1in [19].Remark 11.
It is well known that most of the studies in the literature have focused to derive the bounds for the maximum and minimum eigenvalues, the trace, and the determinant for the solution of the CARE (4); however, the matrix solution bounds are quite restriction. Among the mentioned bounds, the matrix solution bounds are the most useful and efficient because other bounds that are dependent on eigenvalue can be derived directly from matrix solution bounds via monotonicity.By using Theorem9, we can derive the following result immediately.Corollary 12.
Assume thatQis symmetric positive definite and there exists a unique symmetric positive semidefinite solutionPto the CARE (4). ThenPsatisfies the following upper eigenvalue bounds:
(35)λi(P)≤minελi(Pu1(ε,A,Q,R))=Pu11≤λi(Pu1(ε,A,Q,R)),tr(P)≤∑i=1nPu11≤minεtr(Pu1(ε,A,Q,R))≤tr(Pu1(ε,A,Q,R)),det(P)≤∏i=1nPu11≤minεdet(Pu1(ε,A,Q,R))≤det(Pu1(ε,A,Q,R)),
whereεsatisfies (18) andPu1is defined by (16).By establishing the more general form than the matrix identity used in Theorem9 for the CARE (4), one gets the following upper bounds.Theorem 13.
LetXbe any symmetric positive definite matrix. Then the unique symmetric positive semidefinite solutionPto the CARE (4) has the following upper bound
(36)P≤M1-1/2{M11/2[[ηs1(A)+λ1(X1)]2Q-X1+λ1(X1-1)×[ηs1(A)+λ1(X1)]2I]×M11/2}1/2M1-1/2≡Pu2(η,X1,A,Q,R),
where the positive definite matrixX1is chosen so that
(37)M1≡ATX1-1A+R>0,
andηis defined by (14).Proof.
By adding and subtractingX1+PATX1-1APto the CARE (4), we can get
(38)(PAT+X1)X1-1(PAT+X1)T-PRP+Q-X1-PATX1-1AP=0
which is equivalent to
(39)0<P(R+ATX1-1A)P<=Q-X1+(PAT+X1)X1-1(PAT+X1)T.
Introducing Lemmas 1, 2, 4, 5, and 8, respectively, to (39) gives
(40)0<P(R+ATX1-1A)P=Q-X1+(PAT+X1)X1-1(PAT+X1)T≤Q-X1+λ1(X1-1)[(PAT+X1)(PAT+X1)T]≤Q-X1+λ1(X1-1)[s12(PAT+X1)]I≤Q-X1+λ1(X1-1)[s1(PAT)+λ1(X1)]2I≤Q-X1+λ1(X1-1)[λ1(P)s1(A)+λ1(X1)]2I≤Q-X1+λ1(X1-1)[ηs1(A)+λ1(X1)]2I.
By the definition (37) ofM1and pre- and postmultiplyingM1to (40) yields
(41)(M11/2PM11/2)2=M11/2PM1PM11/2≤M11/2[Q-X1+λ1(X1-1)[ηs1(A)+λ1(X1)]2I]M11/2.
Solving this inequality forPshows the upper bound (36).
This builds the proof.Remark 14.
Note that for the upper bound (36), the matricesRandQdon not have to be nonsingular. This means that the upper bound proposed by Theorem 13 can always be computed without any condition for positive definite matrixX1which arbitrarily is selected.From Theorem13, we have the following corollaries.Corollary 15.
The positive semidefinite solutionPto the CARE (4) has
(42)P≤{1λn(M1)[Q-X1≤{1λn(M1)[Q+λ1(X1-1)[ηs1(A)+λ1(X1)]2I]1λn(M1)}1/2≡Pu21,
whereηandM1for the positive definite matrixX1are defined by (14) and (37), respectively.Proof.
Applying Lemma1 to the right side of (41) and solving it with regard toPgive the upper boundPu21.Corollary 16.
The solutionPto the CARE (4) satisfies the following upper eigenvalue bounds:
(43)λi(P)≤minλi(Pu2(η,X1,A,Q,R))=Pu2*≤λi(Pu2(η,X1,A,Q,R)),tr(P)≤∑i=1nPu2*≤mintr(Pu2(η,X1,A,Q,R))≤tr(Pu2(η,X1,A,Q,R)),det(P)≤∏i=1nPu2*≤mindet(Pu2(η,X1,A,Q,R))≤det(Pu2(η,X1,A,Q,R)),
whereηis defined by (14) and the positive matrixX1is selected so as to satisfy the definition (37), respectively.Theorem 17.
LetPbe the positive semidefinite solution of the CARE (4). ThenPhas the upper bound
(44)P≤M2-1/2{M21/2[Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I(X2-1)X22]M21/2}1/2M2-1/2≡Pu3(φ2,X2,A,Q,R),
where the positive definite matrixX2is chosen so that
(45)M2≡R-AT[λ1(X2-1)I-X2-1]A>0,
andφ2is defined by
(46)φ2≡{1λn(M2)λ1[Q-X2+λ1(X2-1)X22]≡+κ22(X2)s12(A)λn2(M2)}1/2+κ2(X2)s1(A)λn(M2).Proof.
By the use of the equality (39), from Lemmas 1 and 2, we can write
(47)P(R+ATX2-1A)P=Q-X2+(PAT+X2)X2-1(PAT+X2)T≤Q-X2+λ1(X2-1)[PATAP+PATX2+X2AP+X22].
Having applied Lemmas 1, 6, and 5, respectively, to the part ofPATX2+X2APin (47), since the following inequalities hold:
(48)PATX2+X2AP≤λ1(PATX2+X2AP)I≤|λ1(PATX2+X2AP)|I≤2s1(PATX2)≤2λ1(P)s1(A)λ1(X2)I
via the definition ofM2from (47), we arrive at
(49)P[R-AT(λ1(X2-1)I-X2-1)A]P=PM2P≤Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)λ1(P)I.
Applying Lemmas 1 and 3 to (49), we have
(50)P2≤1λn(M2)[Q-X2+λ1(X2-1)X22]+2κ2(X2)s1(A)λn(M2)λ1(P)I≤1λn(M2)λ1[Q-X2+λ1(X2-1)X22]I+2κ2(X2)s1(A)λn(M2)λ1(P)I.
Then,
(51)λ1(P2)≤1λn(M2)λ1[Q-X2+λ1(X2-1)X22]≤+2κ2(X2)s1(A)λn(M2)λ1(P).
Solving (51) with respect toλ1(P)gives
(52)λ1(P)≤{κ22(X2)s12(A)λn2(M2)1λn(M2)λ1[Q-X2+λ1(X2-1)X22]+κ22(X2)s12(A)λn2(M2)}1/2+κ2(X2)s1(A)λn(M2)≡φ2.
Substitutingφ2into (49), we get
(53)PM2P≤Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I.
Pre- and postmultiplyingM21/2to (53) leads to
(54)(M21/2PM21/2)2≤M21/2[Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I(X2-1)]M21/2.
Therefore, by the nonsingularity ofM2, the upper matrix bound (44) is directly obtained by solving (54) with respect toP.
The proof is finished.According to Theorem17, we can propose the following corollaries.Corollary 18.
The positive semidefinite solutionPto the CARE (4) satisfies
(55)P≤{1λn(M2)[Q-X2+λ1(X2-1)X22+2κ2(X2)s1(A)φ2I(X2-1)]1λn(M2)}1/2≡Pu31,
where the positive definite matricesX2, andM2and the positive constantφ2are defined by (45) and (46), respectively.Proof.
Substitutingφ2into (50), having solved (50) regard toP, we obtain the upper bound (55).Corollary 19.
The positive semidefinite solutionPto the CARE (4) has the following eigenvalue upper bounds:
(56)λi(P)≤minλi(Pu3(φ2,X2,A,Q,R))=Pu3*≤λi(Pu3(φ2,X2,A,Q,R))tr(P)≤∑i=1nPu3*≤mintr(Pu3(φ2,X2,A,Q,R))≤tr(Pu3(φ2,X2,A,Q,R))det(P)≤∏i=1nPu3*≤mindet(Pu3(φ2,X2,A,Q,R))≤det(Pu3(φ2,X2,A,Q,R)),
where the positive definite matricesX2andM2and the positive constantφ2are defined by (45) and (46), respectively.As considered a diverse matrix identity, in the case that the matrixRis nonsingular, we can derive the following alternative upper bounds for the solution of the CARE (4).Theorem 20.
If the positive definite matrixPis a unique solution matrix of the CARE (4), then
(57)P≤{1λn(M3)[[η+s1(ATX3)]2Q-ATX3A+λ1(X3-1)×[η+s1(ATX3)]2I]1λn(M3)}1/2≡Pu4(η,X3,A,Q,R),
whereX3is a positive constant matrix such thatM3≡R+X3-1>0andηis defined by (14).Proof.
When the termPX3-1P+ATX3Ais added and subtracted from the CARE (4), we can write
(58)(P+ATX3)X3-1(P+ATX3)T-PRP+Q-ATX3A-PX3-1P=0
which is equivalent to
(59)0<P(R+X3-1)P=Q-ATX3A+(P+ATX3)X3-1(P+ATX3)T.
By the use of Lemmas 1, 2, 4, and 8 for the right side of the above equation, respectively, we obtain
(60)0<P(R+X3-1)P≤Q-ATX3A+λ1(X3-1)(P+ATX3)(P+ATX3)T≤Q-ATX3A+λ1(X3-1)s12(P+ATX3)I≤Q-ATX3A+λ1(X3-1)[λ1(P)+s1(ATX3)]2I≤Q-ATX3A+λ1(X3-1)[η+s1(ATX3)]2I
and by the application of Lemma 1 to the termP(R+X3-1)Pof (60), we can write
(61)P2≤1λn(R+X3-1)≤×[Q-ATX3A+λ1(X3-1)[η+s1(ATX3)]2I].
Therefore, if the above inequality is solved with respect toP, we arrive at the upper boundPu4.
Thus, the proof is established.Theorem 21.
LetPbe the positive semidefinite solution of the CARE (4). Then
(62)P≤{1λn(M4)[Q-ATX4A+λ1(X4-1)ATX42A+2s1(A)κ2(X4)φ3I(X4-1)]1λn(M4)}1/2≡Pu5(φ3,X4,A,Q,R),
where the positive definite matrixX4is selected such that
(63)M4≡R-[λ1(X4-1)I-X4-1]>0
and the nonnegative constantφ3is defined by
(64)φ3≡{κ22(X4)s12(A)λn2(M4)1λn(M4)λ1[Q-ATX4A+λ1(X4-1)ATX42A]+κ22(X4)s12(A)λn2(M4)}1/2+κ2(X4)s1(A)λn(M4).Proof.
Consider (58). From Lemma 1, we can easily write
(65)0<P(R+X4-1)P≤Q-ATX4A+λ1(X4-1)(P+ATX4)(P+ATX4)T
and then via the inequality obtained by using Lemmas 1, 6, and 5, respectively,
(66)PX4A+ATX4P≤2λ1(P)s1(A)λ1(X4)
and the definition (63) ofM4, from (65), we have
(67)0<P[R-(λ1(X4-1)I-X4-1)]P=PM4P≤Q-ATX4A+λ1(X4-1)ATX42A+2s1(A)κ2(X4)λ1(P)I.
By the use of Lemmas 1 and 2, it is obtained that
(68)P2≤1λn(M4)≤×[Q-ATX4A+λ1(X4-1)ATX42A≤+2s1(A)κ2(X4)λ1(P)I(X4-1)AT]
and thus applying Lemma 3 to (68) yields
(69)λ1(P2)≤1λn(M4)≤×{λ1[Q-ATX4A+λ1(X4-1)ATX42A]≤+2s1(A)κ2(X4)λ1(P)[Q-ATX4A+λ1(X4-1)ATX42A]}.
As solving (69) according toλ1(P), one can reach the nonnegative constantφ3is defined by (64). If it is substitutedφ3into (68), then
(70)P2≤1λn(M4)[Q-ATX4A+λ1(X4-1)ATX42A≤+2s1(A)κ2(X4)φ3I(X4-1)].
Thus, solving the inequality (70) derives the upper bound (62) for the solutionPof the CARE (4).
This concludes the proof of the theorem.Corollary 22.
The solutionPto the CARE (4) has the following eigenvalue bounds forj=4,5:
(71)λi(P)≤minλi(Puj)=Puj*≤λi(Puj),tr(P)≤∑i=1nPuj*≤mintr(Puj)≤tr(Puj),det(P)≤∏i=1nPuj*≤mindet(Puj)≤det(Puj).Remark 23.
Chen and Lee in [16] indicated in it is hard or impossible to determine the best matrix bound among the parallel results. Since we find that it is difficult to compare the tightness of our results to the parallel result in [18], we will only make the comparisons on an example.


## 3. Numerical Example

In this section, we will give a numerical example to demonstrate the effectiveness of the proposed results of this paper.Example 1. Consider the CARE (4) with
(72)A=(-1010),Q=(1008),R=(94416).
Chooseε=0.002, then using (16) shows the following upper matrix bound:
(73)P≤(0.8232-0.0001-0.00011.2853)≡Pu1,tr(Pu1)=2.1085,det(Pu1)=1.0580.
The upper bound (55) gives
(74)P≤(0.81730.00680.00681.2815)≡Pu31,tr(Pu31)=2.0988,det(Pu31)=1.0473,
withX2=(100.10.110), and the upper bound (62) gives
(75)P≤(0.8070001.2750)≡Pu5,tr(Pu5)=2.082,det(Pu5)=1.0289,
withX4=(100.10.110).Using Theorem3.1of [19], we obtain the following upper matrix bound:
(76)P≤(1.23280.00020.00021.2884)≡Pu[19],tr(Pu[19])=2.5212,det(Pu[19])=1.5883.
The bounds(18)and(23)proposed in [18] are
(77)P≤(1.7135001.0627)≡Pu1[18],tr(Pu1[18])=2.7762,det(Pu1[18])=1.8209,P≤(1.25270.00500.00501.6025)≡Pu2[18],tr(Pu2[18])=2.8552,det(Pu2[18])=2.0074,
forR¯1=X4.By a simple computation, we have(78)Pu5≤Pu31≤Pu[19]≤Pu2[18],Pu5≤Pu1≤P[19],tr(Pu5)≤tr(Pu31)≤tr(Pu1)≤tr(Pu[19]),≤tr(Pu1[18])≤tr(Pu2[18]),det(Pu5)≤det(Pu31)≤det(Pu1)≤det(Pu[19]),≤det(Pu1[18])≤det(Pu2[18])
which means that our upper bounds give more precise solution estimates than the results given by Theorem3.1in [19] and Theorems2and3in [18] for this case.


## 4. Conclusion

In this paper, new upper matrix bounds for the solution of the CARE are improved by using some linear algebraic techniques and matrix inequalities. A numerical example is given to show that the solution upper bounds presented in this paper are sharper than some results in the literature.



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*Source: 792782-2013-11-28.xml*