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Unified, improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation
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Journal of the Franklin Institute 350 (2013) 1634–1648 www.elsevier.com/locate/jfranklin
Unified, improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation Jianhong Xun Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA Received 27 October 2012; received in revised form 1 February 2013; accepted 20 March 2013 Available online 26 April 2013
Abstract We develop a broad framework that leads to a unified, improved matrix upper bound on the positive semidefinite solution of the continuous coupled algebraic Riccati equation (CCARE). Our approach was largely inspired by recent works of Lee and of Davies, Shi, and Wiltshire. Our main result tightens the needed upper bounds on relevant dominant eigenvalues. Besides, it applies to a larger number of situations since a key restriction here is weaker as compared to those in Davies et al. Numerical evidence is also provided to demonstrate the behavior of the new matrix upper bound. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction For i; j∈f1; 2; …; sg ¼ S, where s≥2, let Ai ; Qi ∈Rnn , Bi ∈Rnm , and δi;j ≥0 with ∑j∈S\fig δi;j 40 be given. In particular, Qi are assumed to be (symmetric) positive semidefinite for all i. In this paper, we are concerned with an upper bound for the positive semidefinite solution Pi ∈Rnn of the continuous coupled algebraic Riccati equation (CCARE): ATi Pi þ Pi Ai −Pi Bi BTi Pi þ ∑ δi;j Pj ¼ −Qi ;
1≤i≤s:
ð1Þ
j∈S\fig
We refer the reader to [1,11] for necessary and sufficient conditions on the existence and uniqueness of this positive semidefinite solution Pi.
n
Tel.: +1 618 453 6510; fax: +1 618 453 5300. E-mail address: [email protected]
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.03.015
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The CCARE plays a central role in the optimal control of the so-called jump linear system, which, in the notation of [1], is given by d xðtÞ ¼ AðrðtÞÞxðtÞ þ BðrðtÞÞuðtÞ; dt where r(t) is a finite state Markovian jump process on S, AðrðtÞÞ∈Rnn , BðrðtÞÞ∈Rnm , and xðtÞ∈Rn and uðtÞ∈Rm are the state and control vectors, respectively; see [13] for detailed background material on jump linear systems. It should be mentioned here that there are other relevant formulations for control systems in literature, such as interval systems [3], LQG regulator systems [15], semilinear fractional evolution systems [16,17], and LQ dynamic compensator systems [18]. It has been an important and fairly new question, see [4,6,9], to bound the positive semidefinite solution of the CCARE. As pointed out in [4,6], besides theoretical considerations, such estimates of the solution serve practical purposes as well; for example, in problems such as time-delay system controller design, stability analysis, robust stabilization, and convergence of numerical algorithms, it often suffices to have sufficiently tight estimates of the solution of the CCARE rather than the exact solution. In [12], two matrix upper bounds on the positive semidefinite solution of the non-coupled continuous algebraic Riccati equation (CARE), i.e. for the case when s¼ 1, were first established. These results were recently generalized to CCARE in [6]. We comment that a majority of the relevant existing results deal only with the case of non-coupled algebraic Riccati equations; see, for example, [7,8] and the references therein. Meanwhile, the extension of such results on the CARE to the case of CCARE is not straightforward in general. As far as we can see, the matrix upper bounds as in [6] represent the best estimates available to the CCARE. Observe that [4] only provides upper bounds on eigenvalues, while [9] only applies to the discrete case as a key identity there, i.e. [9, (5)], becomes trivial for the continuous case. However, there are several questions that remain to be investigated concerning the results in [6]. First, what is the relationship between the matrix upper bounds, i.e. [6, Theorems 2.1 & 2.2], on the positive semidefinite solution of the CCARE? Second, can we identify conditions under which one matrix upper bound performs better than the other? Furthermore, is there a best possible matrix upper bound within the framework of [6]? Third, the dominant eigenvalues of Pi's are estimated in [6] using a sum ∑i∈S λ1 ðPi Þ, yet what we really need for constructing matrix upper bounds are bounds on each individual dominant eigenvalue, i.e. λ1 ðPi Þ. This suggests that the results of [6] may be strengthened by estimating these dominant eigenvalues individually. We shall address the above questions in this paper. Specifically, we shall demonstrate that the matrix upper bounds in [6] can be refined as follows: (i) We propose a unifying, broader framework that connects the seemingly separate results in [6, Theorems 2.1 & 2.2]; see Theorem 3.3 and its corollaries. (ii) Within this broader framework, new, tighter matrix upper bounds can be accomplished. In particular, we provide a monotonicity result, i.e. Theorem 3.5, which not only shows that [6, Theorem 2.2] produces a tighter estimate than [6, Theorem 2.1] but also partially answers the question in Eq. (26) about a best possible upper bound. (iii) Our method leads directly to an estimate of each individual dominant eigenvalue λ1 ðPi Þ and thus to improved matrix upper bounds than [6, Theorems 2.1 & 2.2]. Besides, the key condition for our unified matrix upper bound to hold is less restrictive in comparison with the corresponding ones in [6]. Also see Remarks 3.3 and 3.4.
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2. Preliminaries Let us begin with a brief description about the notation we shall use in the sequel. For any A∈Rnn , we denote by λi ðAÞ its i-th eigenvalue and by ρðAÞ ¼ max1≤i≤n fjλi ðAÞjg the spectral radius of A. Given a symmetric A∈Rnn , its eigenvalues are always arranged in nonincreasing order λ1 ðAÞ≥λ2 ðAÞ≥⋯≥λn ðAÞ; we write A≻0 (A≽0, resp.) if A is positive definite (positive semidefinite, resp.); the notation A≻B (A≽B, resp.) means A−B≻0 (A−B≽0, resp.).1 Besides, given any A ¼ ½ai;j ; B ¼ ½bi;j ∈Rnm , the notation A≥0 means ai;j ≥0 for all i; j, whereas A≥B represents A−B≥0. Next, we recall that A ¼ ½ai;j ∈Rnn is called a Z-matrix, written as A∈Znn , provided ai;j ≤0 for all i≠j. If A∈Znn is nonsingular with A−1 ≥0, then A is said to be a (nonsingular) M-matrix. The first lemma below is a consequence of Sylvester's law of inertia for congruent symmetric matrices. Lemma 2.1 (Horn and Johnson [10, p. 470]). Let A; B∈Rnn be symmetric such that A≽B (A≻B, resp.). Then for any nonsingular C∈Rnn , C T AC≽C T BCðC T AC≻C T BC; resp:Þ: We shall also need the particular case from the following lemma. Lemma 2.2 (Horn and Johnson [10, p. 471]). Let A; B∈Rnn with A≽0 and B≻0. Then B≽A if and only if ρðB−1 AÞ≤1. In particular, λ1 ðAÞI≽A. The next lemma is attributed to H. Weyl: Lemma 2.3 (Horn and Johnson [10, p. 181]). Assume that A; B∈Rnn are symmetric. Then for any 1≤i≤n, λi ðAÞ þ λn ðBÞ≤λi ðA þ BÞ≤λi ðAÞ þ λ1 ðBÞ: In particular, (i) λ1 ðA þ BÞ≤λ1 ðAÞ þ λ1 ðBÞ. (ii) A≽B implies λ1 ðAÞ≥λ1 ðBÞ.
Continuing, we quote two more lemmas regarding eigenvalues—these lemmas are due to Ky Fan and A. Ostrowski, respectively. Lemma 2.4 (Marshall et al. [14, p. 324]). Given any A∈Rnn , T A þA Re½λi ðAÞ≤λ1 2 for all 1≤i≤n. 1
This defines the so-called Löwner partial order over the set of positive semidefinite matrices.
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Lemma 2.5 (Horn and Johnson [10, p. 224]). Let A; B∈Rnn with A being symmetric and B being nonsingular. Then for each 1≤i≤n, there exists θi 40 such that λn ðBT BÞ≤θi ≤λ1 ðBT BÞ and λi ðBT ABÞ ¼ θi λi ðAÞ: The last lemma here provides a characterization of M-matrices. Lemma 2.6 (Berman and Plemmons [2, p. 137]). Assume that A∈Znn . Then A is an M-matrix if and only if: (i) A has all positive diagonal entries. (ii) There is a diagonal matrix D with all positive diagonal entries such that AD is strictly diagonally dominant. In particular, A∈Znn is an M-matrix when A has all positive diagonal entries and A itself is strictly diagonally dominant. 3. Main results We are now in a position to develop results on the unified upper bound on the positive semidefinite solution Pi of the CCARE in Eq. (1). Following [6,12], we always assume in what follows that for each i∈S, there exists αi 40 such that ATi þ Ai ≺2αi Bi BTi :
ð2Þ
For convenience, we set Ci ¼ Ai −αi Bi BTi . Clearly, the above condition is equivalent to C Ti þ C i ≺0. Lemma 3.1. For any γ i o0, under condition (2), C i þ γ i I is nonsingular. Proof. By Lemma 2.4, for any 1≤j≤n, 2 Re½λj ðCi þ γ i IÞ≤λ1 ðC Ti þ C i þ 2γ i IÞ ¼ λ1 ðCTi þ C i Þ þ 2γ i o0: This implies λj ðCi þ γ i IÞ≠0 for all j, and thus the conclusion follows.
□
Lemma 3.2. Let γ i o0 and βi 4γ i . Moreover, suppose that condition (2) holds. Then λ1 fðCi þ γ i IÞ−T ½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞIðCi þ γ i IÞ−1 go1: Proof. By Lemma 2.3, it suffices to show that ðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 ≺I; which is equivalent to ðC i þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞI≺ðC i þ γ i IÞT ðCi þ γ i IÞ by Lemma 2.1. Note that ðC i þ γ i IÞT ðCi þ γ i IÞ−½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞI ¼ ðγ i −βi ÞðC Ti þ C i Þ≻0: This completes the proof.
□
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Continuing, we set si ¼
λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 o0: λ1 ½ðCi þ γ i IÞ−T ðC Ti þ C i ÞðCi þ γ i IÞ−1
Observe that the denominator λ1 ½ðC i þ γ i IÞ−T ðCTi þ C i ÞðC i þ γ i IÞ−1 o0 due to CTi þ Ci ≺0 and Lemma 2.1. We now define F ¼ ½f i;j ∈Zss and g∈Rs as follows: 3 2 −1 δ1;2 δ1;3 ⋯ δ1;s 2 3 s1 s1 6δ −1 δ2;3 ⋯ δ2;s 7 7 6 2;1 s2 6 7 s 7 6 2 6 76 ⋮ ⋱ ⋱ ⋱ ⋮ 7 F¼6 76 7 ⋱ 4 56 ⋱ ⋱ δs−1;s 7 5 4 ⋮ ss δs;1 ⋯ ⋯ δs;s−1 s−1 s 3 2 1 δ1;2 s1 δ1;3 s1 ⋯ δ1;s s1 6δ s 1 δ s ⋯ δ2;s s2 7 2;3 2 7 6 2;1 2 7 6 7 6 ⋮ ⋱ ⋱ ⋱ ⋮ ¼6 ð3Þ 7 6 ⋮ ⋱ ⋱ δs−1;s ss−1 7 5 4 δs;1 ss ⋯ ⋯ δs;s−1 ss 1 and gi ¼ −
λ1 ½ðCi þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðCi þ γ i IÞ−1 40: λ1 ½ðCi þ γ i IÞ−T ðC Ti þ Ci ÞðCi þ γ i IÞ−1
ð4Þ
In addition, we let δi ¼ ∑j∈S\fig δi;j 40. According to Lemma 2.6, we see immediately the following result, whose proof is omitted: Theorem 3.1. Under condition (2), if each γ i o0 is such that μi ¼ λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1 þ δi λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 o0;
1≤i≤s;
ð5Þ viz. 1 þ δi si 40;
1≤i≤s;
then F in Eq. (3) is an M-matrix. Using condition (5), we may develop alternative conditions for F to be an M-matrix. The following conclusion provides one such condition. Theorem 3.2. Under condition (2), let γ i o0 be such that νi ¼ λ1 ðC Ti þ C i Þλn ½ðC i þ γ i IÞðCi þ γ i IÞT þ δi λ1 ½ðC i þ γ i IÞðC i þ γ i IÞT o0; where 1≤i≤s. Then F in Eq. (3) is an M-matrix. Proof. By Lemma 2.5, there exists θ1 with θ1 ≥λn ½ðCi þ γ i IÞ−T ðC i þ γ i IÞ−1 40
ð6Þ
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such that λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1 ¼ θ1 λ1 ðCTi þ C i Þ: From Eq. (6), we see that λ1 ðCTi þ C i Þλn ½ðC i þ γ i IÞ−T ðCi þ γ i IÞ−1 þ δi λ1 ½ðC i þ γ i IÞ−T ðCi þ γ i IÞ−1 o0; which implies λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1 þ δi λ1 ½ðCi þ γ i IÞ−T ðCi þ γ i IÞ−1 o0; i.e. Eq. (5). Hence F is an M-matrix.
□
Remark 3.1. Obviously, Theorem 3.1 implies Theorem 3.2, but not the converse. Incidentally, the condition in Eq. (6) can be reformulated in terms of the condition number κ 2 ðÞ: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ1 ðCTi þ C i Þ κ2 ðC i þ γ i IÞo − : δi From Theorem 3.1, the next result is straightforward and thus its proof will be omitted. We comment that the key for this result is F −1 ≥0. Lemma 3.3. Under conditions (2) and (5), the matrix inequality ð7Þ
Fx≤g; where F and g are given by Eqs. (3) and (4), respectively, has solution x≤F −1 g: Before proceeding, we write the solution in Lemma 3.3 as ξ. Hence, we have Fξ ¼ g
viz: ξ ¼ F −1 g:
ð8Þ
We now state our unified matrix upper bound as follows. Theorem 3.3. Suppose that there exists αi 40 such that condition (2) is satisfied for all i∈S. For each i∈S, let βi ; γ i be such that γ i o0, γ i oβi ≤−γ i , and condition (5) holds too. Then the positive semidefinite solution Pi, 1≤i≤s, of the CCARE in Eq. (1) are bounded by (
Pi ⪯ðC i þ γ i IÞ
−T
!)
ξi ½ðC i þ βi IÞ ðC i þ T
βi IÞ−ðβ2i −γ 2i ÞIþðβi −γ i Þ
∑ δi;j ξj I þ Qi þ
α2i Bi BTi
ðC i þ γ i IÞ−1 ¼ U i
j∈S\fig
ð9Þ where ξi are given in Eq. (8). Proof. Using Eq. (1), we can easily verify that ðC i þ γ i IÞT Pi ðC i þ γ i IÞ þ ðβi −γ i ÞðPi −αi IÞBi BTi ðPi −αi IÞ ¼ ðC i þ βi IÞ Pi ðCi þ T
βi IÞ−ðβ2i −γ 2i ÞPi
þ ðβi −γ i Þ
∑ δi;j Pj þ Qi þ j∈S\fig
which leads to ðC i þ γ i IÞT Pi ðC i þ γ i IÞ⪯ðCi þ βi IÞT Pi ðC i þ βi IÞ−ðβ2i −γ 2i ÞPi
! α2i Bi BTi
;
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! þðβi −γ i Þ
∑ δi;j Pj þ Qi þ α2i Bi BTi : j∈S\fig
By Lemma 2.1, it follows Pi ⪯ðCi þ γ i IÞ−T ðC i þ βi IÞT Pi ðCi þ βi IÞ−ðβ2i −γ 2i ÞPi !# þðβi −γ i Þ
∑ δi;j Pj þ Qi þ α2i Bi BTi
ðC i þ γ i IÞ−1 :
j∈S\fig
This, according to Lemma 2.2, implies Pi ⪯ðCi þ γ i IÞ−T λ1 ðPi Þ½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞI !) þðβi −γ i Þ
∑ δi;j λ1 ðPj ÞI þ Qi þ α2i Bi BTi
ðC i þ γ i IÞ−1 :
ð10Þ
j∈S\fig
Now, using Lemma 2.3, we arrive at λ1 ðPi Þ≤λ1 ðPi Þλ1 ðCi þ γ i IÞ−T ðC i þ βi IÞT ðC i þ βi IÞ −ðβ2i −γ 2i ÞI ðC i þ γ i IÞ−1 g þðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðCi þ γ i IÞ−1 ∑ δi;j λ1 ðPj Þ −T
þðβi −γ i Þλ1 ½ðC i þ γ i IÞ ðQi þ
j∈S\fig 2 T αi Bi Bi ÞðC i
þ γ i IÞ−1 :
For convenience, we denote f~ i;i ¼ 1−λ1 fðCi þ γ i IÞ−T ½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2 −γ 2 ÞIðCi þ γ i IÞ−1 g; i
f~ i;j ¼ −ðβi −γ i Þδi;j λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 ;
i
i≠j;
ð11Þ ð12Þ ð13Þ
and g~ i ¼ ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðC i þ γ i IÞ−1 : By Lemma 3.2, f~ i;i 40 for all i. Furthermore, we claim f~ i;i ¼ −ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðC T þ C i ÞðC i þ γ i IÞ−1 ; i
ð14Þ ð15Þ
since ðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 ¼ ðCi þ γ i IÞ−T f½C i þ γ i I þ ðβi −γ i ÞIT ½C i þ γ i I þ ðβi −γ i ÞI−ðβ2i −γ 2i ÞIgðC i þ γ i IÞ−1 ¼ ðCi þ γ i IÞ−T ðC i þ γ i IÞT ðC i þ γ i IÞ þ ðβi −γ i Þ½ðC i þ γ i IÞT þ ðC i þ γ i IÞ −2γ i ðβi −γ i ÞI ðC i þ γ i IÞ−1 ¼ I þ ðβi −γ i ÞðC i þ γ i IÞ−T ðCTi þ C i ÞðC i þ γ i IÞ−1 : −1 Hence, Eq. (11) is in fact equivalent to Eq. (7) with the ith inequality being scaled with f~ i;i and xi ¼ λ1 ðPi Þ for each i∈S. Finally, by Eq. (8), we obtain that for each i∈S, λ1 ðPi Þ≤ξi : Consequently, the estimate above of λ1 ðPi Þ together with Eq. (10) yield directly the matrix upper bound in Eq. (9). □
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Regarding Eqs. (3)–(6), (8) and (9), we remark as follows: Remark 3.2. The estimate of Pi in Eq. (9) hinges upon ξ as given in Eq. (8). It is interesting to notice that F, g, and ξ depend on the αi 's and γ i 's, yet they are independent of the βi 's. Similarly, both conditions (5) and (6) are independent of βi . In [6, Theorems 2.1 & 2.2], two matrix upper bounds were established. In fact, these two bounds can now be thought of as special, more relaxed cases of our Theorem 3.3. Using our unifying framework, we shall prove these two cases together. Corollary 3.1 (cf. Davies et al. [6, Theorem 2.1]). Let βi ¼ 0 and γ i ¼ −1 for all i∈S. Assume that conditions (2) and (5) are satisfied. Then, we have " # Pi ⪯ðC i −IÞ−T ξi ðCTi C i þ IÞ þ ∑ δi;j ξj IþQi þ α2i Bi BTi ðCi −IÞ−1 ;
1≤i≤s;
ð16Þ
j∈S\fig
where ξi are given in Eq. (8). Compared to η in [6, Theorem 2.1], these ξi lead to a tighter upper bound in Eq. (16). Moreover, condition (5) is less restrictive than [6, condition (2.1)]. Corollary 3.2 (cf. Davies et al. [6, Theorem 2.2]). Let βi ¼ 1 and γ i ¼ −1 for all i∈S. Assume that conditions (2) and (5) are satisfied. Then, we have " # Pi ⪯ðC i −IÞ−T ξi ðCi þ IÞT ðC i þ IÞ þ 2 ∑ δi;j ξj Iþ2ðQi þ α2i Bi BTi Þ ðC i −IÞ−1 ;
1≤i≤s;
j∈S\fig
ð17Þ where ξi are given in Eq. (8). Compared to φ in [6, Theorem 2.2], these ξi lead to a tighter upper bound in Eq. (17). Moreover, condition (5) is less restrictive than [6, condition (2.14)]. Proof. First, we show that condition (5) is less restrictive than the corresponding conditions in [6]. Note that condition (5) can be written as f~ i;i þ ∑f~ i;j 40;
1≤i≤s;
j≠i
where f~ i;j are given in Eqs. (12) and (13); i.e. condition (5) is equivalent to that, for each i, 1−λ1 fðC i þ γ i IÞ−T ½ðC i þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞIðCi þ γ i IÞ−1 g −ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 ∑ δi;j 40:
ð18Þ
j∈S\fig
On the other hand, by [6, (2.6) & (2.16)], we see that in our notation, [6, (2.1) & (2.14)] can be unified as ∑ λ1 fðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 g
−T −1 þ ∑ ðβi −γ i Þλ1 ½ðC i þ γ i IÞ ðC i þ γ i IÞ ðs−1Þ max δi;j o1;
i∈S
i∈S
i;j∈S;j≠i
which implies 1−∑ λ1 fðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 g i∈S
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−∑ ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 ∑ δi;j 40: i∈S
ð19Þ
j∈S\fig
Clearly, Eq. (19) is more restrictive than Eq. (18), viz. Eq. (5). We also point out here that Eq. (19) can be expressed as 1−∑ ð1−f~ i;i Þ þ ∑ ∑ f~ i;j 40: i∈S
i∈Sj∈S\fig
Next, we show that ξi are strictly smaller than η and φ in [6] with the respective values of βi and γ i . Observe that η and φ can be unified as ∑i∈S g~ i ~ 1−∑i∈S ð1−f i;i Þ þ ∑i∈S f~ i;j =δi;j ðs−1Þmaxi;j∈S;j≠i δi;j ∑i∈S g~ i ≥ ; ~ 1−∑i∈S ð1−f i;i Þ þ ∑i∈S ∑j∈S\fig f~ i;j
η¼
ð20Þ
where g~ i are given by Eq. (14). On the other hand, we know from the proof of Theorem 3.3 that ξi −ð1−f~ i;i Þξi þ ∑ f~ i;j ξj ¼ g~ i ;
1≤i≤s;
j∈S\fig
which yields ∑ ξi −∑ ð1−f~ i;i Þξi þ ∑ ∑ f~ i;j ξj ¼ ∑ g~ i : i∈S
i∈S
i∈Sj∈S\fig
This implies
i∈S
!
∑ ξi −∑ ð1−f~ i;i Þ∑ ξi þ
∑ ∑ f~ i;j ∑ ξi ≤∑ g~ i ;
i∈S
i∈S
i∈Sj∈S\fig
∑ ξi ≤
∑i∈S g~ i : 1−∑i∈S ð1−f~ i;i Þ þ ∑i∈S ∑j∈S\fig f~ i;j
i∈S
i∈S
i∈S
i.e.
i∈S
ð21Þ
Comparing Eqs. (20) and (21), we arrive at ∑ ξi ≤η:
ð22Þ
i∈S
This completes the proof of Corollaries 3.1 and 3.2.
□
Remark 3.3. According to Eqs. (18) and (19), the difference between condition (5) and [6, (2.1) & (2.14)] can be formulated as that for some yi ≥0, 1≤i≤s, the former requires only each yi o1, while the latter requires ∑i∈S yi o1. Clearly, our condition (5) is less restrictive and, therefore, our Theorem 3.3 applies to a larger number of problems than [6, Theorems 2.1 & 2.2]. Remark 3.4. Notice that both η and φ in [6] can be written as in Eq. (20). The inequality (22) shows that each ξi is strictly smaller than η. In terms of dominant eigenvalues of Pi's, the approach in [6] produces upper bounds only on ∑i∈S λ1 ðPi Þ, see [6, (2.13) & (2.15)], which are then used as bounds on λ1 ðPi Þ. Our method here, see the proof of Theorem 3.3, generates upper bounds directly on each individual λ1 ðPi Þ. This leads to better estimates of these dominant
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eigenvalues and thus to tighter bounds on Pi. In Section 4, we shall provide numerical evidence to illustrate a significant improvement in the sharpness of the resulting bounds. We consider next the case when αi and γ i are fixed for all i∈S. For convenience, in this case, we denote the matrix upper bound in Eq. (9) by ( U i ðβi Þ ¼ ðC i þ γ i IÞ−T ξi ½ðCi þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞI !) þðβi −γ i Þ
∑ δi;j ξj I þ Qi þ
α2i Bi BTi
ðCi þ γ i IÞ−1 :
ð23Þ
j∈S\fig
Then, we have the following results: Theorem 3.4. For each i∈S, let αi 40 and γ i o0 be fixed such that both conditions (2) and (5) ð2Þ are satisfied. Let γ i oβð1Þ i oβi ≤−γ i for all i. Then for each U i ðβ i Þ in Eq. (23), we have ð1Þ U i ðβð2Þ i Þ⪯U i ðβi Þ if and only if V i ¼ ξi ðC Ti þ C i Þ þ ∑ δi;j ξj I þ Qi þ α2i Bi BTi ⪯0:
ð24Þ
j∈S\fig
Proof. Set
!
U^ i ðβi Þ ¼ ξi ½ðC i þ βi IÞ ðC i þ T
βi IÞ−ðβ2i −γ 2i ÞI
þ ðβi −γ i Þ
∑ δi;j ξj I þ Qi þ
α2i Bi BTi
:
j∈S\fig
^ ð1Þ By Lemma 2.1, it suffices to show that U^ i ðβð2Þ i Þ⪯U i ðβi Þ if and only if condition (24) holds. It can be easily verified that " # ð2Þ ð1Þ ð2Þ ð1Þ T 2 T ^ ^ U i ðβi Þ−U i ðβi Þ ¼ ðβi −βi Þ ξi ðCi þ C i Þ þ ∑ δi;j ξj I þ Qi þ αi Bi Bi : j∈S\fig
The conclusion now follows.
□
Theorem 3.5. For each i∈S, let αi 40 and γ i o0 be fixed such that both conditions (2) and (5) are satisfied. Then, for each i and any βi such that γ i oβi ≤−γ i , we have U i ðβi Þ≽U i ð−γ i Þ:
ð25Þ
Proof. It suffices to show that condition (24) holds under the assumptions (2) and (5). By Lemma 3.3 and Eq. (8), we see that for all i, ξi þ si ∑ δi;j ξj ¼ gi : j∈S\fig
This implies ξi þ ∑ δi;j ξj j∈S\fig
λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 λ1 ½ðC i þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðC i þ γ i IÞ−1 ¼ − ; λ1 ½ðCi þ γ i IÞ−T ðC Ti þ Ci ÞðCi þ γ i IÞ−1 λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1
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i.e.
" λ1 ½ξi ðC i þ γ i IÞ−T ðCTi þ C i ÞðC i þ γ i IÞ−1 þ λ1
# ∑ δi;j ξj ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 j∈S\fig
þ λ1 ½ðC i þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðC i þ γ i IÞ−1 ¼ 0: Using Lemma 2.3, we arrive at ( " −T
λ1 ðCi þ γ i IÞ
ξi ðC Ti
#
þ C i Þ þ ∑ δi;j ξj IþQi þ
α2i Bi BTi
) −1
ðCi þ γ i IÞ
≤0;
j∈S\fig
and, subsequently, condition (24) holds according to Lemma 2.1.
□
Theorem 3.5 shows that, in terms of the Löwner partial order, the best upper bound in the form of Eq. (23) is attained at βi ¼ −γ i because condition (24) indeed follows directly from conditions (2) and (5). This result implies that the bound in Corollary 3.2 is tighter than the bound in Corollary 3.1. More generally, it provides the best possible upper bound for arbitrary yet fixed αi and γ i within our unified framework here. Specifically, based upon Eq. (25), this upper bound can be expressed explicitly as " !# Pi ⪯ðCi þ γ i IÞ−T ξi ðC i −γ i IÞT ðC i −γ i IÞ−2γ i
∑ δi;j ξj I þ Qi þ α2i Bi BTi
ðC i þ γ i IÞ−1
j∈S\fig
¼ U ni :
ð26Þ
Having presented a new, unified matrix upper bound in this section, we feel it necessary to further comment on this result. Remark 3.5. Theorem 3.3 provides not only a unifying framework for existing results, but also a possible way of constructing new, tighter matrix upper bounds—also refer to the next section for numerical evidence. It is a difficult problem in general, however, to analyze the effects of parameters αi , βi , and γ i on the bound (9). We shall give a few examples in the next section regarding the effects of varying βi and γ i . Remark 3.6. In literature, see [6,12], αi are usually assumed to satisfy αi 40. This restriction, however, may be relaxed to αi ∈R and the results here remain valid, except in Eq. (4) we have gi ≥0 instead.
4. Numerical results Example 4.1. In [6], one example of the CCARE as in Eq. (1) was considered, in which
1 2 1 1 −3 2 A1 ¼ ; A2 ¼ ; B1 ¼ ; B2 ¼ ; 0 3 2 −2 2 −4
4 3 6 2 δ1;2 ¼ δ2;1 ¼ 0:1; Q1 ¼ and Q2 ¼ : 3 4 2 3 Its positive definite solution is given by
J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648
P1 ¼
4:4860 1:7725 1:7725 1:1080
and
P2 ¼
0:6207
0:2185
0:2185
0:2873
1645
:
Based on fixed α1 ¼ 3, α2 ¼ 0:1, and γ 1 ¼ γ 2 ¼ −1, [6, condition (2.14)] is satisfied, but there are no αi values such that [6, condition (2.1)] holds and, consequently, [6, Theorem 2.1] fails. This is one example showing that the results in [6] can be too restrictive for certain situations. Moreover, [6, Theorem 2.2] leads to
262:7582 −53:5631 P1 ⪯ ¼ P1u2 −53:5631 93:9401 and
P2 ⪯
185:0088
−83:8766
−83:8766
273:6983
¼ P2u2 :
Using Example 4.1 with the same αi , we present here numerical results concerning the new, unified matrix upper bound in the previous section. First, we point out that, being independent from βi by Remark 3.2, both conditions (5) and (6) hold with the above values of αi and γ i . Specifically, we have in this case that: μ1 ¼ −0:2591;
μ2 ¼ −0:2449;
ν1 ¼ −0:3134 and ν2 ¼ −60:4574:
Both Corollaries 3.1 and 3.2, therefore, apply to Example 4.1. The resulting matrix upper bounds are
14:1207 1:0451 2:2519 0:0791 P1 ⪯ ¼ U 1 ð0Þ and P2 ⪯ ¼ U 2 ð0Þ ð27Þ 1:0451 10:0952 0:0791 2:1362 from Eq. (16), and
12:5018 P1 ⪯ 2:0902
2:0902 ¼ U 1 ð1Þ 4:4509
and
1:8045 0:1583 P2 ⪯ ¼ U 2 ð1Þ 0:1583 1:5730
ð28Þ
from Eq. (17). These bounds are much tighter than the respective Piu2. In fact, with the given αi values, numerical experiment indicates that conditions (5) and (6) are satisfied over a wide range of γ i values. In Fig. 1, we show the values of μi and νi as γ 1 ¼ γ 2 vary over the interval [−5, 0). The solid and dashed curves represent cases i¼ 1 and i¼ 2, respectively. Note that negative values of μi and νi indicate the satisfaction of conditions (5) and (6). It can be seen that condition (5) always holds, while the relaxed condition (6) fails on ν1 when γ i are approximately between −0.81 and 0. In the rest of this section, for simplicity, we again denote Ui the unified matrix upper bound in Eq. (9). Fig. 2 illustrates how the eigenvalues of U i −U i ð1Þ, i¼ 1, 2, change when γ 1 ¼ γ 2 ¼ −1 are fixed but β1 ¼ β2 vary on the interval ð−1; 1. The solid curves represent λ1 ½U i −U i ð1Þ, whereas the dashed curves represent λ2 ½U i −U i ð1Þ. The decreasingness of these curves indicates that the upper bounds U1 and U2 become tighter with the growth of β1 and β2 . In fact, by Theorems 3.4 and 3.5, we have
−10:7420 9:6183 −49:6880 50:2185 V1 ¼ and V 2 ¼ ; 9:6183 −21:7902 50:2185 −58:6880
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J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648 0
50
−0.05
0
−0.1
−50
−0.15
−100
−0.2
−150
−0.25
−200
−0.3
−250
−0.35 −5
−4
−3
−2
−1
−300 −5
0
−4
−3
−2
−1
0
Fig. 1. The values of μi and νi with varying γ 1 ¼ γ 2 .
12
1.4
10
1.2 1
8
0.8
6
0.6
4
0.4
2
0.2
0 −1
−0.5
0
0.5
0 −1
1
−0.5
0
0.5
1
Fig. 2. The eigenvalues of U i −U i ð1Þ at fixed γ 1 ¼ γ 2 ¼ −1 with varying β1 ¼ β2 .
which are negative definite. This guarantees that at fixed γ 1 ¼ γ 2 ¼ −1, the larger β1 and β2 , the better U1 and U2. It also explains the reason that for Example 4.1, Corollary 3.2 produces better results than Corollary 3.1. Fig. 3 illustrates how the eigenvalues of U i −U i ð0Þ, i¼ 1, 2, change when β1 ¼ β2 ¼ 0 are fixed and γ 1 ¼ γ 2 vary on the interval [−5, 0). The solid curves represent λ1 ½U i −U i ð0Þ, whereas the dashed curves represent λ2 ½U i −U i ð0Þ. We note from these results that upper bounds tighter than those from Corollary 3.1 can be achieved if γ i fall in the range γ i o−1. When β1 ¼ β2 ¼ 0 and γ 1 ¼ γ 2 ¼ −2, for example, we find from Theorem 3.3 that " # −T T 2 T Pi ⪯ðCi −2IÞ ξi ðC i Ci þ 4IÞ þ 2 ∑ δi;j ξj I þ 2ðQi þ αi Bi Bi Þ ðC i −2IÞ−1 ¼ U~ i ; j∈S\fig
and, numerically,
10:9228 U~ 1 ¼ 2:1704
2:1704 7:6979
and
U~ 2 ¼
1:7453
0:1942
0:1942
1:4952
;
ð29Þ
which are clearly tighter, in terms of the Löwner partial order, than U i ð0Þ as in Eq. (27) from Corollary 3.1 and are also tighter, in the Frobenius norm ∥U i −Pi ∥F , than U i ð1Þ as in Eq. (28) from Corollary 3.2.
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4
50
3
40
1647
2
30
1
20
0
10
−1
0 −10 −5
−4
−3
−2
−1
0
−2 −5
−4
−3
−2
−1
0
Fig. 3. The eigenvalues of U i −U i ð0Þ at fixed β1 ¼ β2 ¼ 0 with varying γ 1 ¼ γ 2 .
According to Theorem 3.5, the upper bounds in Eq. (29) can be further improved by pushing βi toward −γ i ¼ 2. Specifically, using Eq. (26), we see that
1:4129 0:3883 9:8111 4:3408 n n ~ U1 ¼ ≺U 1 and U 2 ¼ ≺U~ 2 ; 0:3883 0:9127 4:3408 3:3613 which, by Theorem 3.5, are the best possible matrix upper bounds at fixed α1 ¼ 3, α2 ¼ 0:1, and γ i ¼ −2. 5. Conclusions The CCARE plays an important role in control theory. Especially, bounds on the positive semidefinite solution of the CCARE can be applied to various control problems. However, there have been only a few relevant results [5,6,9] in the literature. Much more work in this regard is still needed. Motivated by [6,12], we develop in this work a general framework, which produces unified and improved matrix upper bounds on the solution of the CCARE. We also show that such unified bounds not only include but also strengthen the upper bounds in [6]. Numerical evidence confirms that much tighter upper bounds can be achieved within the general framework here. Moreover, we weaken the key restriction in [6] corresponding to Eq. (5). Consequently, our results apply to a larger number of situations. Regarding future work, we remark that as pointed out in [6], the main goals are still to find bounds for the solution of the CCARE that are less restrictive, more concise, and more accurate. Besides, there are a few other problems along the same line of inquiry, such as extensions of the unifying method here to the discrete coupled algebraic Riccati equation (DCARE) and iterative methods for refining the unified matrix upper bound; see [6]. With our general framework, we have obtained some preliminary theoretical results regarding the convergence of the iterative algorithm in [6]. We shall present these results in a separate treatise. Acknowledgments The author thanks the anonymous reviewers for their careful reading of this paper and useful comments, which have helped improve the presentation of the results.
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References [1] H. Abou-Kandil, G. Freiling, G. Jank, Solution and asymptotic behaviour of coupled Riccati equations in jump linear systems, IEEE Transactions on Automatic Control 39 (1994) 1631–1636. [2] A. Berman, R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994. [3] G. Cai, C. Hu, G. Duan, Efficient LMI-based quadratic stability and stabilization of parameter-dependent interval systems with applications, International Journal of Innovative Computing, Information and Control 8 (2012) 1943–1954. [4] A. Czornik, A. Swierniak, Lower bounds on the solution of coupled algebraic Riccati equation, Automatica 37 (2001) 619–624. [5] A. Czornik, A. Swierniak, Upper bounds on the solution of coupled algebraic Riccati Equation, Journal of Inequalities and Applications 6 (2001) 373–385. [6] R. Davies, P. Shi, R. Wiltshire, Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations, Automatica 44 (2008) 1088–1096. [7] R. Davies, P. Shi, R. Wiltshire, New upper matrix bounds for the solution of the continuous algebraic Riccati matrix equation, International Journal of Control, Automation, and Systems 6 (2008) 776–784. [8] R. Davies, P. Shi, R. Wiltshire, New upper solution bounds of the discrete algebraic Riccati matrix equation, Journal of Computational and Applied Mathematics 213 (2008) 307–315. [9] L. Gao, A. Xue, Y. Sun, Matrix bounds for the coupled algebraic Riccati equation, in: Proceedings of the Fourth World Congress on Intelligent Control and Automation, 2002, pp. 180–183. [10] R. Horn, C. Johnson, Matrix Analysis, Cambridge University Press, 1990. [11] Y. Ji, H. Chizeck, Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control 35 (1990) 777–788. [12] C.-H. Lee, New upper bounds of the continuous algebraic Riccati matrix equation, IEEE Transactions on Automatic Control 51 (2006) 330–334. [13] M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, 1990. [14] A. Marshall, I. Olkin, B. Arnold, Inequalities: Theory of Majorization and Its Applications, 2nd ed., Springer, 2011. [15] Y. Takeuchi, Optimization of the observations for stationary LQG stochastic control systems under a quadratic criterion, International Journal of Innovative Computing, Information and Control 8 (2012) 2313–2328. [16] J. Wang, Z. Fan, Y. Zhou, Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, Journal of Optimization Theory and Applications 154 (2012) 292–302. [17] J. Wang, Y. Zhou, W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Systems & Control Letters 61 (2012) 472–476. [18] G. Zhang, L. Liu, W. Liu, Linear quadratic optimal control based on dynamic compensation, International Journal of Innovative Computing, Information and Control 8 (2012) 3743–3754.
Journal of the Franklin Institute 350 (2013) 1634–1648 www.elsevier.com/locate/jfranklin
Unified, improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation Jianhong Xun Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA Received 27 October 2012; received in revised form 1 February 2013; accepted 20 March 2013 Available online 26 April 2013
Abstract We develop a broad framework that leads to a unified, improved matrix upper bound on the positive semidefinite solution of the continuous coupled algebraic Riccati equation (CCARE). Our approach was largely inspired by recent works of Lee and of Davies, Shi, and Wiltshire. Our main result tightens the needed upper bounds on relevant dominant eigenvalues. Besides, it applies to a larger number of situations since a key restriction here is weaker as compared to those in Davies et al. Numerical evidence is also provided to demonstrate the behavior of the new matrix upper bound. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction For i; j∈f1; 2; …; sg ¼ S, where s≥2, let Ai ; Qi ∈Rnn , Bi ∈Rnm , and δi;j ≥0 with ∑j∈S\fig δi;j 40 be given. In particular, Qi are assumed to be (symmetric) positive semidefinite for all i. In this paper, we are concerned with an upper bound for the positive semidefinite solution Pi ∈Rnn of the continuous coupled algebraic Riccati equation (CCARE): ATi Pi þ Pi Ai −Pi Bi BTi Pi þ ∑ δi;j Pj ¼ −Qi ;
1≤i≤s:
ð1Þ
j∈S\fig
We refer the reader to [1,11] for necessary and sufficient conditions on the existence and uniqueness of this positive semidefinite solution Pi.
n
Tel.: +1 618 453 6510; fax: +1 618 453 5300. E-mail address: [email protected]
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.03.015
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The CCARE plays a central role in the optimal control of the so-called jump linear system, which, in the notation of [1], is given by d xðtÞ ¼ AðrðtÞÞxðtÞ þ BðrðtÞÞuðtÞ; dt where r(t) is a finite state Markovian jump process on S, AðrðtÞÞ∈Rnn , BðrðtÞÞ∈Rnm , and xðtÞ∈Rn and uðtÞ∈Rm are the state and control vectors, respectively; see [13] for detailed background material on jump linear systems. It should be mentioned here that there are other relevant formulations for control systems in literature, such as interval systems [3], LQG regulator systems [15], semilinear fractional evolution systems [16,17], and LQ dynamic compensator systems [18]. It has been an important and fairly new question, see [4,6,9], to bound the positive semidefinite solution of the CCARE. As pointed out in [4,6], besides theoretical considerations, such estimates of the solution serve practical purposes as well; for example, in problems such as time-delay system controller design, stability analysis, robust stabilization, and convergence of numerical algorithms, it often suffices to have sufficiently tight estimates of the solution of the CCARE rather than the exact solution. In [12], two matrix upper bounds on the positive semidefinite solution of the non-coupled continuous algebraic Riccati equation (CARE), i.e. for the case when s¼ 1, were first established. These results were recently generalized to CCARE in [6]. We comment that a majority of the relevant existing results deal only with the case of non-coupled algebraic Riccati equations; see, for example, [7,8] and the references therein. Meanwhile, the extension of such results on the CARE to the case of CCARE is not straightforward in general. As far as we can see, the matrix upper bounds as in [6] represent the best estimates available to the CCARE. Observe that [4] only provides upper bounds on eigenvalues, while [9] only applies to the discrete case as a key identity there, i.e. [9, (5)], becomes trivial for the continuous case. However, there are several questions that remain to be investigated concerning the results in [6]. First, what is the relationship between the matrix upper bounds, i.e. [6, Theorems 2.1 & 2.2], on the positive semidefinite solution of the CCARE? Second, can we identify conditions under which one matrix upper bound performs better than the other? Furthermore, is there a best possible matrix upper bound within the framework of [6]? Third, the dominant eigenvalues of Pi's are estimated in [6] using a sum ∑i∈S λ1 ðPi Þ, yet what we really need for constructing matrix upper bounds are bounds on each individual dominant eigenvalue, i.e. λ1 ðPi Þ. This suggests that the results of [6] may be strengthened by estimating these dominant eigenvalues individually. We shall address the above questions in this paper. Specifically, we shall demonstrate that the matrix upper bounds in [6] can be refined as follows: (i) We propose a unifying, broader framework that connects the seemingly separate results in [6, Theorems 2.1 & 2.2]; see Theorem 3.3 and its corollaries. (ii) Within this broader framework, new, tighter matrix upper bounds can be accomplished. In particular, we provide a monotonicity result, i.e. Theorem 3.5, which not only shows that [6, Theorem 2.2] produces a tighter estimate than [6, Theorem 2.1] but also partially answers the question in Eq. (26) about a best possible upper bound. (iii) Our method leads directly to an estimate of each individual dominant eigenvalue λ1 ðPi Þ and thus to improved matrix upper bounds than [6, Theorems 2.1 & 2.2]. Besides, the key condition for our unified matrix upper bound to hold is less restrictive in comparison with the corresponding ones in [6]. Also see Remarks 3.3 and 3.4.
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2. Preliminaries Let us begin with a brief description about the notation we shall use in the sequel. For any A∈Rnn , we denote by λi ðAÞ its i-th eigenvalue and by ρðAÞ ¼ max1≤i≤n fjλi ðAÞjg the spectral radius of A. Given a symmetric A∈Rnn , its eigenvalues are always arranged in nonincreasing order λ1 ðAÞ≥λ2 ðAÞ≥⋯≥λn ðAÞ; we write A≻0 (A≽0, resp.) if A is positive definite (positive semidefinite, resp.); the notation A≻B (A≽B, resp.) means A−B≻0 (A−B≽0, resp.).1 Besides, given any A ¼ ½ai;j ; B ¼ ½bi;j ∈Rnm , the notation A≥0 means ai;j ≥0 for all i; j, whereas A≥B represents A−B≥0. Next, we recall that A ¼ ½ai;j ∈Rnn is called a Z-matrix, written as A∈Znn , provided ai;j ≤0 for all i≠j. If A∈Znn is nonsingular with A−1 ≥0, then A is said to be a (nonsingular) M-matrix. The first lemma below is a consequence of Sylvester's law of inertia for congruent symmetric matrices. Lemma 2.1 (Horn and Johnson [10, p. 470]). Let A; B∈Rnn be symmetric such that A≽B (A≻B, resp.). Then for any nonsingular C∈Rnn , C T AC≽C T BCðC T AC≻C T BC; resp:Þ: We shall also need the particular case from the following lemma. Lemma 2.2 (Horn and Johnson [10, p. 471]). Let A; B∈Rnn with A≽0 and B≻0. Then B≽A if and only if ρðB−1 AÞ≤1. In particular, λ1 ðAÞI≽A. The next lemma is attributed to H. Weyl: Lemma 2.3 (Horn and Johnson [10, p. 181]). Assume that A; B∈Rnn are symmetric. Then for any 1≤i≤n, λi ðAÞ þ λn ðBÞ≤λi ðA þ BÞ≤λi ðAÞ þ λ1 ðBÞ: In particular, (i) λ1 ðA þ BÞ≤λ1 ðAÞ þ λ1 ðBÞ. (ii) A≽B implies λ1 ðAÞ≥λ1 ðBÞ.
Continuing, we quote two more lemmas regarding eigenvalues—these lemmas are due to Ky Fan and A. Ostrowski, respectively. Lemma 2.4 (Marshall et al. [14, p. 324]). Given any A∈Rnn , T A þA Re½λi ðAÞ≤λ1 2 for all 1≤i≤n. 1
This defines the so-called Löwner partial order over the set of positive semidefinite matrices.
J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648
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Lemma 2.5 (Horn and Johnson [10, p. 224]). Let A; B∈Rnn with A being symmetric and B being nonsingular. Then for each 1≤i≤n, there exists θi 40 such that λn ðBT BÞ≤θi ≤λ1 ðBT BÞ and λi ðBT ABÞ ¼ θi λi ðAÞ: The last lemma here provides a characterization of M-matrices. Lemma 2.6 (Berman and Plemmons [2, p. 137]). Assume that A∈Znn . Then A is an M-matrix if and only if: (i) A has all positive diagonal entries. (ii) There is a diagonal matrix D with all positive diagonal entries such that AD is strictly diagonally dominant. In particular, A∈Znn is an M-matrix when A has all positive diagonal entries and A itself is strictly diagonally dominant. 3. Main results We are now in a position to develop results on the unified upper bound on the positive semidefinite solution Pi of the CCARE in Eq. (1). Following [6,12], we always assume in what follows that for each i∈S, there exists αi 40 such that ATi þ Ai ≺2αi Bi BTi :
ð2Þ
For convenience, we set Ci ¼ Ai −αi Bi BTi . Clearly, the above condition is equivalent to C Ti þ C i ≺0. Lemma 3.1. For any γ i o0, under condition (2), C i þ γ i I is nonsingular. Proof. By Lemma 2.4, for any 1≤j≤n, 2 Re½λj ðCi þ γ i IÞ≤λ1 ðC Ti þ C i þ 2γ i IÞ ¼ λ1 ðCTi þ C i Þ þ 2γ i o0: This implies λj ðCi þ γ i IÞ≠0 for all j, and thus the conclusion follows.
□
Lemma 3.2. Let γ i o0 and βi 4γ i . Moreover, suppose that condition (2) holds. Then λ1 fðCi þ γ i IÞ−T ½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞIðCi þ γ i IÞ−1 go1: Proof. By Lemma 2.3, it suffices to show that ðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 ≺I; which is equivalent to ðC i þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞI≺ðC i þ γ i IÞT ðCi þ γ i IÞ by Lemma 2.1. Note that ðC i þ γ i IÞT ðCi þ γ i IÞ−½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞI ¼ ðγ i −βi ÞðC Ti þ C i Þ≻0: This completes the proof.
□
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Continuing, we set si ¼
λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 o0: λ1 ½ðCi þ γ i IÞ−T ðC Ti þ C i ÞðCi þ γ i IÞ−1
Observe that the denominator λ1 ½ðC i þ γ i IÞ−T ðCTi þ C i ÞðC i þ γ i IÞ−1 o0 due to CTi þ Ci ≺0 and Lemma 2.1. We now define F ¼ ½f i;j ∈Zss and g∈Rs as follows: 3 2 −1 δ1;2 δ1;3 ⋯ δ1;s 2 3 s1 s1 6δ −1 δ2;3 ⋯ δ2;s 7 7 6 2;1 s2 6 7 s 7 6 2 6 76 ⋮ ⋱ ⋱ ⋱ ⋮ 7 F¼6 76 7 ⋱ 4 56 ⋱ ⋱ δs−1;s 7 5 4 ⋮ ss δs;1 ⋯ ⋯ δs;s−1 s−1 s 3 2 1 δ1;2 s1 δ1;3 s1 ⋯ δ1;s s1 6δ s 1 δ s ⋯ δ2;s s2 7 2;3 2 7 6 2;1 2 7 6 7 6 ⋮ ⋱ ⋱ ⋱ ⋮ ¼6 ð3Þ 7 6 ⋮ ⋱ ⋱ δs−1;s ss−1 7 5 4 δs;1 ss ⋯ ⋯ δs;s−1 ss 1 and gi ¼ −
λ1 ½ðCi þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðCi þ γ i IÞ−1 40: λ1 ½ðCi þ γ i IÞ−T ðC Ti þ Ci ÞðCi þ γ i IÞ−1
ð4Þ
In addition, we let δi ¼ ∑j∈S\fig δi;j 40. According to Lemma 2.6, we see immediately the following result, whose proof is omitted: Theorem 3.1. Under condition (2), if each γ i o0 is such that μi ¼ λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1 þ δi λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 o0;
1≤i≤s;
ð5Þ viz. 1 þ δi si 40;
1≤i≤s;
then F in Eq. (3) is an M-matrix. Using condition (5), we may develop alternative conditions for F to be an M-matrix. The following conclusion provides one such condition. Theorem 3.2. Under condition (2), let γ i o0 be such that νi ¼ λ1 ðC Ti þ C i Þλn ½ðC i þ γ i IÞðCi þ γ i IÞT þ δi λ1 ½ðC i þ γ i IÞðC i þ γ i IÞT o0; where 1≤i≤s. Then F in Eq. (3) is an M-matrix. Proof. By Lemma 2.5, there exists θ1 with θ1 ≥λn ½ðCi þ γ i IÞ−T ðC i þ γ i IÞ−1 40
ð6Þ
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such that λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1 ¼ θ1 λ1 ðCTi þ C i Þ: From Eq. (6), we see that λ1 ðCTi þ C i Þλn ½ðC i þ γ i IÞ−T ðCi þ γ i IÞ−1 þ δi λ1 ½ðC i þ γ i IÞ−T ðCi þ γ i IÞ−1 o0; which implies λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1 þ δi λ1 ½ðCi þ γ i IÞ−T ðCi þ γ i IÞ−1 o0; i.e. Eq. (5). Hence F is an M-matrix.
□
Remark 3.1. Obviously, Theorem 3.1 implies Theorem 3.2, but not the converse. Incidentally, the condition in Eq. (6) can be reformulated in terms of the condition number κ 2 ðÞ: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ1 ðCTi þ C i Þ κ2 ðC i þ γ i IÞo − : δi From Theorem 3.1, the next result is straightforward and thus its proof will be omitted. We comment that the key for this result is F −1 ≥0. Lemma 3.3. Under conditions (2) and (5), the matrix inequality ð7Þ
Fx≤g; where F and g are given by Eqs. (3) and (4), respectively, has solution x≤F −1 g: Before proceeding, we write the solution in Lemma 3.3 as ξ. Hence, we have Fξ ¼ g
viz: ξ ¼ F −1 g:
ð8Þ
We now state our unified matrix upper bound as follows. Theorem 3.3. Suppose that there exists αi 40 such that condition (2) is satisfied for all i∈S. For each i∈S, let βi ; γ i be such that γ i o0, γ i oβi ≤−γ i , and condition (5) holds too. Then the positive semidefinite solution Pi, 1≤i≤s, of the CCARE in Eq. (1) are bounded by (
Pi ⪯ðC i þ γ i IÞ
−T
!)
ξi ½ðC i þ βi IÞ ðC i þ T
βi IÞ−ðβ2i −γ 2i ÞIþðβi −γ i Þ
∑ δi;j ξj I þ Qi þ
α2i Bi BTi
ðC i þ γ i IÞ−1 ¼ U i
j∈S\fig
ð9Þ where ξi are given in Eq. (8). Proof. Using Eq. (1), we can easily verify that ðC i þ γ i IÞT Pi ðC i þ γ i IÞ þ ðβi −γ i ÞðPi −αi IÞBi BTi ðPi −αi IÞ ¼ ðC i þ βi IÞ Pi ðCi þ T
βi IÞ−ðβ2i −γ 2i ÞPi
þ ðβi −γ i Þ
∑ δi;j Pj þ Qi þ j∈S\fig
which leads to ðC i þ γ i IÞT Pi ðC i þ γ i IÞ⪯ðCi þ βi IÞT Pi ðC i þ βi IÞ−ðβ2i −γ 2i ÞPi
! α2i Bi BTi
;
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J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648
! þðβi −γ i Þ
∑ δi;j Pj þ Qi þ α2i Bi BTi : j∈S\fig
By Lemma 2.1, it follows Pi ⪯ðCi þ γ i IÞ−T ðC i þ βi IÞT Pi ðCi þ βi IÞ−ðβ2i −γ 2i ÞPi !# þðβi −γ i Þ
∑ δi;j Pj þ Qi þ α2i Bi BTi
ðC i þ γ i IÞ−1 :
j∈S\fig
This, according to Lemma 2.2, implies Pi ⪯ðCi þ γ i IÞ−T λ1 ðPi Þ½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞI !) þðβi −γ i Þ
∑ δi;j λ1 ðPj ÞI þ Qi þ α2i Bi BTi
ðC i þ γ i IÞ−1 :
ð10Þ
j∈S\fig
Now, using Lemma 2.3, we arrive at λ1 ðPi Þ≤λ1 ðPi Þλ1 ðCi þ γ i IÞ−T ðC i þ βi IÞT ðC i þ βi IÞ −ðβ2i −γ 2i ÞI ðC i þ γ i IÞ−1 g þðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðCi þ γ i IÞ−1 ∑ δi;j λ1 ðPj Þ −T
þðβi −γ i Þλ1 ½ðC i þ γ i IÞ ðQi þ
j∈S\fig 2 T αi Bi Bi ÞðC i
þ γ i IÞ−1 :
For convenience, we denote f~ i;i ¼ 1−λ1 fðCi þ γ i IÞ−T ½ðC i þ βi IÞT ðC i þ βi IÞ−ðβ2 −γ 2 ÞIðCi þ γ i IÞ−1 g; i
f~ i;j ¼ −ðβi −γ i Þδi;j λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 ;
i
i≠j;
ð11Þ ð12Þ ð13Þ
and g~ i ¼ ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðC i þ γ i IÞ−1 : By Lemma 3.2, f~ i;i 40 for all i. Furthermore, we claim f~ i;i ¼ −ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðC T þ C i ÞðC i þ γ i IÞ−1 ; i
ð14Þ ð15Þ
since ðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 ¼ ðCi þ γ i IÞ−T f½C i þ γ i I þ ðβi −γ i ÞIT ½C i þ γ i I þ ðβi −γ i ÞI−ðβ2i −γ 2i ÞIgðC i þ γ i IÞ−1 ¼ ðCi þ γ i IÞ−T ðC i þ γ i IÞT ðC i þ γ i IÞ þ ðβi −γ i Þ½ðC i þ γ i IÞT þ ðC i þ γ i IÞ −2γ i ðβi −γ i ÞI ðC i þ γ i IÞ−1 ¼ I þ ðβi −γ i ÞðC i þ γ i IÞ−T ðCTi þ C i ÞðC i þ γ i IÞ−1 : −1 Hence, Eq. (11) is in fact equivalent to Eq. (7) with the ith inequality being scaled with f~ i;i and xi ¼ λ1 ðPi Þ for each i∈S. Finally, by Eq. (8), we obtain that for each i∈S, λ1 ðPi Þ≤ξi : Consequently, the estimate above of λ1 ðPi Þ together with Eq. (10) yield directly the matrix upper bound in Eq. (9). □
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Regarding Eqs. (3)–(6), (8) and (9), we remark as follows: Remark 3.2. The estimate of Pi in Eq. (9) hinges upon ξ as given in Eq. (8). It is interesting to notice that F, g, and ξ depend on the αi 's and γ i 's, yet they are independent of the βi 's. Similarly, both conditions (5) and (6) are independent of βi . In [6, Theorems 2.1 & 2.2], two matrix upper bounds were established. In fact, these two bounds can now be thought of as special, more relaxed cases of our Theorem 3.3. Using our unifying framework, we shall prove these two cases together. Corollary 3.1 (cf. Davies et al. [6, Theorem 2.1]). Let βi ¼ 0 and γ i ¼ −1 for all i∈S. Assume that conditions (2) and (5) are satisfied. Then, we have " # Pi ⪯ðC i −IÞ−T ξi ðCTi C i þ IÞ þ ∑ δi;j ξj IþQi þ α2i Bi BTi ðCi −IÞ−1 ;
1≤i≤s;
ð16Þ
j∈S\fig
where ξi are given in Eq. (8). Compared to η in [6, Theorem 2.1], these ξi lead to a tighter upper bound in Eq. (16). Moreover, condition (5) is less restrictive than [6, condition (2.1)]. Corollary 3.2 (cf. Davies et al. [6, Theorem 2.2]). Let βi ¼ 1 and γ i ¼ −1 for all i∈S. Assume that conditions (2) and (5) are satisfied. Then, we have " # Pi ⪯ðC i −IÞ−T ξi ðCi þ IÞT ðC i þ IÞ þ 2 ∑ δi;j ξj Iþ2ðQi þ α2i Bi BTi Þ ðC i −IÞ−1 ;
1≤i≤s;
j∈S\fig
ð17Þ where ξi are given in Eq. (8). Compared to φ in [6, Theorem 2.2], these ξi lead to a tighter upper bound in Eq. (17). Moreover, condition (5) is less restrictive than [6, condition (2.14)]. Proof. First, we show that condition (5) is less restrictive than the corresponding conditions in [6]. Note that condition (5) can be written as f~ i;i þ ∑f~ i;j 40;
1≤i≤s;
j≠i
where f~ i;j are given in Eqs. (12) and (13); i.e. condition (5) is equivalent to that, for each i, 1−λ1 fðC i þ γ i IÞ−T ½ðC i þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞIðCi þ γ i IÞ−1 g −ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 ∑ δi;j 40:
ð18Þ
j∈S\fig
On the other hand, by [6, (2.6) & (2.16)], we see that in our notation, [6, (2.1) & (2.14)] can be unified as ∑ λ1 fðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 g
−T −1 þ ∑ ðβi −γ i Þλ1 ½ðC i þ γ i IÞ ðC i þ γ i IÞ ðs−1Þ max δi;j o1;
i∈S
i∈S
i;j∈S;j≠i
which implies 1−∑ λ1 fðC i þ γ i IÞ−T ½ðCi þ βi IÞT ðC i þ βi IÞ−ðβ2i −γ 2i ÞIðC i þ γ i IÞ−1 g i∈S
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−∑ ðβi −γ i Þλ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 ∑ δi;j 40: i∈S
ð19Þ
j∈S\fig
Clearly, Eq. (19) is more restrictive than Eq. (18), viz. Eq. (5). We also point out here that Eq. (19) can be expressed as 1−∑ ð1−f~ i;i Þ þ ∑ ∑ f~ i;j 40: i∈S
i∈Sj∈S\fig
Next, we show that ξi are strictly smaller than η and φ in [6] with the respective values of βi and γ i . Observe that η and φ can be unified as ∑i∈S g~ i ~ 1−∑i∈S ð1−f i;i Þ þ ∑i∈S f~ i;j =δi;j ðs−1Þmaxi;j∈S;j≠i δi;j ∑i∈S g~ i ≥ ; ~ 1−∑i∈S ð1−f i;i Þ þ ∑i∈S ∑j∈S\fig f~ i;j
η¼
ð20Þ
where g~ i are given by Eq. (14). On the other hand, we know from the proof of Theorem 3.3 that ξi −ð1−f~ i;i Þξi þ ∑ f~ i;j ξj ¼ g~ i ;
1≤i≤s;
j∈S\fig
which yields ∑ ξi −∑ ð1−f~ i;i Þξi þ ∑ ∑ f~ i;j ξj ¼ ∑ g~ i : i∈S
i∈S
i∈Sj∈S\fig
This implies
i∈S
!
∑ ξi −∑ ð1−f~ i;i Þ∑ ξi þ
∑ ∑ f~ i;j ∑ ξi ≤∑ g~ i ;
i∈S
i∈S
i∈Sj∈S\fig
∑ ξi ≤
∑i∈S g~ i : 1−∑i∈S ð1−f~ i;i Þ þ ∑i∈S ∑j∈S\fig f~ i;j
i∈S
i∈S
i∈S
i.e.
i∈S
ð21Þ
Comparing Eqs. (20) and (21), we arrive at ∑ ξi ≤η:
ð22Þ
i∈S
This completes the proof of Corollaries 3.1 and 3.2.
□
Remark 3.3. According to Eqs. (18) and (19), the difference between condition (5) and [6, (2.1) & (2.14)] can be formulated as that for some yi ≥0, 1≤i≤s, the former requires only each yi o1, while the latter requires ∑i∈S yi o1. Clearly, our condition (5) is less restrictive and, therefore, our Theorem 3.3 applies to a larger number of problems than [6, Theorems 2.1 & 2.2]. Remark 3.4. Notice that both η and φ in [6] can be written as in Eq. (20). The inequality (22) shows that each ξi is strictly smaller than η. In terms of dominant eigenvalues of Pi's, the approach in [6] produces upper bounds only on ∑i∈S λ1 ðPi Þ, see [6, (2.13) & (2.15)], which are then used as bounds on λ1 ðPi Þ. Our method here, see the proof of Theorem 3.3, generates upper bounds directly on each individual λ1 ðPi Þ. This leads to better estimates of these dominant
J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648
1643
eigenvalues and thus to tighter bounds on Pi. In Section 4, we shall provide numerical evidence to illustrate a significant improvement in the sharpness of the resulting bounds. We consider next the case when αi and γ i are fixed for all i∈S. For convenience, in this case, we denote the matrix upper bound in Eq. (9) by ( U i ðβi Þ ¼ ðC i þ γ i IÞ−T ξi ½ðCi þ βi IÞT ðCi þ βi IÞ−ðβ2i −γ 2i ÞI !) þðβi −γ i Þ
∑ δi;j ξj I þ Qi þ
α2i Bi BTi
ðCi þ γ i IÞ−1 :
ð23Þ
j∈S\fig
Then, we have the following results: Theorem 3.4. For each i∈S, let αi 40 and γ i o0 be fixed such that both conditions (2) and (5) ð2Þ are satisfied. Let γ i oβð1Þ i oβi ≤−γ i for all i. Then for each U i ðβ i Þ in Eq. (23), we have ð1Þ U i ðβð2Þ i Þ⪯U i ðβi Þ if and only if V i ¼ ξi ðC Ti þ C i Þ þ ∑ δi;j ξj I þ Qi þ α2i Bi BTi ⪯0:
ð24Þ
j∈S\fig
Proof. Set
!
U^ i ðβi Þ ¼ ξi ½ðC i þ βi IÞ ðC i þ T
βi IÞ−ðβ2i −γ 2i ÞI
þ ðβi −γ i Þ
∑ δi;j ξj I þ Qi þ
α2i Bi BTi
:
j∈S\fig
^ ð1Þ By Lemma 2.1, it suffices to show that U^ i ðβð2Þ i Þ⪯U i ðβi Þ if and only if condition (24) holds. It can be easily verified that " # ð2Þ ð1Þ ð2Þ ð1Þ T 2 T ^ ^ U i ðβi Þ−U i ðβi Þ ¼ ðβi −βi Þ ξi ðCi þ C i Þ þ ∑ δi;j ξj I þ Qi þ αi Bi Bi : j∈S\fig
The conclusion now follows.
□
Theorem 3.5. For each i∈S, let αi 40 and γ i o0 be fixed such that both conditions (2) and (5) are satisfied. Then, for each i and any βi such that γ i oβi ≤−γ i , we have U i ðβi Þ≽U i ð−γ i Þ:
ð25Þ
Proof. It suffices to show that condition (24) holds under the assumptions (2) and (5). By Lemma 3.3 and Eq. (8), we see that for all i, ξi þ si ∑ δi;j ξj ¼ gi : j∈S\fig
This implies ξi þ ∑ δi;j ξj j∈S\fig
λ1 ½ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 λ1 ½ðC i þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðC i þ γ i IÞ−1 ¼ − ; λ1 ½ðCi þ γ i IÞ−T ðC Ti þ Ci ÞðCi þ γ i IÞ−1 λ1 ½ðC i þ γ i IÞ−T ðC Ti þ C i ÞðC i þ γ i IÞ−1
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i.e.
" λ1 ½ξi ðC i þ γ i IÞ−T ðCTi þ C i ÞðC i þ γ i IÞ−1 þ λ1
# ∑ δi;j ξj ðC i þ γ i IÞ−T ðC i þ γ i IÞ−1 j∈S\fig
þ λ1 ½ðC i þ γ i IÞ−T ðQi þ α2i Bi BTi ÞðC i þ γ i IÞ−1 ¼ 0: Using Lemma 2.3, we arrive at ( " −T
λ1 ðCi þ γ i IÞ
ξi ðC Ti
#
þ C i Þ þ ∑ δi;j ξj IþQi þ
α2i Bi BTi
) −1
ðCi þ γ i IÞ
≤0;
j∈S\fig
and, subsequently, condition (24) holds according to Lemma 2.1.
□
Theorem 3.5 shows that, in terms of the Löwner partial order, the best upper bound in the form of Eq. (23) is attained at βi ¼ −γ i because condition (24) indeed follows directly from conditions (2) and (5). This result implies that the bound in Corollary 3.2 is tighter than the bound in Corollary 3.1. More generally, it provides the best possible upper bound for arbitrary yet fixed αi and γ i within our unified framework here. Specifically, based upon Eq. (25), this upper bound can be expressed explicitly as " !# Pi ⪯ðCi þ γ i IÞ−T ξi ðC i −γ i IÞT ðC i −γ i IÞ−2γ i
∑ δi;j ξj I þ Qi þ α2i Bi BTi
ðC i þ γ i IÞ−1
j∈S\fig
¼ U ni :
ð26Þ
Having presented a new, unified matrix upper bound in this section, we feel it necessary to further comment on this result. Remark 3.5. Theorem 3.3 provides not only a unifying framework for existing results, but also a possible way of constructing new, tighter matrix upper bounds—also refer to the next section for numerical evidence. It is a difficult problem in general, however, to analyze the effects of parameters αi , βi , and γ i on the bound (9). We shall give a few examples in the next section regarding the effects of varying βi and γ i . Remark 3.6. In literature, see [6,12], αi are usually assumed to satisfy αi 40. This restriction, however, may be relaxed to αi ∈R and the results here remain valid, except in Eq. (4) we have gi ≥0 instead.
4. Numerical results Example 4.1. In [6], one example of the CCARE as in Eq. (1) was considered, in which
1 2 1 1 −3 2 A1 ¼ ; A2 ¼ ; B1 ¼ ; B2 ¼ ; 0 3 2 −2 2 −4
4 3 6 2 δ1;2 ¼ δ2;1 ¼ 0:1; Q1 ¼ and Q2 ¼ : 3 4 2 3 Its positive definite solution is given by
J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648
P1 ¼
4:4860 1:7725 1:7725 1:1080
and
P2 ¼
0:6207
0:2185
0:2185
0:2873
1645
:
Based on fixed α1 ¼ 3, α2 ¼ 0:1, and γ 1 ¼ γ 2 ¼ −1, [6, condition (2.14)] is satisfied, but there are no αi values such that [6, condition (2.1)] holds and, consequently, [6, Theorem 2.1] fails. This is one example showing that the results in [6] can be too restrictive for certain situations. Moreover, [6, Theorem 2.2] leads to
262:7582 −53:5631 P1 ⪯ ¼ P1u2 −53:5631 93:9401 and
P2 ⪯
185:0088
−83:8766
−83:8766
273:6983
¼ P2u2 :
Using Example 4.1 with the same αi , we present here numerical results concerning the new, unified matrix upper bound in the previous section. First, we point out that, being independent from βi by Remark 3.2, both conditions (5) and (6) hold with the above values of αi and γ i . Specifically, we have in this case that: μ1 ¼ −0:2591;
μ2 ¼ −0:2449;
ν1 ¼ −0:3134 and ν2 ¼ −60:4574:
Both Corollaries 3.1 and 3.2, therefore, apply to Example 4.1. The resulting matrix upper bounds are
14:1207 1:0451 2:2519 0:0791 P1 ⪯ ¼ U 1 ð0Þ and P2 ⪯ ¼ U 2 ð0Þ ð27Þ 1:0451 10:0952 0:0791 2:1362 from Eq. (16), and
12:5018 P1 ⪯ 2:0902
2:0902 ¼ U 1 ð1Þ 4:4509
and
1:8045 0:1583 P2 ⪯ ¼ U 2 ð1Þ 0:1583 1:5730
ð28Þ
from Eq. (17). These bounds are much tighter than the respective Piu2. In fact, with the given αi values, numerical experiment indicates that conditions (5) and (6) are satisfied over a wide range of γ i values. In Fig. 1, we show the values of μi and νi as γ 1 ¼ γ 2 vary over the interval [−5, 0). The solid and dashed curves represent cases i¼ 1 and i¼ 2, respectively. Note that negative values of μi and νi indicate the satisfaction of conditions (5) and (6). It can be seen that condition (5) always holds, while the relaxed condition (6) fails on ν1 when γ i are approximately between −0.81 and 0. In the rest of this section, for simplicity, we again denote Ui the unified matrix upper bound in Eq. (9). Fig. 2 illustrates how the eigenvalues of U i −U i ð1Þ, i¼ 1, 2, change when γ 1 ¼ γ 2 ¼ −1 are fixed but β1 ¼ β2 vary on the interval ð−1; 1. The solid curves represent λ1 ½U i −U i ð1Þ, whereas the dashed curves represent λ2 ½U i −U i ð1Þ. The decreasingness of these curves indicates that the upper bounds U1 and U2 become tighter with the growth of β1 and β2 . In fact, by Theorems 3.4 and 3.5, we have
−10:7420 9:6183 −49:6880 50:2185 V1 ¼ and V 2 ¼ ; 9:6183 −21:7902 50:2185 −58:6880
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50
−0.05
0
−0.1
−50
−0.15
−100
−0.2
−150
−0.25
−200
−0.3
−250
−0.35 −5
−4
−3
−2
−1
−300 −5
0
−4
−3
−2
−1
0
Fig. 1. The values of μi and νi with varying γ 1 ¼ γ 2 .
12
1.4
10
1.2 1
8
0.8
6
0.6
4
0.4
2
0.2
0 −1
−0.5
0
0.5
0 −1
1
−0.5
0
0.5
1
Fig. 2. The eigenvalues of U i −U i ð1Þ at fixed γ 1 ¼ γ 2 ¼ −1 with varying β1 ¼ β2 .
which are negative definite. This guarantees that at fixed γ 1 ¼ γ 2 ¼ −1, the larger β1 and β2 , the better U1 and U2. It also explains the reason that for Example 4.1, Corollary 3.2 produces better results than Corollary 3.1. Fig. 3 illustrates how the eigenvalues of U i −U i ð0Þ, i¼ 1, 2, change when β1 ¼ β2 ¼ 0 are fixed and γ 1 ¼ γ 2 vary on the interval [−5, 0). The solid curves represent λ1 ½U i −U i ð0Þ, whereas the dashed curves represent λ2 ½U i −U i ð0Þ. We note from these results that upper bounds tighter than those from Corollary 3.1 can be achieved if γ i fall in the range γ i o−1. When β1 ¼ β2 ¼ 0 and γ 1 ¼ γ 2 ¼ −2, for example, we find from Theorem 3.3 that " # −T T 2 T Pi ⪯ðCi −2IÞ ξi ðC i Ci þ 4IÞ þ 2 ∑ δi;j ξj I þ 2ðQi þ αi Bi Bi Þ ðC i −2IÞ−1 ¼ U~ i ; j∈S\fig
and, numerically,
10:9228 U~ 1 ¼ 2:1704
2:1704 7:6979
and
U~ 2 ¼
1:7453
0:1942
0:1942
1:4952
;
ð29Þ
which are clearly tighter, in terms of the Löwner partial order, than U i ð0Þ as in Eq. (27) from Corollary 3.1 and are also tighter, in the Frobenius norm ∥U i −Pi ∥F , than U i ð1Þ as in Eq. (28) from Corollary 3.2.
J. Xu / Journal of the Franklin Institute 350 (2013) 1634–1648 60
4
50
3
40
1647
2
30
1
20
0
10
−1
0 −10 −5
−4
−3
−2
−1
0
−2 −5
−4
−3
−2
−1
0
Fig. 3. The eigenvalues of U i −U i ð0Þ at fixed β1 ¼ β2 ¼ 0 with varying γ 1 ¼ γ 2 .
According to Theorem 3.5, the upper bounds in Eq. (29) can be further improved by pushing βi toward −γ i ¼ 2. Specifically, using Eq. (26), we see that
1:4129 0:3883 9:8111 4:3408 n n ~ U1 ¼ ≺U 1 and U 2 ¼ ≺U~ 2 ; 0:3883 0:9127 4:3408 3:3613 which, by Theorem 3.5, are the best possible matrix upper bounds at fixed α1 ¼ 3, α2 ¼ 0:1, and γ i ¼ −2. 5. Conclusions The CCARE plays an important role in control theory. Especially, bounds on the positive semidefinite solution of the CCARE can be applied to various control problems. However, there have been only a few relevant results [5,6,9] in the literature. Much more work in this regard is still needed. Motivated by [6,12], we develop in this work a general framework, which produces unified and improved matrix upper bounds on the solution of the CCARE. We also show that such unified bounds not only include but also strengthen the upper bounds in [6]. Numerical evidence confirms that much tighter upper bounds can be achieved within the general framework here. Moreover, we weaken the key restriction in [6] corresponding to Eq. (5). Consequently, our results apply to a larger number of situations. Regarding future work, we remark that as pointed out in [6], the main goals are still to find bounds for the solution of the CCARE that are less restrictive, more concise, and more accurate. Besides, there are a few other problems along the same line of inquiry, such as extensions of the unifying method here to the discrete coupled algebraic Riccati equation (DCARE) and iterative methods for refining the unified matrix upper bound; see [6]. With our general framework, we have obtained some preliminary theoretical results regarding the convergence of the iterative algorithm in [6]. We shall present these results in a separate treatise. Acknowledgments The author thanks the anonymous reviewers for their careful reading of this paper and useful comments, which have helped improve the presentation of the results.
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