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On the iterative refinement of matrix upper bounds for the solution of continuous coupled algebraic Riccati equations
Automatica 49 (2013) 2168–2175
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On the iterative refinement of matrix upper bounds for the solution of continuous coupled algebraic Riccati equations✩ Jianhong Xu, MingQing Xiao 1 Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA
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Article history: Received 31 October 2012 Accepted 12 March 2013 Available online 6 May 2013
In this paper, we develop a new approach for the iterative refinement algorithm along with fully mathematical justifications, which sharpens matrix solution upper bounds for the well-known continuous coupled algebraic Riccati equation (CCARE). Our methodology is studied within a broader framework recently proposed in Xu (in press). Several desirable properties of the algorithm are then established, which include the strict monotonicity and convergence of the sequences generated by the algorithm, thus strengthening the existing results in the literature for computing tighter matrix upper solution bounds for the CCARE. In addition, new admissibility conditions within the general framework are proposed, which guarantee tighter matrix upper solution bounds to be computed from the algorithm. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Coupled Riccati equation Matrix bound Iterative algorithm
1. Introduction Let m, n, s ∈ N such that m ≥ 1, n ≥ 2, and s ≥ 2. Set S = {1, 2, . . . , s}. Suppose that for each i ∈ S , Ai , Qi ∈ Rn×n , and Bi ∈ Rn×m are all given. Moreover, Qi are assumed to be (symmetric) positive semidefinite. For any i, j ∈ S such that i ̸= j, let δi,j ≥ 0, together with j∈S \{i} δi,j > 0 for each i ∈ S. The so-called continuous coupled algebraic Riccati equation, or CCARE for short, in unknowns Pi ∈ Rn×n can then be formulated as ATi Pi + Pi Ai − Pi Bi BTi Pi +
δi,j Pj = −Qi ,
(1)
j∈S \{i}
where i = 1, 2, . . . , s. We shall also write the solution here of the CCARE as Pi . In this work, we are concerned with the iterative algorithms in Davies, Shi, and Wiltshire (2008) for refining certain matrix upper bounds on the solution Pi , provided that Pi are positive semidefinite. We refer the interested reader to Abou-Kandil, Freiling, and Jank (1994), Ji and Chizeck (1990), Xiao and Basar (2001), and Xiao (2005) (and references therein) for necessary and sufficient conditions on the existence and uniqueness of such a positive semidefinite solution. The CCARE plays an important role in a wide variety of problems relevant to control theory; see, for example, Davies et al. (2008)
✩ The research of the second author was supported in part by NSF Grant 1021203. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (J. Xu), [email protected] (M. Xiao). 1 Tel.: +1 618 453 6572; fax: +1 618 453 5300.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.03.022
and the references therein. The question of bounding the positive semidefinite solution of the CCARE has been an area of current research endeavors. It has been pointed out in Davies et al. (2008) that, besides theoretical merits, such solution bounds serve practical purposes as well. Specifically, in problems such as stability analysis, time-delay system controller design, optimal control, robust stabilization, and convergence of numerical algorithms, it often suffices or is crucial to obtain a sufficiently tight solution bound of the CCARE instead of the exact solution, which usually is not available in applications and hence numerical approximation becomes necessary. In the literature, see Czornik and Swierniak (2001), Davies et al. (2008), and Gao, Xue, and Sun (2002), solution bounds for the CCARE fall into several categories, which include the extremal eigenvalues, trace, determinant, along with Löwner (partial) order on the set of positive semidefinite matrices. Among these types, matrix bounds in terms of the Löwner order turn out to be the most general one since the other bounds can be derived subsequently (e.g. see Berman & Plemmons, 1994, and Marshall, Olkin, & Arnold, 2011). For this reason, we shall focus here on matrix upper bounds. Motivated by Davies et al. (2008), and Lee (2006), we have developed a general framework in Xu (in press) that leads to a unified matrix upper bound on the solution of the CCARE. With this framework, we have also shown in Xu (in press) that the two main results in Davies et al. (2008, Theorems 2.1 and 2.2) can be thought of as special cases of our unified result. We, however, have not investigated in Xu (in press) an important algorithmic aspect of Davies et al. (2008) due to the length of that paper. In fact, two iterative algorithms were proposed in Davies et al. (2008) along with numerical evidence indicating that matrix upper bounds on the solution of the CCARE
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
may be noticeably refined by the algorithms. Incidentally, their method is an extension of the corresponding algorithms in Lee (2006), in which the non-coupled continuous algebraic Riccati equation, or CARE for short, i.e. the scenario with s = 1 and without the coupling terms j∈S \{i} δi,j Pj , was considered. As a continuing work that is built upon (Davies et al., 2008; Lee, 2006; Xu, in press), we shall study here an iterative refinement algorithm for the unified matrix upper bound in Xu (in press) concerning the CCARE. Specifically, we shall show that: (i) The two seemingly separate iterative refinement algorithms proposed in Davies et al. (2008) are special cases of our unified algorithm. (ii) This unified algorithm, which is developed using a framework broader than that of Davies et al. (2008), possesses a number of desirable properties. In particular, under certain admissible conditions, it generates strictly monotonically decreasing2 and convergent sequences of positive semidefinite matrix upper bounds, thus providing a better numerical approximation. (iii) There is numerical evidence which demonstrates the unified algorithm is able to produce very tight matrix upper bounds. We comment that our methodology in this work is in a similar spirit as Davies et al. (2008) and Lee (2006); our approach, however, extends the existing results in a unifying and flexible manner with different, improved linear algebra techniques. 2. Preliminaries First of all, we describe briefly some notation being used in the rest of this paper. For any A ∈ Rn×n , we denote by λi (A) its ith eigenvalue and by ρ(A) = max1≤i≤n {|λi (A)|} the spectral radius of A. For any symmetric A ∈ Rn×n , its eigenvalues are always arranged in nonincreasing order, i.e.
λ1 (A) ≥ λ2 (A) ≥ · · · ≥ λn (A); we write A ≻ 0 (A ≽ 0, resp.) if, in addition, A is positive definite (positive semidefinite, resp.). Given A, B ∈ Rn×n , both being symmetric, the expression A ≻ B (A ≽ B, resp.) means A − B ≻ 0 (A − B ≽ 0, resp.). Besides, for a sequence {Ak } ⊂ Rn×n with Ak being all symmetric, we say that it is strictly monotonically decreasing if Ak ≻ Ak+1 for all k. Finally, given any A = [ai,j ], B = [bi,j ] ∈ Rn×m , the notation A ≥ 0 means ai,j ≥ 0 for all i, j; whereas A ≥ B represents A − B ≥ 0. The first lemma is one of the consequences of Sylvester’s law of inertia for congruent symmetric matrices. Lemma 2.1 (Horn & Johnson, 1990, p. 470). Let A, B ∈ Rn×n be both symmetric such that A ≽ B (A ≻ B, resp.). Then for any nonsingular C ∈ Rn×n , C AC ≽ C BC (C AC ≻ C BC , resp.). T
T
T
T
2169
Finally, we recall that for any A = [ai,j ] ∈ Rn×m , its Frobenius norm is defined to be
n m ∥A∥F = a2i,j = tr(AT A). j =1 i =1
Especially, when A ∈ Rn×n is symmetric, we have
n ∥A∥F = λ2i (A) i=1
by the facts that A is orthogonally diagonalizable (Horn & Johnson, 1990, p. 107) and that ∥ · ∥F is orthogonally invariant (Horn & Johnson, 1990, p. 292). It is clear that Lemma 2.2 leads to the following lemma. Lemma 2.3. Let A, B ∈ Rn×n be positive semidefinite and A ≽ B. Then
∥A∥F ≥ ∥B∥F . 3. Unified matrix upper bound In this section, we first introduce some of the results in Xu (in press) regarding the unified matrix upper bound, which provide a starting point for our new results in this paper. Consider the CCARE in the form of (1) with the positive semidefinite solution Pi . Following Davies et al. (2008) and Lee (2006), we suppose that for each i ∈ S, there always exists some αi > 0 such that ATi + Ai ≺ 2αi Bi BTi .
(2)
For brevity, we write Ci = Ai − α Obviously, the above condition is equivalent to CiT + Ci ≺ 0. It can be shown that, under condition (2), for any γi < 0, Ci + γi I is nonsingular (Xu, in press, Lemma 3.1). Let us define T i Bi Bi .
σi =
λ1 (Ci + γi I )−T (Ci + γi I )−1 < 0. λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1
(3)
Observe that the denominator
λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1 < 0 because of CiT + Ci ≺ 0 and Lemma 2.1. We now introduce F ∈ Rs×s and g = [g1 , g2 , . . . , gs ]T ∈ Rs as
1 δ2,1 σ2 . . F = . . . . δs,1 σs
δ1,2 σ1 1
..
.
···
δ1,3 σ1 δ2,3 σ2 .. .
··· ··· .. .
δ1,s σ1 δ2,s σ2 .. .
. ···
. δs,s−1 σs
δs−1,s σs−1
..
..
(4)
1
and The next lemma follows from the Courant–Fischer theorem. Lemma 2.2 (Horn & Johnson, 1990, p. 182). Let A, B ∈ Rn×n both be symmetric with A ≽ B. Then
λi (A) ≥ λi (B) for all 1 ≤ i ≤ n.
λ1 (Ci + γi I )−T (Qi + αi2 Bi BTi )(Ci + γi I )−1 > 0. gi = − λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1
(5)
In addition, we denote
δi =
δi,j > 0.
(6)
j∈S \{i}
In what follows, according to σi defined from (3), by condition (7) we refer to 2 This is interpreted in the sense of the Löwner order.
1 + δi σi > 0,
1 ≤ i ≤ s,
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J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
i.e.
In the rest of this section, we shall prove two new results relevant to the unified iterative refinement algorithm.
τi = λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1 + δi λ1 (Ci + γi I )−T (Ci + γi I )−1 < 0
(7)
for 1 ≤ i ≤ s. Continuing, we set up the linear system
Theorem 3.2. Let αi > 0 be such that condition (2) holds for all i ∈ S. For each i ∈ S, suppose that βi , γi are such that γi < 0 and βi ≥ γi . Then
Pi ≼ (Ci + γi I )
(Ci + βi I )T Pi (Ci + βi I ) − (βi2 − γi2 )Pi
−T
F ξ = g, where F and g are given by (4) and (5), respectively. Under conditions (2) and (7), according to Xu (in press, Theorem 3.1), we know that, F −1 exists, F −1 ≥ 0, and, consequently,
+ (βi − γi )
δi,j Pj + Qi + α
(Ci + γi I )−1 .
2 T i Bi Bi
(12)
j∈S \{i}
ξ = F −1 g ≥ 0.
(8)
With all these preceding preparatory developments in hand, our unified matrix upper bound in Xu (in press) for the solution of the CCARE can be stated as follows. Theorem 3.1 (Xu, in press, Theorem 3.3). Assume that there exist
αi > 0 such that condition (2) is satisfied for all i ∈ S. In addition, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and condition (7) holds as well. Then the positive semidefinite solution Pi , 1 ≤ i ≤ s, of the CCARE in (1) is bounded by
Pi ≼ (Ci + γi I )−T ξi (Ci + βi I )T (Ci + βi I ) − (βi2 − γi2 )I
+ (βi − γi )
δi,j ξj I + Qi + α
2 T i Bi Bi
(Ci + γi I )−1
Proof. Using (1), we can easily verify that
(Ci + γi I )T Pi (Ci + γi I ) + (βi − γi )(Pi − αi I )Bi BTi (Pi − αi I ) = (Ci + βi I )T Pi (Ci + βi I ) − (βi2 − γi2 )Pi 2 T + (βi − γi ) δi,j Pj + Qi + αi Bi Bi ,
(13)
j∈S \{i}
which leads to
(Ci + γi I )T Pi (Ci + γi I ) ≼ (Ci + βi I )T Pi (Ci + βi I ) − (βi2 − γi2 )Pi 2 T δi,j Pj + Qi + αi Bi Bi . + (βi − γi ) j∈S \{i}
j∈S \{i}
It is clear now that (12) follows from the above and Lemma 2.1.
= Ui ,
(9)
where ξi are given in (8). The unified matrix upper bound as in (9) will be utilized as a starting point for the unified iterative refinement algorithm that we shall explore in Section 4. Numerical examples in this regard will include the following two special cases: Corollary 3.1 (Xu, in press, Corollary 3.1, cf. Davies et al., 2008, Theorem 2.1). Let βi = 0 and γi = −1 for all i ∈ S. Assume that conditions (2) and (7) are satisfied. Then, we have
The next result establishes a connection between Ui and ξi . Essentially, it provides a necessary and sufficient condition under which each ξi serves as an upper bound on the spectrum of the corresponding Ui . Theorem 3.3. Suppose that there exist αi > 0 such that condition (2) is satisfied for all i ∈ S. Besides, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and condition (7) holds too. Then for each i ∈ S, Ui ≺ ξi I , where Ui are given by (9), if and only if
Pi ≼ (Ci − I )−T ξi (CiT Ci + I ) +
δi,j ξj I
Vi = ξi (CiT + Ci ) +
j∈S \{i}
δi,j ξj I + Qi + αi2 Bi BTi ≺ 0.
(14)
j∈S \{i}
+ Qi + αi2 Bi BTi (Ci − I )−1 ,
1 ≤ i ≤ s,
(10)
where ξi are given in (8). Compared to η in Davies et al. (2008, Theorem 2.1), these ξi lead to a tighter upper bound in (10). Moreover, condition (7) is less restrictive than Davies et al. (2008, condition (2.1)).
Proof. From (9), we calculate ξi I − Ui :
ξi I − Ui = (Ci + γi I )
−T
ξi (Ci + γi I )T (Ci + γi I )
− ξi (Ci + βi I ) (Ci + βi I ) − (β − γ )I T
Corollary 3.2 (Xu, in press, Corollary 3.2, cf. Davies et al., 2008, Theorem 2.2). Let βi = 1 and γi = −1 for all i ∈ S. Assume that conditions (2) and (7) are satisfied. Then, we have
Pi ≼ (Ci − I )−T ξi (Ci + I )T (Ci + I ) + 2
2 i
− (βi − γi )
δi,j ξj I + Qi + α
2 T i Bi Bi
(Ci + γi I )−1
j∈S \{i}
δi,j ξj I
= −(βi − γi )(Ci + γi I )
j∈S \{i}
−T
ξi (CiT + Ci ) +
+ 2 Qi + αi2 Bi BTi (Ci − I )−1 ,
2 i
δi,j ξj I
j∈S \{i}
1 ≤ i ≤ s,
(11)
+ Qi + α
2 T i Bi Bi
where ξi are given in (8). Compared to ϕ in Davies et al. (2008, Theorem 2.2), these ξi lead to a tighter upper bound in (11). Moreover, condition (7) is less restrictive than Davies et al. (2008, condition (2.14)).
(Ci + γi I )−1 .
On letting either side be positive definite, the conclusion now follows from Lemma 2.1.
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
Condition (14) turns out to be a key ingredient for studying the unified iterative refinement algorithm in Section 4. It has been pointed out in Xu (in press) that (4), (5), (8), and condition (7) are independent of the βi ’s. Similarly, condition (14) here is independent of the βi ’s too. We mention also that in Xu (in press, Theorem 3.5) it has been shown that conditions (2) and (7) imply
ξi (CiT + Ci ) +
δi,j ξj I + Qi + αi2 Bi BTi ≼ 0.
Proof. We show the above conclusion by induction. First of all, we (0) (0) observe for all i ∈ S , Pi ≼ Ui by the choice of Ui . To prove (16), (k)
therefore, it suffices to have Pi ≼ Ui for all i. Using (13) and (15), we obtain (k)
Ui
j∈S \{i}
− (βi2 − γi2 ) Ui(k−1) − Pi δi,j Uj(k−1) − Pj + (βi − γi ) j∈S \{i}
Based on results from Section 3, we now present an iterative algorithm capable of refining the unified matrix upper bound as in (9), yet more general than the ones proposed in Davies et al. (2008) and Lee (2006)—for CCARE and CARE, respectively. Our approach here not only is in a unifying manner but also contains new components. In particular, we shall show new conditions that guarantee tighter matrix upper bounds being computed through this unified iterative refinement algorithm. Our unified iterative refinement algorithm, which is described below, is based on estimate (12) in Theorem 3.2. This algorithm extends, by choosing the values of βi and γi accordingly, to the two seemingly separate iterative refinement algorithms in Davies et al. (2008, Algorithms 2.1 and 2.2). Algorithm 4.1 (Davies et al., 2008, Algorithms 2.1 and 2.2). For each (0) i ∈ S, let Ui be the matrix upper bound given in (9). Then, for k = 1, 2, . . . , we iterate according to Ui
(Ci + βi I )T Ui(k−1) (Ci + βi I )
− (βi2 − γi2 )Ui(k−1) + (βi − γi ) (k−1) 2 T × δi,j Uj + Qi + αi Bi Bi (Ci + γi I )−1 ,
+ (βi − γi )(Pi − αi I )
Bi BTi
(k)
which implies Pi ≼ Ui
The claim that Ui are positive semidefinite follows easily from Pi ≽ 0 and Lemma 2.2—or equivalently the transitivity of the Löwner order. Theorem 4.2. Suppose that there exist αi > 0 such that condition (2) is satisfied for any i ∈ S. In addition, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and both conditions (7) and (14) hold too. Then, for each i ∈ S, Algorithm 4.1 yields a sequence of positive semidefinite matrices
(k)
Ui
k∈N
(k+1)
⊂ Rn×n
≺ Ui(k) for all k = 0, 1, 2, . . . ,
(15)
(1)
Proof. Again, we make use of induction. First, we show Ui Ui , i ∈ S. From (9) and (15), we find Ui
(0)
− Ui
= (Ci + γi I )
It is natural to ask the question as whether Algorithm 4.1 yields matrix upper bounds that are strictly tighter than the bound given in (9). To fully address this question, we demonstrate the following two results concerning some important properties of this algorithm. Theorem 4.1. Suppose that there exist αi > 0 such that condition (2) is satisfied for any i ∈ S. Furthermore, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and condition (7) holds as well. Then for each i ∈ S, Algorithm 4.1 yields a sequence of positive semidefinite matrices
k∈N
⊂ Rn×n
(k)
for all k = 0, 1, 2, . . . .
(Ci + βi I )T Ui(0) − ξi I
j∈S \{i}
(0)
However, we know Ui ≺ ξi I for all i by Theorem 3.3. Thus, together with Lemma 2.1, (17) is true when k = 0. (k)
(k−1)
(k+1)
(k)
Next, we assume Ui ≺ Ui for all i and show Ui ≺ Ui for all i. This is easy to verify since, according to (15), we see (k+1)
Ui
(k)
− Ui
= (Ci + γi I )
−T
(Ci + βi I )T Ui(k) − Ui(k−1)
× (Ci + βi I ) − (βi2 − γi2 ) Ui(k) − Ui(k−1) (k) (k−1) + (βi − γi ) δi,j Uj − Uj (Ci + γi I )−1 j∈S \{i}
≺ 0.
that satisfies Pi ≼ U i
−T
× (Ci + βi I ) − (βi2 − γi2 ) Ui(0) − ξi I (0) + (βi − γi ) δi,j Uj − ξj I (Ci + γi I )−1 .
fall within a prescribed tolerance for some matrix norm ∥ · ∥. Remark 4.1. Algorithm 4.1 reduces to Davies et al. (2008, Algorithm 2.1) when βi = 0 and γi = −1, and to Davies et al. (2008, Algorithm 2.2) when βi = 1 and γi = −1. It, however, is more general and flexible in light of the adjustable parameters αi , βi , and γi .
≺
(0)
(1)
(k) (k−1) Ui − Ui
(k)
(17)
i.e. such a sequence is strictly monotonically decreasing.
j∈S \{i}
Ui
by Lemma 2.1. Hence, (16) holds.
(k)
Ui
where i = 1, 2, . . . , s, until
(Pi − αi I ) (Ci + γi I )−1 ,
that satisfies
= (Ci + γi I )
(Ci + βi I )T (Ui(k−1) − Pi )(Ci + βi I )
− Pi = (Ci + γi I )
4. Unified iterative refinement algorithm
−T
(k−1)
for all i whenever Pi ≼ Ui
−T
Due to the issue of strict monotonicity for the tightening of matrix upper bounds in the next section, however, we shall adopt the new condition (14), which is slightly stronger than the one above.
(k)
2171
(k)
(16)
Finally, the positive semidefiniteness of Ui rem 4.1. The proof is now complete.
follows from Theo-
2172
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
Theorems 4.1 and 4.2 justify that, assuming the same conditions as in Theorem 3.1, Algorithm 4.1 always produces sequences of upper bounds for the solution of the CCARE; and moreover, under the extra condition (14), each iterate in any such sequence is strictly tighter than the preceding one. In the light of (16) and (17), we prove the following monotone convergence result. Such a result may exist in the literature and hence we do not claim its originality here. It appears, nevertheless, that no sources of reference has been quoted; see, for example, Ivanov, Hasanov, and Minchev (2001, p. 30). Theorem 4.3. Let {Wk }k∈N ⊂ Rn×n be a given sequence of symmetric matrices. Assume that there exists a symmetric W0 ∈ Rn×n such that for all k, Wk+1 ≼ Wk
and
Theorem 4.4. Suppose that there exist αi > 0 such that condition (2) is satisfied for all i ∈ S. Moreover, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and both conditions (7) and (14) hold too. Then, for each i ∈ S, there exists some positive semidefinite U i ∈ Rn×n with U i ≽ Pi such that Algorithm 4.1 yields a strictly monotonically decreasing sequence of positive semidefinite matrices
(k)
Ui
k∈N
⊂ Rn×n
that converges to U i . Besides, U i are the unique positive semidefinite solution to
CiT U i + U i Ci +
δi,j U j + Qi + αi2 Bi BTi = 0,
(19)
j∈S \{i}
where i = 1, 2, . . . , s.
Wk ≽ W0 .
Then {Wk }k∈N converges to a unique symmetric W ∈ Rn×n . Proof. Consider the finite dimensional normed linear space Rn×n ,
∥ · ∥F . Note that for any k,
Proof. Under the conditions of this theorem, we know that Algorithm 4.1 converges, strictly monotonically, to some unique U i according to Theorems 4.1 and 4.2 and Corollary 4.1. Observe that the limit case of (15) is of the form
0 ≼ Wk − W0 ≼ W1 − W0 .
U i = (Ci + γi I )−T (Ci + βi I )T U i (Ci + βi I ) − (βi2 − γi2 )U i
This, according to Lemma 2.3, implies that for any k,
∥Wk − W0 ∥F ≤ ∥W1 − W0 ∥F .
+ (βi − γi )
Hence, {Wk }k∈N is bounded. By the Bolzano–Weierstrass Theorem, we know that there exists a subsequence Wkj that converges j∈N
to a unique W ∈ Rn×n in the sense lim Wkj − W = 0.
F
j→∞
(18)
It is clear that W is symmetric too. Besides, Wkj ≽ W for all j ∈ N, which implies Wk ≽ W for all k ∈ N. We claim now that limk→∞ Wk = W . For any ϵ > 0, by (18), there exists some kJ such that
Wk − W < ϵ. J F
δi,j U j + Qi + α
2 T i Bi Bi
(Ci + γi I )−1 , (20)
j∈S \{i}
where i = 1, 2, . . . , s. It can be readily verified that (20) and (19) are equivalent to each other. This completes the proof. It should be pointed out that (19) is independent of both βi and γi . Numerical evidence, however, indicates that it is possible to obtain very tight matrix upper bounds by varying αi ; see the last example in this section. Finally, in order to illustrate the performance of Algorithm 4.1, we turn to an example from Davies et al. (2008) and Xu (in press). For this example, assuming the CCARE in the form of (1), we have
1 A1 = 2
A2 = 2 B2 = ,
1 , 0
B1 =
Consequently, we obtain that for any k ≥ kJ ,
1 , −2
−3
2 , −4
2
3
4 3
3 , 4
6 2
Wk − W < ϵ F
δ1,2 = δ2,1 = 0.1,
because of Wk − W ≼ WkJ − W and Lemma 2.3. This finishes the proof.
Then, the positive definite solution of (1) is known to be
It is quite straightforward to notice that if, additionally, Wk , k ≥ 0, in the preceding result are all positive semidefinite, then so must be the limit W . We formally state this more restrictive case as:
P1 =
Corollary 4.1. Let {Wk }k∈N ⊂ Rn×n be a given sequence of positive semidefinite matrices. Assume that there exists a positive semidefinite W0 ∈ Rn×n such that for all k, Wk+1 ≼ Wk
and
Wk ≽ W0 .
Then {Wk }k∈N converges to a unique positive semidefinite W ∈ Rn×n . We comment here that a similar conclusion holds when Wk+1 ≺ Wk for all k ∈ N as the case in Theorem 4.2. Combining (15), Theorems 4.1 and 4.2, and Corollary 4.1 together, we arrive at the conclusion below.
4.4860 1.7725
Q1 =
1.7725 1.1080
and Q2 =
0.6207 0.2185
and P2 =
2 . 3
0.2185 . 0.2873
(21)
In what follows, we show three numerical examples, all using the above CCARE, but with different αi , βi , and γi . For these examples, the stopping criterion in Algorithm 4.1 is set as
(k) (k−1) Ui − Ui < 10−3 . F
To better demonstrate the strict monotonicity, we graph the eigen(k) (k−1) values of Ui − Ui in all the figures. Example 4.1. Let α1 = 3, α2 = 0.1, β1 = β2 = 0, and γ1 = γ2 = −1. In this case, condition (2) is satisfied with C1T
−4.0000 + C1 = 3.0000
3.0000 −4.0000
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
2173
and C2T
−6.8000 + C2 = 2.8000
2.8000 , −9.8000
which are both negative definite. In addition, condition (7) holds with
τ1 = −0.2591 < 0 and τ2 = −0.2449 < 0. Corollary 3.1 now leads to 14.1207 1.0451
1.0451 = U1(0) 10.0952
P1 ≼
and 2.2519 0.0791
P2 ≼
0.0791 = U2(0) . 2.1362
For condition (14), we have
V1 =
−49.6880 50.2185
50.2185 −58.6880
and
−10.7420 V2 = 9.6183
9.6183 , −21.7902
which are both negative definite. Algorithm 4.1, therefore, can be expected to yield strictly monotonically decreasing, convergent sequences of tighter upper bounds. (k) (k−1) Fig. 1 illustrates the eigenvalues of Ui − Ui during iteration
(k)
using Algorithm 4.1. The solid curves represent λ1 Ui
(k)
whereas the dashed curves represent λ2 Ui
− Ui(k−1) ,
− Ui(k−1) . The re-
fined matrix upper bounds on the solution Pi are (23)
U1
7.4648 4.1830
=
4.1830 3.1048
(23)
and U2
=
1.2439 0.5944
0.5944 0.5190
after 23 iterations. These refined upper bounds appear to be much (0) (0) tighter than U1 and U2 , respectively. Example 4.2. Let α1 = 3, α2 = 0.1, β1 = β2 = 1, and γ1 = γ2 = −1. In this case, we have exactly the same CiT + Ci , τi , and Vi as in
(k)
Fig. 1. The eigenvalues of Ui γ1 = γ2 = −1.
and
Example 4.1. Corollary 3.2 now leads to
P1 ≼
12.5018 2.0902
C2T + C2 =
2.0902 = U1(0) 4.4509
1.8045 0.1583
0.1583 = U2(0) . 1.5730
−2.0000 , −17.0000
τ1 = −0.1883 < 0 and τ2 = −0.1491 < 0. (k)
Fig. 2 illustrates the eigenvalues of Ui
− Ui(k−1) during iteration (k)
using Algorithm 4.1. The solid curves represent λ1 Ui
(k)
whereas the dashed curves represent λ2 Ui
− Ui(k−1) ,
− Ui(k−1) . The
refined matrix upper bounds on the solution Pi are 7.4644 (11) U1 = 4.1832
4.1832 3.1046
1.2434 (11) U2 = 0.5953
and
0.5953 0.5176
after 11 iterations, which are very close to the outcome in Example 4.1 due to Theorem 4.4, viz. U i being independent of βi and γi . Example 4.3. Let α1 = 5, α2 = 0.5, β1 = β2 = 0, and γ1 = γ2 = −2. In this case, condition (2) is satisfied with C1T
−10.0000 −2.0000
which are both negative definite. In addition, condition (7) holds with
and P2 ≼
− Ui(k−1) at α1 = 3, α2 = 0.1, β1 = β2 = 0, and
−8.0000 + C1 = 3.0000
3.0000 −4.0000
For condition (14), we have V1 =
−26.2753 23.7672
23.7672 −23.5857
and
−2.7015 V2 = 1.4212
1.4212 , −11.7272
which are both negative definite. Algorithm 4.1, therefore, can be expected to yield strictly monotonically decreasing, convergent sequences of tighter upper bounds. (k)
(k−1)
Fig. 3 illustrates the eigenvalues of Ui − Ui
during iteration (k) (k−1) using Algorithm 4.1. The solid curves represent λ1 Ui − Ui , (k) (k−1) whereas the dashed curves represent λ2 Ui − Ui . The
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J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
(k)
Fig. 2. The eigenvalues of Ui γ1 = γ2 = −1.
− Ui(k−1) at α1 = 3, α2 = 0.1, β1 = β2 = 1, and
refined matrix upper bounds on the solution Pi are (15)
U1
=
4.5979 1.9262
1.9262 1.9710
and
(15)
U2
0.7063 0.1995
=
0.1995 0.2990
after 15 iterations. Incidentally, as shown in Xu (in press), with the same βi and γi but α1 = 3 and α2 = 0.1, Theorem 3.1 leads to 10.9228 2.1704
P1 ≼
2.1704 = U1 7.6979
and 1.7453 0.1942
P2 ≼
0.1942 = U2 , 1.4952
(0)
which are tighter than Ui in Example 4.1. By varying αi , however, we see that the refined upper bounds appear to be much tighter than U1 and U2 , respectively, although all these involve the same βi and γi values. In fact, Ui(15) from this example are very close to the exact solution given in (21). 5. Conclusion In this note, we establish an iterative refinement algorithm arising from the unified matrix upper bound in Xu (in press)
(k)
Fig. 3. The eigenvalues of Ui γ1 = γ2 = −2.
− Ui(k−1) at α1 = 5, α2 = 0.5, β1 = β2 = 0, and
for the solution of the CCARE. Such a unifying, yet more flexible, algorithm not only includes the two iterative refinement algorithms in Davies et al. (2008) but also can be treated rigorously in terms of its convergence and refinement properties. In particular, we propose a new key condition, viz. (14); which, together with two other admissible conditions from Xu (in press), guarantee strictly monotonically decreasing and convergent sequences of matrix upper bounds being computed from this algorithm. It is interesting to observe that the iterative refinement algorithm here converges to the unique positive semidefinite solution of the modified equation as given in (19), which entails some limitation on the tightness of the resulting matrix upper bounds, even though a noticeable refinement can be seen from numerical results. In the meantime, this observation leads to an intriguing question as to how close the computed matrix upper bounds are to the exact solution of the CCARE. A related question will be whether the algorithm can be further improved to reduce such a gap that is induced intrinsically by (19). References Abou-Kandil, H., Freiling, G., & Jank, G. (1994). Solution and asymptotic behaviour of coupled Riccati equations in jump linear systems. IEEE Transactions on Automatic Control, 39, 1631–1636. Berman, A., & Plemmons, R. (1994). Nonnegative matrices in the mathematical sciences. SIAM.
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175 Czornik, A., & Swierniak, A. (2001). Upper bounds on the solution of coupled algebraic Riccati Equation. Journal of Inequalities and Applications, 6, 373–385. Davies, R., Shi, P., & Wiltshire, R. (2008). Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations. Automatica, 44, 1088–1096. Gao, L., Xue, A., & Sun, Y. (2002). Matrix bounds for the coupled algebraic Riccati equation. In Proceedings of the fourth world congress on intelligent control and automation (pp. 180–183). Horn, R., & Johnson, C. (1990). Matrix analysis. Cambridge University Press. Ivanov, I., Hasanov, V., & Minchev, B. (2001). On matrix equations X ± A∗ X −2 A = I. Linear Algebra and its Applications, 326, 27–44. Ji, Y., & Chizeck, H. (1990). Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control. IEEE Transactions on Automatic Control, 35, 777–788. Lee, C.-H. (2006). New upper bounds of the continuous algebraic Riccati matrix equation. IEEE Transactions on Automatic Control, 51, 330–334. Marshall, A., Olkin, I., & Arnold, B. (2011). Inequalities: theory of majorization and its applications (2nd ed.). Springer. Xiao, M. (2005). Optimal control of nonlinear systems with controlled transitions. Nonlinear Dynamics and Systems Theory, 5, 177–188. Xiao, M., & Basar, T. (2001). Viscosity solution of a class of coupled HJB equations. Journal of Inequalities and Applications, 6, 519–545. Xu, J. Unified, Improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation, Journal of the Franklin Institute (2013) (in press).
2175 Jianhong Xu got his Ph.D. in Mathematics from the University of Connecticut in 2003. He is currently an Associate Professor at the Department of Mathematics, Southern Illinois University at Carbondale. His research interests include numerical analysis, numerical linear algebra, matrix theory and applications.
MingQing Xiao got his Ph.D. in 1997 from the University of Illinois at Urbana-Champaign. He had been a visiting research assistant professor at University of California at Davis from 1997 to 1999. He currently is a professor at the Department of Mathematics, Southern Illinois University at Carbondale. Dr. Xiao is a senior Member of IEEE and his research interests include dynamical system theory, nonlinear observer design, and control of distributed parameter systems.
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Brief paper
On the iterative refinement of matrix upper bounds for the solution of continuous coupled algebraic Riccati equations✩ Jianhong Xu, MingQing Xiao 1 Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA
article
abstract
info
Article history: Received 31 October 2012 Accepted 12 March 2013 Available online 6 May 2013
In this paper, we develop a new approach for the iterative refinement algorithm along with fully mathematical justifications, which sharpens matrix solution upper bounds for the well-known continuous coupled algebraic Riccati equation (CCARE). Our methodology is studied within a broader framework recently proposed in Xu (in press). Several desirable properties of the algorithm are then established, which include the strict monotonicity and convergence of the sequences generated by the algorithm, thus strengthening the existing results in the literature for computing tighter matrix upper solution bounds for the CCARE. In addition, new admissibility conditions within the general framework are proposed, which guarantee tighter matrix upper solution bounds to be computed from the algorithm. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Coupled Riccati equation Matrix bound Iterative algorithm
1. Introduction Let m, n, s ∈ N such that m ≥ 1, n ≥ 2, and s ≥ 2. Set S = {1, 2, . . . , s}. Suppose that for each i ∈ S , Ai , Qi ∈ Rn×n , and Bi ∈ Rn×m are all given. Moreover, Qi are assumed to be (symmetric) positive semidefinite. For any i, j ∈ S such that i ̸= j, let δi,j ≥ 0, together with j∈S \{i} δi,j > 0 for each i ∈ S. The so-called continuous coupled algebraic Riccati equation, or CCARE for short, in unknowns Pi ∈ Rn×n can then be formulated as ATi Pi + Pi Ai − Pi Bi BTi Pi +
δi,j Pj = −Qi ,
(1)
j∈S \{i}
where i = 1, 2, . . . , s. We shall also write the solution here of the CCARE as Pi . In this work, we are concerned with the iterative algorithms in Davies, Shi, and Wiltshire (2008) for refining certain matrix upper bounds on the solution Pi , provided that Pi are positive semidefinite. We refer the interested reader to Abou-Kandil, Freiling, and Jank (1994), Ji and Chizeck (1990), Xiao and Basar (2001), and Xiao (2005) (and references therein) for necessary and sufficient conditions on the existence and uniqueness of such a positive semidefinite solution. The CCARE plays an important role in a wide variety of problems relevant to control theory; see, for example, Davies et al. (2008)
✩ The research of the second author was supported in part by NSF Grant 1021203. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (J. Xu), [email protected] (M. Xiao). 1 Tel.: +1 618 453 6572; fax: +1 618 453 5300.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.03.022
and the references therein. The question of bounding the positive semidefinite solution of the CCARE has been an area of current research endeavors. It has been pointed out in Davies et al. (2008) that, besides theoretical merits, such solution bounds serve practical purposes as well. Specifically, in problems such as stability analysis, time-delay system controller design, optimal control, robust stabilization, and convergence of numerical algorithms, it often suffices or is crucial to obtain a sufficiently tight solution bound of the CCARE instead of the exact solution, which usually is not available in applications and hence numerical approximation becomes necessary. In the literature, see Czornik and Swierniak (2001), Davies et al. (2008), and Gao, Xue, and Sun (2002), solution bounds for the CCARE fall into several categories, which include the extremal eigenvalues, trace, determinant, along with Löwner (partial) order on the set of positive semidefinite matrices. Among these types, matrix bounds in terms of the Löwner order turn out to be the most general one since the other bounds can be derived subsequently (e.g. see Berman & Plemmons, 1994, and Marshall, Olkin, & Arnold, 2011). For this reason, we shall focus here on matrix upper bounds. Motivated by Davies et al. (2008), and Lee (2006), we have developed a general framework in Xu (in press) that leads to a unified matrix upper bound on the solution of the CCARE. With this framework, we have also shown in Xu (in press) that the two main results in Davies et al. (2008, Theorems 2.1 and 2.2) can be thought of as special cases of our unified result. We, however, have not investigated in Xu (in press) an important algorithmic aspect of Davies et al. (2008) due to the length of that paper. In fact, two iterative algorithms were proposed in Davies et al. (2008) along with numerical evidence indicating that matrix upper bounds on the solution of the CCARE
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
may be noticeably refined by the algorithms. Incidentally, their method is an extension of the corresponding algorithms in Lee (2006), in which the non-coupled continuous algebraic Riccati equation, or CARE for short, i.e. the scenario with s = 1 and without the coupling terms j∈S \{i} δi,j Pj , was considered. As a continuing work that is built upon (Davies et al., 2008; Lee, 2006; Xu, in press), we shall study here an iterative refinement algorithm for the unified matrix upper bound in Xu (in press) concerning the CCARE. Specifically, we shall show that: (i) The two seemingly separate iterative refinement algorithms proposed in Davies et al. (2008) are special cases of our unified algorithm. (ii) This unified algorithm, which is developed using a framework broader than that of Davies et al. (2008), possesses a number of desirable properties. In particular, under certain admissible conditions, it generates strictly monotonically decreasing2 and convergent sequences of positive semidefinite matrix upper bounds, thus providing a better numerical approximation. (iii) There is numerical evidence which demonstrates the unified algorithm is able to produce very tight matrix upper bounds. We comment that our methodology in this work is in a similar spirit as Davies et al. (2008) and Lee (2006); our approach, however, extends the existing results in a unifying and flexible manner with different, improved linear algebra techniques. 2. Preliminaries First of all, we describe briefly some notation being used in the rest of this paper. For any A ∈ Rn×n , we denote by λi (A) its ith eigenvalue and by ρ(A) = max1≤i≤n {|λi (A)|} the spectral radius of A. For any symmetric A ∈ Rn×n , its eigenvalues are always arranged in nonincreasing order, i.e.
λ1 (A) ≥ λ2 (A) ≥ · · · ≥ λn (A); we write A ≻ 0 (A ≽ 0, resp.) if, in addition, A is positive definite (positive semidefinite, resp.). Given A, B ∈ Rn×n , both being symmetric, the expression A ≻ B (A ≽ B, resp.) means A − B ≻ 0 (A − B ≽ 0, resp.). Besides, for a sequence {Ak } ⊂ Rn×n with Ak being all symmetric, we say that it is strictly monotonically decreasing if Ak ≻ Ak+1 for all k. Finally, given any A = [ai,j ], B = [bi,j ] ∈ Rn×m , the notation A ≥ 0 means ai,j ≥ 0 for all i, j; whereas A ≥ B represents A − B ≥ 0. The first lemma is one of the consequences of Sylvester’s law of inertia for congruent symmetric matrices. Lemma 2.1 (Horn & Johnson, 1990, p. 470). Let A, B ∈ Rn×n be both symmetric such that A ≽ B (A ≻ B, resp.). Then for any nonsingular C ∈ Rn×n , C AC ≽ C BC (C AC ≻ C BC , resp.). T
T
T
T
2169
Finally, we recall that for any A = [ai,j ] ∈ Rn×m , its Frobenius norm is defined to be
n m ∥A∥F = a2i,j = tr(AT A). j =1 i =1
Especially, when A ∈ Rn×n is symmetric, we have
n ∥A∥F = λ2i (A) i=1
by the facts that A is orthogonally diagonalizable (Horn & Johnson, 1990, p. 107) and that ∥ · ∥F is orthogonally invariant (Horn & Johnson, 1990, p. 292). It is clear that Lemma 2.2 leads to the following lemma. Lemma 2.3. Let A, B ∈ Rn×n be positive semidefinite and A ≽ B. Then
∥A∥F ≥ ∥B∥F . 3. Unified matrix upper bound In this section, we first introduce some of the results in Xu (in press) regarding the unified matrix upper bound, which provide a starting point for our new results in this paper. Consider the CCARE in the form of (1) with the positive semidefinite solution Pi . Following Davies et al. (2008) and Lee (2006), we suppose that for each i ∈ S, there always exists some αi > 0 such that ATi + Ai ≺ 2αi Bi BTi .
(2)
For brevity, we write Ci = Ai − α Obviously, the above condition is equivalent to CiT + Ci ≺ 0. It can be shown that, under condition (2), for any γi < 0, Ci + γi I is nonsingular (Xu, in press, Lemma 3.1). Let us define T i Bi Bi .
σi =
λ1 (Ci + γi I )−T (Ci + γi I )−1 < 0. λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1
(3)
Observe that the denominator
λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1 < 0 because of CiT + Ci ≺ 0 and Lemma 2.1. We now introduce F ∈ Rs×s and g = [g1 , g2 , . . . , gs ]T ∈ Rs as
1 δ2,1 σ2 . . F = . . . . δs,1 σs
δ1,2 σ1 1
..
.
···
δ1,3 σ1 δ2,3 σ2 .. .
··· ··· .. .
δ1,s σ1 δ2,s σ2 .. .
. ···
. δs,s−1 σs
δs−1,s σs−1
..
..
(4)
1
and The next lemma follows from the Courant–Fischer theorem. Lemma 2.2 (Horn & Johnson, 1990, p. 182). Let A, B ∈ Rn×n both be symmetric with A ≽ B. Then
λi (A) ≥ λi (B) for all 1 ≤ i ≤ n.
λ1 (Ci + γi I )−T (Qi + αi2 Bi BTi )(Ci + γi I )−1 > 0. gi = − λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1
(5)
In addition, we denote
δi =
δi,j > 0.
(6)
j∈S \{i}
In what follows, according to σi defined from (3), by condition (7) we refer to 2 This is interpreted in the sense of the Löwner order.
1 + δi σi > 0,
1 ≤ i ≤ s,
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J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
i.e.
In the rest of this section, we shall prove two new results relevant to the unified iterative refinement algorithm.
τi = λ1 (Ci + γi I )−T (CiT + Ci )(Ci + γi I )−1 + δi λ1 (Ci + γi I )−T (Ci + γi I )−1 < 0
(7)
for 1 ≤ i ≤ s. Continuing, we set up the linear system
Theorem 3.2. Let αi > 0 be such that condition (2) holds for all i ∈ S. For each i ∈ S, suppose that βi , γi are such that γi < 0 and βi ≥ γi . Then
Pi ≼ (Ci + γi I )
(Ci + βi I )T Pi (Ci + βi I ) − (βi2 − γi2 )Pi
−T
F ξ = g, where F and g are given by (4) and (5), respectively. Under conditions (2) and (7), according to Xu (in press, Theorem 3.1), we know that, F −1 exists, F −1 ≥ 0, and, consequently,
+ (βi − γi )
δi,j Pj + Qi + α
(Ci + γi I )−1 .
2 T i Bi Bi
(12)
j∈S \{i}
ξ = F −1 g ≥ 0.
(8)
With all these preceding preparatory developments in hand, our unified matrix upper bound in Xu (in press) for the solution of the CCARE can be stated as follows. Theorem 3.1 (Xu, in press, Theorem 3.3). Assume that there exist
αi > 0 such that condition (2) is satisfied for all i ∈ S. In addition, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and condition (7) holds as well. Then the positive semidefinite solution Pi , 1 ≤ i ≤ s, of the CCARE in (1) is bounded by
Pi ≼ (Ci + γi I )−T ξi (Ci + βi I )T (Ci + βi I ) − (βi2 − γi2 )I
+ (βi − γi )
δi,j ξj I + Qi + α
2 T i Bi Bi
(Ci + γi I )−1
Proof. Using (1), we can easily verify that
(Ci + γi I )T Pi (Ci + γi I ) + (βi − γi )(Pi − αi I )Bi BTi (Pi − αi I ) = (Ci + βi I )T Pi (Ci + βi I ) − (βi2 − γi2 )Pi 2 T + (βi − γi ) δi,j Pj + Qi + αi Bi Bi ,
(13)
j∈S \{i}
which leads to
(Ci + γi I )T Pi (Ci + γi I ) ≼ (Ci + βi I )T Pi (Ci + βi I ) − (βi2 − γi2 )Pi 2 T δi,j Pj + Qi + αi Bi Bi . + (βi − γi ) j∈S \{i}
j∈S \{i}
It is clear now that (12) follows from the above and Lemma 2.1.
= Ui ,
(9)
where ξi are given in (8). The unified matrix upper bound as in (9) will be utilized as a starting point for the unified iterative refinement algorithm that we shall explore in Section 4. Numerical examples in this regard will include the following two special cases: Corollary 3.1 (Xu, in press, Corollary 3.1, cf. Davies et al., 2008, Theorem 2.1). Let βi = 0 and γi = −1 for all i ∈ S. Assume that conditions (2) and (7) are satisfied. Then, we have
The next result establishes a connection between Ui and ξi . Essentially, it provides a necessary and sufficient condition under which each ξi serves as an upper bound on the spectrum of the corresponding Ui . Theorem 3.3. Suppose that there exist αi > 0 such that condition (2) is satisfied for all i ∈ S. Besides, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and condition (7) holds too. Then for each i ∈ S, Ui ≺ ξi I , where Ui are given by (9), if and only if
Pi ≼ (Ci − I )−T ξi (CiT Ci + I ) +
δi,j ξj I
Vi = ξi (CiT + Ci ) +
j∈S \{i}
δi,j ξj I + Qi + αi2 Bi BTi ≺ 0.
(14)
j∈S \{i}
+ Qi + αi2 Bi BTi (Ci − I )−1 ,
1 ≤ i ≤ s,
(10)
where ξi are given in (8). Compared to η in Davies et al. (2008, Theorem 2.1), these ξi lead to a tighter upper bound in (10). Moreover, condition (7) is less restrictive than Davies et al. (2008, condition (2.1)).
Proof. From (9), we calculate ξi I − Ui :
ξi I − Ui = (Ci + γi I )
−T
ξi (Ci + γi I )T (Ci + γi I )
− ξi (Ci + βi I ) (Ci + βi I ) − (β − γ )I T
Corollary 3.2 (Xu, in press, Corollary 3.2, cf. Davies et al., 2008, Theorem 2.2). Let βi = 1 and γi = −1 for all i ∈ S. Assume that conditions (2) and (7) are satisfied. Then, we have
Pi ≼ (Ci − I )−T ξi (Ci + I )T (Ci + I ) + 2
2 i
− (βi − γi )
δi,j ξj I + Qi + α
2 T i Bi Bi
(Ci + γi I )−1
j∈S \{i}
δi,j ξj I
= −(βi − γi )(Ci + γi I )
j∈S \{i}
−T
ξi (CiT + Ci ) +
+ 2 Qi + αi2 Bi BTi (Ci − I )−1 ,
2 i
δi,j ξj I
j∈S \{i}
1 ≤ i ≤ s,
(11)
+ Qi + α
2 T i Bi Bi
where ξi are given in (8). Compared to ϕ in Davies et al. (2008, Theorem 2.2), these ξi lead to a tighter upper bound in (11). Moreover, condition (7) is less restrictive than Davies et al. (2008, condition (2.14)).
(Ci + γi I )−1 .
On letting either side be positive definite, the conclusion now follows from Lemma 2.1.
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
Condition (14) turns out to be a key ingredient for studying the unified iterative refinement algorithm in Section 4. It has been pointed out in Xu (in press) that (4), (5), (8), and condition (7) are independent of the βi ’s. Similarly, condition (14) here is independent of the βi ’s too. We mention also that in Xu (in press, Theorem 3.5) it has been shown that conditions (2) and (7) imply
ξi (CiT + Ci ) +
δi,j ξj I + Qi + αi2 Bi BTi ≼ 0.
Proof. We show the above conclusion by induction. First of all, we (0) (0) observe for all i ∈ S , Pi ≼ Ui by the choice of Ui . To prove (16), (k)
therefore, it suffices to have Pi ≼ Ui for all i. Using (13) and (15), we obtain (k)
Ui
j∈S \{i}
− (βi2 − γi2 ) Ui(k−1) − Pi δi,j Uj(k−1) − Pj + (βi − γi ) j∈S \{i}
Based on results from Section 3, we now present an iterative algorithm capable of refining the unified matrix upper bound as in (9), yet more general than the ones proposed in Davies et al. (2008) and Lee (2006)—for CCARE and CARE, respectively. Our approach here not only is in a unifying manner but also contains new components. In particular, we shall show new conditions that guarantee tighter matrix upper bounds being computed through this unified iterative refinement algorithm. Our unified iterative refinement algorithm, which is described below, is based on estimate (12) in Theorem 3.2. This algorithm extends, by choosing the values of βi and γi accordingly, to the two seemingly separate iterative refinement algorithms in Davies et al. (2008, Algorithms 2.1 and 2.2). Algorithm 4.1 (Davies et al., 2008, Algorithms 2.1 and 2.2). For each (0) i ∈ S, let Ui be the matrix upper bound given in (9). Then, for k = 1, 2, . . . , we iterate according to Ui
(Ci + βi I )T Ui(k−1) (Ci + βi I )
− (βi2 − γi2 )Ui(k−1) + (βi − γi ) (k−1) 2 T × δi,j Uj + Qi + αi Bi Bi (Ci + γi I )−1 ,
+ (βi − γi )(Pi − αi I )
Bi BTi
(k)
which implies Pi ≼ Ui
The claim that Ui are positive semidefinite follows easily from Pi ≽ 0 and Lemma 2.2—or equivalently the transitivity of the Löwner order. Theorem 4.2. Suppose that there exist αi > 0 such that condition (2) is satisfied for any i ∈ S. In addition, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and both conditions (7) and (14) hold too. Then, for each i ∈ S, Algorithm 4.1 yields a sequence of positive semidefinite matrices
(k)
Ui
k∈N
(k+1)
⊂ Rn×n
≺ Ui(k) for all k = 0, 1, 2, . . . ,
(15)
(1)
Proof. Again, we make use of induction. First, we show Ui Ui , i ∈ S. From (9) and (15), we find Ui
(0)
− Ui
= (Ci + γi I )
It is natural to ask the question as whether Algorithm 4.1 yields matrix upper bounds that are strictly tighter than the bound given in (9). To fully address this question, we demonstrate the following two results concerning some important properties of this algorithm. Theorem 4.1. Suppose that there exist αi > 0 such that condition (2) is satisfied for any i ∈ S. Furthermore, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and condition (7) holds as well. Then for each i ∈ S, Algorithm 4.1 yields a sequence of positive semidefinite matrices
k∈N
⊂ Rn×n
(k)
for all k = 0, 1, 2, . . . .
(Ci + βi I )T Ui(0) − ξi I
j∈S \{i}
(0)
However, we know Ui ≺ ξi I for all i by Theorem 3.3. Thus, together with Lemma 2.1, (17) is true when k = 0. (k)
(k−1)
(k+1)
(k)
Next, we assume Ui ≺ Ui for all i and show Ui ≺ Ui for all i. This is easy to verify since, according to (15), we see (k+1)
Ui
(k)
− Ui
= (Ci + γi I )
−T
(Ci + βi I )T Ui(k) − Ui(k−1)
× (Ci + βi I ) − (βi2 − γi2 ) Ui(k) − Ui(k−1) (k) (k−1) + (βi − γi ) δi,j Uj − Uj (Ci + γi I )−1 j∈S \{i}
≺ 0.
that satisfies Pi ≼ U i
−T
× (Ci + βi I ) − (βi2 − γi2 ) Ui(0) − ξi I (0) + (βi − γi ) δi,j Uj − ξj I (Ci + γi I )−1 .
fall within a prescribed tolerance for some matrix norm ∥ · ∥. Remark 4.1. Algorithm 4.1 reduces to Davies et al. (2008, Algorithm 2.1) when βi = 0 and γi = −1, and to Davies et al. (2008, Algorithm 2.2) when βi = 1 and γi = −1. It, however, is more general and flexible in light of the adjustable parameters αi , βi , and γi .
≺
(0)
(1)
(k) (k−1) Ui − Ui
(k)
(17)
i.e. such a sequence is strictly monotonically decreasing.
j∈S \{i}
Ui
by Lemma 2.1. Hence, (16) holds.
(k)
Ui
where i = 1, 2, . . . , s, until
(Pi − αi I ) (Ci + γi I )−1 ,
that satisfies
= (Ci + γi I )
(Ci + βi I )T (Ui(k−1) − Pi )(Ci + βi I )
− Pi = (Ci + γi I )
4. Unified iterative refinement algorithm
−T
(k−1)
for all i whenever Pi ≼ Ui
−T
Due to the issue of strict monotonicity for the tightening of matrix upper bounds in the next section, however, we shall adopt the new condition (14), which is slightly stronger than the one above.
(k)
2171
(k)
(16)
Finally, the positive semidefiniteness of Ui rem 4.1. The proof is now complete.
follows from Theo-
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J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
Theorems 4.1 and 4.2 justify that, assuming the same conditions as in Theorem 3.1, Algorithm 4.1 always produces sequences of upper bounds for the solution of the CCARE; and moreover, under the extra condition (14), each iterate in any such sequence is strictly tighter than the preceding one. In the light of (16) and (17), we prove the following monotone convergence result. Such a result may exist in the literature and hence we do not claim its originality here. It appears, nevertheless, that no sources of reference has been quoted; see, for example, Ivanov, Hasanov, and Minchev (2001, p. 30). Theorem 4.3. Let {Wk }k∈N ⊂ Rn×n be a given sequence of symmetric matrices. Assume that there exists a symmetric W0 ∈ Rn×n such that for all k, Wk+1 ≼ Wk
and
Theorem 4.4. Suppose that there exist αi > 0 such that condition (2) is satisfied for all i ∈ S. Moreover, for each i ∈ S, let βi , γi be such that γi < 0, γi < βi ≤ −γi , and both conditions (7) and (14) hold too. Then, for each i ∈ S, there exists some positive semidefinite U i ∈ Rn×n with U i ≽ Pi such that Algorithm 4.1 yields a strictly monotonically decreasing sequence of positive semidefinite matrices
(k)
Ui
k∈N
⊂ Rn×n
that converges to U i . Besides, U i are the unique positive semidefinite solution to
CiT U i + U i Ci +
δi,j U j + Qi + αi2 Bi BTi = 0,
(19)
j∈S \{i}
where i = 1, 2, . . . , s.
Wk ≽ W0 .
Then {Wk }k∈N converges to a unique symmetric W ∈ Rn×n . Proof. Consider the finite dimensional normed linear space Rn×n ,
∥ · ∥F . Note that for any k,
Proof. Under the conditions of this theorem, we know that Algorithm 4.1 converges, strictly monotonically, to some unique U i according to Theorems 4.1 and 4.2 and Corollary 4.1. Observe that the limit case of (15) is of the form
0 ≼ Wk − W0 ≼ W1 − W0 .
U i = (Ci + γi I )−T (Ci + βi I )T U i (Ci + βi I ) − (βi2 − γi2 )U i
This, according to Lemma 2.3, implies that for any k,
∥Wk − W0 ∥F ≤ ∥W1 − W0 ∥F .
+ (βi − γi )
Hence, {Wk }k∈N is bounded. By the Bolzano–Weierstrass Theorem, we know that there exists a subsequence Wkj that converges j∈N
to a unique W ∈ Rn×n in the sense lim Wkj − W = 0.
F
j→∞
(18)
It is clear that W is symmetric too. Besides, Wkj ≽ W for all j ∈ N, which implies Wk ≽ W for all k ∈ N. We claim now that limk→∞ Wk = W . For any ϵ > 0, by (18), there exists some kJ such that
Wk − W < ϵ. J F
δi,j U j + Qi + α
2 T i Bi Bi
(Ci + γi I )−1 , (20)
j∈S \{i}
where i = 1, 2, . . . , s. It can be readily verified that (20) and (19) are equivalent to each other. This completes the proof. It should be pointed out that (19) is independent of both βi and γi . Numerical evidence, however, indicates that it is possible to obtain very tight matrix upper bounds by varying αi ; see the last example in this section. Finally, in order to illustrate the performance of Algorithm 4.1, we turn to an example from Davies et al. (2008) and Xu (in press). For this example, assuming the CCARE in the form of (1), we have
1 A1 = 2
A2 = 2 B2 = ,
1 , 0
B1 =
Consequently, we obtain that for any k ≥ kJ ,
1 , −2
−3
2 , −4
2
3
4 3
3 , 4
6 2
Wk − W < ϵ F
δ1,2 = δ2,1 = 0.1,
because of Wk − W ≼ WkJ − W and Lemma 2.3. This finishes the proof.
Then, the positive definite solution of (1) is known to be
It is quite straightforward to notice that if, additionally, Wk , k ≥ 0, in the preceding result are all positive semidefinite, then so must be the limit W . We formally state this more restrictive case as:
P1 =
Corollary 4.1. Let {Wk }k∈N ⊂ Rn×n be a given sequence of positive semidefinite matrices. Assume that there exists a positive semidefinite W0 ∈ Rn×n such that for all k, Wk+1 ≼ Wk
and
Wk ≽ W0 .
Then {Wk }k∈N converges to a unique positive semidefinite W ∈ Rn×n . We comment here that a similar conclusion holds when Wk+1 ≺ Wk for all k ∈ N as the case in Theorem 4.2. Combining (15), Theorems 4.1 and 4.2, and Corollary 4.1 together, we arrive at the conclusion below.
4.4860 1.7725
Q1 =
1.7725 1.1080
and Q2 =
0.6207 0.2185
and P2 =
2 . 3
0.2185 . 0.2873
(21)
In what follows, we show three numerical examples, all using the above CCARE, but with different αi , βi , and γi . For these examples, the stopping criterion in Algorithm 4.1 is set as
(k) (k−1) Ui − Ui < 10−3 . F
To better demonstrate the strict monotonicity, we graph the eigen(k) (k−1) values of Ui − Ui in all the figures. Example 4.1. Let α1 = 3, α2 = 0.1, β1 = β2 = 0, and γ1 = γ2 = −1. In this case, condition (2) is satisfied with C1T
−4.0000 + C1 = 3.0000
3.0000 −4.0000
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
2173
and C2T
−6.8000 + C2 = 2.8000
2.8000 , −9.8000
which are both negative definite. In addition, condition (7) holds with
τ1 = −0.2591 < 0 and τ2 = −0.2449 < 0. Corollary 3.1 now leads to 14.1207 1.0451
1.0451 = U1(0) 10.0952
P1 ≼
and 2.2519 0.0791
P2 ≼
0.0791 = U2(0) . 2.1362
For condition (14), we have
V1 =
−49.6880 50.2185
50.2185 −58.6880
and
−10.7420 V2 = 9.6183
9.6183 , −21.7902
which are both negative definite. Algorithm 4.1, therefore, can be expected to yield strictly monotonically decreasing, convergent sequences of tighter upper bounds. (k) (k−1) Fig. 1 illustrates the eigenvalues of Ui − Ui during iteration
(k)
using Algorithm 4.1. The solid curves represent λ1 Ui
(k)
whereas the dashed curves represent λ2 Ui
− Ui(k−1) ,
− Ui(k−1) . The re-
fined matrix upper bounds on the solution Pi are (23)
U1
7.4648 4.1830
=
4.1830 3.1048
(23)
and U2
=
1.2439 0.5944
0.5944 0.5190
after 23 iterations. These refined upper bounds appear to be much (0) (0) tighter than U1 and U2 , respectively. Example 4.2. Let α1 = 3, α2 = 0.1, β1 = β2 = 1, and γ1 = γ2 = −1. In this case, we have exactly the same CiT + Ci , τi , and Vi as in
(k)
Fig. 1. The eigenvalues of Ui γ1 = γ2 = −1.
and
Example 4.1. Corollary 3.2 now leads to
P1 ≼
12.5018 2.0902
C2T + C2 =
2.0902 = U1(0) 4.4509
1.8045 0.1583
0.1583 = U2(0) . 1.5730
−2.0000 , −17.0000
τ1 = −0.1883 < 0 and τ2 = −0.1491 < 0. (k)
Fig. 2 illustrates the eigenvalues of Ui
− Ui(k−1) during iteration (k)
using Algorithm 4.1. The solid curves represent λ1 Ui
(k)
whereas the dashed curves represent λ2 Ui
− Ui(k−1) ,
− Ui(k−1) . The
refined matrix upper bounds on the solution Pi are 7.4644 (11) U1 = 4.1832
4.1832 3.1046
1.2434 (11) U2 = 0.5953
and
0.5953 0.5176
after 11 iterations, which are very close to the outcome in Example 4.1 due to Theorem 4.4, viz. U i being independent of βi and γi . Example 4.3. Let α1 = 5, α2 = 0.5, β1 = β2 = 0, and γ1 = γ2 = −2. In this case, condition (2) is satisfied with C1T
−10.0000 −2.0000
which are both negative definite. In addition, condition (7) holds with
and P2 ≼
− Ui(k−1) at α1 = 3, α2 = 0.1, β1 = β2 = 0, and
−8.0000 + C1 = 3.0000
3.0000 −4.0000
For condition (14), we have V1 =
−26.2753 23.7672
23.7672 −23.5857
and
−2.7015 V2 = 1.4212
1.4212 , −11.7272
which are both negative definite. Algorithm 4.1, therefore, can be expected to yield strictly monotonically decreasing, convergent sequences of tighter upper bounds. (k)
(k−1)
Fig. 3 illustrates the eigenvalues of Ui − Ui
during iteration (k) (k−1) using Algorithm 4.1. The solid curves represent λ1 Ui − Ui , (k) (k−1) whereas the dashed curves represent λ2 Ui − Ui . The
2174
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175
(k)
Fig. 2. The eigenvalues of Ui γ1 = γ2 = −1.
− Ui(k−1) at α1 = 3, α2 = 0.1, β1 = β2 = 1, and
refined matrix upper bounds on the solution Pi are (15)
U1
=
4.5979 1.9262
1.9262 1.9710
and
(15)
U2
0.7063 0.1995
=
0.1995 0.2990
after 15 iterations. Incidentally, as shown in Xu (in press), with the same βi and γi but α1 = 3 and α2 = 0.1, Theorem 3.1 leads to 10.9228 2.1704
P1 ≼
2.1704 = U1 7.6979
and 1.7453 0.1942
P2 ≼
0.1942 = U2 , 1.4952
(0)
which are tighter than Ui in Example 4.1. By varying αi , however, we see that the refined upper bounds appear to be much tighter than U1 and U2 , respectively, although all these involve the same βi and γi values. In fact, Ui(15) from this example are very close to the exact solution given in (21). 5. Conclusion In this note, we establish an iterative refinement algorithm arising from the unified matrix upper bound in Xu (in press)
(k)
Fig. 3. The eigenvalues of Ui γ1 = γ2 = −2.
− Ui(k−1) at α1 = 5, α2 = 0.5, β1 = β2 = 0, and
for the solution of the CCARE. Such a unifying, yet more flexible, algorithm not only includes the two iterative refinement algorithms in Davies et al. (2008) but also can be treated rigorously in terms of its convergence and refinement properties. In particular, we propose a new key condition, viz. (14); which, together with two other admissible conditions from Xu (in press), guarantee strictly monotonically decreasing and convergent sequences of matrix upper bounds being computed from this algorithm. It is interesting to observe that the iterative refinement algorithm here converges to the unique positive semidefinite solution of the modified equation as given in (19), which entails some limitation on the tightness of the resulting matrix upper bounds, even though a noticeable refinement can be seen from numerical results. In the meantime, this observation leads to an intriguing question as to how close the computed matrix upper bounds are to the exact solution of the CCARE. A related question will be whether the algorithm can be further improved to reduce such a gap that is induced intrinsically by (19). References Abou-Kandil, H., Freiling, G., & Jank, G. (1994). Solution and asymptotic behaviour of coupled Riccati equations in jump linear systems. IEEE Transactions on Automatic Control, 39, 1631–1636. Berman, A., & Plemmons, R. (1994). Nonnegative matrices in the mathematical sciences. SIAM.
J. Xu, M. Xiao / Automatica 49 (2013) 2168–2175 Czornik, A., & Swierniak, A. (2001). Upper bounds on the solution of coupled algebraic Riccati Equation. Journal of Inequalities and Applications, 6, 373–385. Davies, R., Shi, P., & Wiltshire, R. (2008). Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations. Automatica, 44, 1088–1096. Gao, L., Xue, A., & Sun, Y. (2002). Matrix bounds for the coupled algebraic Riccati equation. In Proceedings of the fourth world congress on intelligent control and automation (pp. 180–183). Horn, R., & Johnson, C. (1990). Matrix analysis. Cambridge University Press. Ivanov, I., Hasanov, V., & Minchev, B. (2001). On matrix equations X ± A∗ X −2 A = I. Linear Algebra and its Applications, 326, 27–44. Ji, Y., & Chizeck, H. (1990). Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control. IEEE Transactions on Automatic Control, 35, 777–788. Lee, C.-H. (2006). New upper bounds of the continuous algebraic Riccati matrix equation. IEEE Transactions on Automatic Control, 51, 330–334. Marshall, A., Olkin, I., & Arnold, B. (2011). Inequalities: theory of majorization and its applications (2nd ed.). Springer. Xiao, M. (2005). Optimal control of nonlinear systems with controlled transitions. Nonlinear Dynamics and Systems Theory, 5, 177–188. Xiao, M., & Basar, T. (2001). Viscosity solution of a class of coupled HJB equations. Journal of Inequalities and Applications, 6, 519–545. Xu, J. Unified, Improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation, Journal of the Franklin Institute (2013) (in press).
2175 Jianhong Xu got his Ph.D. in Mathematics from the University of Connecticut in 2003. He is currently an Associate Professor at the Department of Mathematics, Southern Illinois University at Carbondale. His research interests include numerical analysis, numerical linear algebra, matrix theory and applications.
MingQing Xiao got his Ph.D. in 1997 from the University of Illinois at Urbana-Champaign. He had been a visiting research assistant professor at University of California at Davis from 1997 to 1999. He currently is a professor at the Department of Mathematics, Southern Illinois University at Carbondale. Dr. Xiao is a senior Member of IEEE and his research interests include dynamical system theory, nonlinear observer design, and control of distributed parameter systems.