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[15] K. Sugimoto, “Partial pole placement by lq regulators: An inverse problem approach,” IEEE Trans. on Automatic Control, vol. 43, no. 5, pp. 1769–1776, May 1998. [16] J. Daafouz and J. Bernussou, “Parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties,” Syst. Control Lett., vol. 43, pp. 355–359, 2001. [17] J. Daafouz and G. Millerioux, “Poly-quadratic stability and global chaos synchronization of discrete time hybrid systems,” Special Issue Math. Comput. Simulation, vol. 58, pp. 295–307, Mar. 2002. [18] M. C. de Oliveira, D. P. Farias, and J. C. Geromel. LMI solver. [Online]. Available: http://www.dt.fee.unicamp.br/ carvalho/software.html [19] G. Millerioux, G. Bloch, J. M. Amigo, A. Bastos, and F. Anstett, “Real-time video communication secured by a chaotic key stream cipher,” in Proc. IEEE 16th Eur. Conf. Circuits Theory Design, ECCTD’03, Krakow, Poland, Sept. 1–4, 2003, pp. 245–248. [20] G. Millerioux and J. Daafouz, “Unknown input observers for messageembedded chaos synchronization of discrete-time systems,” Int. J. Bifurcation Chaos, vol. 14, no. 4, Apr. 2004. , “Unknown input observers for switched linear discrete time sys[21] tems,” presented at the Amer. Control Conf. ACC’2004, Boston, MA, June 30–July3 2004.
On
Model Reduction Using LMIs
Yoshio Ebihara and Tomomichi Hagiwara Abstract—In this note, we deal with the problem of approximating by an th-order a given th-order linear time-invariant system system where . It is shown that lower bounds of the norm of the associated error system can be analyzed by using linear matrix ineqaulity (LMI)-related techniques. These lower bounds are given in terms of the Hankel singular values of the system and coincide with those obtained in the previous studies where the analysis of the Hankel operators plays a central role. Thus, this note provides an alternative proof for those lower bounds via simple algebraic manipulations related to LMIs. Moreover, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the problem is essentially convex and the optimal reduced-order models can be constructed via LMI optimization. Index Terms—
model reduction, linear matrix inequalities (LMIs).
I. INTRODUCTION The H1 model reduction has been a central topic in control theory. Given a linear time-invariant (LTI) system G of McMillan degree n, the problem is to find a system Gr of McMillan degree r that minimizes the H1 norm kG 0 Gr k1 where r < n. Intuitively, model reduction can be done by removing the states from G that are of little effect on the system input-output characteristics. The well-known balanced truncation method [5], [16], [17] has been developed to achieve this. On the other hand, in the optimal Hankel norm approximation
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method [5], the problem has been dealt with more rigorously by analyzing the Hankel operator of G. It has been shown that the Hankel norm of the error incurred in approximating G by Gr is at least as large as the (r + 1)-st largest Hankel singular value of G, and that we can obtain Gr that achieves this lower bound by following the all-pass embedding procedure [5]. These two methods provide constructive ways for model reduction. One significant achievement is that upper bounds and lower bounds of the error have been gained in an analytic form in terms of the Hankel singular values [5], [17]. From the viewpoints of the LMI-based H1 controller synthesis, the H1 model reduction problem is difficult since it can be regarded as a special case of the reduced-order controller synthesis. In stark contrast with the full-order cases, the reduced-order problems are considered to be bilinear matrix ineqaulities (BMIs) and still remain open to this date [3], [9]. Although some effective local algorithms for the computation of reduced-order controllers have been developed [4], [7], [10], we cannot evaluate the resulting H1 cost rigorously due to the lack of analytic results on the achievable performance by the reduced-order controllers. Hence, it is of great importance to establish ways for computing strict lower bounds of the H1 cost. The goal of this note is to show that, when dealing with the H1 model reduction problems, we can readily obtain lower bounds of the H1 cost by using the well-established LMI-related techniques. The Parrott’s Lemma [2], [14], which plays a key role in the LMI-based H1 controller synthesis [3], [9], [12], [13], leads us to two matrix inequalities that are closely related to the Lyapunov equalities with respect to the controllability and observability Gramians [5], [17]. With these matrix inequalities and the results from the balanced realization [5], [17], it follows that the lower bounds are given in terms of the Hankel singular values. These lower bounds are exactly the same as those obtained in the optimal Hankel approximation method [5]. Thus, this note provides an alternative proof for those lower bounds via simple algebraic manipulations related to LMIs. Moreover, in the case where we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the H1 model reduction problem is essentially convex, and that the optimal reduced-order models can be constructed by solving LMI feasibility/optimization problems. We use the following notations in this note. In and 0n;m denote respectively the identity matrix of dimension n and the zero matrix of dimension n2m; the dimensions are omitted when they can be inferred from the context. For a matrix A 2 n2n , HefAg is a shorthand notation for A + AT . For a symmetric matrix A, we denote by triplet (In0 (A), In0 (A), In+ (A)) the numbers of its strictly negative, zero, and strictly positive eigenvalues, respectively. Furthermore, Sn denotes the set of n 2 n positive–definite matrices. The following lemma is used in the subsequent discussions. Lemma 1 [11]: For given two symmetric matrices A 2 n2n and B 2 n2n , A < B holds only if i (A) < i (B ) (i = 1; 1 1 1 ; n) where i (A) denotes the ith-largest eigenvalue of A.
R
R
R
II. BALANCED REALIZATION AND LMI-BASED MODEL REDUCTION Manuscript received October 1, 2003; revised April 2, 2004. Recommended by Associate Editor E. Jonckheere. This work supported in part by the Ministry of Education, Culture, Sports, Science and Technology of Japan under Grant-in-Aid for Young Scientists (B), 15760314. The authors are with the Department of Electrical Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2004.831116
Let us consider a system G(s) 2 its minimal realization G(s)
=
A
B
C
D
0018-9286/04$20.00 © 2004 IEEE
;
B
A
RH1 of McMillan degree
2 Rn2n
2 Rn2p ;
C
2 Rq2n ;
D
n
2 Rq2p:
and
(1)
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In the sequel, we assume that the realization in (1) is already balanced, i.e., its controllability and observability Gramians are equal and diagonal [5], [17]. Denoting the balanced Gramians by 6 , we have
A6 + 6AT + BBT = 0 6A + AT 6 + C T C = 0
(2a) (2b)
6 = diag 1 Ik ; 1 1 1 ; l Ik ; l+1 Ik ; 1 1 1 ; m Ik 1 > 1 1 1 > l > l+1 > 1 1 1 > m > 0: (3) Note that ki is the multiplicity of i and k1 + 1 1 1 + km = n. The diagonal entries of 6 are called the Hankel singular values of the system G(s) [16]. Suppose l l+1 . Then, the balanced realization implies that those states corresponding to l+1 ; 1 1 1 ; m are less controllable and observable than those states corresponding to 1 ; . . . ; l . Hence, truncating states corresponding to l+1 ; . . . ; m will not lose
much information about the system input–output characteristics. The balanced truncation method simply applies this truncation operation to G(s) and obtains a reduced-order model Gr (s) of McMillan degree r := k1 + 1 1 1 + kl [5]. It has been shown that the resulting model Gr (s) is stable. Moreover, the approximation error is proved to be bounded by the following formula [5]:
kG(s) 0 Gr (s)k1 2(l+1 + 1 1 1 + m ):
(4)
Although the balanced truncation method is highly constructive, it is deficient in the sense that the resulting reduced-order models are not necessarily optimal with respect to the H1 cost. To overcome this, in the framework of the LMIs, the H1 optimal models have been sought by means of the bounded real lemma [1]. Indeed, if we denote the state space matrices of Gr (s) by (Ar , Br , Cr , Dr ), then the optimal models can be sought by minimizing 2 subject to the matrix inequalities shown in (5) at the bottom of the page. Unfortunately, however, the aforementioned inequalities are not LMIs with respect to P11 , P12 , P22 and Ar , Br , Cr , Dr since bilinear terms occur. Thus, the H1 model reduction problems are essentially nonconvex problems represented by BMIs and, hence, computing globally optimal solutions remains open to this date [6]. Nevertheless, (5) is still useful to obtain suboptimal solutions via the coordinate-based decent methods [8], [10]. Indeed, by constraining the variables Ar and Br to be constant, the inequalities in (5) are linear with respect to P , Cr and Dr . Also, if we fix P12 and P22 to be constant, the inequalities in (5) come to be LMIs with respect to P11 , Ar , Br , Cr and Dr . By minimizing 2 using the freedom of unfixed variables iteratively, we can obtain suboptimal solutions for the H1 model reduction problems. III. MAIN RESULTS A. Analysis of Lower Bounds Using LMI-Related Techniques Now, we are in a position to state the main results of the note. The first result concerns lower bounds of the H1 cost incurred in approximating G(s) by Gr (s). To derive the lower bounds, we follow the standard procedure for the LMI-based H1 controller synthesis. Applying the Parrott’s Lemma [2], [14] to (5), we readily
P = PP11T PP12 >0 12 22
obtain the following theorem that forms an important basis for the analysis of the lower bounds. Theorem 1: Let us consider a system G(s) 2 RH1 of McMillan degree n and its minimal realization
B : G(s) = CA D (6) Then, there exists a Gr (s) 2 RH1 of McMillan degree at most r that satisfies kG(s) 0 Gr (s)k1 < if and only if there exist X11 2 Sn , P11 2 Sn , P12 2 Rn2r and P22 2 Sr satisfying the following matrix
inequalities.
AX11 + X11 AT + 12 BBT < 0 P11 A + AT P11 + C T C < 0 01 X11 = P11 0 P12 P2201P12T :
(7b)
kG(s) 0 Gr (s)k1 r+1 :
(8)
(7a)
(7c)
Proof: See the Appendix section for the proof. Q : E: D : Condition (7) is still nonconvex due to (7c). This equality constraint commonly arises in the general reduced-order H1 controller synthesis [3], [9] and prevents us from reducing those synthesis problems into LMIs. It is known that this equality constraint can be recast into a rank constraint on the variables X11 and P11 and, hence, in the previous works, research efforts have been made mainly on establishing efficient computation methods for solving those rank-constrained-LMIs [4], [7], [10]. On the other hand, studies on seeking for analytic results deduced by the rank-constrained-LMIs are rare, and research in this direction would be an important topic in the future. In this note, we are dealing with a special case of the reduced-order H1 controller synthesis problems, i.e., the H1 model reduction problem. It follows that we can fully rely on the results from the balanced realization. Indeed, by noting that the first two inequalities in (7) are closely related to the Lyapunov equalities (2) for the balanced Gramian, we can show that lower bounds of the H1 cost can be given in terms of the Hankel singular values. In the following corollary, we neglect the multiplicity of the Hankel singular values of G(s) given in (3) and denote them by 1 1 1 1 r r+1 1 1 1 n > 0 for the ease of our statements. Corollary 1: Let us consider a system G(s) 2 RH1 of McMillan degree n with the Hankel singular values 1 1 1 1 r r+1 1 1 1 n > 0. Then, for all Gr (s) 2 RH1 of McMillan degree less than or equal to r , we have
Proof: To prove the assertion, we show that (7) does not hold if
r+1 . From (2) and the first two inequalities in (7), we readily obtain
A X11 0 12 6 + X11 0 12 6 AT < 0 T (P11 0 6 )A + A (P11 0 6 ) < 0: (9) 2 Since A is stable, it follows that X11 0 (1= )6 > 0 and P11 06 > 0. With these inequalities and (7c), we see that the following condition is necessary for (7) to hold: 6 0 2 6 01 < P P 01P T : (10)
12 22 12
P11 A + AT P11 AT P12 + P12 Ar P11 B + P12 Br CT T T 3 P22 Ar + Ar P22 P12 B + P22 Br 0CrT < 0: 3 3 0 2 I DT 0 DrT 3 3 3 0I
(5)
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If r+1 , however, we see from the diagonal entries of 60 2 6 01 that In0 (6 0 2 6 01 ) n 0 r 0 1 whereas it is apparent that 01 P12 T ) n 0 r . Thus, from Lemma 1, the condition (10) In0 (P12 P22 Q:E:D: cannot be satisfied if r+1 . This completes the proof. The lower bound given in Corollary 1 is exactly the same as those obtained in the optimal Hankel norm approximation method [5], [16]. In these previous works, the Hankel operator of G(s) and its Hankel norm is analyzed in detail and the lower bound is derived for approximation errors measured by the Hankel norm. In stark contrast, we derive here the lower bound by directly working on the H1 norm of the associated error systems. Simple algebraic manipulations related to the LMIs and basic results form linear algebra are enough to arrive at the lower bound. B. Optimal H1 Model Reduction via LMI Optimization
In the preceding subsection, we have shown that kG(s) 0G (s)k +1 holds for all G (s) 2 RH1 of McMillan degree less than or equal to r . The goal of this subsection is to show that, in the case where r
r
r
we reduce the system order by the multiplicity of the smallest Hankel singular value, i.e., if r = n 0 km , this lower bound is indeed the infimum and the optimal reduced-order model that attains this infimum can be obtained via LMI optimization. To this end, let us again focus on the Lyapunov equalities in (2). Then, it is a direct consequence that the 2 )6; 6 ) satisfies the following equalities corresponding to pair ((1=m (7a) and (7b) with = m , respectively.
A 12 6 + 12 6A + 12 BB = 0 6A + A 6 + C C = 0: T
T
m
m
(11a)
m
T
T
(11b)
Moreover, in relation to (7c), it is important to note that the pair 2 )6; 6 ) satisfies ((1=m
01
6 0 P12 P2201 P12
1
m2 6 =
T
(12)
with
P12 =
I0 n
0k
P22 = diag
From Lemma 2 and Corollary 1, we can conclude that m is the infimum of kG(s) 0 Gn0k (s)k1 . The proof of the above lemma heavily relies on the equalities (11) and (12) (see the Appendix section). These equalities are obtained particularly for r = n 0 km , and unfortunately, similar equalities are not easily available in other cases. Due to this fact, our discussion here is rather restrictive, and we cannot say anything on the strictness of the lower bounds given in Corollary 1 when r < n 0 km . The results in Lemma 2 coincide with those obtained in the optimal Hankel norm approximation method (see, e.g., [16]). In that method, a way to construct the optimal reduced-order model Gn0k (s) that achieves the infimal approximation error has been given by means of the all-pass embedding procedure. In the rest of section, we show that the optimal reduced-order models can be constructed also via LMI optimization. One important implication of the proof of Lemma 2 is that, in the case where r = n0km , we can fix the matrix variable P12 in (7) to be constant as in (13) without introducing any conservatism. If P12 is fixed, however, the matrix inequalities in (7) turn out to LMIs. Once the matrix variables (P11 , P12 , P22 ) that satisfy (7) can be found, the optimal reduced-order models can be reconstructed by solving (5) for (Ar , Br , Cr , and Dr ). To summarize, the H1 optimal reduced-order models can be obtained by solving LMI optimization/feasibility problems. Theorem 2: The reduced-order model Gn0k (s) of McMillan degree at most n 0 km that minimizes kG(s) 0 Gn0k (s)k1 can be obtained by the following two-step procedure. 1) Minimize 2 subject to the LMIs
P11 P12 Q22 Q22 P12 Q22 > 0 He P11 0P12 Q22 P12 A P11 0P12 Q22 P12 B < 0 0 2 I B P11 0P12 Q22 P12 P11 A + A P11 + C C < 0 (14) are matrix variables where P11 2 S and Q22 2 S 0 . whereas P12 is a constant matrix given by P12 = 0 P P 11 12 For the subsequent step, define P~ = 01 and denote P12 Q22 the optimal value of by opt . 2) Obtain (A , B , and C , D ) by solving the LMI (5), where P is fixed to P~ and to opt . The LMI (14) in the first step follows from (7) by defining Q22 := 01 T
T
T
T
T
T
T
n
k
0k
;n
2 01
1 0 1
m
m
k
m
m
r
> 0:
k
(13)
The equalities in (11) and (12) imply that, in the case where r = n 0 km , the conditions in (7) will be satisfied for = m with X11 = (1=m2 )6 , P11 = 6 and P12 and P22 given in (13), provided that we replace the inequalities in (7) to equalities. Although these arguments are not enough to conclude that m is the infimum of kG(s) 0 Gn0k (s)k1 , the above discussions can be made more rigorous and we are led to the following results. Lemma 2: Let us consider a system G(s) 2 RH1 of McMillan degree n with the Hankel singular values given in (3). Then, for arbitrary > m , there exists a Gn0k (s) 2 RH1 of McMillan degree at most n 0 km that satisfies kG(s) 0 Gn0k (s)k1 < . Q :E :D : Proof: See the Appendix section for the proof.
r
r
I
r
P22 . Analytic formulas in [9], [15] are also useful for the reconstruction of Gr (s) in the second step. It should be noted that the results in Theorem 2 are valid only in the case where (A, B , C ) is balanced, since the choice of P12 depends on the state space realizations. Thus, in other cases, the specific choice of P12 given in Theorem 2 could be a source of conservatism and the optimal reduced-order models might not be obtained. In closing this section, we show that it is possible also to obtain the optimal reduced-order model Gn0k (s) via a one-step LMI optimiza01 , tion procedure. By the similarity transformation Ar := P22 Ar P22 Br := P22 Br and Cr := Cr P2201 , we see that there exist (Ar , Br , Cr , and Dr ) that satisfy (5) for some P > 0 if and only if (15), as shown at the bottom of the page, holds. By the congruence transfor-
P11 A + A P11 A P12 + P12 P2201A P22 P11 B + P12 P2201B C 3 A P22 + P22 A P12 B + B 0P22 C 3 3 0 2 I D 0D 3 3 3 0I T
k
T
I ;...;
2 01 01 0 01 I
n
T
r
r
T r
T
r
T
r
T
T r T r
< 0:
(15)
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01 , we have (16), as mation with diag(I; Q22 ; I; I) where Q22 := P22 shown at the bottom of the page. If the matrix variable P12 is fixed to be constant, the aforementioned inequality is an LMI with respect to ~r := Q22 B r , Cr , the matrix variables P11 , Q22 and A~r := Q22 Ar , B Dr . Once these variables have been found, the optimal reduced-order models can be reconstructed by Gr (s) =
01 A~r Q01B~r Q22 22 : Dr Cr
where
A B1 B2 C1 D11 D12 C2 D21 D22 G
(17)
The matrix inequality (16) as well as (7) clearly indicate that the nonconvexity of the problem stems from the bilinear terms with respect to the matrix variable P12 . Hence, if we can fix P12 without introducing any conservatism as in Theorem 2, we are able to obtain globally optimal solutions via LMI optimization.
HefP Ag B1T P
P > 0;
C1
[13], we arrived at two matrix inequalities with nonconvex equality constraints that commonly occur in the general reduced-order H1 controller synthesis. With these inequalities and the particular results from the balanced realization, it turns out that the lower bounds are given in terms of the Hankel singular values. Moreover, in the case where we reduce the system order by the multiplicity of the smallest Hankel singular value, we prove that the problem is essentially convex and the H1 optimal reduced-order models can be obtained by solving LMI optimization problems. These results are not completely new and coincide with those obtained in the optimal Hankel norm approximation method [5]. Our novel contribution is showing alternative proofs for those results via recently developed LMI-related techniques. Recall that the H1 model reduction problem is a special case of the reduced-order H1 controller synthesis problems. It should be noted that those results on the lower bounds of the H1 cost and the optimal solutions for a specific order case have not been gained in the general reduced-order H1 controller synthesis setting. It is not yet clear to us whether the LMI-based techniques explored in this note can be extended to handle the general reduced-order H1 controller synthesis. This topic is currently under investigation.
P B1 0 2 I
D11
P B2 0p;r+q
+He
In this note, we applied the well-established LMI techniques to the
0 B 0 0 0 D Ir 0 0 Ip Br : Dr
0 Ir 0 0 0
0 0 0I q 0 0 (19)
Then, the matrix inequalities in (5) come to
IV. CONCLUSION
H1 model reduction problems so that we can obtain lower bounds of the H1 cost incurred in the approximation. Following the standard procedure for the LMI-based H1 controller synthesis [3], [9], [12],
A 0 = C 0 0 Ar = Cr
D12
C1 D11 0I T
T
G [C2 D21
0r+p;q ]
< 0: (20)
The conditions in (7) are now derived from (20) by eliminating the variable G . Indeed, we see from the Parrott’s Lemma [2], [14] that (20) holds if and only if there exists P 2 Sn+r such that
?
P B2 0p;r+q
L(P ) D12 C2 ? D21 L(P ) T
T
0q;r+p
P B2 0p;r+q
?
D12 C2 ? D21 T
T
0: T
T
(24b) (24c)
">
Then, to prove Lemma 2, it is enough to show that for any 0, there exists 11 2 Sn satisfying (24) with 12 and 22 given in (13). To 0 of the following Riccati this end, let us first consider a solution 0 is small enough: equation, which does exist if
P
P 5>
Q>
P
5A + AT 5 + 21m 5BBT 5 + Q = 0:
(25)
P := 6 + "5 satisfies (24b), since we have from
Then, we see that 11 (11b) and (25) that
(6 + "5 )A + AT (6 + "5 ) + CC T = 0" 21 5BB T 5 + Q < 0: m
(26)
Condition (24c) is also satisfied since (12) indicates that
6 + "5 0 P12 P2201 P12T = "5 + m2 6 01 > 0:
(27)
On the other hand, the left-hand side of (24a) comes to be
He 6 + "5 0 P12 P2201 P12T A + ( 1+ ")2 m T 0 1 T 2 6 + "5 0 P12 P22 P12 BB 2 6 + "5 0 P12 P2201P12T (28a) " 2 0 1 T T 2 0 1 = ( + ")2 m 6 BB 5 + 5BB m 6 m 2 2 + ( "+ ")2 5BB T 5 0 22 (m " ++"")2 m m m 0 1 2 0 1 T 2 2 m 6 BB m 6 + "(5A + AT 5 ) (28b) 2 2 2 m" + " T 2 0 1 m = 0 2 ( + ")2 2 + " 5 0 m 6 BB m
m
m
2 2 2mm+ " 5 0 m2 6 01 2 0 2m(2"m + ") 5BBT 5 0 "Q < 0
(28c)
where in deriving (28b) from (28a) we use (27) and the following equality condition that results from (11a):
m2 6 01 A + AT m2 6 01 + m2 6 01 BBT 6 01 = 0:
(29)
Furthermore, (28c) is readily derived from (28b) by using (25) and completing the square. Thus, by observing that 11 = + 0 satisfies (24) with 12 and 22 given in (13), the proof is completed. QED
P
P
P
6 "5 > : : :
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REFERENCES [1] S. P. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1998. [2] C. Davis, W. M. Kahan, and H. F. Weinberger, “Norm-preserving dilations and their applications to optimal error bounds,” SIAM J. Numer. Anal., vol. 19, no. 3, pp. 445–469, 1982. [3] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to control,” Int. J. Robust Nonlinear Control, vol. 4, no. 4, pp. 421–448, 1994. [4] L. E. Ghaoui, F. Oustry, and M. AitRami, “A cone complementary linearization algorithm for static output-feedback and related problems,” IEEE Trans. Automat. Contr., vol. 42, pp. 1171–1176, Aug. 1997. [5] M. Green and D. Limebeer, Linear Robust Control. Upper Saddle River, NJ: Prentice-Hall, 1995. model reduction via linear matrix [6] K. M. Grigoriadis, “Optimal inequalities: continuous- and discrete-time cases,” Syst. Control Lett., vol. 26, no. 5, pp. 321–333, 1995. [7] K. M. Grigoriadis and R. E. Skelton, “Low-order control design for LMI problems using alternating projection methods,” Automatica, vol. 32, no. 8, pp. 1117–1125, 1996. [8] A. Helmersson, “Model reduction using LMI’s,” in Proc. IEEE Conf. Decision Control, Lake Buene Vista, FL, 1994, pp. 3217–3222. con[9] T. Iwasaki and R. E. Skelton, “All controllers for the general trol problem: LMI existence conditions and state space formulas,” Automatica, vol. 30, no. 8, pp. 1307–1317, 1994. [10] T. Iwasaki, “The dual iteration for fixed-order control,” IEEE Trans. Automat. Contr., vol. 44, pp. 783–788, Apr. 1999. [11] S. Kodama and N. Suda, Matrix Theory for System Control. Tokyo, Japan: Corona, 1978. [12] A. Packard, K. Zhou, P. Pandey, and G. Becker, “A collection of robust control problems leading to LMI’s,” in Proc. IEEE Conf. Decision Control, Brighton, U.K., 1991, pp. 1245–1250. [13] A. Packard, K. Zhou, P. Pandey, J. Leonhardson, and G. Balas, “Optimal, constant I/O similarity scaling for full-information and state-feedback control problems,” Syst. Control Lett., vol. 19, no. 4, pp. 271–280, 1992. [14] S. Parrott, “On a quotient norm and the Sz.-Nagy-Foias lifting theorem,” J. Functional Anal., vol. 30, pp. 311–328, 1978. [15] R. E. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design. New York: Taylor Francis, 1997. [16] K. Zhou, K. Glover, and J. C. Doyle, Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [17] K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice-Hall, 1998.
