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From transfer matrices to realizations: Convergence properties and parametrization of robustness analysis conditions
Systems & Control Letters 62 (2013) 632–642
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From transfer matrices to realizations: Convergence properties and parametrization of robustness analysis conditions Carsten W. Scherer a , İ. Emre Köse b,∗ a
Department of Mathematics, University of Stuttgart, Germany
b
Department of Mechanical Eng., Boğazici University, Istanbul, Turkey
article
abstract
info
Article history: Received 3 February 2011 Received in revised form 28 December 2012 Accepted 9 April 2013 Available online 1 June 2013
In various branches of systems and control theory, one is confronted with the need for approximating transfer functions by a sequence of FIR expansions in the H∞ -norm. The approximating sequence grows in its McMillan degree, while the limiting transfer matrix has a finite number of poles. Considering the corresponding state-space realizations of the approximating sequence and its limit, it is of interest to understand the limiting behavior of the realization matrices. This paper investigates this behavior and, thus, provides an answer for the continuous-time counterpart of FIR expansions and exponential convergence. In a similar vein, it is well-understood how to translate frequency-domain inequalities for transfer matrices into LMIs for realizations by using the Kalman–Yakubovich–Popov (KYP) Lemma. However, it is often less clear how obviously valid manipulations of the frequency-domain inequalities lift into operations on the solutions of the corresponding LMIs. The paper provides some novel insights on such questions with applications to robustness analysis. © 2013 Elsevier B.V. All rights reserved.
Keywords: Convergence of realizations Linear matrix inequalities Robustness analysis
0. Notation We denote the extended imaginary axis For a transfer matrix G with realizationG(s) A + D we use G∗ (s) = G(−s)T , G = C
A C
B D
iR ∪ {∞} by C0 . = C (sI − A)−1 B B and Gss := D
. If A has no eigenvalues in C0 and if P = P T , the
frequency-domain inequality (FDI) G∗ PG ≺ 0 on C0 is, by the Kalman–Yakubovich–Popov (KYP) lemma [1], equivalent to the existence of X = X T such that L (X , P , Gss ) :=
XA + (XA)T (XB)T
XB 0
+
CT DT
T
X 0 0
0 0 P
I A C
=
I A C
0 B D
0 X 0
P C 0 B D
D
≺ 0.
(1)
Aψν Cψν
Bψν Dψν
→
Aψ Cψ
Bψ Dψ
for the
• Even if the McMillan degrees of Aψν and Aψ coincide, convergence might be destroyed by ν -dependent coordinate changes for the realization of ψν . This motivates to choose both realizations with particular extra properties.
exp
is said to converge exponentially to ψ , expressed as ψν −→ ψ , if
0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.04.001
Suppose that ψν is a sequence in RH∞ with ψν → ψ ∈ RH∞ in the H∞ -norm. If we choose minimal realizations (Aψν , Bψν , Cψν , Dψν ) of ψν and (Aψ , Bψ , Cψ , Dψ ) of ψ , it is clearly realization matrices. There are various stumbling blocks:
Corresponding author. Tel.: +90 535 747 2623. E-mail address: [email protected] (İ.E. Köse).
1. Introduction
generally impossible to conclude
We say that (1) certifies G∗ PG ≺ 0 on C0 or that X is a KYPcertificate for this FDI. For reasons of space, we drop ‘‘on C0 ’’ for FDIs and use the placeholder ⋆ for matrices that can be inferred by symmetry or are of no relevance. A sequence ψν in a normed space
∗
there exist K ≥ 0 and η ∈ (0, 1) such that ∥ψν − ψ∥ ≤ K ην for all ν . We use this notion for proper and stable transfer matrices and the H∞ -norm ∥ · ∥∞ , as well as for real matrices and the spectral norm σmax (·) = ∥ · ∥. We denote the set of proper and stable k×m rational matrices of dimension k × m by RH∞ .
• Typically, Aψν grows unbounded in size which implies that
we cannot apply simple compactness arguments in order to enforce convergence of Aψν , Bψν , Cψν . • The eigenvalues of Aψν and its Jordan structure are not (necessarily) related to those of A.
In view of these troubles it is a natural idea to choose balanced realizations and to consider those realization matrices that emerge through model reduction of ψν by truncation in order to enforce equality of the McMillan degrees. In many concrete problems,
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
though, the approximating sequence is chosen with a fixed polepair (Aψν , Bψν ) and with coefficient matrices Cψν , Dψν that are obtained by optimization. This would e.g. be a typical situation in multi-objective H2 /H∞ -control problems by applying the Youla parametrization (as discussed for example in [2]). Therefore, it is desirable to investigate the question of convergence for realizations that are input-balanced only. The purpose of this paper is twofold. First, we extract the asymptotic properties of the realization matrices (Aψν , Bψν , Cψν , Dψν ) for input-balanced realizations in case that the convergence of ψν to ψ is exponential in the H∞ -norm. In particular, we show that there exists a state-transformation Tψν such that a compatibly-dimensioned portion of the realization (Tψ−ν1 Aψν Tψν , Tψ−ν1 Bψν , Cψν Tψν , Dψν ) converges exponentially to (Aψ , Bψ , Cψ , Dψ ), while the remaining modes are canceled asymptotically. This result establishes the state-space counterpart of the properties of convergent sequences of transfer matrices in the H∞ -norm. As an application, if ψν → ψ and P2 ≺ P1 , we construct a KYP-certificate for the FDI ψν∗ P2 ψν − ψ ∗ P1 ψ ≺ 0 which is valid for all large ν . This is essential for the remaining results in the paper. The second purpose of the paper is related to robustness analysis of uncertain systems. It is well-known [3] that the verification of robust stability for an uncertain dynamical system in the standard LFT form involves a search over RH∞ for frequencydependent (i.e., ‘‘dynamic’’) matrices D(jω) = D(jω)∗ ≻ 0 such that a certain FDI is satisfied. In the LMI approach to the problem, the search over D is replaced by the search over ψ with D = ψ ∗ ψ in a finite dimensional subspace of RH∞ . Choosing a sequence Sν of such subspaces that is dense in RH∞ , one can find a sequence of transfer functions ψν ∈ Sν which approaches ψ for ν → ∞ in the H∞ -norm. If ψ satisfies the robustness FDI, it is quite obvious that the FDI will persist to hold for ψν and sufficiently large ν . In this sense, the transition from ψ to ψν can be carried out in the frequency domain without any difficulty. However, the relation of the corresponding KYP-certificates with realizations of ψ and ψν is quite intricate and plays a fundamental role in a recently proposed gain-scheduling synthesis algorithm [4]. Thus, the second of our main results provides essential insights into relations between such KYP certificates. Several technical tools are developed in obtaining these fundamental results. They are related to manipulations on frequency-domain inequalities and the resulting operations on the corresponding KYP certificates. For instance, if G and H1 = H1∗ , H2 = H2∗ are transfer matrices of compatible dimensions, the FDIs H2 ≼ H1 and G∗ H1 G ≺ 0 clearly imply G∗ H2 G ≺ 0. It is then natural to ask how the KYP certificates of the first two are related to that of the third. We provide simple, yet clarifying, answers to this and similar questions, which we believe are of independent interest. The paper is organized as follows. Section 2 contains the main result of the paper, Theorem 1, on the asymptotic properties of the realization matrices of exponentially converging transfer matrices. In Section 3 we discuss how to translate two FDI manipulations, namely ordered substitution and taking the Schur-complement, into operations on the corresponding KYP certificates. Section 4 illustrates the application of our technical insights to robust analysis by dynamic multipliers. Simple numerical examples illustrate our results. Finally, we collect auxiliary facts and proofs in the Appendices A and B respectively. 2. Convergence of realizations 2.1. Preliminaries—approximation in RH∞ [5,2] that the real span of For p > 0 it is well-known s−p 2 s−p 3 s−p 1, s+p , ( s+p ) , ( s+p ) , . . . is dense in RH∞ . Hence, with bν (s) :=
633
ν T
s−p s−p and for ψ ∈ RH∞ , there exists a ··· s+p s+p sequence λν ∈ Rν+1 such that λTν bν converges to ψ in the H∞ 1
exp
norm; one can even assure exponential convergence, i.e., λTν bν −→ ψ . Consequently, if d ∈ N and ⊗ denotes the Kronecker product, d×d any ψ ∈ RH∞ can be approximated by ψν := Λν (Id ⊗ bν ) exp
with Λν ∈ Rd×d(ν+1) such that ψν −→ ψ . Let us introduce the realization
−p .. . 0 := 0 0 . . √.
··· .. . ··· ··· ···
bν =
Abν Cbν
Bbν Dbν
..
−2p .. . −p √0
2p
1 1
2p
1
.. √.
.
···
2p
√ − 2p .. √. − 2p .. .
(2)
which is stable, minimal and input-balanced, i.e., Abν + ATbν +
Bbν BTbν = 0. Then the realization
ψν =
Id ⊗ Abν Λν (Id ⊗ Cbν )
Id ⊗ Bbν Λν (Id ⊗ Dbν )
(3)
of Λν (Id ⊗ bν ) is as well stable and input-balanced. It has been argued in [2] how to extract information about the McMillan degree of ψ , denoted as k, from the sequence ψν : in the ordered list of Hankel-singular values of ψν , k is equal to the index of the first Hankel-singular value that converges to zero. However, it has not been clarified in the literature how one can extract more detailed information about the realization matrices (Aψ , Bψ , Cψ , Dψ ) of ψ from those of ψν systematically. Since (3) is input-balanced, the squared Hankel singular values of ψν are equal to the eigenvalues of the corresponding observability Gramian. Let us perform an orthogonal statecoordinate change on (3) such that the observability Gramian is given by
Qψν =
Qψ1ν 0
0 Qψ2ν
with Qψ1ν ∈ Rk×k
and Qψ1ν ≽ Qψ2ν ≽ 0.
(4)
The resulting realization of ψν remains input-balanced and is denoted and partitioned as
Aψν Cψν
Bψν Dψν
A11 ψν
A12 ψν
B1ψν
= A21 ψν
A22 ψν Cψ2 ν
B2ψν Dψν
1
Cψν
k×k with A11 . (5) ψν ∈ R
Let us stress that the dimension and, by Lemma 8, the norm of Aψν both grow at most linearly in ν . The same hold for Bψν since the realization is input-balanced. Note that this realization can be determined with the mere knowledge of k and does not rely on any further properties of the limiting matrix ψ . We now prove that it rather explicitly displays information about the realization 1 1 of ψ , since a state-coordinate change of (A11 ψν , Bψν , Cψν , Dψν ) guarantees convergence to (Aψ , Bψ , Cψ , Dψ ), while the influence 12 of the eigenvalues of A22 ψν gets asymptotically canceled since Aψν and Cψ2 ν converge to zero. This is a consequence of Theorem 1. 2.2. Main result Theorem 1. Suppose that ψ
Aψν Cψν
Bψν Dψν
=
Aψ Cψ
Bψ Dψ
and ψν
=
are stable, input-balanced realizations. Let the
634
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
former be minimal and of degree k, while the latter is partitioned as in (5), has the observability Gramian (4), and the dimension of Aψν as well as ∥Aψν ∥ grow at most polynomially in ν . Finally, let Qψ and Sψν be the unique solutions of ATψ Qψ + Qψ Aψ + CψT Cψ = 0 and
exp
ATψν Sψν + Sψν Aψ + CψT ν Cψ = 0,
(6)
and define Tψν as the upper k × k block of
Sψν Qψ−1 .
exp
If ψν −→ ψ
then, for all sufficiently large ν, Tψν is non-singular, both Tψν and Tψ−ν1
are bounded, and
Tψ−ν1 A11 ψν Tψν Cψ1 ν Tψν
Tψ−ν1 B1ψν Dψν
exp
−→
The left-upper block of (10) is QΨν − Hν with Hν := Sψν Rψν + (Sψν Rψν )T − RTψν Qψ Rψν . If exploiting Sψν Qψ−1 SψT ν ≼ Qψν we infer ∥ Qψ−1 SψT ν − Qψ Rψν ∥2 ≤ λmax (Sψν Qψ−1 SψT ν − Hν )
Aψ Cψ
Bψ Dψ
≤ λmax (Qψν − Hν ) ≤ λν −→ 0.
(12)
Therefore exp
T Sψ − Qψ Rψν −→ 0 ν
(13) exp
exp
T T and thus Sψ RT − Qψ Rψν RTψν −→ 0 which implies Sψ RT −→ Qψ ν ψν ν ψν
T in view of (11). Since (Sψ − Qψ Rψν )(SψT ν − Qψ Rψν )T = SψT ν Sψν − ν
as well as
exp
T T Sψ RT Q − Qψ Rψν Sψ + Qψ Rψν RTψν Qψ −→ 0 we hence get ν ψν ψ ν
A12 exp ψν 2 −→ 0. Cψ
exp
(7)
ν
T Sψ S −→ (Qψ )2 ≻ 0. ν ψν
(14)
T From (12) we can as well extract ∥Qψν − Sψν Qψ−1 Sψ ∥ ≤ ∥Qψν − ν exp
It remains a conjecture that the results are valid even if the convergence ψν → ψ is not exponential. Proof. Step 1: Preparation. Define Gν = ψν − Dψν and G = ψ − Dψ .
T Hν ∥ + ∥Hν − Sψν Qψ−1 Sψ ∥ −→ 0, i.e., ν T Qψν − Sψν Qψ−1 Sψ = ν
Qψ1ν 0
0 Qψ2ν
exp
Since ∥ψν (∞) − ψ(∞)∥ = ∥Dψν − Dψ ∥ −→ 0 we infer from ∥Gν −G∥∞ = ∥ψν −ψ −(Dψν −Dψ )∥∞ ≤ ∥ψν −ψ∥∞ +∥Dψν −Dψ ∥
−
exp
that ∥Gν − G∥∞ −→ 0. In the sequel the ordered Hankel-singular values of a stable transfer matrix H are denoted in the usual fashion by σ1 (H ) ≥ σ2 (H ) ≥ σ3 (H ) ≥ · · · . Consider the errortransfer matrix G − Gν = Aψν 0 Bψν 0 Aψ Bψ . Since Aψν + ATψν + Bψν BTψν = 0 and Cψν −Cψ 0 Aψ + ATψ + Bψ BTψ = 0, the positive semi-definite controllability Gramian of G − Gν has the particular structure as in
Aψν 0
+
0 Aψ Bψν Bψ
RTψν I
I
Rψν
Bψν Bψ
T
+
RTψν I
I
Rψν
Aψν 0
0 Aψ
Aψν 0
+
0 Aψ
T
I 0
RTψ ν R˜ ψν
(8)
−Sψν
λν := λmax
Aψν 0
Qψ
I 0
CT 0 + ψνT Aψ −C ψ
RTψν
R˜ ψν
Cψν −Cψ = 0. (9)
RTψ ν I
I Rψν
=
I
Rψν
0 R˜ ψ
ν
I − Rψν RTψν . Hence,
Qψν −SψT ν
−Sψν
exp
−→ 0.
(15)
Let us further observe
0 =
T
I T Qψ−1 Sψ ν
(9)
I T Qψ−1 Sψ ν
= ATψν (Qψν − Sψν Qψ−1 SψT ν ) + (Qψν − Sψν Qψ−1 SψT ν )Aψν + (Cψν − Cψ Qψ−1 SψT ν )T (Cψν − Cψ Qψ−1 SψT ν ). Due to (15) and the polynomial growth of Aψν , we infer Cψν − exp
Cψν Sψν −→ Cψ Qψ .
(16) T
T
T
The equation for Sψν in (6) implies Aψ Sψν Sψν + Sψν Aψν Sψν + exp
T CψT Cψν Sψν = 0. Note that ATψ Sψ S + CψT Cψν Sψν −→ ATψ Qψ2 + ν ψν T T T Cψ Cψ Qψ = (Aψ Qψ + Cψ Cψ )Qψ = −Qψ Aψ Qψ . Therefore
(17)
From (8) we similarly infer Aψ (I −Rψν RTψν )+(I −Rψν RTψν )ATψ +(Bψ − exp
Rψν Bψν )(BTψ − BTψν RTψν ) = 0. Hence, due to (11), Rψν Bψν −→ Bψ . By (13) (and since Bψν grows at most polynomially, because (Aψν , Bψν ) exp
T B − Qψ Rψν Bψν −→ 0 and thus is input-balanced), Sψ ν ψν exp
T Sψ B −→ Qψ Bψ . ν ψν
Qψ exp
I
Rψν
0 R˜ ψν
(18)
2 2 Step 2: Proof of convergence of A12 ψν , Cψν and Qψν . Since G has McMillan degree k, we recall that
(10)
Considering the right-lower block, since Qψ ≻ 0 we infer from exp λν ≥ λmax (R˜ ψν Qψ R˜ ψν ) = ∥ Qψ R˜ ψν ∥2 ≥ 0 that Qψ R˜ ψν −→ 0 exp and thus R˜ ψν −→ 0 and thus Rψν RTψν −→ I .
T
exp
= σ1 (Gν − G)2 ≼ ∥Gν − G∥2∞ −→ 0.
exp
1 Sψ ν 2 Sψ ν
T Sψ A S −→ Qψ Aψ Qψ . ν ψν ψν
Qψ
for R˜ ψν :=
exp
= 0.
Due to I − Rψν RTψν ≽ 0 we have
Qψ−1
exp
Qψν −SψT ν
−Sψν
Qψν −SψT ν
T −→ 0 and thus Cψν Sψν − Cψ Qψ−1 SψT ν Sψν −→ 0 and Cψ Qψ−1 Sψ ν thus, by (14),
T
With Qψ and Sψν satisfying (6), its positive semi-definite observability Gramian is given as in
1 Sψ ν 2 Sψ ν
(11)
σk+l (Gν ) ≤ σk+l (Gν − G) ≤ ∥Gν − G∥∞ for l = 1, 2, . . . . (19) 11 Aψν B1ψν Let us define G1ν := which results from Gν by 1
Cψν Dψν truncation. One can check that the standard proof of the H∞ norm error bound [3] carries over to the current situation, which
dim(Aψ
)−k
implies ∥Gν − G1ν ∥∞ ≤ 2 l=1 ν σk+l (Gν ). Due to (19), the exp right-hand side is bounded by dim(Aψν )∥Gν − G∥∞ −→ 0, which
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
635
exp
allows to conclude ∥Gν − G1ν ∥∞ −→ 0. Therefore ∥G − G1ν ∥∞ ≤ exp
∥G − Gν ∥∞ + ∥Gν − G1ν ∥∞ −→ 0. For l = 1, . . . , k this implies exp |σl (G) − σl (G1ν )| ≤ ∥G − G1ν ∥∞ −→ 0. Therefore, σl (G1ν ) =
λl (Qψ1ν ) → σl (G) > 0 and we can infer the existence of 0 <
γ ≤ Γ such that γ I ≼ Qψ1ν ≼ Γ I for all large ν . Moreover, since Qψ1ν has dimension k, we have σk+1 (Gν ) = λmax (Qψ2ν ). By (19) exp we infer λmax (Qψ2ν ) ≤ ∥G − Gν ∥∞ and thus Qψ2ν −→ 0. From 2 22 2 T 2 T 2 (A22 ψν ) Qψν + Qψν Aψν + (Cψν ) Cψν = 0 and since ∥Aψν ∥ grows at exp
T 1 most polynomially, we conclude Cψ2 ν −→ 0. From (A11 ψν ) Qψν +
1 T 1 1 2 T 1 Qψ1ν A11 ≤ ∥(A11 ψν + (Cψν ) Cψν = 0, we infer ∥Cψν ∥ ψν ) Qψν + 1 11 1 11 1 Qψν Aψν ∥ ≤ 2∥Qψν ∥ ∥Aψν ∥ ≤ 2Γ ∥Aψν ∥ such that ∥Cψν ∥ grows at
most polynomially. Hence ( implies
)
T 2 A21 ψν Qψν
+
1
Qψν A12 ψν
+ (Cψν ) Cψν = 0 1
T
2
A21 ψν 12 exp
∥ also grows at most polynomially this finally implies
that Aψν −→ 0. We have proven the second statement in (7). Step 3: Completion of proof. Let us now exploit (15). Since exp
exp
2 Qψ2ν −→ 0 and since Qψ ≻ 0, we infer that Sψ −→ 0. Due to (14) ν exp
T 1 T 1 and Sψ S = (Sψ ) Sψν + (Sψ2 ν )T Sψ2 ν we conclude (Sψ1 ν )T Sψ1 ν −→ ν ψν ν (Qψ )2 . We infer that Sψ1 ν is bounded. Since Sψ1 ν is square, we can also
1 −1 extract that (Sψ ) exists for all large ν and is bounded as well. Let ν us finally recall from (16)–(18) that T Sψ A S ν ψν ψν
1 1 T 12 2 = (Sψ1 ν )T A11 ψν (Sψν ) + (Sψν ) Aψν (Sψν ) 2 T 21 1 2 T 22 + (Sψν ) Aψν (Sψν ) + (Sψν ) Aψν (Sψ2 ν )
exp
−→ Qψ Aψ Qψ , exp
and
exp
1 Cψν Sψν = Cψ1 ν (Sψ ) + Cψ2 ν (Sψ2 ν ) −→ Cψ Qψ . ν
exp
exp
−→ Qψ Aψ Qψ , (Sψ1 ν )T B1ψν
exp
−→
1 Qψ Bψ , Cψ1 ν (Sψ ) −→ Cψ Qψ and thus ν exp
−1 1 [Qψ−1 (Sψ1 ν )T ]A11 ψν [Sψν Qψ ] −→ Aψ , exp
exp
[Qψ−1 (Sψ1 ν )T ]B1ψν −→ Bψ and Cψ1 ν [Sψ1 ν Qψ−1 ] −→ Cψ . 1 Now note that Tψν = Sψ Q −1 . Hence the proof is complete if ν ψ exp
1 T we can show that Tψ−ν1 − Qψ−1 (Sψ ) −→ 0. Indeed, we have ν
already established that Tψν and Tψ−ν1 are bounded for all large ν ; 1 T thus it suffices to observe that Qψ−1 (Sψ ) Tψν = Qψ−1 (Sψ1 ν )T Sψ1 ν Qψ−1 ν exp
−→ I.
Aψ
X +X
Abν
0
0
Aψ
BT BTψ X bν T λν Cbν − Cψ
X
Bbν
Bψ
−γ I λTν Dbν − Dψ
CbT λν ν −CψT
≺ 0. DTb λν − DTψ ν −γ I
(21)
We can hence compute γν := infλν ∈Rν+1 ∥λTν bν − ψ∥∞ by minimizing γ over all X = X T ∈ R(ν+2)×(ν+2) , λν ∈ Rν+1 and γ ∈ R exp
that satisfy (21). Due to optimality of γν we can conclude γν −→ 0. Semi-logarithmic plots of the approximation error γν and the two errors related to (7) over ν are given in Fig. 1; the curves do indeed reveal exponential convergence as claimed. E.g., for ν = 6 we obtain λTν = (0 −0.01 0.07 −0.06 2.43 2.87 1) from which no information can be extracted. However, the realizations of λTν bν after orthogonally diagonalizing the observability Gramian and that after having performed the coordinate changes as proposed in Theorem 1 are given by 2.66 0 −0.05 0 0.39 0.3 −0.1
−6 −3 −1.82 0 13.36 10.3 3.76 −1.09 0.51 0.78 0 −5.7 −4.39 −1.42
−5.14 −4.91 1.65 0
−12.09 −9.32 −3.48
0 0 −0.14 1.04 1.01 1.57 0 0 0 −0.14 1.04 1.01 1.57 0
0 0 −1.05 0 −4.48 0 0 0 0 −1.05 0 −4.48 0 0
0 0 1.02 4.48 −7.44 −11.5 0 0 0 1.02 4.48 −7.44 −11.5 0
0 0 0 0 0.03 −4.42 0 0 0 0 0 0.03 −4.42 0
−3.46 −0.1 −0.53 0 , 3.86 2.97 1 1.48 3.13 −0.53 0 3.86 2.97 1
.
Both realizations do indeed admit the claimed structure (after rounding), while the second one allows to directly read-off the input-balanced realization of ψ in (20).
d×d Let us now suppose that ψ ∈ RH∞ is invertible on C0 . Choose ψν as in Section 2.1 and with the properties as formulated in Theorem 1. If two symmetric matrices satisfy P2 − P1 ≺ 0, we conclude from the trivial inequality ψ ∗ (P2 − P1 )ψ = ψ ∗ P2 ψ − ψ ∗ P1 ψ ≺ 0 that, for all large ν ,
and ψ(s) =
(s − 1)(s − 6) (s + 2)(s + 4)
with the input-balanced realization
ψ=
0
T
2.4. Construction of a certificate for approximation
For a numerical experiment let us consider
0
2.3. Numerical example I
p=3
Abν
T 1 T 1 Sψ B = (Sψ ) Bψν + (Sψ2 ν )T B2ψν −→ −Qψ Bψ , ν ψν ν
1 T 11 This leads to (Sψ ) Aψν (Sψ1 ν ) ν
In view of the bounded real lemma, the inequality ∥λTν bν − ψ∥∞ < γ holds iff there exists X = X T with
1 −1 21 T 2 1 −1 1 T 2 ∥A12 ψν ∥ = ∥−(Qψν ) (Aψν ) Qψν − (Qψν ) (Cψν ) Cψν ∥ 1 21 ∥Aψν ∥ ∥Qψ2ν ∥ + ∥Cψ1 ν ∥ ∥Cψ2 ν ∥ . ≤ γ
Since ∥
Fig. 1. Plots of the approximation error γν (blue) and the errors in (7) (green, red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
−1.09 0.51 −1.42
−5.14 −4.91 −3.48
1.48 3.13 1
.
(20)
ψν ψ
∗
P2 0
0 −P 1
ψν ≺ 0. ψ
(22)
636
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
Let us now construct certificates for this FDI with nice asymptotic properties for ν → ∞. Theorem 2. For any P ≺ 0 there exist K ≻ 0, L = LT and a sequence Kν ≻ 0 with limν→∞ ∥Kν ∥ = 0 and such that ∥Kν−1 ∥ grows at most polynomially in ν with the following properties: if the symmetric P1 , P2 satisfy P2 − P1 ≼ P and if β ∈ (0, 1), then for all sufficiently large α there exists some να such that for all ν ≥ να the following LMI certification of (22) holds:
L
α Tψ−νT KTψ−ν1
0
−α Tψ−νT K
0
Kν
0
−α KTψ−ν1
0
αK + β L
,
A11 ψν
A12 ψν
0
B1ψν
21 Aψν 0 1 Cψν
A22 ψν
0
B2ψν
0
Aψ
Bψ
Cψ2 ν
0
Dψν
0
Cψ
Dψ
0
P2
0
0
−P1
+ ν A21 ψν
B2ψν
A21 ψν
≺ 0.
+ Cψ
Dψ
T
B2ψν
T
+I ≺0
(24)
LBψ 0
(25)
We abbreviate P˜ := diag(P2 , −P1 ) and assume w.l.o.g. that Tψν = I. If we add in (23) the first row/column to the third and left-/right-multiply the second one with Uν := Kν−1 , which is a congruence transformation, we obtain (by exploiting that (1) formally extends to non-symmetric X and non-square A):
αK
L 0 0
A11 ψν −α K 0 0 , P˜ , 1 Cψν βL 0
0
21 Aψν + 0 0
+ CψT P2 Cψ
(
)
T A21 ψν
A12 ψν Uν
A11 ψν
B1ψν
0
Aψ
Cψ2 ν Uν
Cψ1 ν
Bψ Dψν
0
Cψ
0
0
Dψ
T (A21 ψν )
0
≺ 0. 0
0
0
(Bψν ) 2
T
B2ψν
β(ATψ L + LAψ ) β BTψ L CψT P2 Cψ
CψT P2 Dψ
CψT (P2 − P1 )Cψ
CψT (P2 − P1 )Dψ .
Dψ (P2 − P1 )Cψ
(27)
Dψ (P2 − P1 )Dψ
T
T
0 A21 ψν
B2ψν
0
(B2ψν )T
0
≺ 0.
0
0
(28)
0
1 21 Aψν
A21 ψν
B2ψν
ϵ0 + ϵ0 I ≼ Vν + ν A21 ψν
A21 ψν
B2ψν
A21 ψν
A21 ψν
B2ψν
B2ψν
T
T
+ I ≺ 0.
If Hν denotes the first matrix in (26) partitioned into Hνij , we now claim that there exists some να ≥ ν0 such that for all ν ≥ να the inequality Hν − E ≺ 0 is true; then (26) holds for all ν ≥ να in view of (28), which finishes the proof. It remains to prove the claim above. Since Uν grows at most polynomially in ν , we infer from (7) that ∥A12 ψν Uν ∥ → 0 and
∥Cψ2 ν Uν ∥ → 0. This in turn implies ∥Hνij ∥ → 0 for i = 2 or j = 2. ϵ One can hence increase ν0 such that Hν22 ≼ ∥Hν22 ∥I ≼ 20 I for all ν ≥ ϵ0 22 22 ν0 . In view of Hν −ϵ0 I ≼ − 2 I ≺ 0 and thus −(Hν −ϵ0 I )−1 ≼ ϵ20 I, the inequality Hν − E ≺ 0 is implied by (Schur) 11 13 14 Hν
Hν
Hν31
Hν33
Hν41
Hν43
2
ϵ0
Hν12
Hν Hν34 + ϵ0 I Hν44
Hν32 Hν21
Hν23
Hν24
≺ 0.
(29)
42
Hν
The limit of the left-hand side of (29) for ν → ∞ just equals (27) +ϵ0 I and is hence negative definite because of (27) ≼ −2ϵ0 I; therefore, there exists some να ≥ ν0 such that (29) is valid for all ν ≥ να .
A21 ψν
0 β LBψ 0
22 T Indeed, with the abbreviation Vν := A22 ψν Uν + Uν (Aψν ) , by taking the Schur complement and exploiting (24), we infer for all large ν that
+
22 T A22 ψν Uν + Uν (Aψν )
T (A21 ψν ) 22 T A22 ψν Uν + Uν (Aψν ) 21 T (Aψν )
0
A21 ψν 0
Dψ ≺ 0.
CψT P2 Cψ
E+
Note that L = LT exists due to the KYP-lemma since, trivially, ψ ∗ P ψ ≺ 0. (The matrix L can even be taken to be positive definite, but this is not required in the sequel.)
Bψ Bψ Dψ Dψ
0
0 0
=
0
ATψ L + LAψ BTψ L
P Cψ
α(ATψ K + KAψ )
(23)
having the described properties. Moreover, let us fix K ≻ 0 and L = LT to satisfy ATψ K + KAψ ≺ 0,
−α K βL
0
Vν +
Aψ Aψ Cψ Cψ
Since P2 − P1 ≼ P ≺ 0 and β ∈ (0, 1), we can exploit (25) to infer that the right-lower 2 × 2-block of (27) is negative definite. Due to ATψ K + KAψ ≺ 0, there exists some ϵ0 ∈ (0, 1) with (27) ≼ −2ϵ0 I for all large α . Fix any such α . If E := diag(−ϵ0 I , ϵ0 I , −ϵ0 I , −ϵ0 I ) let us establish that there exists ν0 ∈ N0 such that for all ν ≥ ν0 one has
,
T
αK
L
Dψ P2 Cψ
Aψ 0 , P˜ , Cψ 0
T
2 Proof. Since ∥A21 ψν ∥, ∥Bψν ∥ grow at most polynomially in ν , we can apply Lemma 9 and choose Kν with
−1 −1 A22 A22 ψν Kν + Kν ψν
2.5. Numerical example II (26)
After canceling the 2nd block row/column of the l.h.s. in (26), (7) (with Tψν = I) implies that the limit of the resulting matrix for ν → ∞ is
Consider a sequence λν ∈ Rν+1 as constructed in Section 2.3 exp with λTν bν = ψν −→ ψ = 5ss++55 using p = 3. Then the FDI
∗ 1 0 ψν ≺ 0 holds for all ψν∗ ψν − 1.05 ψ ∗ ψ = ψψν 0 −1.05 ψ sufficiently large ν . Let us now determine certificates of the nonstrict versions of these FDIs by computing the stabilizing solutions
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
of the Riccati equations that correspond to the respective LMIs; for the underlying state-space realizations we assume that the coordinate changes of Theorem 1 have been performed, which implies Tψν = I. For ν = 3 and ν = 6, this computation leads to the certificates 93.92 −0.07 −0.14 −94.43 0.02 0 0.08 −0.07 and −0.14 0 0.03 0.15 −94.43 0.08 0.15 95 84.0016 0 −84.0017
0
−84.0017
0 , 84.0018
05×5 0
with structures approximately shared with that of the certificate in (23).
In this section let us consider transfer matrices G, Ψo , Ψn without poles on C0 and real symmetric matrices Po , Pn . Suppose that the FDI G∗ Ψo∗ Po Ψo G ≺ 0 is valid. If the ‘‘old’’ transfer matrix Ψo∗ Po Ψo is related to the ‘‘new’’ one Ψn∗ Pn Ψn as in
Ψn∗ Pn Ψn
Ψo∗ Po Ψo ∗ Ψn Pn ⇐⇒ Ψo 0
≼
0 −P o
Ψn ≼ 0, Ψo
(30)
we can trivially combine these FDIs to infer G∗ Ψn∗ Pn Ψn G ≺ 0. The main purpose of this section is to understand this process in the state-space. More precisely, we reveal how solutions of the LMIs corresponding to G∗ Ψo∗ Po Ψo G ≺ 0 and to (30) can be glued together in order to obtain a solution of the LMI corresponding to G∗ Ψn∗ Pn Ψn G ≺ 0. The result itself and its proof are formulated in state-space only; the link with the motivating considerations for FDIs is established through the correspondences
A C
G=
Ψn =
B D
An 0 Cn
,
Ψo =
Ano Ao Cno
Bn Bo Dn
Ao Co
Bo Do
,
L
X12 X22
, Po ,
Corollary 4. Under the hypotheses of Theorem 3 and if Ano = 0 and Cno = 0, the block X11 +R22 is non-singular and the Schur complement of the certificate in (33) with respect to the (2, 2)-block, namely
L
Ao 0 Co
Bo C A Do C
Bo D B Do D
An 0 , Cn 0
R11
L
R21
R22
0
≺0
T , Pn ,
0
−P o
Ano Ao Cno Co
0
An 0 0 Cn
R12 X11 + R22 X21 Ano Ao 0 Cno
Bn C Bo C A Dn C
Bn D B Dn D
≼ 0.
(34)
If the left-upper block of (32) is negative definite, then (34) is strict. Moreover, if we define T with the perturbed version of R under the hypothesis in (c), then (34) is strict as well. As a non-trivial application of these results, let us consider the obvious FDI equivalence I ∗ 0 (⋆) 0 0
⇐⇒
0 ψ 0 0 I I 0 0
0 I 0 0
0 0 0 I
ψ φ −1
∗
0
−I
0 I
0
φ∗ = 0
ψ ∗ψ I
I
φφ
∗
≻0
I
ψ φ −1
(35)
R11 L R21
Bn Bo ≼ 0. Dn Do
0 X12 X22
Aψ 0 Cψ 0 0 0
I 0 , 0 0
R12 R22
0 I 0 0
0 0 0 I
0 0 , I 0
0 −ATφ 0 −BTφ 0 0
Bψ 0 Dψ 0 I 0
0 CφT 0 DTφ 0 I
≻ 0,
(36)
Γ11 L Γ21
Then the following statements are true: (a) Inequalities (31) and (32) imply
R 11 L R21
Bn C A Dn C
(31)
(32)
An 0 Cn
R12 P , n
−R12 (X11 + R22 )−1 X12 , X11 − X21 (X11 + R22 )−1 X12
satisfies
and
R11 − R12 (X11 + R22 )−1 R21 −X21 (X11 + R22 )−1 R12
for square transfer matrices ψ and φ , the latter being invertible. With natural realizations, the LMIs that certify these two FDIs read, respectively, as
Theorem 3. Let X and R satisfy X11 X21
In case that Ano = 0 and Cno = 0, the realization of Ψn just equals (An , Bn , Cn , Dn ) by unobservability. Lemma 10 in Appendix A then allows to establish a link between LMI (33) and the corresponding version for this smaller-sized realization which leads to the following result.
= −ψ ∗ ψ + (φφ ∗ )−1 ≺ 0
and the resulting natural realizations for Ψo G, col(Ψn , Ψo ) as well as Ψn G.
(b) If the left-upper block of (32) in the natural partition is negative definite, then (33) is strict. (c) Suppose that An has no eigenvalues on C0 and choose Rn = RTn with ATn Rn + Rn An ≺ 0. For all sufficiently small ϵ > 0, the strict version of (33) holds if R is replaced with R + diag(ϵ Rn , 0).
T :=
3. ‘‘Gluing’’ of certificates in state-space
637
Γ12 Γ22
−I , 0
0 I
,
, Pn ,
Bn D Bo D ≼ 0. B Dn D
Aψ 0 Cψ 0
(33)
0 1 Aφ − Bφ D− φ Cφ 0 1 −D− φ Cφ
Bψ 1 Bφ D− φ ≺ 0. Dψ 1 D− φ
(37)
The following theorem clarifies the relation between R and Γ , again directly in the state-space.
638
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
in X = X T by using the KYP lemma. As in Section 2.4 let us now suppose that ψν approximates ψ . The corresponding FDI reads as
Fig. 2. Description of uncertainty.
Theorem 5. Suppose Dφ is square with det(Dφ ) ̸= 0 and Aφ − 1 0 Bφ D− φ Cφ has no eigenvalues in C . (a) If R satisfies (36), then (37) is feasible, R22 is non-singular and
Γ11 Γ21
Γ12 Γ22
:=
1 R12 R− 22 R21 − R11 1 R− 22 R21
1 R12 R− 22 −1 R22
R11 R21
R12 R22
:=
−1 Γ12 Γ22 Γ21 − Γ11 −1 Γ22 Γ21
−1 Γ12 Γ22 −1 Γ22
(39)
(43) exp
In the case of ψν −→ ψ , it is obvious that (40) implies (42). Given an LMI certificate X of (40) as in (41), it is much less clear how one can construct a certificate Xν of (42) as in (43). This is the purpose of the following result which is based on a combination of Theorem 2 and Corollary 4. Theorem 6. Suppose X satisfies (41). Then there exist K ≻ 0, L = LT and a sequence Kν ≻ 0 with limν→∞ ∥Kν ∥ = 0 with the following properties: for any small β ∈ (0, 1) and any sufficiently large α there exists a να such that for all ν ≥ να , (43) is satisfied by
satisfies the non-strict version of (36). 4. Application to robustness analysis with D-scales
α Tψ−νT KTψ−ν1
To recap, we have shown how to construct
• a certificate for ψν∗ P2 ψν − ψ ∗ P1 ψ ≺ 0 for sufficiently large ν , where P2 ≺ P1 and ψν → ψ , and • a certificate for G∗ Ψn∗ Pn Ψn G ≺ 0 using any certificates of G∗ Ψo∗ Po Ψo G ≺ 0 and Ψn∗ Pn Ψn ≺ Ψo∗ Po Ψo .
0
Xν =
0 0 0
We now discuss the application of these insights to robustness analysis, where the roles played by Ψn , Ψo , G will become clear from the context. This application is an essential ingredient of the solution of the gain-scheduled synthesis problem using frequencydependent D-scales in [4].
×
In analyzing uncertain dynamical interconnections, it is customary to decompose the system into two subsystems as in Fig. 2, where M represents the nominal linear time-invariant stable system, while all uncertainties, be they parametric, dynamic, nonlinear, etc., are collected in the operator ∆. When ∆ is stable, LTI and norm-bounded by 1, it is sufficient for ˆ M ψˆ −1 ) < robust stability of the interconnection in Fig. 2 if σmax (ψ 0 ˆ 1 on C , where ψ is a non-singular frequency-dependent matrix without poles on C0 that commutes with ∆ [3,6]. This condition ˆ ∗ ψˆ M ≺ ψˆ ∗ ψˆ . By factorizing can equivalently be written as M ∗ ψ
ψˆ ψˆ •×• = ψ ∗ ψ ≻ 0 for some square ψ ∈ RH∞ , we obtain the FDI ∗ ψM I 0 ψM ψ 0 −I ψ ∗ ∗ M ψ ψ 0 M = ≺ 0. (40) I 0 −ψ ∗ ψ I ∗
This can now be converted, with obvious notations for realizations of ψ, M, into the matrix inequality
L X ,
I 0
0 −I
,
Aψ 0 0 Cψ 0
0 Aψ 0 0 Cψ
Bψ C 0 A Dψ C 0
Bψ D Bψ B Dψ D Dψ
≺ 0
0 0
0 0
0 0
0 0 0
α Tψ−νT KTψ−ν1
0 Kν 0
X33
0 0
−α KTψ−ν1
0 0
0 0 X31
−α KTψ−ν1
×
0 0
X12 α K + β L + X22
X21
0 X32
α K + β L + X11
−1
−α Tψ−νT K
0
0
0
X13
0
0
−α Tψ−νT K
0
X23
.
(44)
1
Proof. For all δ ∈ (0, 1) close to 1, (41) implies L X
ψM ψ ss
δ
I
0
0
−δ I
≺ 0. If we apply Theorem 2 for (P2 , P1 ) = (I , 1δ I )
and (P2 , P1 ) = (−I , −δ I ) (e.g. with P = (δ − 1)I ≺ 0), we can diagonally combine the certificates for ψν∗ ψν − 1δ ψ ∗ ψ and −ψν∗ ψν + δψ ∗ ψ ≺ 0 to obtain the certificate α Tψ−νT KTψ−ν1 0 0 0 −α Tψ−νT K 0 0 Kν 0 0 0 0 − T − 1 − T 0 0 α T KT 0 0 −α T K ψν ψν ψν 0 0 0 Kν 0 0 − α KT −1 0 0 0 αK + β L 0 ψν
0
0
ψν 0 ψ
for
(41)
0 Kν
0
−
4.1. Main result
(42)
which is again equivalent to the existence of Xν = XνT such that Aψν 0 Bψν C Bψν D 0 Aψν 0 Bψν I 0 0 A B L Xν , , 0 ≺ 0. 0 −I C 0 Dψν C Dψν D ψν 0 Cψν 0 Dψν
(38)
satisfies the non-strict version of (37). (b) If Γ satisfies (37), then (36) is feasible, Γ22 is non-singular and
∗ ψν M I 0 ψν M ψν 0 −I ψν ∗ ∗ M ψν ψν 0 M = ≺ 0, I 0 −ψν∗ ψν I
0
−α KTψ−ν1 ∗
I 0 0 ψν 0 0 ψ 0
0
0
0 −I
0 0
0 0
− 1δ I 0
Corollary 4 then finishes the proof.
αK + β L
0 ψν 0 0
0 ψ 0 δI
0
ψν ≺ 0. 0 ψ
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642 41.03
0
0
X :=
0
37.03
0
−29.27
−29.27 23.14
−24.65
σmax (ψν M ψν−1 )
Fig. 3. Plots of σmax (ψ M ψ ) and over frequency for ν = 1, 2, 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) −1
Remark 7. It is remarkable to observe that Xν in (44) converges for α → ∞ and β → 0 to Tψ−νT X11 Tψ−ν1 0
−T T X21 T −1 ψν ψν 0 X31 Tψ−ν1
0 Kν
Tψ−νT X12 Tψ−ν1 0
0 0
0 0 0
Tψ−νT X22 Tψ−ν1 0 0
0 Kν 0
Tψ−νT X13 0
.
Hence one can construct a certificate of (42) which is, approximately and up to the congruence transformation with Tψν , just a slightly perturbed as well as diagonally augmented version of X . This relation will be illustrated in the following numerical example. 4.2. Numerical example III For a minimum-phase g ∈ RH∞ and γ ∈ (0, 1), consider the stable transfer matrix
γ /g
0 γg
M =
ψ :=
with ψ M ψ −1 =
0
0
γ
γ
0
for
g 0
0 . 1
We clearly have ∥M ∥∞ = γ max{∥g ∥∞ , ∥g −1 ∥∞ } and ∥ψ M ψ −1 ∥∞ = γ < 1, which implies that (40) is valid. Now consider a exp
sequence λν ∈ Rν+1 as constructed in Section 2.3 with λTν bν −→ g. With ψν := diag(λTν bν , 1) we infer (42) for sufficiently large ν . Note that the latter FDI reads, for Nν = λν λTν , as
bν 0
∗
0 N M ν 1 0
bν
0 1
0
bν
0
0 1 0
0 1
0 1 M × bν 0 ≺ 0. 0
0 N − ν 0
0
41.74
0
−24.65
0
,
0
14.81
0
−0.01
0
0
0
0
0
0.01
0
0
0
0
0
−0.01
0
0.01
0
0
0
0
0
0
0
37.91
0
−0.01
−29.9
0
0
0
0
0.01
0
0
−25.12
0
0.01 0 0
0
0
0
0.01
0.01
0
0
0
−0.01 −29.9
0
0
0
0.01
23.59
0
−25.12
0
0.01
0
0
0
0
15.13
in accordance with Remark 7. For ν = 6 the certificate for (42) should lead to a better approximation of X , which is indeed confirmed by inspection:
41.03 0
0 0
0 0
0 0
X6 =
0 05×5
0 0
0 0
37.03 0
0 05×5
−29.27
0
0 0
−29.27
0 0
23.14 0
−24.65
0
0
−24.65 0 0 . 0 0 14.81
5. Conclusions We have investigated the asymptotic behavior of realizations of transfer functions belonging to sequences converging exponentially in the H∞ -norm. When input-balanced realizations are used for elements of the sequence, our main result indicates that compatibly-dimensioned portions of the realizations converge to the realization of the limit through a coordinate transformation. We provided some technical tools of independent interest which allow to lift manipulations of frequency-domain inequalities into the related state-space LMIs. For the purpose of illustration, these basic facts have been applied to robust stability conditions for uncertain systems with parametrized dynamic D-scales. All presented results happen to be crucial in the solution of the gainscheduling controller synthesis problem by dynamic D-scales as recently proposed in [4]. Acknowledgments Our thanks go to Joost Veenman for suggesting a simplification of the formulation and the proof of Theorem 3. Moreover the author C.W. Scherer would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.
X3 =
0 0 X33
639
(45)
Appendix A. Auxiliary facts
1
Since Nν enters affinely, one can verify the existence of some Nν = Nν T satisfying (45) through solving a genuine LMI feasibility problem after applying the KYP-lemma. This is the procedure to numerically handle robustness analysis conditions as discussed in detail in [4]. Let us now provide some numerical results for p = 3, g (s) = 5s+5 , γ = 0.95, whose simplicity allows us to display the s+5 certificates explicitly. For ν = 1, 2, 3, Fig. 3 shows the resulting plots of σmax (ψ M ψ −1 ) and σmax (ψν M ψν−1 ) over frequency. This reveals that the FDI (42) holds for ν ≥ 3. Let us now determine certificates for the non-strict versions of (40) and (42) as in Section 2.5. For ν = 3 we obtain
Lemma 8. Aψν in (3) satisfies ∥Aψν ∥ ≤ p(2ν − 1). Moreover, there exist symmetric Pψν and Qψν with I ≼ Pψν ≼ ν I ,
Qψν ≽
p 2
I
and
Aψν Pψν + Pψν ATψν + Qψν = 0.
(A.1)
Proof. Let Iν , Jν , Eν be the identity matrix, the upper-triangular Jordan block with eigenvalue zero and the matrix with all ones on and above the diagonal, all of dimension ν . Note that Eν = (Iν − Jν )−1 ; since ∥Iν − Jν ∥ ≤ ∥Iν ∥ + ∥Jν ∥ = 2, we infer (Eν EνT )−1 = (Iν − Jν )T (Iν − Jν ) ≼ 4I and thus Eν EνT ≽ 14 I.
640
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
He
X11 X21
X12 X22
A1 A21
0 A2
+
C1T PC1 0
0 0
X11 X21
X12 X22
⋆
B1 B2
+
C1 PD 0
≺ 0.
(A.6)
T
D PD Box I.
He
Xs A1 −1 X22 X21 A1 + A21
0 −1 A2 X22
+
C1T PC1 0
0 0
Xs B1 −1 X22 X21 B1 + B2
⋆
+
C1 PD 0
≺ 0.
(A.7)
DT PD Box II.
AT1 Xs + Xs A1 + C1T PC1 A21
⋆
1
ϵ
AT21
1 −1 T (A2 R− 2 + R2 A2 )
Xs B1 B2
+
C1 PD 0
≺ 0.
(A.8)
DT PD Box III.
ν−1
For Abν in (2) we have ∥Abν ∥ = ∥ − pIν − 2p µ=1 Jνµ ∥ ≤ p + 2p(ν − 1) = p(2ν − 1). Hence ∥I ⊗ Abν ∥ ≤ p(2ν − 1); because Aψν results from I ⊗ Abν by a unitary similarity transformation, the first statement is proved. For µ = 0, 1, . . . , ν − 1 one easily checks that Jνµ Abν = Abν Jνµ . Since the realization (2) is input-balanced, we conclude Abν Jνµ (Jνµ )T + Jνµ (Jνµ )T Abν + Jνµ Bbν BTbν (Jνµ )T = 0. By summation this
ν−1
+ Qν = 0 for Pν := (J µ )T J µ = ν−1 µ µ=0 µν νT diag(ν, ν − 1, . . . , 2, 1) and Qν := = µ=0 (Jν Bbν )(Jν Bbν ) p 2pEν EνT . Note that, trivially, I ≼ Pν ≼ ν I and Qν = 2pEν EνT ≽ 2 I. We can directly infer (Id ⊗ Abν )(Id ⊗ Pν ) + (Id ⊗ Pν )(Id ⊗ Abν )T + Id ⊗ Qν = 0. Again, since Aψν results from I ⊗ Abν implies Abν Pν +
Pν ATbν
by a unitary similarity transformation, an orthogonal congruence transformations of Id ⊗ Pν and Id ⊗ Qν leads to Pψν , Qψν satisfying the equation in (A.1). The bounds on Pν , Qν continue to hold for Id ⊗ Pν , Id ⊗ Qν and hence also for Pψν , Qψν , which proves the inequalities in (A.1).
With the Schur complement Uνs = Uν22 − Uν21 (Uν11 )−1 Uν12 , the Lyapunov inequality in (A.3) implies s s 22 T 21 11 −1 ν s A22 ψν Uν + Uν (Aψν ) + Pν + I − [Uν (Uν ) A12 Uν ] s T − [Uν21 (Uν11 )−1 A12 ν Uν ] ≼ 0.
The first inequality in (A.3) implies that Uν21 and Uν22 grow at most polynomially in ν . By the second one, Uν11 ≽ ν I and hence (Uν11 )−1 is bounded. With the third inequality in (A.3) we infer Uν22 ≽ Uνs ≽ ν I; hence Uνs grows at most polynomially in exp
ν ν −1 (U11 ) ν and satisfies (Uνs )−1 → 0. Since Aν12 −→ 0 we infer U21 ν ν ν −1 ν ν exp A12 Us −→ 0. For all large ν we conclude −I +[U21 (U11 ) A12 Usν ]+ ν ν −1 ν [U21 (U11 ) A12 Usν ]T ≺ 0. Hence (A.4) implies (A.2) for Kν := s −1 (Uν ) and all large ν .
Lemma 10. For P = P T consider the two LMIs
L
Lemma 9. Suppose (Aψν , Bψν , Cψν , Dψν ) is constructed as in Section 2.1 and let Pν be a sequence of symmetric matrices of the same dimension as A22 ψν that grows at most polynomially in ν . Then there exists some Kν ≻ 0 such that and such that, for all large ν ,
∥Kν−1 ∥ grows at most polynomially in ν
−1 −1 22 T A22 ψν Kν + Kν (Aψν ) + Pν ≺ 0 and ∥Kν ∥ → 0.
(A.2)
Proof. Let Pψν and Qψν be the sequences from lemma (A.1). Define Nν = diag(0k×k , Pν + I ) and choose a scalar sequence qν ≥ ν that grows at most polynomially and satisfies qν Qψν ≽ Nν ; this is possible since Qψν is bounded away from zero. For Uν = qν Pψν we hence infer Aψν Uν + Uν ATψν + Nν = qν (Aψν Pψν + Pψν ATψν )+ Nν = Nν − qν Qψν ≼ 0. Therefore
ν qν I ≽ Aψν
Uν12
Uν21
Uν22
11
12
Uν
21
Uν
+
Uν11
Uν
≽ νI ≽
22
+
Uν
0
0
0
Pν + I
0
νI 12
Uν
21
Uν22
Uν
0
0
11
Uν
≼ 0.
L
X11 X21 Xs , P ,
X12 X22
A1 C1
, P, B1 D
A1 A21 C1
ATψν
(A.3)
0 A2 0
B1 B2 D
≺ 0 and
≺ 0.
(A.5)
(a) If X satisfies the first LMI then X22 is invertible and Xs := X11 − −1 X12 X22 X22 satisfies the second. (b) Suppose A2 has no eigenvalues in C0 and choose R2 = RT2 with AT2 R2 + R2 A2 ≺ 0. If Xs satisfies the second LMI then X := Xs 0
0
ϵ R2
satisfies the first for all sufficiently small ϵ > 0.
(c) If X satisfies the non-strict version of the first LMI and X22 is −1 invertible then Xs := X11 − X12 X22 X22 satisfies the non-strict version of the second LMI. Proof. With He(A) := AT + A, the first LMI in (A.5) reads as Eq. (A.6) given in Box I. By considering the (2, 2)-block we get AT2 X22 + X22 A2 ≺ 0 and hence X22 is invertible. A congruence transformation with diag
and
(A.4)
I −1 −X22 X21
0 −1 X22
, I leads to Eq. (A.7)
given in Box II. Canceling the 2nd block row/column in (A.7) leads to the second LMI in (A.5). This proves (a) and, similarly, (c). To 1 −1 T show (b) note that R2 is invertible and satisfies A2 R− 2 + R2 A2 ≺ 0. With X21 = 0 and X22 = ϵ R2 for ϵ > 0, the inequality (A.7) reads as Eq. (A.8) given in Box III. If Xs is a solution of the second LMI in (A.5), we can render (A.8) satisfied by choosing ϵ > 0 sufficiently small. Since (A.8) is equivalent to (A.6), we have proved (b).
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
Appendix B. Proofs
641
which implies that X11 + R22 is non-singular. If we apply Lemma 10 to (33) (which is possible due to Ano = 0, Cno = 0), we infer that T satisfies (34). The analogues of (b) and (c) are direct consequences.
B.1. Proof of Theorem 3 Under the hypothesis of (c), due to (31) and Lemma 10(b) we conclude for all sufficiently small ϵ > 0 that An 0 L diag (ϵ Rn , X ) , Po , 0 0
Ano Ao 0 Co
Bn C Bo C A Do C
Bn D B o D ≺ 0. B Do D
B.3. Proof of Theorem 5 The steps in the proof are motivated by the observation that the inequalities in (35) can be expressed as 0 ∗ −I (∗) 0 0
−I
0 ∗ −I (∗) 0 0
−I
This trivially implies
0 0 0
A n 0 L diag (ϵ Rn , X ) , diag (0, Po ) , 0
Cn 0
Ano Ao 0 Cno Co
Bn C Bo C A Dn C Do C
Bn D Bo D B ≺ 0. (B.1)
Dn D Do D
=
A congruence transformation of (32) with H := diag(I , I , (C D)) and an augmentation of R leads to
A n 0 L diag (R, 0) , diag (Pn , −Po ) , 0
Cn 0
Ano Ao 0 Cno Co
Bn C Bo C A Dn C Do C
Bn D B o D B ≼ 0. (B.2)
Dn D Do D
R11 + ϵ Rn R21 L 0
An 0 0 Cn 0
Ano Ao 0 Cno Co
R12 X11 + R22 X21 Bn C Bo C A Dn C Do C
Bn D Bo D B Dn D Do D
0 X12 X22
, diag (Pn , 0) ,
(B.3)
A trivial simplification yields the strict version of (33) which proves (c). The proof of (a) proceeds similarly for ϵ = 0 and does not require any hypothesis on An . Now denote the left-hand side of (32) by L and partition it into a 3 × 3-block matrix with left-upper block L11 . Then the kernel of the l.h.s. of (B.2) is ker(H T LH ). If L˜ ≼ 0 and Lˆ ≼ 0 denote the left-hand sides of (B.1) and (B.3) for ϵ = 0, then ker(Lˆ ) = ker(H T LH )∩ ker(L˜ ). Note that L˜ carries a 4 × 4-partition and, since (31) is strict, we have ker(L˜ ) = {col(x1 , 0, 0, 0) : x1 free}. Using L ≼ 0 and due to the particular structure of H, this allows to conclude ker(Lˆ ) = {col(x1 , 0, 0, 0) : L11 x1 = 0} which proves (b).
0 0 −I 0
0 0 0 φ −1 ψ 0 −I 0
We use the notation from Appendix B.1. If Ano = 0 and Cno = 0 and if we partition Lˆ in an obvious way we have Lˆ 22 = ATo (X11 + R22 ) + (X11 + R22 )Ao ≼ 0. If Lˆ 22 x2 = 0 then x =
ˆ = 0 and thus Lx ˆ = 0; since ker(Lˆ ) col(0, x2 , 0, 0) satisfies xT Lx = {col(x1 , 0, 0, 0) : L11 x1 = 0} we infer x = 0 and conclude that Lˆ 22 has no kernel. Therefore ATo (X11 + R22 ) + (X11 + R22 )Ao ≺ 0
φ
−φ −∗ −I
−∗
0 I
≺ 0 and
I 0 0 I
≺0
transforming it with
(B.4)
I 00
0 I 0 −T Dφ BTφ
0 0 I
0 0 by 0 −T Dφ
0
congruence and
performing simple permutations, we indeed get
−A T φi 0 0 T Bφ i 0 0
0 −I , 0 0
−R21 −R11
0
0
Aψ 0 0
Bψ I 0
Cψ 0
Dψ 0
−I 0 0 0
0 0 −I 0
0 0 , 0 −I
−CφT i 0 0 ≺ 0. DTφ i
(B.5)
0 I
One easily verifies by direct computation that
0 L −I Aφ i
B.2. Proof of Corollary 4
0
with identical left-hand sides. The first FDI is related to (36). To −1 1 see this introduce Aφ i := Aφ − Bφ D− φ Cφ , Bφ i := Bφ Dφ , Cφ i := 1 1 := D− −D− φ . After multiplying (36) with −1, φ Cφ and Dφ i
−R22 L −R12
≺ 0.
−ψ ∗ ψ −φ −1
I 0 0 0 ψ 0 −I 0
0
Since the outer factors in (B.1) and (B.2) are identical, we can just add (B.1) and (B.2) to get
0 0 0
0 0 −I 0
0 −I 0 0 −I , 0 0 0
−I 0 0 0 0 0 0 0
0 0
0
0 −ATφ i
Bφ i 0
0 Cφ i 0 0 0 0
0 0 Cψ 0 0 BTφ i
0 Dφ i Dψ 0 I 0
0 0
Cψ 0
Dψ 0
0 0 −I 0 0 0 0 0
0 0 0 −I 0 0 0 0
0 −CφT i I 0 0 = 0. I 0 DTφ i 0 I
0 0 0 0 0 I 0 0
0 0 0 0 I 0 0 0
0 0 0 0 0 0 I 0
0 0 0 0 0 0 0 I
,
(B.6)
642
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
Now we apply Corollary 4 to (B.5) and (B.6). For this purpose
≺ 0. Then ΓΓˆ11 21 (−R22 )−1 −I −R21 satisfies ˆ
we choose Rn with ATφ i Rn + Rn Aφ i
ϵ Rn 0
0 −R11
−
−I −R12
Γˆ 11 L Γˆ 21
A φi 0 0 Cφ i 0 0
Γˆ 12 Γˆ 22
0
−I
0
−I , 0
0 0
0 −I
0
0
0
0
Bφ i
0
Aψ 0
Bψ 0
0 I
0 Cψ
Dφ i Dψ
0
0
0
Γˆ 12 Γˆ 22
:=
Γˆ 12 Γˆ 22
Aφ i 0 −Cφ i Cφ i 0 −Cφ i
0 Aψ 0 0 Cψ 0
0 , 0
−I
−I
0 −I , 0 0
ˆ Γ11 L Γˆ 21
Bφ i Bψ −Dφ i Dφ i Dψ −Dφ i
0 0 0 0 0 I 0 0 I
0 0 −I 0
0 0 , 0 −I
≺ 0
(B.8)
and the non-strict version for ϵ = 0. Cancellation of the fourth row and column, simplifications and permutations prove statement (a). Finally, (b) is shown by reversing the steps.
≺ 0 0 0
(B.7)
I
for all small ϵ > 0 and the non-strict version for ϵ = 0. Note that this is the LMI corresponding to the second FDI in (B.4). A simple congruence transformation yields
References [1] A. Rantzer, On the Kalman–Yakubovich–Popov lemma, Systems and Control Letters 28 (1996) 7–10. [2] C.W. Scherer, Multiobjective H2 /H∞ control, IEEE Transactions on Automatic Control 40 (1995) 1054–1062. [3] K. Zhou, J. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, 1996. [4] C.W. Scherer, I.E. Köse, Control synthesis using dynamic D-scales, IEEE Transactions on Automatic Control 57 (2012) 2219–2234. [5] A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, 1985. [6] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control 42 (1997) 819–830.
Contents lists available at SciVerse ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
From transfer matrices to realizations: Convergence properties and parametrization of robustness analysis conditions Carsten W. Scherer a , İ. Emre Köse b,∗ a
Department of Mathematics, University of Stuttgart, Germany
b
Department of Mechanical Eng., Boğazici University, Istanbul, Turkey
article
abstract
info
Article history: Received 3 February 2011 Received in revised form 28 December 2012 Accepted 9 April 2013 Available online 1 June 2013
In various branches of systems and control theory, one is confronted with the need for approximating transfer functions by a sequence of FIR expansions in the H∞ -norm. The approximating sequence grows in its McMillan degree, while the limiting transfer matrix has a finite number of poles. Considering the corresponding state-space realizations of the approximating sequence and its limit, it is of interest to understand the limiting behavior of the realization matrices. This paper investigates this behavior and, thus, provides an answer for the continuous-time counterpart of FIR expansions and exponential convergence. In a similar vein, it is well-understood how to translate frequency-domain inequalities for transfer matrices into LMIs for realizations by using the Kalman–Yakubovich–Popov (KYP) Lemma. However, it is often less clear how obviously valid manipulations of the frequency-domain inequalities lift into operations on the solutions of the corresponding LMIs. The paper provides some novel insights on such questions with applications to robustness analysis. © 2013 Elsevier B.V. All rights reserved.
Keywords: Convergence of realizations Linear matrix inequalities Robustness analysis
0. Notation We denote the extended imaginary axis For a transfer matrix G with realizationG(s) A + D we use G∗ (s) = G(−s)T , G = C
A C
B D
iR ∪ {∞} by C0 . = C (sI − A)−1 B B and Gss := D
. If A has no eigenvalues in C0 and if P = P T , the
frequency-domain inequality (FDI) G∗ PG ≺ 0 on C0 is, by the Kalman–Yakubovich–Popov (KYP) lemma [1], equivalent to the existence of X = X T such that L (X , P , Gss ) :=
XA + (XA)T (XB)T
XB 0
+
CT DT
T
X 0 0
0 0 P
I A C
=
I A C
0 B D
0 X 0
P C 0 B D
D
≺ 0.
(1)
Aψν Cψν
Bψν Dψν
→
Aψ Cψ
Bψ Dψ
for the
• Even if the McMillan degrees of Aψν and Aψ coincide, convergence might be destroyed by ν -dependent coordinate changes for the realization of ψν . This motivates to choose both realizations with particular extra properties.
exp
is said to converge exponentially to ψ , expressed as ψν −→ ψ , if
0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.04.001
Suppose that ψν is a sequence in RH∞ with ψν → ψ ∈ RH∞ in the H∞ -norm. If we choose minimal realizations (Aψν , Bψν , Cψν , Dψν ) of ψν and (Aψ , Bψ , Cψ , Dψ ) of ψ , it is clearly realization matrices. There are various stumbling blocks:
Corresponding author. Tel.: +90 535 747 2623. E-mail address: [email protected] (İ.E. Köse).
1. Introduction
generally impossible to conclude
We say that (1) certifies G∗ PG ≺ 0 on C0 or that X is a KYPcertificate for this FDI. For reasons of space, we drop ‘‘on C0 ’’ for FDIs and use the placeholder ⋆ for matrices that can be inferred by symmetry or are of no relevance. A sequence ψν in a normed space
∗
there exist K ≥ 0 and η ∈ (0, 1) such that ∥ψν − ψ∥ ≤ K ην for all ν . We use this notion for proper and stable transfer matrices and the H∞ -norm ∥ · ∥∞ , as well as for real matrices and the spectral norm σmax (·) = ∥ · ∥. We denote the set of proper and stable k×m rational matrices of dimension k × m by RH∞ .
• Typically, Aψν grows unbounded in size which implies that
we cannot apply simple compactness arguments in order to enforce convergence of Aψν , Bψν , Cψν . • The eigenvalues of Aψν and its Jordan structure are not (necessarily) related to those of A.
In view of these troubles it is a natural idea to choose balanced realizations and to consider those realization matrices that emerge through model reduction of ψν by truncation in order to enforce equality of the McMillan degrees. In many concrete problems,
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
though, the approximating sequence is chosen with a fixed polepair (Aψν , Bψν ) and with coefficient matrices Cψν , Dψν that are obtained by optimization. This would e.g. be a typical situation in multi-objective H2 /H∞ -control problems by applying the Youla parametrization (as discussed for example in [2]). Therefore, it is desirable to investigate the question of convergence for realizations that are input-balanced only. The purpose of this paper is twofold. First, we extract the asymptotic properties of the realization matrices (Aψν , Bψν , Cψν , Dψν ) for input-balanced realizations in case that the convergence of ψν to ψ is exponential in the H∞ -norm. In particular, we show that there exists a state-transformation Tψν such that a compatibly-dimensioned portion of the realization (Tψ−ν1 Aψν Tψν , Tψ−ν1 Bψν , Cψν Tψν , Dψν ) converges exponentially to (Aψ , Bψ , Cψ , Dψ ), while the remaining modes are canceled asymptotically. This result establishes the state-space counterpart of the properties of convergent sequences of transfer matrices in the H∞ -norm. As an application, if ψν → ψ and P2 ≺ P1 , we construct a KYP-certificate for the FDI ψν∗ P2 ψν − ψ ∗ P1 ψ ≺ 0 which is valid for all large ν . This is essential for the remaining results in the paper. The second purpose of the paper is related to robustness analysis of uncertain systems. It is well-known [3] that the verification of robust stability for an uncertain dynamical system in the standard LFT form involves a search over RH∞ for frequencydependent (i.e., ‘‘dynamic’’) matrices D(jω) = D(jω)∗ ≻ 0 such that a certain FDI is satisfied. In the LMI approach to the problem, the search over D is replaced by the search over ψ with D = ψ ∗ ψ in a finite dimensional subspace of RH∞ . Choosing a sequence Sν of such subspaces that is dense in RH∞ , one can find a sequence of transfer functions ψν ∈ Sν which approaches ψ for ν → ∞ in the H∞ -norm. If ψ satisfies the robustness FDI, it is quite obvious that the FDI will persist to hold for ψν and sufficiently large ν . In this sense, the transition from ψ to ψν can be carried out in the frequency domain without any difficulty. However, the relation of the corresponding KYP-certificates with realizations of ψ and ψν is quite intricate and plays a fundamental role in a recently proposed gain-scheduling synthesis algorithm [4]. Thus, the second of our main results provides essential insights into relations between such KYP certificates. Several technical tools are developed in obtaining these fundamental results. They are related to manipulations on frequency-domain inequalities and the resulting operations on the corresponding KYP certificates. For instance, if G and H1 = H1∗ , H2 = H2∗ are transfer matrices of compatible dimensions, the FDIs H2 ≼ H1 and G∗ H1 G ≺ 0 clearly imply G∗ H2 G ≺ 0. It is then natural to ask how the KYP certificates of the first two are related to that of the third. We provide simple, yet clarifying, answers to this and similar questions, which we believe are of independent interest. The paper is organized as follows. Section 2 contains the main result of the paper, Theorem 1, on the asymptotic properties of the realization matrices of exponentially converging transfer matrices. In Section 3 we discuss how to translate two FDI manipulations, namely ordered substitution and taking the Schur-complement, into operations on the corresponding KYP certificates. Section 4 illustrates the application of our technical insights to robust analysis by dynamic multipliers. Simple numerical examples illustrate our results. Finally, we collect auxiliary facts and proofs in the Appendices A and B respectively. 2. Convergence of realizations 2.1. Preliminaries—approximation in RH∞ [5,2] that the real span of For p > 0 it is well-known s−p 2 s−p 3 s−p 1, s+p , ( s+p ) , ( s+p ) , . . . is dense in RH∞ . Hence, with bν (s) :=
633
ν T
s−p s−p and for ψ ∈ RH∞ , there exists a ··· s+p s+p sequence λν ∈ Rν+1 such that λTν bν converges to ψ in the H∞ 1
exp
norm; one can even assure exponential convergence, i.e., λTν bν −→ ψ . Consequently, if d ∈ N and ⊗ denotes the Kronecker product, d×d any ψ ∈ RH∞ can be approximated by ψν := Λν (Id ⊗ bν ) exp
with Λν ∈ Rd×d(ν+1) such that ψν −→ ψ . Let us introduce the realization
−p .. . 0 := 0 0 . . √.
··· .. . ··· ··· ···
bν =
Abν Cbν
Bbν Dbν
..
−2p .. . −p √0
2p
1 1
2p
1
.. √.
.
···
2p
√ − 2p .. √. − 2p .. .
(2)
which is stable, minimal and input-balanced, i.e., Abν + ATbν +
Bbν BTbν = 0. Then the realization
ψν =
Id ⊗ Abν Λν (Id ⊗ Cbν )
Id ⊗ Bbν Λν (Id ⊗ Dbν )
(3)
of Λν (Id ⊗ bν ) is as well stable and input-balanced. It has been argued in [2] how to extract information about the McMillan degree of ψ , denoted as k, from the sequence ψν : in the ordered list of Hankel-singular values of ψν , k is equal to the index of the first Hankel-singular value that converges to zero. However, it has not been clarified in the literature how one can extract more detailed information about the realization matrices (Aψ , Bψ , Cψ , Dψ ) of ψ from those of ψν systematically. Since (3) is input-balanced, the squared Hankel singular values of ψν are equal to the eigenvalues of the corresponding observability Gramian. Let us perform an orthogonal statecoordinate change on (3) such that the observability Gramian is given by
Qψν =
Qψ1ν 0
0 Qψ2ν
with Qψ1ν ∈ Rk×k
and Qψ1ν ≽ Qψ2ν ≽ 0.
(4)
The resulting realization of ψν remains input-balanced and is denoted and partitioned as
Aψν Cψν
Bψν Dψν
A11 ψν
A12 ψν
B1ψν
= A21 ψν
A22 ψν Cψ2 ν
B2ψν Dψν
1
Cψν
k×k with A11 . (5) ψν ∈ R
Let us stress that the dimension and, by Lemma 8, the norm of Aψν both grow at most linearly in ν . The same hold for Bψν since the realization is input-balanced. Note that this realization can be determined with the mere knowledge of k and does not rely on any further properties of the limiting matrix ψ . We now prove that it rather explicitly displays information about the realization 1 1 of ψ , since a state-coordinate change of (A11 ψν , Bψν , Cψν , Dψν ) guarantees convergence to (Aψ , Bψ , Cψ , Dψ ), while the influence 12 of the eigenvalues of A22 ψν gets asymptotically canceled since Aψν and Cψ2 ν converge to zero. This is a consequence of Theorem 1. 2.2. Main result Theorem 1. Suppose that ψ
Aψν Cψν
Bψν Dψν
=
Aψ Cψ
Bψ Dψ
and ψν
=
are stable, input-balanced realizations. Let the
634
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
former be minimal and of degree k, while the latter is partitioned as in (5), has the observability Gramian (4), and the dimension of Aψν as well as ∥Aψν ∥ grow at most polynomially in ν . Finally, let Qψ and Sψν be the unique solutions of ATψ Qψ + Qψ Aψ + CψT Cψ = 0 and
exp
ATψν Sψν + Sψν Aψ + CψT ν Cψ = 0,
(6)
and define Tψν as the upper k × k block of
Sψν Qψ−1 .
exp
If ψν −→ ψ
then, for all sufficiently large ν, Tψν is non-singular, both Tψν and Tψ−ν1
are bounded, and
Tψ−ν1 A11 ψν Tψν Cψ1 ν Tψν
Tψ−ν1 B1ψν Dψν
exp
−→
The left-upper block of (10) is QΨν − Hν with Hν := Sψν Rψν + (Sψν Rψν )T − RTψν Qψ Rψν . If exploiting Sψν Qψ−1 SψT ν ≼ Qψν we infer ∥ Qψ−1 SψT ν − Qψ Rψν ∥2 ≤ λmax (Sψν Qψ−1 SψT ν − Hν )
Aψ Cψ
Bψ Dψ
≤ λmax (Qψν − Hν ) ≤ λν −→ 0.
(12)
Therefore exp
T Sψ − Qψ Rψν −→ 0 ν
(13) exp
exp
T T and thus Sψ RT − Qψ Rψν RTψν −→ 0 which implies Sψ RT −→ Qψ ν ψν ν ψν
T in view of (11). Since (Sψ − Qψ Rψν )(SψT ν − Qψ Rψν )T = SψT ν Sψν − ν
as well as
exp
T T Sψ RT Q − Qψ Rψν Sψ + Qψ Rψν RTψν Qψ −→ 0 we hence get ν ψν ψ ν
A12 exp ψν 2 −→ 0. Cψ
exp
(7)
ν
T Sψ S −→ (Qψ )2 ≻ 0. ν ψν
(14)
T From (12) we can as well extract ∥Qψν − Sψν Qψ−1 Sψ ∥ ≤ ∥Qψν − ν exp
It remains a conjecture that the results are valid even if the convergence ψν → ψ is not exponential. Proof. Step 1: Preparation. Define Gν = ψν − Dψν and G = ψ − Dψ .
T Hν ∥ + ∥Hν − Sψν Qψ−1 Sψ ∥ −→ 0, i.e., ν T Qψν − Sψν Qψ−1 Sψ = ν
Qψ1ν 0
0 Qψ2ν
exp
Since ∥ψν (∞) − ψ(∞)∥ = ∥Dψν − Dψ ∥ −→ 0 we infer from ∥Gν −G∥∞ = ∥ψν −ψ −(Dψν −Dψ )∥∞ ≤ ∥ψν −ψ∥∞ +∥Dψν −Dψ ∥
−
exp
that ∥Gν − G∥∞ −→ 0. In the sequel the ordered Hankel-singular values of a stable transfer matrix H are denoted in the usual fashion by σ1 (H ) ≥ σ2 (H ) ≥ σ3 (H ) ≥ · · · . Consider the errortransfer matrix G − Gν = Aψν 0 Bψν 0 Aψ Bψ . Since Aψν + ATψν + Bψν BTψν = 0 and Cψν −Cψ 0 Aψ + ATψ + Bψ BTψ = 0, the positive semi-definite controllability Gramian of G − Gν has the particular structure as in
Aψν 0
+
0 Aψ Bψν Bψ
RTψν I
I
Rψν
Bψν Bψ
T
+
RTψν I
I
Rψν
Aψν 0
0 Aψ
Aψν 0
+
0 Aψ
T
I 0
RTψ ν R˜ ψν
(8)
−Sψν
λν := λmax
Aψν 0
Qψ
I 0
CT 0 + ψνT Aψ −C ψ
RTψν
R˜ ψν
Cψν −Cψ = 0. (9)
RTψ ν I
I Rψν
=
I
Rψν
0 R˜ ψ
ν
I − Rψν RTψν . Hence,
Qψν −SψT ν
−Sψν
exp
−→ 0.
(15)
Let us further observe
0 =
T
I T Qψ−1 Sψ ν
(9)
I T Qψ−1 Sψ ν
= ATψν (Qψν − Sψν Qψ−1 SψT ν ) + (Qψν − Sψν Qψ−1 SψT ν )Aψν + (Cψν − Cψ Qψ−1 SψT ν )T (Cψν − Cψ Qψ−1 SψT ν ). Due to (15) and the polynomial growth of Aψν , we infer Cψν − exp
Cψν Sψν −→ Cψ Qψ .
(16) T
T
T
The equation for Sψν in (6) implies Aψ Sψν Sψν + Sψν Aψν Sψν + exp
T CψT Cψν Sψν = 0. Note that ATψ Sψ S + CψT Cψν Sψν −→ ATψ Qψ2 + ν ψν T T T Cψ Cψ Qψ = (Aψ Qψ + Cψ Cψ )Qψ = −Qψ Aψ Qψ . Therefore
(17)
From (8) we similarly infer Aψ (I −Rψν RTψν )+(I −Rψν RTψν )ATψ +(Bψ − exp
Rψν Bψν )(BTψ − BTψν RTψν ) = 0. Hence, due to (11), Rψν Bψν −→ Bψ . By (13) (and since Bψν grows at most polynomially, because (Aψν , Bψν ) exp
T B − Qψ Rψν Bψν −→ 0 and thus is input-balanced), Sψ ν ψν exp
T Sψ B −→ Qψ Bψ . ν ψν
Qψ exp
I
Rψν
0 R˜ ψν
(18)
2 2 Step 2: Proof of convergence of A12 ψν , Cψν and Qψν . Since G has McMillan degree k, we recall that
(10)
Considering the right-lower block, since Qψ ≻ 0 we infer from exp λν ≥ λmax (R˜ ψν Qψ R˜ ψν ) = ∥ Qψ R˜ ψν ∥2 ≥ 0 that Qψ R˜ ψν −→ 0 exp and thus R˜ ψν −→ 0 and thus Rψν RTψν −→ I .
T
exp
= σ1 (Gν − G)2 ≼ ∥Gν − G∥2∞ −→ 0.
exp
1 Sψ ν 2 Sψ ν
T Sψ A S −→ Qψ Aψ Qψ . ν ψν ψν
Qψ
for R˜ ψν :=
exp
= 0.
Due to I − Rψν RTψν ≽ 0 we have
Qψ−1
exp
Qψν −SψT ν
−Sψν
Qψν −SψT ν
T −→ 0 and thus Cψν Sψν − Cψ Qψ−1 SψT ν Sψν −→ 0 and Cψ Qψ−1 Sψ ν thus, by (14),
T
With Qψ and Sψν satisfying (6), its positive semi-definite observability Gramian is given as in
1 Sψ ν 2 Sψ ν
(11)
σk+l (Gν ) ≤ σk+l (Gν − G) ≤ ∥Gν − G∥∞ for l = 1, 2, . . . . (19) 11 Aψν B1ψν Let us define G1ν := which results from Gν by 1
Cψν Dψν truncation. One can check that the standard proof of the H∞ norm error bound [3] carries over to the current situation, which
dim(Aψ
)−k
implies ∥Gν − G1ν ∥∞ ≤ 2 l=1 ν σk+l (Gν ). Due to (19), the exp right-hand side is bounded by dim(Aψν )∥Gν − G∥∞ −→ 0, which
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
635
exp
allows to conclude ∥Gν − G1ν ∥∞ −→ 0. Therefore ∥G − G1ν ∥∞ ≤ exp
∥G − Gν ∥∞ + ∥Gν − G1ν ∥∞ −→ 0. For l = 1, . . . , k this implies exp |σl (G) − σl (G1ν )| ≤ ∥G − G1ν ∥∞ −→ 0. Therefore, σl (G1ν ) =
λl (Qψ1ν ) → σl (G) > 0 and we can infer the existence of 0 <
γ ≤ Γ such that γ I ≼ Qψ1ν ≼ Γ I for all large ν . Moreover, since Qψ1ν has dimension k, we have σk+1 (Gν ) = λmax (Qψ2ν ). By (19) exp we infer λmax (Qψ2ν ) ≤ ∥G − Gν ∥∞ and thus Qψ2ν −→ 0. From 2 22 2 T 2 T 2 (A22 ψν ) Qψν + Qψν Aψν + (Cψν ) Cψν = 0 and since ∥Aψν ∥ grows at exp
T 1 most polynomially, we conclude Cψ2 ν −→ 0. From (A11 ψν ) Qψν +
1 T 1 1 2 T 1 Qψ1ν A11 ≤ ∥(A11 ψν + (Cψν ) Cψν = 0, we infer ∥Cψν ∥ ψν ) Qψν + 1 11 1 11 1 Qψν Aψν ∥ ≤ 2∥Qψν ∥ ∥Aψν ∥ ≤ 2Γ ∥Aψν ∥ such that ∥Cψν ∥ grows at
most polynomially. Hence ( implies
)
T 2 A21 ψν Qψν
+
1
Qψν A12 ψν
+ (Cψν ) Cψν = 0 1
T
2
A21 ψν 12 exp
∥ also grows at most polynomially this finally implies
that Aψν −→ 0. We have proven the second statement in (7). Step 3: Completion of proof. Let us now exploit (15). Since exp
exp
2 Qψ2ν −→ 0 and since Qψ ≻ 0, we infer that Sψ −→ 0. Due to (14) ν exp
T 1 T 1 and Sψ S = (Sψ ) Sψν + (Sψ2 ν )T Sψ2 ν we conclude (Sψ1 ν )T Sψ1 ν −→ ν ψν ν (Qψ )2 . We infer that Sψ1 ν is bounded. Since Sψ1 ν is square, we can also
1 −1 extract that (Sψ ) exists for all large ν and is bounded as well. Let ν us finally recall from (16)–(18) that T Sψ A S ν ψν ψν
1 1 T 12 2 = (Sψ1 ν )T A11 ψν (Sψν ) + (Sψν ) Aψν (Sψν ) 2 T 21 1 2 T 22 + (Sψν ) Aψν (Sψν ) + (Sψν ) Aψν (Sψ2 ν )
exp
−→ Qψ Aψ Qψ , exp
and
exp
1 Cψν Sψν = Cψ1 ν (Sψ ) + Cψ2 ν (Sψ2 ν ) −→ Cψ Qψ . ν
exp
exp
−→ Qψ Aψ Qψ , (Sψ1 ν )T B1ψν
exp
−→
1 Qψ Bψ , Cψ1 ν (Sψ ) −→ Cψ Qψ and thus ν exp
−1 1 [Qψ−1 (Sψ1 ν )T ]A11 ψν [Sψν Qψ ] −→ Aψ , exp
exp
[Qψ−1 (Sψ1 ν )T ]B1ψν −→ Bψ and Cψ1 ν [Sψ1 ν Qψ−1 ] −→ Cψ . 1 Now note that Tψν = Sψ Q −1 . Hence the proof is complete if ν ψ exp
1 T we can show that Tψ−ν1 − Qψ−1 (Sψ ) −→ 0. Indeed, we have ν
already established that Tψν and Tψ−ν1 are bounded for all large ν ; 1 T thus it suffices to observe that Qψ−1 (Sψ ) Tψν = Qψ−1 (Sψ1 ν )T Sψ1 ν Qψ−1 ν exp
−→ I.
Aψ
X +X
Abν
0
0
Aψ
BT BTψ X bν T λν Cbν − Cψ
X
Bbν
Bψ
−γ I λTν Dbν − Dψ
CbT λν ν −CψT
≺ 0. DTb λν − DTψ ν −γ I
(21)
We can hence compute γν := infλν ∈Rν+1 ∥λTν bν − ψ∥∞ by minimizing γ over all X = X T ∈ R(ν+2)×(ν+2) , λν ∈ Rν+1 and γ ∈ R exp
that satisfy (21). Due to optimality of γν we can conclude γν −→ 0. Semi-logarithmic plots of the approximation error γν and the two errors related to (7) over ν are given in Fig. 1; the curves do indeed reveal exponential convergence as claimed. E.g., for ν = 6 we obtain λTν = (0 −0.01 0.07 −0.06 2.43 2.87 1) from which no information can be extracted. However, the realizations of λTν bν after orthogonally diagonalizing the observability Gramian and that after having performed the coordinate changes as proposed in Theorem 1 are given by 2.66 0 −0.05 0 0.39 0.3 −0.1
−6 −3 −1.82 0 13.36 10.3 3.76 −1.09 0.51 0.78 0 −5.7 −4.39 −1.42
−5.14 −4.91 1.65 0
−12.09 −9.32 −3.48
0 0 −0.14 1.04 1.01 1.57 0 0 0 −0.14 1.04 1.01 1.57 0
0 0 −1.05 0 −4.48 0 0 0 0 −1.05 0 −4.48 0 0
0 0 1.02 4.48 −7.44 −11.5 0 0 0 1.02 4.48 −7.44 −11.5 0
0 0 0 0 0.03 −4.42 0 0 0 0 0 0.03 −4.42 0
−3.46 −0.1 −0.53 0 , 3.86 2.97 1 1.48 3.13 −0.53 0 3.86 2.97 1
.
Both realizations do indeed admit the claimed structure (after rounding), while the second one allows to directly read-off the input-balanced realization of ψ in (20).
d×d Let us now suppose that ψ ∈ RH∞ is invertible on C0 . Choose ψν as in Section 2.1 and with the properties as formulated in Theorem 1. If two symmetric matrices satisfy P2 − P1 ≺ 0, we conclude from the trivial inequality ψ ∗ (P2 − P1 )ψ = ψ ∗ P2 ψ − ψ ∗ P1 ψ ≺ 0 that, for all large ν ,
and ψ(s) =
(s − 1)(s − 6) (s + 2)(s + 4)
with the input-balanced realization
ψ=
0
T
2.4. Construction of a certificate for approximation
For a numerical experiment let us consider
0
2.3. Numerical example I
p=3
Abν
T 1 T 1 Sψ B = (Sψ ) Bψν + (Sψ2 ν )T B2ψν −→ −Qψ Bψ , ν ψν ν
1 T 11 This leads to (Sψ ) Aψν (Sψ1 ν ) ν
In view of the bounded real lemma, the inequality ∥λTν bν − ψ∥∞ < γ holds iff there exists X = X T with
1 −1 21 T 2 1 −1 1 T 2 ∥A12 ψν ∥ = ∥−(Qψν ) (Aψν ) Qψν − (Qψν ) (Cψν ) Cψν ∥ 1 21 ∥Aψν ∥ ∥Qψ2ν ∥ + ∥Cψ1 ν ∥ ∥Cψ2 ν ∥ . ≤ γ
Since ∥
Fig. 1. Plots of the approximation error γν (blue) and the errors in (7) (green, red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
−1.09 0.51 −1.42
−5.14 −4.91 −3.48
1.48 3.13 1
.
(20)
ψν ψ
∗
P2 0
0 −P 1
ψν ≺ 0. ψ
(22)
636
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
Let us now construct certificates for this FDI with nice asymptotic properties for ν → ∞. Theorem 2. For any P ≺ 0 there exist K ≻ 0, L = LT and a sequence Kν ≻ 0 with limν→∞ ∥Kν ∥ = 0 and such that ∥Kν−1 ∥ grows at most polynomially in ν with the following properties: if the symmetric P1 , P2 satisfy P2 − P1 ≼ P and if β ∈ (0, 1), then for all sufficiently large α there exists some να such that for all ν ≥ να the following LMI certification of (22) holds:
L
α Tψ−νT KTψ−ν1
0
−α Tψ−νT K
0
Kν
0
−α KTψ−ν1
0
αK + β L
,
A11 ψν
A12 ψν
0
B1ψν
21 Aψν 0 1 Cψν
A22 ψν
0
B2ψν
0
Aψ
Bψ
Cψ2 ν
0
Dψν
0
Cψ
Dψ
0
P2
0
0
−P1
+ ν A21 ψν
B2ψν
A21 ψν
≺ 0.
+ Cψ
Dψ
T
B2ψν
T
+I ≺0
(24)
LBψ 0
(25)
We abbreviate P˜ := diag(P2 , −P1 ) and assume w.l.o.g. that Tψν = I. If we add in (23) the first row/column to the third and left-/right-multiply the second one with Uν := Kν−1 , which is a congruence transformation, we obtain (by exploiting that (1) formally extends to non-symmetric X and non-square A):
αK
L 0 0
A11 ψν −α K 0 0 , P˜ , 1 Cψν βL 0
0
21 Aψν + 0 0
+ CψT P2 Cψ
(
)
T A21 ψν
A12 ψν Uν
A11 ψν
B1ψν
0
Aψ
Cψ2 ν Uν
Cψ1 ν
Bψ Dψν
0
Cψ
0
0
Dψ
T (A21 ψν )
0
≺ 0. 0
0
0
(Bψν ) 2
T
B2ψν
β(ATψ L + LAψ ) β BTψ L CψT P2 Cψ
CψT P2 Dψ
CψT (P2 − P1 )Cψ
CψT (P2 − P1 )Dψ .
Dψ (P2 − P1 )Cψ
(27)
Dψ (P2 − P1 )Dψ
T
T
0 A21 ψν
B2ψν
0
(B2ψν )T
0
≺ 0.
0
0
(28)
0
1 21 Aψν
A21 ψν
B2ψν
ϵ0 + ϵ0 I ≼ Vν + ν A21 ψν
A21 ψν
B2ψν
A21 ψν
A21 ψν
B2ψν
B2ψν
T
T
+ I ≺ 0.
If Hν denotes the first matrix in (26) partitioned into Hνij , we now claim that there exists some να ≥ ν0 such that for all ν ≥ να the inequality Hν − E ≺ 0 is true; then (26) holds for all ν ≥ να in view of (28), which finishes the proof. It remains to prove the claim above. Since Uν grows at most polynomially in ν , we infer from (7) that ∥A12 ψν Uν ∥ → 0 and
∥Cψ2 ν Uν ∥ → 0. This in turn implies ∥Hνij ∥ → 0 for i = 2 or j = 2. ϵ One can hence increase ν0 such that Hν22 ≼ ∥Hν22 ∥I ≼ 20 I for all ν ≥ ϵ0 22 22 ν0 . In view of Hν −ϵ0 I ≼ − 2 I ≺ 0 and thus −(Hν −ϵ0 I )−1 ≼ ϵ20 I, the inequality Hν − E ≺ 0 is implied by (Schur) 11 13 14 Hν
Hν
Hν31
Hν33
Hν41
Hν43
2
ϵ0
Hν12
Hν Hν34 + ϵ0 I Hν44
Hν32 Hν21
Hν23
Hν24
≺ 0.
(29)
42
Hν
The limit of the left-hand side of (29) for ν → ∞ just equals (27) +ϵ0 I and is hence negative definite because of (27) ≼ −2ϵ0 I; therefore, there exists some να ≥ ν0 such that (29) is valid for all ν ≥ να .
A21 ψν
0 β LBψ 0
22 T Indeed, with the abbreviation Vν := A22 ψν Uν + Uν (Aψν ) , by taking the Schur complement and exploiting (24), we infer for all large ν that
+
22 T A22 ψν Uν + Uν (Aψν )
T (A21 ψν ) 22 T A22 ψν Uν + Uν (Aψν ) 21 T (Aψν )
0
A21 ψν 0
Dψ ≺ 0.
CψT P2 Cψ
E+
Note that L = LT exists due to the KYP-lemma since, trivially, ψ ∗ P ψ ≺ 0. (The matrix L can even be taken to be positive definite, but this is not required in the sequel.)
Bψ Bψ Dψ Dψ
0
0 0
=
0
ATψ L + LAψ BTψ L
P Cψ
α(ATψ K + KAψ )
(23)
having the described properties. Moreover, let us fix K ≻ 0 and L = LT to satisfy ATψ K + KAψ ≺ 0,
−α K βL
0
Vν +
Aψ Aψ Cψ Cψ
Since P2 − P1 ≼ P ≺ 0 and β ∈ (0, 1), we can exploit (25) to infer that the right-lower 2 × 2-block of (27) is negative definite. Due to ATψ K + KAψ ≺ 0, there exists some ϵ0 ∈ (0, 1) with (27) ≼ −2ϵ0 I for all large α . Fix any such α . If E := diag(−ϵ0 I , ϵ0 I , −ϵ0 I , −ϵ0 I ) let us establish that there exists ν0 ∈ N0 such that for all ν ≥ ν0 one has
,
T
αK
L
Dψ P2 Cψ
Aψ 0 , P˜ , Cψ 0
T
2 Proof. Since ∥A21 ψν ∥, ∥Bψν ∥ grow at most polynomially in ν , we can apply Lemma 9 and choose Kν with
−1 −1 A22 A22 ψν Kν + Kν ψν
2.5. Numerical example II (26)
After canceling the 2nd block row/column of the l.h.s. in (26), (7) (with Tψν = I) implies that the limit of the resulting matrix for ν → ∞ is
Consider a sequence λν ∈ Rν+1 as constructed in Section 2.3 exp with λTν bν = ψν −→ ψ = 5ss++55 using p = 3. Then the FDI
∗ 1 0 ψν ≺ 0 holds for all ψν∗ ψν − 1.05 ψ ∗ ψ = ψψν 0 −1.05 ψ sufficiently large ν . Let us now determine certificates of the nonstrict versions of these FDIs by computing the stabilizing solutions
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
of the Riccati equations that correspond to the respective LMIs; for the underlying state-space realizations we assume that the coordinate changes of Theorem 1 have been performed, which implies Tψν = I. For ν = 3 and ν = 6, this computation leads to the certificates 93.92 −0.07 −0.14 −94.43 0.02 0 0.08 −0.07 and −0.14 0 0.03 0.15 −94.43 0.08 0.15 95 84.0016 0 −84.0017
0
−84.0017
0 , 84.0018
05×5 0
with structures approximately shared with that of the certificate in (23).
In this section let us consider transfer matrices G, Ψo , Ψn without poles on C0 and real symmetric matrices Po , Pn . Suppose that the FDI G∗ Ψo∗ Po Ψo G ≺ 0 is valid. If the ‘‘old’’ transfer matrix Ψo∗ Po Ψo is related to the ‘‘new’’ one Ψn∗ Pn Ψn as in
Ψn∗ Pn Ψn
Ψo∗ Po Ψo ∗ Ψn Pn ⇐⇒ Ψo 0
≼
0 −P o
Ψn ≼ 0, Ψo
(30)
we can trivially combine these FDIs to infer G∗ Ψn∗ Pn Ψn G ≺ 0. The main purpose of this section is to understand this process in the state-space. More precisely, we reveal how solutions of the LMIs corresponding to G∗ Ψo∗ Po Ψo G ≺ 0 and to (30) can be glued together in order to obtain a solution of the LMI corresponding to G∗ Ψn∗ Pn Ψn G ≺ 0. The result itself and its proof are formulated in state-space only; the link with the motivating considerations for FDIs is established through the correspondences
A C
G=
Ψn =
B D
An 0 Cn
,
Ψo =
Ano Ao Cno
Bn Bo Dn
Ao Co
Bo Do
,
L
X12 X22
, Po ,
Corollary 4. Under the hypotheses of Theorem 3 and if Ano = 0 and Cno = 0, the block X11 +R22 is non-singular and the Schur complement of the certificate in (33) with respect to the (2, 2)-block, namely
L
Ao 0 Co
Bo C A Do C
Bo D B Do D
An 0 , Cn 0
R11
L
R21
R22
0
≺0
T , Pn ,
0
−P o
Ano Ao Cno Co
0
An 0 0 Cn
R12 X11 + R22 X21 Ano Ao 0 Cno
Bn C Bo C A Dn C
Bn D B Dn D
≼ 0.
(34)
If the left-upper block of (32) is negative definite, then (34) is strict. Moreover, if we define T with the perturbed version of R under the hypothesis in (c), then (34) is strict as well. As a non-trivial application of these results, let us consider the obvious FDI equivalence I ∗ 0 (⋆) 0 0
⇐⇒
0 ψ 0 0 I I 0 0
0 I 0 0
0 0 0 I
ψ φ −1
∗
0
−I
0 I
0
φ∗ = 0
ψ ∗ψ I
I
φφ
∗
≻0
I
ψ φ −1
(35)
R11 L R21
Bn Bo ≼ 0. Dn Do
0 X12 X22
Aψ 0 Cψ 0 0 0
I 0 , 0 0
R12 R22
0 I 0 0
0 0 0 I
0 0 , I 0
0 −ATφ 0 −BTφ 0 0
Bψ 0 Dψ 0 I 0
0 CφT 0 DTφ 0 I
≻ 0,
(36)
Γ11 L Γ21
Then the following statements are true: (a) Inequalities (31) and (32) imply
R 11 L R21
Bn C A Dn C
(31)
(32)
An 0 Cn
R12 P , n
−R12 (X11 + R22 )−1 X12 , X11 − X21 (X11 + R22 )−1 X12
satisfies
and
R11 − R12 (X11 + R22 )−1 R21 −X21 (X11 + R22 )−1 R12
for square transfer matrices ψ and φ , the latter being invertible. With natural realizations, the LMIs that certify these two FDIs read, respectively, as
Theorem 3. Let X and R satisfy X11 X21
In case that Ano = 0 and Cno = 0, the realization of Ψn just equals (An , Bn , Cn , Dn ) by unobservability. Lemma 10 in Appendix A then allows to establish a link between LMI (33) and the corresponding version for this smaller-sized realization which leads to the following result.
= −ψ ∗ ψ + (φφ ∗ )−1 ≺ 0
and the resulting natural realizations for Ψo G, col(Ψn , Ψo ) as well as Ψn G.
(b) If the left-upper block of (32) in the natural partition is negative definite, then (33) is strict. (c) Suppose that An has no eigenvalues on C0 and choose Rn = RTn with ATn Rn + Rn An ≺ 0. For all sufficiently small ϵ > 0, the strict version of (33) holds if R is replaced with R + diag(ϵ Rn , 0).
T :=
3. ‘‘Gluing’’ of certificates in state-space
637
Γ12 Γ22
−I , 0
0 I
,
, Pn ,
Bn D Bo D ≼ 0. B Dn D
Aψ 0 Cψ 0
(33)
0 1 Aφ − Bφ D− φ Cφ 0 1 −D− φ Cφ
Bψ 1 Bφ D− φ ≺ 0. Dψ 1 D− φ
(37)
The following theorem clarifies the relation between R and Γ , again directly in the state-space.
638
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
in X = X T by using the KYP lemma. As in Section 2.4 let us now suppose that ψν approximates ψ . The corresponding FDI reads as
Fig. 2. Description of uncertainty.
Theorem 5. Suppose Dφ is square with det(Dφ ) ̸= 0 and Aφ − 1 0 Bφ D− φ Cφ has no eigenvalues in C . (a) If R satisfies (36), then (37) is feasible, R22 is non-singular and
Γ11 Γ21
Γ12 Γ22
:=
1 R12 R− 22 R21 − R11 1 R− 22 R21
1 R12 R− 22 −1 R22
R11 R21
R12 R22
:=
−1 Γ12 Γ22 Γ21 − Γ11 −1 Γ22 Γ21
−1 Γ12 Γ22 −1 Γ22
(39)
(43) exp
In the case of ψν −→ ψ , it is obvious that (40) implies (42). Given an LMI certificate X of (40) as in (41), it is much less clear how one can construct a certificate Xν of (42) as in (43). This is the purpose of the following result which is based on a combination of Theorem 2 and Corollary 4. Theorem 6. Suppose X satisfies (41). Then there exist K ≻ 0, L = LT and a sequence Kν ≻ 0 with limν→∞ ∥Kν ∥ = 0 with the following properties: for any small β ∈ (0, 1) and any sufficiently large α there exists a να such that for all ν ≥ να , (43) is satisfied by
satisfies the non-strict version of (36). 4. Application to robustness analysis with D-scales
α Tψ−νT KTψ−ν1
To recap, we have shown how to construct
• a certificate for ψν∗ P2 ψν − ψ ∗ P1 ψ ≺ 0 for sufficiently large ν , where P2 ≺ P1 and ψν → ψ , and • a certificate for G∗ Ψn∗ Pn Ψn G ≺ 0 using any certificates of G∗ Ψo∗ Po Ψo G ≺ 0 and Ψn∗ Pn Ψn ≺ Ψo∗ Po Ψo .
0
Xν =
0 0 0
We now discuss the application of these insights to robustness analysis, where the roles played by Ψn , Ψo , G will become clear from the context. This application is an essential ingredient of the solution of the gain-scheduled synthesis problem using frequencydependent D-scales in [4].
×
In analyzing uncertain dynamical interconnections, it is customary to decompose the system into two subsystems as in Fig. 2, where M represents the nominal linear time-invariant stable system, while all uncertainties, be they parametric, dynamic, nonlinear, etc., are collected in the operator ∆. When ∆ is stable, LTI and norm-bounded by 1, it is sufficient for ˆ M ψˆ −1 ) < robust stability of the interconnection in Fig. 2 if σmax (ψ 0 ˆ 1 on C , where ψ is a non-singular frequency-dependent matrix without poles on C0 that commutes with ∆ [3,6]. This condition ˆ ∗ ψˆ M ≺ ψˆ ∗ ψˆ . By factorizing can equivalently be written as M ∗ ψ
ψˆ ψˆ •×• = ψ ∗ ψ ≻ 0 for some square ψ ∈ RH∞ , we obtain the FDI ∗ ψM I 0 ψM ψ 0 −I ψ ∗ ∗ M ψ ψ 0 M = ≺ 0. (40) I 0 −ψ ∗ ψ I ∗
This can now be converted, with obvious notations for realizations of ψ, M, into the matrix inequality
L X ,
I 0
0 −I
,
Aψ 0 0 Cψ 0
0 Aψ 0 0 Cψ
Bψ C 0 A Dψ C 0
Bψ D Bψ B Dψ D Dψ
≺ 0
0 0
0 0
0 0
0 0 0
α Tψ−νT KTψ−ν1
0 Kν 0
X33
0 0
−α KTψ−ν1
0 0
0 0 X31
−α KTψ−ν1
×
0 0
X12 α K + β L + X22
X21
0 X32
α K + β L + X11
−1
−α Tψ−νT K
0
0
0
X13
0
0
−α Tψ−νT K
0
X23
.
(44)
1
Proof. For all δ ∈ (0, 1) close to 1, (41) implies L X
ψM ψ ss
δ
I
0
0
−δ I
≺ 0. If we apply Theorem 2 for (P2 , P1 ) = (I , 1δ I )
and (P2 , P1 ) = (−I , −δ I ) (e.g. with P = (δ − 1)I ≺ 0), we can diagonally combine the certificates for ψν∗ ψν − 1δ ψ ∗ ψ and −ψν∗ ψν + δψ ∗ ψ ≺ 0 to obtain the certificate α Tψ−νT KTψ−ν1 0 0 0 −α Tψ−νT K 0 0 Kν 0 0 0 0 − T − 1 − T 0 0 α T KT 0 0 −α T K ψν ψν ψν 0 0 0 Kν 0 0 − α KT −1 0 0 0 αK + β L 0 ψν
0
0
ψν 0 ψ
for
(41)
0 Kν
0
−
4.1. Main result
(42)
which is again equivalent to the existence of Xν = XνT such that Aψν 0 Bψν C Bψν D 0 Aψν 0 Bψν I 0 0 A B L Xν , , 0 ≺ 0. 0 −I C 0 Dψν C Dψν D ψν 0 Cψν 0 Dψν
(38)
satisfies the non-strict version of (37). (b) If Γ satisfies (37), then (36) is feasible, Γ22 is non-singular and
∗ ψν M I 0 ψν M ψν 0 −I ψν ∗ ∗ M ψν ψν 0 M = ≺ 0, I 0 −ψν∗ ψν I
0
−α KTψ−ν1 ∗
I 0 0 ψν 0 0 ψ 0
0
0
0 −I
0 0
0 0
− 1δ I 0
Corollary 4 then finishes the proof.
αK + β L
0 ψν 0 0
0 ψ 0 δI
0
ψν ≺ 0. 0 ψ
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642 41.03
0
0
X :=
0
37.03
0
−29.27
−29.27 23.14
−24.65
σmax (ψν M ψν−1 )
Fig. 3. Plots of σmax (ψ M ψ ) and over frequency for ν = 1, 2, 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) −1
Remark 7. It is remarkable to observe that Xν in (44) converges for α → ∞ and β → 0 to Tψ−νT X11 Tψ−ν1 0
−T T X21 T −1 ψν ψν 0 X31 Tψ−ν1
0 Kν
Tψ−νT X12 Tψ−ν1 0
0 0
0 0 0
Tψ−νT X22 Tψ−ν1 0 0
0 Kν 0
Tψ−νT X13 0
.
Hence one can construct a certificate of (42) which is, approximately and up to the congruence transformation with Tψν , just a slightly perturbed as well as diagonally augmented version of X . This relation will be illustrated in the following numerical example. 4.2. Numerical example III For a minimum-phase g ∈ RH∞ and γ ∈ (0, 1), consider the stable transfer matrix
γ /g
0 γg
M =
ψ :=
with ψ M ψ −1 =
0
0
γ
γ
0
for
g 0
0 . 1
We clearly have ∥M ∥∞ = γ max{∥g ∥∞ , ∥g −1 ∥∞ } and ∥ψ M ψ −1 ∥∞ = γ < 1, which implies that (40) is valid. Now consider a exp
sequence λν ∈ Rν+1 as constructed in Section 2.3 with λTν bν −→ g. With ψν := diag(λTν bν , 1) we infer (42) for sufficiently large ν . Note that the latter FDI reads, for Nν = λν λTν , as
bν 0
∗
0 N M ν 1 0
bν
0 1
0
bν
0
0 1 0
0 1
0 1 M × bν 0 ≺ 0. 0
0 N − ν 0
0
41.74
0
−24.65
0
,
0
14.81
0
−0.01
0
0
0
0
0
0.01
0
0
0
0
0
−0.01
0
0.01
0
0
0
0
0
0
0
37.91
0
−0.01
−29.9
0
0
0
0
0.01
0
0
−25.12
0
0.01 0 0
0
0
0
0.01
0.01
0
0
0
−0.01 −29.9
0
0
0
0.01
23.59
0
−25.12
0
0.01
0
0
0
0
15.13
in accordance with Remark 7. For ν = 6 the certificate for (42) should lead to a better approximation of X , which is indeed confirmed by inspection:
41.03 0
0 0
0 0
0 0
X6 =
0 05×5
0 0
0 0
37.03 0
0 05×5
−29.27
0
0 0
−29.27
0 0
23.14 0
−24.65
0
0
−24.65 0 0 . 0 0 14.81
5. Conclusions We have investigated the asymptotic behavior of realizations of transfer functions belonging to sequences converging exponentially in the H∞ -norm. When input-balanced realizations are used for elements of the sequence, our main result indicates that compatibly-dimensioned portions of the realizations converge to the realization of the limit through a coordinate transformation. We provided some technical tools of independent interest which allow to lift manipulations of frequency-domain inequalities into the related state-space LMIs. For the purpose of illustration, these basic facts have been applied to robust stability conditions for uncertain systems with parametrized dynamic D-scales. All presented results happen to be crucial in the solution of the gainscheduling controller synthesis problem by dynamic D-scales as recently proposed in [4]. Acknowledgments Our thanks go to Joost Veenman for suggesting a simplification of the formulation and the proof of Theorem 3. Moreover the author C.W. Scherer would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.
X3 =
0 0 X33
639
(45)
Appendix A. Auxiliary facts
1
Since Nν enters affinely, one can verify the existence of some Nν = Nν T satisfying (45) through solving a genuine LMI feasibility problem after applying the KYP-lemma. This is the procedure to numerically handle robustness analysis conditions as discussed in detail in [4]. Let us now provide some numerical results for p = 3, g (s) = 5s+5 , γ = 0.95, whose simplicity allows us to display the s+5 certificates explicitly. For ν = 1, 2, 3, Fig. 3 shows the resulting plots of σmax (ψ M ψ −1 ) and σmax (ψν M ψν−1 ) over frequency. This reveals that the FDI (42) holds for ν ≥ 3. Let us now determine certificates for the non-strict versions of (40) and (42) as in Section 2.5. For ν = 3 we obtain
Lemma 8. Aψν in (3) satisfies ∥Aψν ∥ ≤ p(2ν − 1). Moreover, there exist symmetric Pψν and Qψν with I ≼ Pψν ≼ ν I ,
Qψν ≽
p 2
I
and
Aψν Pψν + Pψν ATψν + Qψν = 0.
(A.1)
Proof. Let Iν , Jν , Eν be the identity matrix, the upper-triangular Jordan block with eigenvalue zero and the matrix with all ones on and above the diagonal, all of dimension ν . Note that Eν = (Iν − Jν )−1 ; since ∥Iν − Jν ∥ ≤ ∥Iν ∥ + ∥Jν ∥ = 2, we infer (Eν EνT )−1 = (Iν − Jν )T (Iν − Jν ) ≼ 4I and thus Eν EνT ≽ 14 I.
640
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
He
X11 X21
X12 X22
A1 A21
0 A2
+
C1T PC1 0
0 0
X11 X21
X12 X22
⋆
B1 B2
+
C1 PD 0
≺ 0.
(A.6)
T
D PD Box I.
He
Xs A1 −1 X22 X21 A1 + A21
0 −1 A2 X22
+
C1T PC1 0
0 0
Xs B1 −1 X22 X21 B1 + B2
⋆
+
C1 PD 0
≺ 0.
(A.7)
DT PD Box II.
AT1 Xs + Xs A1 + C1T PC1 A21
⋆
1
ϵ
AT21
1 −1 T (A2 R− 2 + R2 A2 )
Xs B1 B2
+
C1 PD 0
≺ 0.
(A.8)
DT PD Box III.
ν−1
For Abν in (2) we have ∥Abν ∥ = ∥ − pIν − 2p µ=1 Jνµ ∥ ≤ p + 2p(ν − 1) = p(2ν − 1). Hence ∥I ⊗ Abν ∥ ≤ p(2ν − 1); because Aψν results from I ⊗ Abν by a unitary similarity transformation, the first statement is proved. For µ = 0, 1, . . . , ν − 1 one easily checks that Jνµ Abν = Abν Jνµ . Since the realization (2) is input-balanced, we conclude Abν Jνµ (Jνµ )T + Jνµ (Jνµ )T Abν + Jνµ Bbν BTbν (Jνµ )T = 0. By summation this
ν−1
+ Qν = 0 for Pν := (J µ )T J µ = ν−1 µ µ=0 µν νT diag(ν, ν − 1, . . . , 2, 1) and Qν := = µ=0 (Jν Bbν )(Jν Bbν ) p 2pEν EνT . Note that, trivially, I ≼ Pν ≼ ν I and Qν = 2pEν EνT ≽ 2 I. We can directly infer (Id ⊗ Abν )(Id ⊗ Pν ) + (Id ⊗ Pν )(Id ⊗ Abν )T + Id ⊗ Qν = 0. Again, since Aψν results from I ⊗ Abν implies Abν Pν +
Pν ATbν
by a unitary similarity transformation, an orthogonal congruence transformations of Id ⊗ Pν and Id ⊗ Qν leads to Pψν , Qψν satisfying the equation in (A.1). The bounds on Pν , Qν continue to hold for Id ⊗ Pν , Id ⊗ Qν and hence also for Pψν , Qψν , which proves the inequalities in (A.1).
With the Schur complement Uνs = Uν22 − Uν21 (Uν11 )−1 Uν12 , the Lyapunov inequality in (A.3) implies s s 22 T 21 11 −1 ν s A22 ψν Uν + Uν (Aψν ) + Pν + I − [Uν (Uν ) A12 Uν ] s T − [Uν21 (Uν11 )−1 A12 ν Uν ] ≼ 0.
The first inequality in (A.3) implies that Uν21 and Uν22 grow at most polynomially in ν . By the second one, Uν11 ≽ ν I and hence (Uν11 )−1 is bounded. With the third inequality in (A.3) we infer Uν22 ≽ Uνs ≽ ν I; hence Uνs grows at most polynomially in exp
ν ν −1 (U11 ) ν and satisfies (Uνs )−1 → 0. Since Aν12 −→ 0 we infer U21 ν ν ν −1 ν ν exp A12 Us −→ 0. For all large ν we conclude −I +[U21 (U11 ) A12 Usν ]+ ν ν −1 ν [U21 (U11 ) A12 Usν ]T ≺ 0. Hence (A.4) implies (A.2) for Kν := s −1 (Uν ) and all large ν .
Lemma 10. For P = P T consider the two LMIs
L
Lemma 9. Suppose (Aψν , Bψν , Cψν , Dψν ) is constructed as in Section 2.1 and let Pν be a sequence of symmetric matrices of the same dimension as A22 ψν that grows at most polynomially in ν . Then there exists some Kν ≻ 0 such that and such that, for all large ν ,
∥Kν−1 ∥ grows at most polynomially in ν
−1 −1 22 T A22 ψν Kν + Kν (Aψν ) + Pν ≺ 0 and ∥Kν ∥ → 0.
(A.2)
Proof. Let Pψν and Qψν be the sequences from lemma (A.1). Define Nν = diag(0k×k , Pν + I ) and choose a scalar sequence qν ≥ ν that grows at most polynomially and satisfies qν Qψν ≽ Nν ; this is possible since Qψν is bounded away from zero. For Uν = qν Pψν we hence infer Aψν Uν + Uν ATψν + Nν = qν (Aψν Pψν + Pψν ATψν )+ Nν = Nν − qν Qψν ≼ 0. Therefore
ν qν I ≽ Aψν
Uν12
Uν21
Uν22
11
12
Uν
21
Uν
+
Uν11
Uν
≽ νI ≽
22
+
Uν
0
0
0
Pν + I
0
νI 12
Uν
21
Uν22
Uν
0
0
11
Uν
≼ 0.
L
X11 X21 Xs , P ,
X12 X22
A1 C1
, P, B1 D
A1 A21 C1
ATψν
(A.3)
0 A2 0
B1 B2 D
≺ 0 and
≺ 0.
(A.5)
(a) If X satisfies the first LMI then X22 is invertible and Xs := X11 − −1 X12 X22 X22 satisfies the second. (b) Suppose A2 has no eigenvalues in C0 and choose R2 = RT2 with AT2 R2 + R2 A2 ≺ 0. If Xs satisfies the second LMI then X := Xs 0
0
ϵ R2
satisfies the first for all sufficiently small ϵ > 0.
(c) If X satisfies the non-strict version of the first LMI and X22 is −1 invertible then Xs := X11 − X12 X22 X22 satisfies the non-strict version of the second LMI. Proof. With He(A) := AT + A, the first LMI in (A.5) reads as Eq. (A.6) given in Box I. By considering the (2, 2)-block we get AT2 X22 + X22 A2 ≺ 0 and hence X22 is invertible. A congruence transformation with diag
and
(A.4)
I −1 −X22 X21
0 −1 X22
, I leads to Eq. (A.7)
given in Box II. Canceling the 2nd block row/column in (A.7) leads to the second LMI in (A.5). This proves (a) and, similarly, (c). To 1 −1 T show (b) note that R2 is invertible and satisfies A2 R− 2 + R2 A2 ≺ 0. With X21 = 0 and X22 = ϵ R2 for ϵ > 0, the inequality (A.7) reads as Eq. (A.8) given in Box III. If Xs is a solution of the second LMI in (A.5), we can render (A.8) satisfied by choosing ϵ > 0 sufficiently small. Since (A.8) is equivalent to (A.6), we have proved (b).
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
Appendix B. Proofs
641
which implies that X11 + R22 is non-singular. If we apply Lemma 10 to (33) (which is possible due to Ano = 0, Cno = 0), we infer that T satisfies (34). The analogues of (b) and (c) are direct consequences.
B.1. Proof of Theorem 3 Under the hypothesis of (c), due to (31) and Lemma 10(b) we conclude for all sufficiently small ϵ > 0 that An 0 L diag (ϵ Rn , X ) , Po , 0 0
Ano Ao 0 Co
Bn C Bo C A Do C
Bn D B o D ≺ 0. B Do D
B.3. Proof of Theorem 5 The steps in the proof are motivated by the observation that the inequalities in (35) can be expressed as 0 ∗ −I (∗) 0 0
−I
0 ∗ −I (∗) 0 0
−I
This trivially implies
0 0 0
A n 0 L diag (ϵ Rn , X ) , diag (0, Po ) , 0
Cn 0
Ano Ao 0 Cno Co
Bn C Bo C A Dn C Do C
Bn D Bo D B ≺ 0. (B.1)
Dn D Do D
=
A congruence transformation of (32) with H := diag(I , I , (C D)) and an augmentation of R leads to
A n 0 L diag (R, 0) , diag (Pn , −Po ) , 0
Cn 0
Ano Ao 0 Cno Co
Bn C Bo C A Dn C Do C
Bn D B o D B ≼ 0. (B.2)
Dn D Do D
R11 + ϵ Rn R21 L 0
An 0 0 Cn 0
Ano Ao 0 Cno Co
R12 X11 + R22 X21 Bn C Bo C A Dn C Do C
Bn D Bo D B Dn D Do D
0 X12 X22
, diag (Pn , 0) ,
(B.3)
A trivial simplification yields the strict version of (33) which proves (c). The proof of (a) proceeds similarly for ϵ = 0 and does not require any hypothesis on An . Now denote the left-hand side of (32) by L and partition it into a 3 × 3-block matrix with left-upper block L11 . Then the kernel of the l.h.s. of (B.2) is ker(H T LH ). If L˜ ≼ 0 and Lˆ ≼ 0 denote the left-hand sides of (B.1) and (B.3) for ϵ = 0, then ker(Lˆ ) = ker(H T LH )∩ ker(L˜ ). Note that L˜ carries a 4 × 4-partition and, since (31) is strict, we have ker(L˜ ) = {col(x1 , 0, 0, 0) : x1 free}. Using L ≼ 0 and due to the particular structure of H, this allows to conclude ker(Lˆ ) = {col(x1 , 0, 0, 0) : L11 x1 = 0} which proves (b).
0 0 −I 0
0 0 0 φ −1 ψ 0 −I 0
We use the notation from Appendix B.1. If Ano = 0 and Cno = 0 and if we partition Lˆ in an obvious way we have Lˆ 22 = ATo (X11 + R22 ) + (X11 + R22 )Ao ≼ 0. If Lˆ 22 x2 = 0 then x =
ˆ = 0 and thus Lx ˆ = 0; since ker(Lˆ ) col(0, x2 , 0, 0) satisfies xT Lx = {col(x1 , 0, 0, 0) : L11 x1 = 0} we infer x = 0 and conclude that Lˆ 22 has no kernel. Therefore ATo (X11 + R22 ) + (X11 + R22 )Ao ≺ 0
φ
−φ −∗ −I
−∗
0 I
≺ 0 and
I 0 0 I
≺0
transforming it with
(B.4)
I 00
0 I 0 −T Dφ BTφ
0 0 I
0 0 by 0 −T Dφ
0
congruence and
performing simple permutations, we indeed get
−A T φi 0 0 T Bφ i 0 0
0 −I , 0 0
−R21 −R11
0
0
Aψ 0 0
Bψ I 0
Cψ 0
Dψ 0
−I 0 0 0
0 0 −I 0
0 0 , 0 −I
−CφT i 0 0 ≺ 0. DTφ i
(B.5)
0 I
One easily verifies by direct computation that
0 L −I Aφ i
B.2. Proof of Corollary 4
0
with identical left-hand sides. The first FDI is related to (36). To −1 1 see this introduce Aφ i := Aφ − Bφ D− φ Cφ , Bφ i := Bφ Dφ , Cφ i := 1 1 := D− −D− φ . After multiplying (36) with −1, φ Cφ and Dφ i
−R22 L −R12
≺ 0.
−ψ ∗ ψ −φ −1
I 0 0 0 ψ 0 −I 0
0
Since the outer factors in (B.1) and (B.2) are identical, we can just add (B.1) and (B.2) to get
0 0 0
0 0 −I 0
0 −I 0 0 −I , 0 0 0
−I 0 0 0 0 0 0 0
0 0
0
0 −ATφ i
Bφ i 0
0 Cφ i 0 0 0 0
0 0 Cψ 0 0 BTφ i
0 Dφ i Dψ 0 I 0
0 0
Cψ 0
Dψ 0
0 0 −I 0 0 0 0 0
0 0 0 −I 0 0 0 0
0 −CφT i I 0 0 = 0. I 0 DTφ i 0 I
0 0 0 0 0 I 0 0
0 0 0 0 I 0 0 0
0 0 0 0 0 0 I 0
0 0 0 0 0 0 0 I
,
(B.6)
642
C.W. Scherer, İ.E Köse / Systems & Control Letters 62 (2013) 632–642
Now we apply Corollary 4 to (B.5) and (B.6). For this purpose
≺ 0. Then ΓΓˆ11 21 (−R22 )−1 −I −R21 satisfies ˆ
we choose Rn with ATφ i Rn + Rn Aφ i
ϵ Rn 0
0 −R11
−
−I −R12
Γˆ 11 L Γˆ 21
A φi 0 0 Cφ i 0 0
Γˆ 12 Γˆ 22
0
−I
0
−I , 0
0 0
0 −I
0
0
0
0
Bφ i
0
Aψ 0
Bψ 0
0 I
0 Cψ
Dφ i Dψ
0
0
0
Γˆ 12 Γˆ 22
:=
Γˆ 12 Γˆ 22
Aφ i 0 −Cφ i Cφ i 0 −Cφ i
0 Aψ 0 0 Cψ 0
0 , 0
−I
−I
0 −I , 0 0
ˆ Γ11 L Γˆ 21
Bφ i Bψ −Dφ i Dφ i Dψ −Dφ i
0 0 0 0 0 I 0 0 I
0 0 −I 0
0 0 , 0 −I
≺ 0
(B.8)
and the non-strict version for ϵ = 0. Cancellation of the fourth row and column, simplifications and permutations prove statement (a). Finally, (b) is shown by reversing the steps.
≺ 0 0 0
(B.7)
I
for all small ϵ > 0 and the non-strict version for ϵ = 0. Note that this is the LMI corresponding to the second FDI in (B.4). A simple congruence transformation yields
References [1] A. Rantzer, On the Kalman–Yakubovich–Popov lemma, Systems and Control Letters 28 (1996) 7–10. [2] C.W. Scherer, Multiobjective H2 /H∞ control, IEEE Transactions on Automatic Control 40 (1995) 1054–1062. [3] K. Zhou, J. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, 1996. [4] C.W. Scherer, I.E. Köse, Control synthesis using dynamic D-scales, IEEE Transactions on Automatic Control 57 (2012) 2219–2234. [5] A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, 1985. [6] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control 42 (1997) 819–830.