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ROBUST STABILITY ANALYSIS TESTS FOR LINEAR TIME-VARYING PERTURBATIONS WITH BOUNDED RATES-OF-VARIATION
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ROBUST STABILITY ANALYSIS TESTS FOR LINEAR TIME-VARYING PERTURBATIONS WITH BOUNDED RATES-OF-VARIATION Hakan K¨ oro˘ glu
Carsten W. Scherer
Delft Center for Systems and Control, Delft University of Technology Mekelweg 2 2628 CD Delft The Netherlands Phone : +31(0)1527{87171,85899} Fax : +31(0)152786679 E-Mail : {h.koroglu,c.w.scherer}@dcsc.tudelft.nl Abstract: Robust stability analysis is considered for linear time-varying perturbations with bounded norms and rates-of-variation, and a possible generalization of the frequency-dependent D-G-scaling test is provided in an IQC framework by favor of a generalized version of the swapping lemma. Moreover, a convexoptimization-based method is developed for testing when a system cannot be robustly stable, by employing a generalized version of the Kalman-Yakubovichc Popov lemma together with a power distribution theorem. Copyright 2006 IFAC Keywords: Robust stability, stability analysis, structured uncertainty, linear matrix inequalities (LMI), integral quadratic constraints (IQC).
1. INTRODUCTION There are natural motivations for considering the problem of robust stability analysis against structured linear time-varying (LTV) perturbations in a robust control setting, since there will be variations with time in any engineering system, some of them being introduced by the designer for adaptation purposes (see e.g. Athans et al. [2005]). Two crucial results by Shamma [1994], Poolla and Tikku [1995] have established that constant/frequency-dependent D-scaling tests can be employed as exact analysis tests in the cases of arbitrarily fast/slow dynamic LTV perturbations respectively. For parametric LTV perturbations, there are numerous works that employ parameterdependent Lyapunov function approach to exploit the information on maximal rates-of-variation in analysis (see e.g. Gahinet et al. [1996], Amato et al. [1997]). However, the parameter-dependent Lyapunov function approach is mainly applicable for parametric perturbations, whereas the unifying approach of Megretski and Rantzer [1997] based on integral quadratic constraints in an operator setting can be employed for analysis against dynamic rate-bounded LTV perturbations as well. In fact, the IQC test of J¨ onsson and Rantzer [1996], based on the so-called swapping lemma,
was extended by K¨oro˘glu and Scherer [2005b,a] for robust stability and performance analysis against dynamic LTV perturbations with specified maximal rates-of-variation. For considering dynamic LTV perturbations instead of, or in addition to, time-varying parametric uncertainties, there are motivations like reducing the computational complexity of analysis or examining the safety and reliability of the designed systems in more uncertain environments. In this paper, we study the robust stability analysis problem, formulated in the next section for structured LTV perturbations as considered also in K¨oro˘glu and Scherer [2005b,a]. We develop in Section 3 an IQC test by generalizing the approach of Helmersson [1999], which was also based on (an implicit version of) the swapping lemma and provided as an improvement to the test of J¨onsson and Rantzer [1996]. As a novel contribution in the direction of conservatism analysis, we provide in Section 4 a convex-optimization-based method, developed by employing a generalized version of the Kalman-Yakubovich-Popov lemma together with the power distribution theorem of Wolodkin and Poolla [1994]. We conclude by some final remarks.
Preprints of the 5th IFAC Symposium on Robust Control Design
ξ
w
M
ROCOND'06, Toulouse, France, July 5-7, 2006
3. AN IQC APPROACH TO ROBUST STABILITY ANALYSIS FOR STRUCTURED LTV PERTURBATIONS
z
∆
ζ
In this section, we will present a test for analyzing when a given system is robustly stable. The derivation is based on the IQC stability theorem and a generalized version of the swapping lemma, which we will summarize before we present the main result of the section. 3.1 IQC-Based Robust Stability Analysis
Fig. 1.Standard setup for robust stability analysis. 2. PROBLEM FORMULATION The problem that we consider in this paper is formulated within the generic setup of Figure 1, where the system is represented by a stable and proper real-rational transfer function, M ∈ η×η RH∞ , interconnected in a feedback configuration with an unknown causal and linear operator, ∆ ∈ Lcη×η , coming out of the set ∆. We are especially interested in block-diagonal structured perturbations, ∆ = [Ili ⊗ ∆i ]κi=1 , diag (Il1 ⊗ ∆1 , . . . , Ilκ ⊗ ∆κ ), that have induced ℓ2 -norms of at most unity, i.e. ∆ ∈ BLη×η , and that have specc ified maximal rates-of-variation, kV∆ ki2 (≤ 2), where V∆ , λ−1 ∆λ − ∆, with λ/λ−1 denoting the right/left shift operators respectively. The feedback interconnection is referred to as wellposed, if the map (I − M ∆) has a causal inverse, (I − M ∆)−1 ∈ Lη×η . We say that the system is c uniformly robustly ℓ2 -stable against ∆, if, moreover, (I − M ∆)−1 i2 is bounded in ∆, i.e. if ∃β ∈ R+ such that k(I−M ∆)−1 ki2 < β, ∀∆ ∈ ∆. If the system in Figure 1 is robustly stable, for any ∆ ∈ ∆ we will have w ∈ ℓη2 , z ∈ ℓη2 , whenever ξ ∈ ℓη2 , ζ ∈ ℓη2 . Based on these notions, the problem that we study is formulated as follows: Problem 1. Consider the feedback system in Figure 1 with ha given causal and stable LTI plant, i AM BM −1 , C − AM )−1 BM + M (λ−1 ) = CM M (λ DM
DM ∈ RHη×η ∞ , subject to causal and bounded structured LTV perturbations ∆ that are contained in the set ∆ = ∆sν ⊂ BLη×η , defined as c i ×mi ; [Ili ⊗ ∆i ]κi=1 : ∆i ∈ BLm c , (1) ∆sν , kV∆i ki2 ≤ νi , i ∈ I; ∆∗i = ∆i , i ∈ Ip where s = {li , mi , ai }κi=1 , (ai = −1, for i ∈ Ip ⊆ I = {1, . . . , κ}; ai = 0, for i ∈ Id , I \Ip , with Ip intended to represent the index set of parametric perturbation channels) and ν = {νi }κi=1 , νi ∈ [0, 2], describe, respectively, the structure and the maximum rate-of-variation of the uncertainty. For later use, define Iv , {i ∈ I : νP i 6= 0}, Ipv , Ip ∩ Iv , Idv , Id ∩ Iv and κp , i∈Ip 1, P κv , i∈Iv 1. Determine whether the feedback system is robustly ℓ2 -stable against ∆sν or not.
With the adjoint of an operator, T ∈ Lm×n , defined as the unique operator, T ∗ ∈ Ln×m , for which hT x, yi =Phx, T ∗ yi , ∀x ∈ ℓn2 , ∀y ∈ ℓm 2 , where T hx, yi , x (t)y(t), and with the positivet semi-definiteness, T ≥ 0, of a self-adjoint operator, T = T ∗ ∈ Lm×m , understood as hT x, xi ≥ 0, ∀x ∈ ℓm 2 , the IQC approach to robust stability analysis can be summarized as follows: Theorem 2. (Megretski and Rantzer [1997]) Let ∗ I Π11 Π12 I η×η ∆Π , ∆ ∈ Lc : ≥ 0 , (2) ∆ Π∗12 Π22 ∆ 2η×2η where the Π = Π∗ ∈ RL∞ and assume that
(i) (M, τ ∆) is well-posed ∀τ ∈ [0, 1] and ∀∆ ∈ ∆. (ii) τ ∆ ∈ ∆Π , ∀τ ∈ [0, 1] and ∀∆ ∈ ∆. M is uniformly robustly ℓ2 -stable against ∆, if, ∀ω ∈ [0, 2π], M (eω ) satisfies ∗ M (eω ) Π11 (eω )Π12 (eω ) M (eω ) < 0. (3) I Π∗12 (eω )Π22 (eω ) I The LMI versions of the IQC tests are usually derived by employing the Kalman-YakubovichPopov lemma, which we provide below in a generalized form from Iwasaki et al. [2005]: Lemma 3. i (Generalized KYP Lemma) Let Φ = h AΦ BΦ Φ ×nΦ ∈ RHm , where AΦ ∈ RkΦ ×kΦ , ∞ CΦ DΦ BΦ ∈ RkΦ ×nΦ , CΦ ∈ RmΦ ×kΦ , DΦ ∈ RmΦ ×nΦ , with AΦ having all its eigenvalues strictly inside the unit disk, and let Q ∈ HmΦ be a Hermitian matrix. With d̟ ∈ [0, 2π] and ̟ ∈ [0.5d̟, 2π − 0.5d̟], the following conditions are equivalent: (i) For all ω ∈ [̟ − 0.5d̟, ̟ + 0.5d̟], ∗ Φ(eω ) Q Φ(eω ) < 0.
(4)
T RΦ LRΦ
(ii) There exists L ∈ L̟,d̟ such that < 0, where I 0 RΦ , AΦ BΦ , (5) CΦ DΦ −P0 ρ1 P1 0 ρ1 P1 P0 − ρ0 P1 0 : 0 0 Q L̟,d̟ , .(6) ̟ ρ = 2 cos(0.5d̟), ρ = e , 0 1 P0 ∈ HkΦ , P1 ∈ Hk+Φ
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
3.2 Generalized Swapping Lemma The following lemma, reformulated from K¨oro˘glu and Scherer [2005b], describes the effect of composing LTI and LTV operators in changing orders and will help to develop the IQC test in the sequel: Lemma 4. (Generalized Swapping Lemma) Consider, in the setup of Problem 1, a ∆r = [Ili ⊗ ∆i ]i∈I ∈h ∆sν , together with H , [Hi ⊗ Imi ]i∈I , i Hi =
AHi BHi CHi DHi
∈ RHq∞i ×li , where AHi ∈
Rki ×ki , BHi ∈ Rki ×li , CHi ∈ Rqi ×ki , DHi ∈ Rqi ×li , with AHi ’s having all their eigenvalues strictly inside the unit disk. Let ∆l , [Iqi ⊗∆i ]i∈I , −1 ηv ×ηv ∆ , where ηv , Pc , [Iki ⊗ νi ∆i ]i∈Iv ∈ Lc k m , and define V such that V V∆ c V T = i∈Iv i i κ [Iki ⊗ V∆i ]i=1 (i.e. V = [Vij ⊗ Ini ×nIv (j) ], Vij = √ νi Iki , if Iv (j) = i; Vij = 0ki ×kIv (j) , otherwise, i = 1, . . . , κ, j = 1, . . . , κv ). With HB , [HiB ⊗ I]i∈I and HC , [HiC ⊗ Imi ]i∈I , where HiB , λ−1 (λ−1 − AHi )−1 BHi , −1
HiC , CHi (λ
−1
− AHi )
,
(7) (8)
we have ∆r H ∆l 0 H HC V = . (9) 0 V∆c V THB 0 I V∆cV THB | {z } | {z } | {z } | {z } Hel
∆er
∆el
Her
3.3 A Swapping-Lemma-Based Robust Stability Analysis Test for Structured LTV Perturbations Generalizing the IQC test of Helmersson [1999], the following theorem provides a method to analyze whether a given system is robustly stable against a set of dynamic LTV perturbations with specified maximal rates-of-variation: Theorem 5. Consider the setup of Problem 1 and assume that (M, τ ∆) is well-posed for all ∆ ∈ ∆sν and τ ∈ [0, 1]. With Hi ∈ RHq∞i ×li being given stable (basis) transfer functions, let Her and Hel be defined as in Lemma 4. Construct the matrix RΦ according to (5) with a (preferably) Her Me minimal realization for Φ = , where Hel Me , M 0η×ηv is an extended version of the original plant, and define the set L as −P P X 0 Y 0 : 0 X 0 Y v v T Y 0 −X 0 T 0 −X 0 Y v v . (10) L, kΦ P ∈S qi X =[X ⊗I ] ,X ∈S i m i∈I i i + ki Xv=[Xvi⊗Imi]i∈Iv,Xvi ∈S+ T qi×qi Y =[Yi ⊗Imi ]i∈I ,Yi = ai Yi ∈R T ki×k i Yv =[Yvi⊗Imi ]i∈Iv ,Yvi= ai Yvi ∈R
Then, M is robustly stable against ∆sν if there T exists an L ∈ L such that RΦ LRΦ < 0. PROOF. The proof is based on establishing the robust stability of Me against ∆r ∆r = [Ili ⊗∆i ]i∈I ∈ ∆sν , ∆e , ∆er = : , V∆cV THB V∆c =[Iki ⊗νi−1 V∆i ]i∈Iv which necessarily implies the robust stability of M against ∆. To obtain a multiplier class for the extended uncertainty set, we first observe that
I τ ∆er
∗
∗ Her Xe Her Her Ye Hel I ∗ T ∗ τ ∆er Hel Ye Her −Hel Xe Hel {z } | Πe
∗ = Her
I τ ∆el
∗
Xe Ye I Her , τ ∆el YeT −Xe {z } | Q
for any ∆er ∈ ∆e and for matrices Xe and Ye of appropriate sizes. It then follows immediately that we will have τ ∆er ∈ ∆Πe , ∀∆er ∈ ∆e and ∀τ ∈ [0, 1], provided that Xe = diag (X, Xv ), and Ye = diag (Y, Yv ), with X, Xv , Y and Yv being chosen as in (10). Theorem 5 then follows from an application of Theorem 2 followed by the employment of the standard KYP lemma. 2 Remark 6. The method identified by Theorem 5 can be viewed as a possible generalization of the D-G scaling test. In case νi = 0, ∀i, the terms V THB and HC V (as well as Xv , Yv ) are void and thus we recover the frequency-dependent D-G scaling test. On the other hand, if we choose Hi = Ili , ∀i ∈ I, these terms are again void and we thus recover the constant D-G scaling test. Remark 7. The number of decision variables in the LMI test of Theorem 5 depend heavily on the McMillan degree of Φ, and thus on the choice of basis transfer functions. The number of extra variables introduced (due to the time-varying channels)P as compared toPthe LTI case is given by nv = i∈Ipv ki2 + 0.5 i∈Idv ki (ki + 1). If we
consider bases of the form Hi = [ I λ · · · λpi ]T , we can find minimal realizations as 0 0 1 I(pi −1) 0 0 AHi BHi = (11) 0 0 1 ⊗ Ili , CHi DHi I(pi −1) 0 0 0 1 0 for which we clearly have qi = (pi + 1)li and ki = pi li . With AH , [AHi ⊗Imi ]κi=1 , BH , [BHi ⊗ Imi ]κi=1 , CH , [CHi ⊗ Imi ]κi=1 and DH , [DHi ⊗ Imi ]κi=1 , we can obtain a (not necessarily -yet most probably- minimal) realization for Φ as
Preprints of the 5th IFAC Symposium on Robust Control Design
AM 0 BH CM A H 0 0 CH Φ= H CM D V TBH CM V TAH 0 0 0 0
0 BM 0 BH DM AH BH 0 DH DM 0 V TBH DM CH DH 0 0
ROCOND'06, Toulouse, France, July 5-7, 2006
0 0 V 0 0 0 Iηv
. (12)
Thus, the total number of decision variables, as well as the number of extra variables as compared to the LTI case, can be observed to be proportional to p2i for the considered basis. We should note that there is significant reduction in computational complexity as compared to the tests in K¨oro˘glu and Scherer [2005b,a]. Moreover, we can expect reduction of conservatism especially in the case of parametric perturbations, since the self-adjointness of the variation operator for selfadjoint perturbations is utilized in the derivation. 4. ESTIMATION OF CONSERVATISM IN ROBUST STABILITY ANALYSIS FOR DYNAMIC LTV PERTURBATIONS
where x ˜ ∈ ℓk2 , z˜ ∈ ℓη2 , and Γx , [ a1x · · · arx ], ajx ∈ Ck ; Γz , [ a1z · · · arz ], ajz ∈ Cη satisfy Γx Ω = AM Γx + BM Γw ,
(18)
Γz
(19)
= CM Γx + DM Γw .
PROOF. By direct substitution of (16) & (17) in (13) & (14). 4.2 Power Distribution Theorem The power distribution theorem of Wolodkin and Poolla [1994] relates the power distribution capability of an LTV operator to its rate-of-variation in the ℓ2 sense, as summarized in the following lemma: Lemma 9. (Wolodkin and Poolla [1994] ) Consider w(t) = 2ℜe {Γw Ωt 1} and z(t) = 2ℜe {Γz Ωt 1}, where Ω , [eωj ]rj=1 , 0 ≤ ω1 < . . . < ωr ≤ π; Γw , [a1w · · · arw ], ajw ∈ Cm ; Γz , [a1z · · · arz ], ajz ∈ Cm . T Define aw , az ∈ Rr as aw , ka1w k · · · karw k , T √ az , ka1z k · · · karz k , where kak , a∗ a, and assume that they satisfy
In this section, we provide a novel test based on convex optimization, for determining when a given system cannot be robustly stable. More specifically, we sketch a procedure to obtain the minimum maximal rates-of-variation for the dynamic uncertainties that will destabilize the system. The derivation is based on sinusoidal signal interpolation with LTI and LTV operators, and in particular the lemmas that will be provided in the following subsections, before the main result is described.
Then, there exists an LTV operator ∆ ∈ BLcm×m with kV∆ k ≤ ν such that ∆z = w + w, ˜ w ˜ ∈ ℓm 2 , if opt and only if ν ≥ ν (Γw , Γz , Ω), where
4.1 LTI System Response to Sinusoidal Inputs The following lemma summarizes the LTI system response to multi-sinusoidal inputs:
4.3 A Test for Lack of Robust Stability Against Dynamic LTV Perturbations
Lemma 8. Let M ∈ RHη×η be a causal and ∞ stable LTI operator that admits a state-space description of the form x(t + 1) = AM x(t) + BM w(t),
(13)
z(t)
(14)
= CM x(t) + DM w(t),
where AM ∈ Rk×k , BM ∈ Rk×η , CM ∈ Rη×k and DM ∈ Rη×η , with AM having all its zeros strictly inside the unit disk. Consider a multi-sinusoidal input, w ∈ ℓη2e , of the form r X w(t) = 2ℜe ajw eωj t = 2ℜe Γw Ωt 1 , (15) j=1
where Ω , [eωj ]rj=1 , with 0 ≤ ω1 < . . . < ωr ≤ π denoting a group of distinct frequencies, Γw , [ a1w · · · arw ], with ajw ∈ Cη denoting a group of amplitude vectors, and 1 , [ 1 · · · 1 ]T . For this input, the state and the output signals will be of the form x(t) = x ˜(t) + 2ℜe Γx Ωt 1 , (16) t z(t) = z˜(t) + 2ℜe Γz Ω 1 , (17)
tr(Γw Γ∗w ) = aTw aw ≤ tr(Γz Γ∗z ) = aTz az .
ν opt ,
inf
(20)
σ(ΩS − SΩ) (21)
S∈Cr×r ,σ(S)≤1,Saz =aw
≤ 2 sin (0.5(ωr − ω1 )) .
(22)
The test that we will provide below has its origins in the alternative proof provided by Poolla and Tikku [1995] for the result of Shamma [1994] on exactness of constant D-scaling test. It was realized in K¨oro˘glu and Scherer [2005a], where robust performance analysis problem was considered, that one can verify lack of robust performance for perturbations with sufficiently fast variations, by employing constant D-scaling test over sufficiently long frequency intervals together with a restricted use of power distribution theorem (up to the upper bound provided by (22)). The main result of this section presents the robust stability analysis version, by also providing an improvement in the form of full use of power distribution theorem, thanks to the employment of semi-definite programming duality. Let us now briefly describe (within the setup of Problem 1 and, for simplicity of presentation, with li = 1, ∀i ∈ I) the starting point, which is basically the reformulation of the constant Dscaling test over a specified interval, in the form
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of an optimization problem, through the use of generalized KYP lemma. Let R Φ be constructed M according to (5) with Φ = , using a minimal I realization for M , and with ρ0 = 2 cos(0.5d̟), d̟ ∈ [0, π]; ρ1 = e̟ , ̟ ∈ [0.5d̟, π − 0.5d̟], let the set L̟,d̟ be defined as −P0 ρ1 P1 ρ1 P1 P0 − ρ0 P1 : X L̟,d̟ , −X . (23) X = [xi ⊗ Imi ]κi=1 , xi ∈ R+ P0 ∈ HkΦ , P1 ∈ Hk+Φ
Consider the minimization problem β opt =
inf
β, (24)
L∈L̟,d̟ , RT LRΦ <βI, P1 >−βI, [xi ]κ >−βI Φ i=1
the optimum value of which can also be obtained as β opt =
k
K∈H+Φ
+η
max k
L(K, Q, [yi ]κi=1 ), (25)
,Q∈H+Φ ,yi ∈R+ ,∀i∈I
where L(K, Q, [yi ]κi=1 ) denotes the Lagrange dual cost L(K, Q, [yi ]κi=1 ) ,
inf
L∈L̟,d̟
l(L, K, Q, [yi ]κi=1 ),(26)
with l(L, K, Q, [yi ]κi=1 ) denoting the Lagrange function defined as l(L, K, Q, [yi ]κi=1 ) , T ℜe tr KRΦ LRΦ −tr(QP1 )−tr ([yi xi ]κi=1 ) + β (1 − ℜe {tr(K) + tr(Q) + tr([yi ]κi=1 )}). (27) Clearly, β opt < 0 means that the constant Dscaling test is not satisfied by M over the frequency interval [̟ − 0.5d̟, ̟ + 0.5d̟]. Taking the infeasibility of the constant D-scaling test over frequency intervals as a starting point, the following theorem specifies the class of perturbations (in particular the least maximal rates-of-variation thereof) against which M cannot be robustly stable. Theorem 10. Suppose that we have β opt < 0 in (24). Let us now consider a possible factorization for the optimal dual multiplier K ◦ of (25) as Γx ∗ ∗ Γx Γw , (28) K◦ = Γw where Γx ∈ Ck×r , Γw ∈ Cη×r with r = rank(K), and let us assume that Γx has full column rank. With Ω = [eωj ]rj=1 , ωj ∈ [̟ − 0.5d̟, ̟ + 0.5d̟], obtained from the eigenvalue decomposition of Λ , (Γ∗x Γx )−1 Γ∗x Γv
(29) −1
= UΛ ΩUΛ−1 , UΛ = (UΛ∗ )
∈ Cr×r , (30)
and with Γw◦ = [ Γ∗w◦ · · · Γ∗wκ◦ ]∗ , Γwi◦ ∈ Cmi ×r ; 1 Γz◦ = [ Γ∗z◦ · · · Γ∗zκ◦ ]∗ , Γzi◦ ∈ Cmi ×r obtained as 1 Γw◦ = Γw UΛ , Γz◦ = (CM Γx + DM Γw )UΛ , the system cannot be robustly ℓ2 -stable against ∆sν , if νi ≥ νiopt , ν opt (Γwi◦ , Γzi◦ , Ω), for i = 1, . . . , κ, where νiopt ’s are obtained from (21).
PROOF. The proof, which we only sketch here for reasons of space, is based on constructing multi-sinusoidal signals w◦ , z ◦ and a contractive LTV operator ∆◦ ∈ ∆sν that maps z ◦ to w◦ (up to some transient component), which is mapped back to z ◦ (up to some transient component again) when applied as input to M . The first step is to show that, the amplitude matrices Γw◦ , Γz◦ and Γx◦ = Γx UΛ satisfy (18) and (19), with the Ω obtained as in the theorem statement. This can be done by employing the semi-definite programming duality theory (see e.g. Boyd and Vandenberghe [2004]). It can also be observed from this investigation that we have tr(Γwi◦ Γ∗w◦ ) ≤ tr(Γzi◦ Γ∗z◦ ), for all i i i = 1, . . . , κ. Recalling Lemma 8, we can easily observe as the next step that, for the multi-sinusoidal signals w◦ (t) = [ w1◦ · · · wκ◦ ] = 2ℜe {Γw◦ Ωt 1}, z ◦ (t) = [ z1◦ · · · zκ◦ ] = 2ℜe {Γz◦ Ωt 1}, we will have M w◦ = z ◦ + z˜◦ , for some z˜◦ (t) ∈ ℓη2 . On the other hand, we can conclude from Lemma 9 that there exist LTV operators ∆◦i , i = 1, . . . , κ with kV∆◦i ki2 ≤ νiopt , for which ∆◦i zi◦ = wi◦ + w ˜i◦ , for i some w ˜i◦ ∈ ℓm . Clearly, the operator formed as 2 ∆◦ = [∆◦i ]κi=1 then satisfies ∆◦ z = w◦ + w ˜◦ , with w ˜ ∈ ℓη2 . If ν opt (Γwi◦ , Γzi◦ , Ω) ≤ νi , ∀i ∈ I, then ∆◦ ∈ ∆sν . Thus showing the existence of an oscillation in the loop in response to finite energy signals establishes the required result, which might be shown more precisely based on similar arguments to the proof of Theorem 3.2 of Poolla and Tikku [1995]. Remark 11. The test of Theorem 10 can be applied on various frequency intervals [̟ − 0.5d̟, ̟ +0.5d̟] to verify lack of robust stability for different maximal rates-of-variation. Referring to (22), we can note that the maximal rates-ofvariation for which the lack of robust stability is to be verified can be controlled via the interval length, d̟. More clearly, as we let d̟ → 0, we will obtain analysis results for slow variations, whereas for d̟ → π, we will investigate robust stability against fast perturbations. 5. ILLUSTRATIVE EXAMPLE In this section, we consider an academic example in which we assume that a plant with realization −0.28 −0.37 0.44 −0.21 −0.11 0.08 0.05 0.43 −0.15 −0.21 0.01 0.03 AM BM 0.07 0.13 0.32 −0.24 −0.02 0.00 = , 0.35 0.31 2.00 −0.84 0.00 0.00 CM DM 1.49 −0.88 0.05 0.00 0.00 0.00 0.27 −2.40
1.00 −7.20
0.00 0.00
is subject to perturbations in two uncertainty channels, ∆1 ∈ L1×1 and ∆2 ∈ BL2×2 . Our obc c jective is to determine maximum allowable norm bound, β1 = max k∆1 ki2 , versus maximum normalized rate-of-variation, ν1 = max kV∆1 ki2 /β1 , in the first channel. Regions where the existence or lack of robust stability of the system is verified for arbitrary and self-adjoint first channel uncertainties with different tests are shown in Figure 2.
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perturbations. The new test offers reduced computational complexity (and reduced conservatism, especially in the case of parametric perturbations) as compared to K¨oro˘glu and Scherer [2005b]. Moreover, a method is developed for estimating the conservatism of this analysis test in the case of dynamic LTV perturbations. Extension of this method to parametric LTV perturbations is challenging and deserves further investigation.
βlb: (∆ =∆* ∈L ) with test of Theorem 5 1 1 1 c lb * β1 : (∆1=∆1∈Lc) with test of Köroglu&Scherer[2005b] lb β1 : (∆1∈Lc) with test of Theorem 5 βub: (∆ ∈L ) with test of Theorem 10 (optimization) 1 1 c ub β1 : (∆1∈Lc) with test of Theorem 10 (sine formula)
4
β1 = max ||∆1||i2 (lower and upper bounds)
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3.9 3.8 3.7 3.6 3.5 3.4 3.3
ACKNOWLEDGEMENT
3.2 3.1 3
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ν = max ||V || /β 1
∆
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i2
1
Fig. 2. Robust stability regions in ν1 − β1 domain. We observe from the figure that, in the case of parametric first channel perturbations, (∆1 = ∆∗1 ∈ Lc1×1 ), the new test provides better results than the test of K¨ oro˘ glu and Scherer [2005b] (with significantly reduced computational complexity). We also note that, when the first channel perturbations are dynamic LTV, in addition to obtaining a lower boundary for the stability region by the test of Theorem 5, we can also obtain an upper boundary for it by the test of Theorem 10. And we moreover realize that the upper boundary determined through optimization (i.e. the lower left boundary of the cloud) is significantly better than the boundary determined by the sine formula. A closer analysis reveals that the lower bounds obtained for the stability margin β1 in the case of dynamic first channel perturbations become exact for ν → 0 and ν ≥ 1.55. Exactness for ν → 0 is already expected from the result of Paganini [1996], which established the exactness of analysis based on mixed constant/frequency-dependent D-scaling test for mixed dynamic perturbations with arbitrarily fast/slow variations. On the other hand, exactness of the lower bounds for ν ≥ 1.55 is established thanks to the method developed in Theorem 10. With this method, we first obtained the value of β1 for which constant D-scaling is not satisfied (over the whole frequency interval [0, π]) as 3.044. Together with this, we also obtained the minimum normalized rate-of-variation in the first channel of a destabilizing dynamic perturbation as 1.55. Note that the second channel was assumed to be dynamic and allowed to vary arbitrarily fast (i.e. ν2 ≤ 2). This allowed us to determine the minimum destabilizing rate-of-variation in the first channel without having to check the ratesof-variation obtained for the second channel also as a result of the application of the method of Theorem 10. 6. CONCLUDING REMARKS We have extended the approach of Helmersson [1999] to develop a generalized version of the frequency-dependent D-G scaling test for robust stability analysis against general structured LTV
Supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
REFERENCES F. Amato, M. Corless, M. Mattei, and R. Setola. A multivariable stability margin in the presence of timevarying, bounded rate gains. Int. Journ. Rob. & Nonl. Cont., 7(2):127–143, Feb. 1997. M. Athans, S. Fekri, and A. Pascoal. Issues on robust adaptive feedback control. In Proc. 16th IFAC World Cong., Prague, Czech Rep., July 2005. S. P. Boyd and L. Vandenberghe. Convex Optimization. Cambridge Uni. Pr., Cambridge, 2004. P. Gahinet, P. Apkarian, and M. Chilali. Affine parameterdependent lyapunov functions and real parametric uncertainty. IEEE Trans. Aut. Cont., 41(3):436–442, March 1996. A. Helmersson. An IQC-based stability criterion for systems with slowly varying parameters. In Proc. 14th IFAC World Cong., pages 525–530, Beijing, China, July 1999. T. Iwasaki, S. Hara, and A. L. Fradkov. Time domain interpretations of frequency domain inequalities on (semi)finite ranges. Syst. & Cont. Lett., 54(7):681– 691, July 2005. U. J¨ onsson and A. Rantzer. Systems with uncertain parameters - time-variations with bounded derivatives. Int. Journ. Rob. & Nonl. Cont., 6(9–10):969–982, Nov. 1996. H. K¨ oro˘ glu and C. W. Scherer. Robust performance analysis for linear slowly time-varying perturbations. In Proc. 44th IEEE Conf. Dec. & Cont., Sevilla, Spain, Dec. 2005a. to appear. H. K¨ oro˘ glu and C. W. Scherer. Robust stability analysis for structured uncertainties with bounded variation rates. In Proc. 16th IFAC World Cong., Prague, Czech Rep., July 2005b. J. L¨ ofberg. YALMIP : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. Available from http://control.ee.ethz.ch/~joloef/yalmip.php. A. Megretski and A. Rantzer. System analysis via integral quadratic constraints. IEEE Trans. Aut. Cont., 42(6): 819–830, June 1997. F. Paganini. Robust Stability under Mixed Time-Varying, Time-Invariant and Parametric Uncertainty. Automatica, 32(10):1381–1392, Oct. 1996. K. Poolla and A. Tikku. Robust performance against timevarying structured perturbations. IEEE Trans. Aut. Cont., 40(9):1589–1602, Sept. 1995. J. S. Shamma. Robust stability with time-varying structured uncertainty. IEEE Trans. Aut. Cont., 39(4):714– 724, April 1994. G. Wolodkin and K. Poolla. Spectral power distribution using time-varying operators. In Proc. American Cont. Conf., pages 3147–3151, Baltimore, ML, June 1994.
ROCOND'06, Toulouse, France, July 5-7, 2006
ROBUST STABILITY ANALYSIS TESTS FOR LINEAR TIME-VARYING PERTURBATIONS WITH BOUNDED RATES-OF-VARIATION Hakan K¨ oro˘ glu
Carsten W. Scherer
Delft Center for Systems and Control, Delft University of Technology Mekelweg 2 2628 CD Delft The Netherlands Phone : +31(0)1527{87171,85899} Fax : +31(0)152786679 E-Mail : {h.koroglu,c.w.scherer}@dcsc.tudelft.nl Abstract: Robust stability analysis is considered for linear time-varying perturbations with bounded norms and rates-of-variation, and a possible generalization of the frequency-dependent D-G-scaling test is provided in an IQC framework by favor of a generalized version of the swapping lemma. Moreover, a convexoptimization-based method is developed for testing when a system cannot be robustly stable, by employing a generalized version of the Kalman-Yakubovichc Popov lemma together with a power distribution theorem. Copyright 2006 IFAC Keywords: Robust stability, stability analysis, structured uncertainty, linear matrix inequalities (LMI), integral quadratic constraints (IQC).
1. INTRODUCTION There are natural motivations for considering the problem of robust stability analysis against structured linear time-varying (LTV) perturbations in a robust control setting, since there will be variations with time in any engineering system, some of them being introduced by the designer for adaptation purposes (see e.g. Athans et al. [2005]). Two crucial results by Shamma [1994], Poolla and Tikku [1995] have established that constant/frequency-dependent D-scaling tests can be employed as exact analysis tests in the cases of arbitrarily fast/slow dynamic LTV perturbations respectively. For parametric LTV perturbations, there are numerous works that employ parameterdependent Lyapunov function approach to exploit the information on maximal rates-of-variation in analysis (see e.g. Gahinet et al. [1996], Amato et al. [1997]). However, the parameter-dependent Lyapunov function approach is mainly applicable for parametric perturbations, whereas the unifying approach of Megretski and Rantzer [1997] based on integral quadratic constraints in an operator setting can be employed for analysis against dynamic rate-bounded LTV perturbations as well. In fact, the IQC test of J¨ onsson and Rantzer [1996], based on the so-called swapping lemma,
was extended by K¨oro˘glu and Scherer [2005b,a] for robust stability and performance analysis against dynamic LTV perturbations with specified maximal rates-of-variation. For considering dynamic LTV perturbations instead of, or in addition to, time-varying parametric uncertainties, there are motivations like reducing the computational complexity of analysis or examining the safety and reliability of the designed systems in more uncertain environments. In this paper, we study the robust stability analysis problem, formulated in the next section for structured LTV perturbations as considered also in K¨oro˘glu and Scherer [2005b,a]. We develop in Section 3 an IQC test by generalizing the approach of Helmersson [1999], which was also based on (an implicit version of) the swapping lemma and provided as an improvement to the test of J¨onsson and Rantzer [1996]. As a novel contribution in the direction of conservatism analysis, we provide in Section 4 a convex-optimization-based method, developed by employing a generalized version of the Kalman-Yakubovich-Popov lemma together with the power distribution theorem of Wolodkin and Poolla [1994]. We conclude by some final remarks.
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ξ
w
M
ROCOND'06, Toulouse, France, July 5-7, 2006
3. AN IQC APPROACH TO ROBUST STABILITY ANALYSIS FOR STRUCTURED LTV PERTURBATIONS
z
∆
ζ
In this section, we will present a test for analyzing when a given system is robustly stable. The derivation is based on the IQC stability theorem and a generalized version of the swapping lemma, which we will summarize before we present the main result of the section. 3.1 IQC-Based Robust Stability Analysis
Fig. 1.Standard setup for robust stability analysis. 2. PROBLEM FORMULATION The problem that we consider in this paper is formulated within the generic setup of Figure 1, where the system is represented by a stable and proper real-rational transfer function, M ∈ η×η RH∞ , interconnected in a feedback configuration with an unknown causal and linear operator, ∆ ∈ Lcη×η , coming out of the set ∆. We are especially interested in block-diagonal structured perturbations, ∆ = [Ili ⊗ ∆i ]κi=1 , diag (Il1 ⊗ ∆1 , . . . , Ilκ ⊗ ∆κ ), that have induced ℓ2 -norms of at most unity, i.e. ∆ ∈ BLη×η , and that have specc ified maximal rates-of-variation, kV∆ ki2 (≤ 2), where V∆ , λ−1 ∆λ − ∆, with λ/λ−1 denoting the right/left shift operators respectively. The feedback interconnection is referred to as wellposed, if the map (I − M ∆) has a causal inverse, (I − M ∆)−1 ∈ Lη×η . We say that the system is c uniformly robustly ℓ2 -stable against ∆, if, moreover, (I − M ∆)−1 i2 is bounded in ∆, i.e. if ∃β ∈ R+ such that k(I−M ∆)−1 ki2 < β, ∀∆ ∈ ∆. If the system in Figure 1 is robustly stable, for any ∆ ∈ ∆ we will have w ∈ ℓη2 , z ∈ ℓη2 , whenever ξ ∈ ℓη2 , ζ ∈ ℓη2 . Based on these notions, the problem that we study is formulated as follows: Problem 1. Consider the feedback system in Figure 1 with ha given causal and stable LTI plant, i AM BM −1 , C − AM )−1 BM + M (λ−1 ) = CM M (λ DM
DM ∈ RHη×η ∞ , subject to causal and bounded structured LTV perturbations ∆ that are contained in the set ∆ = ∆sν ⊂ BLη×η , defined as c i ×mi ; [Ili ⊗ ∆i ]κi=1 : ∆i ∈ BLm c , (1) ∆sν , kV∆i ki2 ≤ νi , i ∈ I; ∆∗i = ∆i , i ∈ Ip where s = {li , mi , ai }κi=1 , (ai = −1, for i ∈ Ip ⊆ I = {1, . . . , κ}; ai = 0, for i ∈ Id , I \Ip , with Ip intended to represent the index set of parametric perturbation channels) and ν = {νi }κi=1 , νi ∈ [0, 2], describe, respectively, the structure and the maximum rate-of-variation of the uncertainty. For later use, define Iv , {i ∈ I : νP i 6= 0}, Ipv , Ip ∩ Iv , Idv , Id ∩ Iv and κp , i∈Ip 1, P κv , i∈Iv 1. Determine whether the feedback system is robustly ℓ2 -stable against ∆sν or not.
With the adjoint of an operator, T ∈ Lm×n , defined as the unique operator, T ∗ ∈ Ln×m , for which hT x, yi =Phx, T ∗ yi , ∀x ∈ ℓn2 , ∀y ∈ ℓm 2 , where T hx, yi , x (t)y(t), and with the positivet semi-definiteness, T ≥ 0, of a self-adjoint operator, T = T ∗ ∈ Lm×m , understood as hT x, xi ≥ 0, ∀x ∈ ℓm 2 , the IQC approach to robust stability analysis can be summarized as follows: Theorem 2. (Megretski and Rantzer [1997]) Let ∗ I Π11 Π12 I η×η ∆Π , ∆ ∈ Lc : ≥ 0 , (2) ∆ Π∗12 Π22 ∆ 2η×2η where the Π = Π∗ ∈ RL∞ and assume that
(i) (M, τ ∆) is well-posed ∀τ ∈ [0, 1] and ∀∆ ∈ ∆. (ii) τ ∆ ∈ ∆Π , ∀τ ∈ [0, 1] and ∀∆ ∈ ∆. M is uniformly robustly ℓ2 -stable against ∆, if, ∀ω ∈ [0, 2π], M (eω ) satisfies ∗ M (eω ) Π11 (eω )Π12 (eω ) M (eω ) < 0. (3) I Π∗12 (eω )Π22 (eω ) I The LMI versions of the IQC tests are usually derived by employing the Kalman-YakubovichPopov lemma, which we provide below in a generalized form from Iwasaki et al. [2005]: Lemma 3. i (Generalized KYP Lemma) Let Φ = h AΦ BΦ Φ ×nΦ ∈ RHm , where AΦ ∈ RkΦ ×kΦ , ∞ CΦ DΦ BΦ ∈ RkΦ ×nΦ , CΦ ∈ RmΦ ×kΦ , DΦ ∈ RmΦ ×nΦ , with AΦ having all its eigenvalues strictly inside the unit disk, and let Q ∈ HmΦ be a Hermitian matrix. With d̟ ∈ [0, 2π] and ̟ ∈ [0.5d̟, 2π − 0.5d̟], the following conditions are equivalent: (i) For all ω ∈ [̟ − 0.5d̟, ̟ + 0.5d̟], ∗ Φ(eω ) Q Φ(eω ) < 0.
(4)
T RΦ LRΦ
(ii) There exists L ∈ L̟,d̟ such that < 0, where I 0 RΦ , AΦ BΦ , (5) CΦ DΦ −P0 ρ1 P1 0 ρ1 P1 P0 − ρ0 P1 0 : 0 0 Q L̟,d̟ , .(6) ̟ ρ = 2 cos(0.5d̟), ρ = e , 0 1 P0 ∈ HkΦ , P1 ∈ Hk+Φ
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3.2 Generalized Swapping Lemma The following lemma, reformulated from K¨oro˘glu and Scherer [2005b], describes the effect of composing LTI and LTV operators in changing orders and will help to develop the IQC test in the sequel: Lemma 4. (Generalized Swapping Lemma) Consider, in the setup of Problem 1, a ∆r = [Ili ⊗ ∆i ]i∈I ∈h ∆sν , together with H , [Hi ⊗ Imi ]i∈I , i Hi =
AHi BHi CHi DHi
∈ RHq∞i ×li , where AHi ∈
Rki ×ki , BHi ∈ Rki ×li , CHi ∈ Rqi ×ki , DHi ∈ Rqi ×li , with AHi ’s having all their eigenvalues strictly inside the unit disk. Let ∆l , [Iqi ⊗∆i ]i∈I , −1 ηv ×ηv ∆ , where ηv , Pc , [Iki ⊗ νi ∆i ]i∈Iv ∈ Lc k m , and define V such that V V∆ c V T = i∈Iv i i κ [Iki ⊗ V∆i ]i=1 (i.e. V = [Vij ⊗ Ini ×nIv (j) ], Vij = √ νi Iki , if Iv (j) = i; Vij = 0ki ×kIv (j) , otherwise, i = 1, . . . , κ, j = 1, . . . , κv ). With HB , [HiB ⊗ I]i∈I and HC , [HiC ⊗ Imi ]i∈I , where HiB , λ−1 (λ−1 − AHi )−1 BHi , −1
HiC , CHi (λ
−1
− AHi )
,
(7) (8)
we have ∆r H ∆l 0 H HC V = . (9) 0 V∆c V THB 0 I V∆cV THB | {z } | {z } | {z } | {z } Hel
∆er
∆el
Her
3.3 A Swapping-Lemma-Based Robust Stability Analysis Test for Structured LTV Perturbations Generalizing the IQC test of Helmersson [1999], the following theorem provides a method to analyze whether a given system is robustly stable against a set of dynamic LTV perturbations with specified maximal rates-of-variation: Theorem 5. Consider the setup of Problem 1 and assume that (M, τ ∆) is well-posed for all ∆ ∈ ∆sν and τ ∈ [0, 1]. With Hi ∈ RHq∞i ×li being given stable (basis) transfer functions, let Her and Hel be defined as in Lemma 4. Construct the matrix RΦ according to (5) with a (preferably) Her Me minimal realization for Φ = , where Hel Me , M 0η×ηv is an extended version of the original plant, and define the set L as −P P X 0 Y 0 : 0 X 0 Y v v T Y 0 −X 0 T 0 −X 0 Y v v . (10) L, kΦ P ∈S qi X =[X ⊗I ] ,X ∈S i m i∈I i i + ki Xv=[Xvi⊗Imi]i∈Iv,Xvi ∈S+ T qi×qi Y =[Yi ⊗Imi ]i∈I ,Yi = ai Yi ∈R T ki×k i Yv =[Yvi⊗Imi ]i∈Iv ,Yvi= ai Yvi ∈R
Then, M is robustly stable against ∆sν if there T exists an L ∈ L such that RΦ LRΦ < 0. PROOF. The proof is based on establishing the robust stability of Me against ∆r ∆r = [Ili ⊗∆i ]i∈I ∈ ∆sν , ∆e , ∆er = : , V∆cV THB V∆c =[Iki ⊗νi−1 V∆i ]i∈Iv which necessarily implies the robust stability of M against ∆. To obtain a multiplier class for the extended uncertainty set, we first observe that
I τ ∆er
∗
∗ Her Xe Her Her Ye Hel I ∗ T ∗ τ ∆er Hel Ye Her −Hel Xe Hel {z } | Πe
∗ = Her
I τ ∆el
∗
Xe Ye I Her , τ ∆el YeT −Xe {z } | Q
for any ∆er ∈ ∆e and for matrices Xe and Ye of appropriate sizes. It then follows immediately that we will have τ ∆er ∈ ∆Πe , ∀∆er ∈ ∆e and ∀τ ∈ [0, 1], provided that Xe = diag (X, Xv ), and Ye = diag (Y, Yv ), with X, Xv , Y and Yv being chosen as in (10). Theorem 5 then follows from an application of Theorem 2 followed by the employment of the standard KYP lemma. 2 Remark 6. The method identified by Theorem 5 can be viewed as a possible generalization of the D-G scaling test. In case νi = 0, ∀i, the terms V THB and HC V (as well as Xv , Yv ) are void and thus we recover the frequency-dependent D-G scaling test. On the other hand, if we choose Hi = Ili , ∀i ∈ I, these terms are again void and we thus recover the constant D-G scaling test. Remark 7. The number of decision variables in the LMI test of Theorem 5 depend heavily on the McMillan degree of Φ, and thus on the choice of basis transfer functions. The number of extra variables introduced (due to the time-varying channels)P as compared toPthe LTI case is given by nv = i∈Ipv ki2 + 0.5 i∈Idv ki (ki + 1). If we
consider bases of the form Hi = [ I λ · · · λpi ]T , we can find minimal realizations as 0 0 1 I(pi −1) 0 0 AHi BHi = (11) 0 0 1 ⊗ Ili , CHi DHi I(pi −1) 0 0 0 1 0 for which we clearly have qi = (pi + 1)li and ki = pi li . With AH , [AHi ⊗Imi ]κi=1 , BH , [BHi ⊗ Imi ]κi=1 , CH , [CHi ⊗ Imi ]κi=1 and DH , [DHi ⊗ Imi ]κi=1 , we can obtain a (not necessarily -yet most probably- minimal) realization for Φ as
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AM 0 BH CM A H 0 0 CH Φ= H CM D V TBH CM V TAH 0 0 0 0
0 BM 0 BH DM AH BH 0 DH DM 0 V TBH DM CH DH 0 0
ROCOND'06, Toulouse, France, July 5-7, 2006
0 0 V 0 0 0 Iηv
. (12)
Thus, the total number of decision variables, as well as the number of extra variables as compared to the LTI case, can be observed to be proportional to p2i for the considered basis. We should note that there is significant reduction in computational complexity as compared to the tests in K¨oro˘glu and Scherer [2005b,a]. Moreover, we can expect reduction of conservatism especially in the case of parametric perturbations, since the self-adjointness of the variation operator for selfadjoint perturbations is utilized in the derivation. 4. ESTIMATION OF CONSERVATISM IN ROBUST STABILITY ANALYSIS FOR DYNAMIC LTV PERTURBATIONS
where x ˜ ∈ ℓk2 , z˜ ∈ ℓη2 , and Γx , [ a1x · · · arx ], ajx ∈ Ck ; Γz , [ a1z · · · arz ], ajz ∈ Cη satisfy Γx Ω = AM Γx + BM Γw ,
(18)
Γz
(19)
= CM Γx + DM Γw .
PROOF. By direct substitution of (16) & (17) in (13) & (14). 4.2 Power Distribution Theorem The power distribution theorem of Wolodkin and Poolla [1994] relates the power distribution capability of an LTV operator to its rate-of-variation in the ℓ2 sense, as summarized in the following lemma: Lemma 9. (Wolodkin and Poolla [1994] ) Consider w(t) = 2ℜe {Γw Ωt 1} and z(t) = 2ℜe {Γz Ωt 1}, where Ω , [eωj ]rj=1 , 0 ≤ ω1 < . . . < ωr ≤ π; Γw , [a1w · · · arw ], ajw ∈ Cm ; Γz , [a1z · · · arz ], ajz ∈ Cm . T Define aw , az ∈ Rr as aw , ka1w k · · · karw k , T √ az , ka1z k · · · karz k , where kak , a∗ a, and assume that they satisfy
In this section, we provide a novel test based on convex optimization, for determining when a given system cannot be robustly stable. More specifically, we sketch a procedure to obtain the minimum maximal rates-of-variation for the dynamic uncertainties that will destabilize the system. The derivation is based on sinusoidal signal interpolation with LTI and LTV operators, and in particular the lemmas that will be provided in the following subsections, before the main result is described.
Then, there exists an LTV operator ∆ ∈ BLcm×m with kV∆ k ≤ ν such that ∆z = w + w, ˜ w ˜ ∈ ℓm 2 , if opt and only if ν ≥ ν (Γw , Γz , Ω), where
4.1 LTI System Response to Sinusoidal Inputs The following lemma summarizes the LTI system response to multi-sinusoidal inputs:
4.3 A Test for Lack of Robust Stability Against Dynamic LTV Perturbations
Lemma 8. Let M ∈ RHη×η be a causal and ∞ stable LTI operator that admits a state-space description of the form x(t + 1) = AM x(t) + BM w(t),
(13)
z(t)
(14)
= CM x(t) + DM w(t),
where AM ∈ Rk×k , BM ∈ Rk×η , CM ∈ Rη×k and DM ∈ Rη×η , with AM having all its zeros strictly inside the unit disk. Consider a multi-sinusoidal input, w ∈ ℓη2e , of the form r X w(t) = 2ℜe ajw eωj t = 2ℜe Γw Ωt 1 , (15) j=1
where Ω , [eωj ]rj=1 , with 0 ≤ ω1 < . . . < ωr ≤ π denoting a group of distinct frequencies, Γw , [ a1w · · · arw ], with ajw ∈ Cη denoting a group of amplitude vectors, and 1 , [ 1 · · · 1 ]T . For this input, the state and the output signals will be of the form x(t) = x ˜(t) + 2ℜe Γx Ωt 1 , (16) t z(t) = z˜(t) + 2ℜe Γz Ω 1 , (17)
tr(Γw Γ∗w ) = aTw aw ≤ tr(Γz Γ∗z ) = aTz az .
ν opt ,
inf
(20)
σ(ΩS − SΩ) (21)
S∈Cr×r ,σ(S)≤1,Saz =aw
≤ 2 sin (0.5(ωr − ω1 )) .
(22)
The test that we will provide below has its origins in the alternative proof provided by Poolla and Tikku [1995] for the result of Shamma [1994] on exactness of constant D-scaling test. It was realized in K¨oro˘glu and Scherer [2005a], where robust performance analysis problem was considered, that one can verify lack of robust performance for perturbations with sufficiently fast variations, by employing constant D-scaling test over sufficiently long frequency intervals together with a restricted use of power distribution theorem (up to the upper bound provided by (22)). The main result of this section presents the robust stability analysis version, by also providing an improvement in the form of full use of power distribution theorem, thanks to the employment of semi-definite programming duality. Let us now briefly describe (within the setup of Problem 1 and, for simplicity of presentation, with li = 1, ∀i ∈ I) the starting point, which is basically the reformulation of the constant Dscaling test over a specified interval, in the form
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of an optimization problem, through the use of generalized KYP lemma. Let R Φ be constructed M according to (5) with Φ = , using a minimal I realization for M , and with ρ0 = 2 cos(0.5d̟), d̟ ∈ [0, π]; ρ1 = e̟ , ̟ ∈ [0.5d̟, π − 0.5d̟], let the set L̟,d̟ be defined as −P0 ρ1 P1 ρ1 P1 P0 − ρ0 P1 : X L̟,d̟ , −X . (23) X = [xi ⊗ Imi ]κi=1 , xi ∈ R+ P0 ∈ HkΦ , P1 ∈ Hk+Φ
Consider the minimization problem β opt =
inf
β, (24)
L∈L̟,d̟ , RT LRΦ <βI, P1 >−βI, [xi ]κ >−βI Φ i=1
the optimum value of which can also be obtained as β opt =
k
K∈H+Φ
+η
max k
L(K, Q, [yi ]κi=1 ), (25)
,Q∈H+Φ ,yi ∈R+ ,∀i∈I
where L(K, Q, [yi ]κi=1 ) denotes the Lagrange dual cost L(K, Q, [yi ]κi=1 ) ,
inf
L∈L̟,d̟
l(L, K, Q, [yi ]κi=1 ),(26)
with l(L, K, Q, [yi ]κi=1 ) denoting the Lagrange function defined as l(L, K, Q, [yi ]κi=1 ) , T ℜe tr KRΦ LRΦ −tr(QP1 )−tr ([yi xi ]κi=1 ) + β (1 − ℜe {tr(K) + tr(Q) + tr([yi ]κi=1 )}). (27) Clearly, β opt < 0 means that the constant Dscaling test is not satisfied by M over the frequency interval [̟ − 0.5d̟, ̟ + 0.5d̟]. Taking the infeasibility of the constant D-scaling test over frequency intervals as a starting point, the following theorem specifies the class of perturbations (in particular the least maximal rates-of-variation thereof) against which M cannot be robustly stable. Theorem 10. Suppose that we have β opt < 0 in (24). Let us now consider a possible factorization for the optimal dual multiplier K ◦ of (25) as Γx ∗ ∗ Γx Γw , (28) K◦ = Γw where Γx ∈ Ck×r , Γw ∈ Cη×r with r = rank(K), and let us assume that Γx has full column rank. With Ω = [eωj ]rj=1 , ωj ∈ [̟ − 0.5d̟, ̟ + 0.5d̟], obtained from the eigenvalue decomposition of Λ , (Γ∗x Γx )−1 Γ∗x Γv
(29) −1
= UΛ ΩUΛ−1 , UΛ = (UΛ∗ )
∈ Cr×r , (30)
and with Γw◦ = [ Γ∗w◦ · · · Γ∗wκ◦ ]∗ , Γwi◦ ∈ Cmi ×r ; 1 Γz◦ = [ Γ∗z◦ · · · Γ∗zκ◦ ]∗ , Γzi◦ ∈ Cmi ×r obtained as 1 Γw◦ = Γw UΛ , Γz◦ = (CM Γx + DM Γw )UΛ , the system cannot be robustly ℓ2 -stable against ∆sν , if νi ≥ νiopt , ν opt (Γwi◦ , Γzi◦ , Ω), for i = 1, . . . , κ, where νiopt ’s are obtained from (21).
PROOF. The proof, which we only sketch here for reasons of space, is based on constructing multi-sinusoidal signals w◦ , z ◦ and a contractive LTV operator ∆◦ ∈ ∆sν that maps z ◦ to w◦ (up to some transient component), which is mapped back to z ◦ (up to some transient component again) when applied as input to M . The first step is to show that, the amplitude matrices Γw◦ , Γz◦ and Γx◦ = Γx UΛ satisfy (18) and (19), with the Ω obtained as in the theorem statement. This can be done by employing the semi-definite programming duality theory (see e.g. Boyd and Vandenberghe [2004]). It can also be observed from this investigation that we have tr(Γwi◦ Γ∗w◦ ) ≤ tr(Γzi◦ Γ∗z◦ ), for all i i i = 1, . . . , κ. Recalling Lemma 8, we can easily observe as the next step that, for the multi-sinusoidal signals w◦ (t) = [ w1◦ · · · wκ◦ ] = 2ℜe {Γw◦ Ωt 1}, z ◦ (t) = [ z1◦ · · · zκ◦ ] = 2ℜe {Γz◦ Ωt 1}, we will have M w◦ = z ◦ + z˜◦ , for some z˜◦ (t) ∈ ℓη2 . On the other hand, we can conclude from Lemma 9 that there exist LTV operators ∆◦i , i = 1, . . . , κ with kV∆◦i ki2 ≤ νiopt , for which ∆◦i zi◦ = wi◦ + w ˜i◦ , for i some w ˜i◦ ∈ ℓm . Clearly, the operator formed as 2 ∆◦ = [∆◦i ]κi=1 then satisfies ∆◦ z = w◦ + w ˜◦ , with w ˜ ∈ ℓη2 . If ν opt (Γwi◦ , Γzi◦ , Ω) ≤ νi , ∀i ∈ I, then ∆◦ ∈ ∆sν . Thus showing the existence of an oscillation in the loop in response to finite energy signals establishes the required result, which might be shown more precisely based on similar arguments to the proof of Theorem 3.2 of Poolla and Tikku [1995]. Remark 11. The test of Theorem 10 can be applied on various frequency intervals [̟ − 0.5d̟, ̟ +0.5d̟] to verify lack of robust stability for different maximal rates-of-variation. Referring to (22), we can note that the maximal rates-ofvariation for which the lack of robust stability is to be verified can be controlled via the interval length, d̟. More clearly, as we let d̟ → 0, we will obtain analysis results for slow variations, whereas for d̟ → π, we will investigate robust stability against fast perturbations. 5. ILLUSTRATIVE EXAMPLE In this section, we consider an academic example in which we assume that a plant with realization −0.28 −0.37 0.44 −0.21 −0.11 0.08 0.05 0.43 −0.15 −0.21 0.01 0.03 AM BM 0.07 0.13 0.32 −0.24 −0.02 0.00 = , 0.35 0.31 2.00 −0.84 0.00 0.00 CM DM 1.49 −0.88 0.05 0.00 0.00 0.00 0.27 −2.40
1.00 −7.20
0.00 0.00
is subject to perturbations in two uncertainty channels, ∆1 ∈ L1×1 and ∆2 ∈ BL2×2 . Our obc c jective is to determine maximum allowable norm bound, β1 = max k∆1 ki2 , versus maximum normalized rate-of-variation, ν1 = max kV∆1 ki2 /β1 , in the first channel. Regions where the existence or lack of robust stability of the system is verified for arbitrary and self-adjoint first channel uncertainties with different tests are shown in Figure 2.
Preprints of the 5th IFAC Symposium on Robust Control Design
perturbations. The new test offers reduced computational complexity (and reduced conservatism, especially in the case of parametric perturbations) as compared to K¨oro˘glu and Scherer [2005b]. Moreover, a method is developed for estimating the conservatism of this analysis test in the case of dynamic LTV perturbations. Extension of this method to parametric LTV perturbations is challenging and deserves further investigation.
βlb: (∆ =∆* ∈L ) with test of Theorem 5 1 1 1 c lb * β1 : (∆1=∆1∈Lc) with test of Köroglu&Scherer[2005b] lb β1 : (∆1∈Lc) with test of Theorem 5 βub: (∆ ∈L ) with test of Theorem 10 (optimization) 1 1 c ub β1 : (∆1∈Lc) with test of Theorem 10 (sine formula)
4
β1 = max ||∆1||i2 (lower and upper bounds)
ROCOND'06, Toulouse, France, July 5-7, 2006
3.9 3.8 3.7 3.6 3.5 3.4 3.3
ACKNOWLEDGEMENT
3.2 3.1 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ν = max ||V || /β 1
∆
1
i2
1
Fig. 2. Robust stability regions in ν1 − β1 domain. We observe from the figure that, in the case of parametric first channel perturbations, (∆1 = ∆∗1 ∈ Lc1×1 ), the new test provides better results than the test of K¨ oro˘ glu and Scherer [2005b] (with significantly reduced computational complexity). We also note that, when the first channel perturbations are dynamic LTV, in addition to obtaining a lower boundary for the stability region by the test of Theorem 5, we can also obtain an upper boundary for it by the test of Theorem 10. And we moreover realize that the upper boundary determined through optimization (i.e. the lower left boundary of the cloud) is significantly better than the boundary determined by the sine formula. A closer analysis reveals that the lower bounds obtained for the stability margin β1 in the case of dynamic first channel perturbations become exact for ν → 0 and ν ≥ 1.55. Exactness for ν → 0 is already expected from the result of Paganini [1996], which established the exactness of analysis based on mixed constant/frequency-dependent D-scaling test for mixed dynamic perturbations with arbitrarily fast/slow variations. On the other hand, exactness of the lower bounds for ν ≥ 1.55 is established thanks to the method developed in Theorem 10. With this method, we first obtained the value of β1 for which constant D-scaling is not satisfied (over the whole frequency interval [0, π]) as 3.044. Together with this, we also obtained the minimum normalized rate-of-variation in the first channel of a destabilizing dynamic perturbation as 1.55. Note that the second channel was assumed to be dynamic and allowed to vary arbitrarily fast (i.e. ν2 ≤ 2). This allowed us to determine the minimum destabilizing rate-of-variation in the first channel without having to check the ratesof-variation obtained for the second channel also as a result of the application of the method of Theorem 10. 6. CONCLUDING REMARKS We have extended the approach of Helmersson [1999] to develop a generalized version of the frequency-dependent D-G scaling test for robust stability analysis against general structured LTV
Supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
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