Available online at ScienceDirect. IFAC PapersOnLine 50-1 (2017)

Transcription

1 Avaiabe onine at ScienceDirect IFAC PapersOnLine 50-1 ( Stabiization of discrete-time switched inear systems: Lyapunov-Metzer inequaities versus S-procedure characterizations A. Kundu J. Daafouz W. P. M. H. Heemes Robert Bosch Centre for Cyber-Physica Systems, Indian Institute of Science Bangaore, India, Centre for Research in Automatic Contro of Nancy, University of Lorraine, France, Contro Systems Technoogy Group, Department of Mechanica Engineering, Eindhoven University of Technoogy, the Netherands, Abstract: In this paper we study connections between Lyapunov-Metzer inequaities and S- procedure characterizations in the context of stabiizing discrete-time switched inear systems using min-switching strategies. We propose two generaized versions of S-procedure characterization aong the ines of the generaized versions of Lyapunov-Metzer inequaities recenty proposed in the iterature. It is shown that the existence of a soution to the generaized version(s of Lyapunov-Metzer inequaities is equivaent to the existence of a soution to the generaized version(s of S-procedure characterization with a restricted choice of the scaar quantities invoved in the atter. This recovers some of our earier wors on the cassica Lyapunov-Metzer inequaities as a specia case. We aso highight and discuss an open question of whether the generaized versions of S-procedure characterization are stricty ess conservative than the generaized versions of Lyapunov-Metzer inequaities, which in turn are equivaent to periodic stabiizabiity as was recenty shown. 2017, IFAC (Internationa Federation of Automatic Contro Hosting by Esevier Ltd. A rights reserved. Keywords: Discrete-time switched inear systems, stabiizabiity, min-switching strategy, Lyapunov-Metzer inequaities, S-procedure characterizations, matrix inequaities 1. INTRODUCTION A switched system consists of two components a famiy of (subsystems and a switching strategy. The switching strategy seects an active subsystem at every instant of time, i.e., the subsystem from the famiy that is currenty determining the state evoution (Liberzon, 2003, In this paper we study stabiization (Lee and Duerud, 2006; Gerome and Coaneri, 2006 of discrete-time switched inear systems using so-caed min-switching strategies. Two we-nown toos in the iterature for the design of minswitching strategies are the Lyapunov-Metzer inequaities and the S-procedure characterization. The Lyapunov-Metzer inequaities proposed in (Gerome and Coaneri, 2006 comprise of a set of biinear matrix inequaities (BMIs invoving the foowing to-be-designed parameters: a set of positive definite matrices, and a specific type of Metzer matrix. If for a given famiy of This wor was carried out during Atreyee Kundu s Postdoctora research at TU/Eindhoven. Atreyee Kundu and Maurice Heemes were supported by the Innovationa Research Incentives Scheme under the VICI grant Wireess contro systems: A new frontier in automation (No awarded by NWO (The Netherands Organisation for Scientific Research and STW (Dutch Technoogy Foundation. systems, the Lyapunov-Metzer inequaities are feasibe, a stabiizing min-switching strategy has been constructed. The Lyapunov-Metzer inequaities are now standard in the iterature for stabiization and performance anaysis of switched systems. Recenty, the Lyapunov-Metzer inequaities are extended to the case of constrained switched inear systems in (Jungers et a., The S-procedure was first introduced in (Lur e and Postniov, 1944 with its theoretica justification in (Yaubovich, It is commony used to ensure that certain quadratic functions ony need to be negative if other quadratic functions are negative, see (Boyd et a., 1994; Feron, 1999; E Ghaoui and Nicuescu, 2000 for a detaied discussion on this topic. The S-procedure characterizations are widey used for anayzing stabiity and performance of piecewise inear systems (the minswitching strategy resuts in a cosed-oop system of this form, see e.g., (Ferrari-Trecate et a., 2002; Johansson and Rantzer, In the context of designing a min-switching strategy, the S-procedure characterization invoves BMIs with the foowing to-be-designed parameters: a set of positive definite matrices, and sets of non-negative scaars. Recenty in (Heemes et a., 2017 we studied connections between Lyapunov-Metzer inequaities and S-procedure characterizations both in the continuous and discrete-time , IFAC (Internationa Federation of Automatic Contro Hosting by Esevier Ltd. A rights reserved. Peer review under responsibiity of Internationa Federation of Automatic Contro /j.ifaco

2 A. Kundu et a. / IFAC PapersOnLine 50-1 ( setting. Amongst others, we showed that in the context of continuous-time switched inear systems the S-procedure characterization is equivaent to the Lyapunov-Metzer inequaities. In this paper we are interested in the stabiization of discrete-time switched inear systems and compare generaized S-procedure characterizations and generaized Lyapunov-Metzer inequaities, which were recenty proposed in the iterature. Indeed, in the recent wor (Fiacchini et a., 2016, 2014 the authors proposed two generaized versions of the Lyapunov-Metzer inequaities, and showed that these are equivaent to periodic stabiizabiity (i.e., the existence of a time periodic switching strategy that resuts in a stabe cosed-oop switched inear system and ead to stabiizing generaized min-switching strategies. Moreover, it was aso estabished that the existence of a soution to the Lyapunov-Metzer inequaities is ony a sufficient condition for the existence of the soution to the generaized Lyapunov-Metzer inequaities. Finay, in (Fiacchini et a., 2016 the authors showed that the existence of a stabiizing switching strategy does not impy periodic stabiizabiity. In this paper we propose two generaized versions of the S-procedure characterization aong the ines of the generaized versions of the Lyapunov-Metzer inequaities presented in (Fiacchini et a., It is shown that the existence of a soution to the generaized Lyapunov- Metzer inequaities is equivaent to the existence of a soution to the generaized S-procedure characterization with a restricted choice of the scaar quantities invoved in the atter. Hence, generaized S-procedure characterizations are never more conservative than the generaized Lyapunov-Metzer inequaities. In fact, we recover some of the discrete-time resuts presented in (Heemes et a., 2017 that stated reated reationships for the cassica Lyapunov-Metzer inequaities and S-procedure characterizations as a specia case. An interesting question that arises from the above set of resuts is that whether the (generaized S-procedure characterizations can go beyond (are ess conservative the (generaized Lyapunov-Metzer inequaities (and hence periodic stabiizabiity. We investigate this question by considering a series of numerica exampes, which seem to hint upon a negative answer to the above question. We provide a preiminary resut that may aid in obtaining such an equivaence, athough in the absence of anaytica toos, it is uncear whether this equivaence is indeed true or not. Apart from presenting these new resuts towards generaizations and a comparative study between two wenown toos for stabiization of discrete-time switched inear systems avaiabe in the iterature, another objective of this paper is to discuss the mentioned open question, thereby hopefuy stimuating many researchers to consider this interesting probem. 2. PRELIMINARY RESULTS In this section we formuate the probem under consideration and cataog the required preiminaries. We consider a famiy of discrete-time inear systems x(t + 1 = A j x(t, j N,t N 0, (1 where x(t R n denotes the vector of states at time t, N = {1, 2,...,N} is an index set, and A j, j N, are constant matrices in R n n.aswitching signa σ : N 0 N seects an active system from the famiy (1 at every instant of time. The switched system generated by the famiy of systems (1 and a switching signa σ is given by x(t + 1 = A σ(t x(t, t N. (2 Definition 1. (Fiacchini et a., 2016, Definition 2 The switched system (2 is (gobay exponentiay stabiizabe if there are scaars c 0 and λ [0, 1, and for a x(0 R n, there exists a switching strategy σ : N N such that the corresponding soution x : N R n satisfies: x(t cλ x(0 for a t N. (3 We are interested in assessing this stabiizabiity property in this paper. In particuar, we wi focus on the stabiization of discrete-time switched inear systems (2 using socaed min-switching strategies. These switching strategies are widey used state-dependent signas defined as σ(t := arg min x(t P j x(t, (4 j N where P j R n n is a positive definite matrix associated to the j-th subsystem, j N. In the context of stabiization of the switched system (2 using a min-switching strategy (4, two we-nown anaysis and design toos are Lyapunov-Metzer inequaities and S-procedure characterizations. We reca them briefy for sef-containedness. 2.1 The Lyapunov-Metzer inequaities A subcass of Metzer matrices pays a crucia roe in Lyapunov-Metzer inequaities. Reca that a Metzer matrix is a square matrix in which a off-diagona components are non-negative. The foowing subcass is of our interest: Definition 2. For N N \{0} the set M N d consists of a square matrices Π R N N satisfying π ij 0 for a i, j N, and the coumn sums satisfying N i=1 π ij = 1 for a j N. Proposition 1. (Gerome and Coaneri, 2006, Theorem 3 Consider the discrete-time switched inear system (2. Suppose that there exist a set of positive definite matrices {P j } j N and a matrix Π M N d such that the foowing set of inequaities is satisfied: ( π ij P i A j P j 0 for a j N. (5 A j i N Then the switched system (2 is stabiizabe. In addition, the cosed-oop switched inear system given by (2 and (4 is gobay exponentiay stabe. Condition (5 is nown as the Lyapunov-Metzer inequaities for guaranteeing stabiizabiity of switched inear system (2 using strategies in (4. These inequaities are widey used in switched systems iterature. Recenty in Fiacchini et a. (2016 the authors proposed two generaized versions of the Lyapunov-Metzer inequaities, and showed their equivaence to LMI-based conditions for stabiizabiity (Fiacchini et a., 2016, Theorem 15 and periodic stabiizabiity, which is defined as foows:

3 3414 A. Kundu et a. / IFAC PapersOnLine 50-1 ( Definition 3. (Fiacchini et a., 2016, Definition 3 The discrete-time switched inear system (2 is caed periodicay stabiizabe if there exists a periodic switching signa σ such that under the system (2 is gobay exponentiay stabe. 1 In the remainder of this section we reca the generaized versions of Lyapunov-Metzer inequaities from Fiacchini et a. (2016. The foowing notations are empoyed: N = {(a 1,...,a a i N,i=1,...,} N [M:N] = N N =M denotes a the possibe sequences of numbers in N of ength from M to N, [M:N] N denotes the number of eements in N [M:N]. Given j =(j 1,...,j in N [1:N], we define A j = i=1 A j i. Proposition 2. (Fiacchini et a., 2016, Proposition 12 Consider the discrete-time switched inear system (2. Suppose that there exist M N, a set of positive definite matrices {P j } j N [1:M], and a matrix Π M N [1:M] d such that the foowing inequaities ( A j π ij P i A j P j 0 for a j N [1:M] (6 i N [1:M] are satisfied. Then the switched system (2 is stabiizabe. In (Fiacchini et a., 2016 the condition (6 are termed as the Lyapunov-Metzer inequaities Generaized I. It recovers condition (5 for M = 1. The condition (6 can be recast as the Lyapunov-Metzer condition (5 by considering the switched system (2 obtained by defining a fictitious subsystem for every matrix A j, j N [1:M], see (Fiacchini et a., 2016, Remar 13 for a detaied discussion. Proposition 3. (Fiacchini et a., 2016, Proposition 14 If for every j N there exist a set of indices K j = {1, 2,...,h j }, h j N, a set of positive definite matrices } K j, and there are π (p,j [0, 1] satisfying {P (j p N p N m, π (p,j m, m K p =1, (7 for a K j, such that the set of inequaities ( A j π (p,j m, P m (p A j P (j 0 (8 m K p is satisfied for a j N, K j. Then the switched system (2 is stabiizabe. Conditions (8 are termed Lyapunov-Metzer inequaities Generaized II. On the one hand, Lyapunov-Metzer inequaities Generaization I are based on increasing the ength of a sequence. On the other hand, Lyapunov- Metzer inequaities Generaized II maintains the ength of the sequence as 1, but increases the eements in the set, K j associated to each subsystem j N, see (Fiacchini et a., 2016, p. 5 for a detaied discussion on this matter. Proposition 3 recovers Proposition 1 with h j = 1 for a j N. of eipsoids determined by P (j 1 We ca the switching function σ periodic, if there exists a T N such that σ(t + T =σ(t for a t N. 2.2 The S-procedure characterization The foowing proposition can be derived using the reasoning as in (Ferrari-Trecate et a., 2002 based on the S- procedure (Lur e and Postniov, 1944; Yaubovich, 1971, see aso (Heemes et a., 2017, Theorem 9 for the detais. Proposition 4. Consider the discrete-time switched inear system (2. Suppose that there exist a set of positive definite matrices {P j } j N, and two sets of non-negative scaars { } i,j, N, =j and { } i,j, N, =i such that the foowing set of inequaities is satisfied: A j P i A j P j (P j P + N, =i N, =j A j (P i P A j for a i, j N. (9 Then the switched system (2 is stabiizabe. In addition, the cosed-oop switched inear system given by (2 and (4 is gobay exponentiay stabe. Condition (9 is nown as the S-procedure characterization for guaranteeing stabiizabiity of switched inear system (2 using strategies in (4. This particuar form for checing if the min-switching strategy (4 is indeed stabiizing for (2 was given in (Heemes et a., It is interesting to note that there are N positive definite matrices and N(N 1 free scaar quantities (in the Metzer matrix invoved in a soution to the Lyapunov-Metzer inequaities (5, whie a soution to the S-procedure characterization (9 contains N positive definite matrices and 2N 2 (N 1 scaar quantities. This is due to the S- procedure characterization requiring N 2 (N 1 scaar quantities { } i,j, N, =j corresponding to the regiona conditions x(t P x(t x(t P j x(t when σ(t = j, and another N 2 (N 1 scaar quantities { } i,j, N, =i corresponding to regiona conditions x(t+1 P x(t+1 x(t+1 P i x(t+1 when σ(t+1 = i. Aso, the S-procedure characterisation has N 2 +N matrix inequaities (of size n nand2n 2 (N 1 scaar inequaities, whie the Lyapunov- Metzer inequaities have N +1 matrix inequaities (of size n n and N 2 scaar inequaities. We next propose two generaized versions of the S- procedure characterization, and estabish their connections to the generaized versions to Lyapunov-Metzer inequaities described in MAIN RESULTS 3.1 S-procedure characterization: Generaization I Proposition 5. Consider the discrete-time switched inear system (2. If there exist M N, a set of positive definite matrices {P j } j N [1:M], and two sets of non-negative scaars { } i,j, N [1:M], =j and { } i,j, N [1:M], =i such that the set of inequaities A j P i A j P j (P j P + N [1:M], =i N [1:M], =j A j (P i P A j for a i, j N [1:M] (10 is satisfied. Then the switched system (2 is stabiizabe. The proof of the above proposition foows under the same set of arguments as for S-procedure characterization (9

4 A. Kundu et a. / IFAC PapersOnLine 50-1 ( considering the switched system (2 with fictitious subsystems corresponding to each matrix A j, j N [1:M]. We omit the proof here for brevity. It is evident that Proposition 5 recovers Proposition 4 for M = 1. A soution to condition (10 invoves N [1:M] positive definite matrices and 2 N [1:M] 2 ( N [1:M] 1 scaars. In contrast, a soution to condition (6 invoves N [1:M] positive definite matrices and N [1:M] ( N [1:M] 1 free scaars. We observe that S-procedure characterization Generaized I presented in Proposition 5 is reated to Lyapunov-Metzer inequaities Generaized I discussed in Proposition 2 as formuated in the next theorem. Theorem 1. Consider the discrete-time switched inear system (2. The foowing are equivaent: (1 There is a soution to the Lyapunov-Metzer inequaities Generaized I (6. (2 There is a soution to S-procedure characterization Generaized I (10 with for a i, j N [1:M] it hods that a = 0 for j, b and c N [1:M], i 1, = β j r for N [1:M] \{i, r}, r N [1:M]. Proof. 1 2: Pic an arbitrary i N [1:M]. Given that there is a soution to (6, we have that ( A j π j P A j P j + A j P i A j A j P i A j 0 N [1:M] (11 By the properties of Π, ( the eft-hand side of the above inequaity is equa to A j π j (P P i A j P j N [1:M], i +A j P i A j. Consequenty, (11 can be written as ( A j P i A j P j A j π j (P i P for a j N [1:M]. N [1:M], i Reca that i N [1:M] was chosen arbitrariy. We therefore concude that (10 hods with = 0 and = π j for a i,j,, N [1:M], j, i. Ceary, N [1:M], i βj i A j = N [1:M], i π j =1 π ij 1, which gives property (b. Moreover, due to the choice of = π j for a i,j, N [1:M], i aso (c hods. 2 1: Given that (10 has a soution with (a, (b and (c satisfied for a i, j N [1:M], we have ( P j A j P i + N [1:M], i (P P i A j (12 for a i, j N [1:M].The right-hand ( side of the above ( inequaity can be written as A j 1 P i + N [1:M], i P N [1:M] i A j. By the hypothesis that N [1:M], i 1 and defining i := 1 0, the above quantity is A j ( N [1:M], i P A j. Substituting this in (12, we obtain for a i, j N [1:M] N [1:M] that A j ( N [1:M] βj i P A j P j 0. By taing now i = 1, we obtain (6 with π j = β j 1 for a j, N [1:M]. Obviousy, the resuting Π M N [1:M] d thereby aso satisfying (6. This competes the proof. Theorem 1 shows that the existence of a soution to Lyapunov-Metzer inequaities Generaized I is equivaent to the existence of a soution to S-procedure characterization Generaized I with the sets of scaars { } i,j, N [1:M], j and { } i,j, N [1:M], i satisfying the conditions a, b, and c in 2 above. 3.2 S-procedure characterization: Generaization II We now proceed towards proposing a second generaized version of S-procedure characterization. Proposition 6. Consider the discrete-time switched inear system (2. Suppose that for every j N there exist K j = {1, 2,...,h j }, h j N, a set of positive definite matrices {P (j } K j, and two sets of non-negative scaars and {α (,j (,i (m,p } i,j,,p N, Kj, K i, m K p,(,j (m,p {β (,j (,i } i,j,,q N, Kj, K i, such that the foowing set n K q,(,i of inequaities is satisfied: A j P (i A j P (j + q N,n K q, (,i p N,m K p, (,j (m,p α (,j (,i (m,p (P (j P m (p β (,j (,i A j (P (i P n (q A j (13 for a i, j N, K j, K i. Then the switched system (2 is stabiizabe. Proposition 6 utiizes a simiar concept as in Proposition 3, i.e., whie the ength of the sequence is ept as 1, the number of eements in the set of eipsoids determined by P (j, j N, is increased. We observe that under a simiar set of restrictions on the scaar quantities α (,j (,i (m,p and as in Theorem 1, S-procedure characterization β (,j (,i Generaized II and Lyapunov-Metzer inequaities Generaized II can be connected. This is formuated in the next theorem. Theorem 2. Consider the discrete-time switched inear system (2. The foowing are equivaent: (1 There is a soution to the Lyapunov-Metzer inequaities Generaized II (8. (2 There is a soution to the S-procedure characterization Generaized II (13 with for a i, j N it hods that a α (,j (,i (m,p = 0 for a p N, K j, K i, m K p, (, j (m, p, b 3470

5 3416 A. Kundu et a. / IFAC PapersOnLine 50-1 ( q N,n K q,(,i β (,j (,i 1, and c β (,j (,i = β (,j (,r for q N \{i, r} and r N. The proof of the above theorem foows aong simiar ines as the proof of Theorem 1. We omit the proof for brevity. Recenty, in Heemes et a. (2017 we showed that the existence of a soution to the Lyapunov-Metzer inequaities (5 is equivaent to the existence of a soution to the S-procedure characterization (9 with a restricted set of choice for the scaars { } i,j, N, i. We reca our resut beow. { } i,j, N, j and Theorem 3. (Heemes et a., 2017, Theorem 10 Consider the discrete-time switched inear system (2. The foowing statements are equivaent: (1 There is a soution to the Lyapunov-Metzer inequaities (5. (2 There is a soution to the S-procedure characterization (9 with for a i, j N it hods that a =0 for j, b N, i for N \{i, r} and r N. 1, and c = β j r Ceary, when M = 1, Theorem 1 recovers our earier resut Theorem 3. Aso, Theorem 2 recovers Theorem 3 as a specia case when K j contains exacty one eement for a j N. Up to this point, we proposed two generaized versions of S-procedure characterizations, and discussed their connections to the generaized versions of Lyapunov-Metzer inequaities, which by the connections estabished in (Fiacchini et a., 2016, V are equivaent to periodic stabiizabiity. 3.3 Discussion As observed in Theorems 1, 2 and 3, the S-procedure characterizations with a restricted choice of the scaar quantities is equivaent to the Lyapunov-Metzer inequaities. Indeed, right at the eve of formuation, S-procedure characterizations invove additiona regiona conditions giving rise to the additiona scaar quantities as aready highighted in 2. In addition, by definition of the Metzer matrix, the scaar quantities invoved in Lyapunov-Metzer inequaities obey certain upper bounds unie the scaar quantities in S-procedure characterization, which are just nown to be non-negative. Figure 1 iustrates the connections among various existing stabiizabiity conditions obtained from Fiacchini et a. (2016 and Heemes et a. (2017 and Theorems 1, 2 and 3 in this paper. 4. OPEN QUESTIONS Since the generaized L-M inequaities are equivaent to S-procedure characterizations with additiona conditions on the scaars (some actuay being 0, a natura question to pose is whether the S-procedure characterizations in the discrete-time setting are stricty ess conservative than the corresponding Lyapunov-Metzer inequaities. In other words, are there famiies of (subsystems that admit soutions to S-procedure characterizations (resp. generaized versions, but do not admit soutions to Lyapunov-Metzer Fig. 1. Connections among stabiizabiity conditions based on Fiacchini et a. (2016; Heemes et a. (2017 and Theorems 1 and 2 in this paper inequaities (resp. generaized versions? We formaize the questions as foows: Question 1. (Cassica version. Is the existence of a soution to the Lyapunov-Metzer inequaities (5 equivaent to the existence of a soution to the S-procedure characterization (9? Question 2. (Generaized version. Is the existence of a soution to the Lyapunov-Metzer inequaities Generaized I (6 (resp. Generaized II (8 equivaent to the existence of a soution to the S-procedure characterization Generaized I (10 (resp. Generaized II (13? These questions are reated to the question mars in Fig. 1. On the one hand, Question 1 is of interest because it pertains to the cassica Lyapunov-Metzer inequaities. In addition, recenty in Heemes et a. (2017 we showed that in the context of continuous-time switched inear systems the cassica S-procedure characterization is equivaent to the cassica Lyapunov-Metzer inequaities. It is therefore a natura question to study whether the same hods for discrete-time switched inear systems. On the other hand, Question 2 is of interest in the stabization of switched inear systems as the generaized Lyapunov-Metzer inequaities are equivaent to periodic stabiizabiity as was shown in Fiacchini et a. (2016 (whie the cassica Lyapunov- Metzer inequaities are not. It is of interest to now if the genearized S-procedure characterizations I-II can actuay go beyond the periodic stabiizabiition. In other words, are there switched systems for which the generaized S- procedure conditions can be true, but the system is not perioidcay stabiizabe. We attempt to shed some ight on these open questions using a set of numerica experiments. 5. NUMERICAL STUDY Prior to presenting the observations from our numerica experiments, it is worth noting that the S-procedure characterizations invove BMIs, and identifying soutions (or their absence to them is a difficut tas. Thus, for the foowing set of resuts we rey on performance of standard BMI soving toos, which empoy approximations. We stic to the simpest case of Lyapunov-Metzer inequaities (5 (which is aso a sufficient condition for the existence of a soution to the generaized Lyapunov-Metzer inequaities (6 and (8, and perform an empirica study. Whie not finding a soution to (5 does not ensure that there is 3471

6 A. Kundu et a. / IFAC PapersOnLine 50-1 ( no soution to (6 (resp. (8, it definitey provides some insights into the subtety of the gap between Lyapunov- Metzer inequaities and S-procedure characterizations in genera. Exampe 4. We generate famiies of random 2 2 matrices with the entries of the matrices ranging in the interva ( a, a. Here, a is piced from the interva (0, uniformy at random. It is ensured that each matrix in the famiy is unstabe by checing the maximum eigenvaue of the randomy generated matrices. The objective is to identify the cases (if any where the Lyapunov-Metzer inequaities (5 do not have a soution, but the S-procedure characterization (9 admits a soution within a pre-specified duration of time. Our observations are isted in Tabe 1. N No. of sampes Soution No soution to (5, sampes to (5 but soution to ( Tabe 1. Data for Exampe 4 The above numerica study hints upon a situation simiar to the case of continuous-time setting, see (Heemes et a., 2017 where the cassica S-procedure characterization turned out to be equivaent to the cassica Lyapunov- Metzer inequaities, thereby providing a positive answer to Question 1. However, a compete proof or a counterexampe to assert the above, is currenty missing. We state a first resut in this direction, the proof of which is omitted for brevity. Theorem 5. Consider the discrete-time switched inear system (2 with N = {1, 2}. Suppose that there exists a soution to the S-procedure characterization (9 with = 0 for a i,j, N, j. Then there exists a soution to the S-procedure characterization (9 with 1 for a i,j, N, i. Moreover, the S-procedure characterization (9 reduces to = 0 for a i,j, N, j and A j P i A j P j A j (P i P A j (14 for a j N and any i N, i. Observe that for N = 2, the Lyapunov-Metzer inequaities (5 invove two inequaities, whie the S-procedure characterization (9 invoves four inequaities. In Theorem 5 we show that for the S-procedure characterization (9 it suffices to sove two inequaities. Consequenty, one of the restrictions on the parameters, i in Theorem 3 is not restrictive at a for N = 2. However, even in the simpest case of S-procedure characterization (9 with = 0 for a i,j, N, 0 a genera anaytica proof for reaxing the conditions on, i,j, N, i does not seem to be obvious beyond N = 2. In particuar, the foowing subquestions are unanswered: If there exists a soution to S-procedure characterization (9 with = 0 for a i,j, N, j, then does there exist a soution to S-procedure characterization (9 with = 0 for a i,j, N, j and N, =i βj i 1 for a i, j N? Moreover, if there exists a soution to S-procedure characterization (9, then does there exist a soution to S-procedure characterization (9 with =0 for a i,j, N, j? If the answers to these are yes, then it is evident that S-procedure characterization (9 is equivaent to Lyapunov-Metzer inequaities (5. Even though Lyapunov-Metzer inequaities (5 are just sufficient for their generaized versions to be satisfied, the proof techniques may carry over as is seen with the proof technique of Theorem 2. REFERENCES Boyd, S., E Ghaoui, L., Feron, E., and Baarishnan, V. (1994. Linear matrix inequaities in system and contro theory, voume 15 of SIAM Studies in Appied Mathematics. Society for Industria and Appied Mathematics (SIAM, Phiadephia, PA. E Ghaoui, L. and Nicuescu, S.I. (2000. Advances in Matrix Inequaity Methods in Contro. Advances in Design and Contro. SIAM. Feron, E. (1999. Nonconvex quadratic programming, semidefinite reaxations and randomization agorithms in information and decision systems. In Advances in Systems Theory, Kuwer Academic Pubishers. Ferrari-Trecate, G., Cuzzoa, F.A., Mignone, D., and Morari, M. (2002. Anaysis of discrete-time piecewise affine and hybrid systems. Automatica, 38(12, Fiacchini, M., Girard, A., and Jungers, M. (2014. On stabiizabiity conditions for discrete-time switched inear systems. In IEEE Conference on Decision and Contro (CDC, Fiacchini, M., Girard, A., and Jungers, M. (2016. On the stabiizabiity of discrete-time switched inear systems: nove conditions and comparisons. IEEE Transactions on Automatic Contro, 61(5, Gerome, J.C. and Coaneri, P. (2006. Stabiity and stabiization of discrete time switched systems. Internat. J. Contro, 79(7, Heemes, W.P.M.H., Kundu, A., and Daafouz, J. (2017. On Lyapunov-metzer inequaities and s-procedure characterisations for the stabiisation of switched inear systems. IEEE Trans. Autom. Contro (to appear. Johansson, M. and Rantzer, A. (1998. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Transactions on Automatic Contro, 43(4, Jungers, M., Girard, A., and Fiacchini, M. (2016. Language constrained stabiization of discrete-time switched inear systems: A Lyapunov-Metzer inequaities approach. In Proceedings of 55th IEEE Conference on Decision and Contro (CDC Lee, J.W. and Duerud, G.E. (2006. Uniform stabiization of discrete-time switched and marovian jump inear systems. Automatica, 42, Liberzon, D. (2003. Switching in systems and contro. Systems & Contro: Foundations & Appications. Birhäuser Boston Inc., Boston, MA. Lur e, A.I. and Postniov, V.N. (1944. On the theory of stabiity of contro systems. Appied Mathematics and Mechanics, in Russian, 8(3. Yaubovich, V.A. (1971. The s-procedure in noninear contro theory. Vestni Leningrad University, in Russian, Engish transation in Vestni Leningrad Univ. 4:73-93,