Linear matrix inequality relaxations and its application to data-driven control design for switched affine systems

Received: 22 July 2022 Revised: 23 March 2023 Accepted: 5 April 2023 DOI: 10.1002/rnc.6712 RESEARCH ARTICLE Linear matrix inequality relaxations and its application to data-driven control design for switched affine systems Alexandre Seuret Carolina Albea Francesco Gordillo Dpto de Ingeniería de Sistemas y Automática, Universidad de Sevilla, Sevilla, Spain Correspondence Alexandre Seuret, Dpto de Ingeniería de Sistemas y Automática, Universidad de Sevilla, Camino de los Descubrimientos, s/n 41092, Sevilla, Spain. Email: [email protected] Funding information MCIN/AEI and FEDER, Grant/Award Numbers: PID2019-109071RB-I00, PID2019-105890RJ-I00; Agence Nationale de la Recherche (ANR)-France, Grant/Award Number: ANR-18-CE40-0022-01 Abstract The problem of data-driven control is addressed here in the context of switched affine systems. This class of nonlinear systems is of particular importance when controlling many types of applications in electronic, biology, medicine and so forth. Still in the view of practical applications, providing an accurate model for this class of systems can be a hard task, and it might be more relevant to work on data issued from some trajectories obtained from experiments and to deploy a new branch of tools to stabilize the systems that are compatible with the processed data. Following the recent concept of data-driven control design, this paper first presents a generic equivalence lemma that shows a matrix constraint based on data, instead of the system parameter. Then, following the concept of robust hybrid limit cycles for uncertain switched affine systems, robust model-based and then data-driven control laws are designed based on a Lyapunov approach. The proposed results are then illustrated and evaluated on an academic example. KEYWORDS data-driven control design, LMI, robust stabilization, switching affine systems 1INTRODUCTION Overthepastdecades, robustcontroltheoryhasbeendevelopedtosolveanalysisandcontrolproblemsforawide classof dynamicalsystems.1-3 Themainadvantagesofthis methodrelyonthepossibility toderive stabilityandstabilization tests as to evaluate and measure the robustness of systems as to control these uncertain systems. These tests are often written in the form of linear matrix inequalities (LMIs) that can be easily and efficiently solved using semi-definite programming.Generally, thisdirectionofresearchhas a commonrequirement,namely knowledge ofamodel, eventhoughsome parametersinthemodelmightbesubjecttouncertaintiesmodeledasnorm-boundedorpolytopicuncertainties.Recently, due to the emergence of artificial intelligence and the possibility of storing numerous data, the area of data-based identification has been enhanced. The main motivation for this new paradigm is to avoid or limit the modeling phase to some extents and to better rely on a set of several experiments providing data in order to compensate for the lack of knowledge on the model. These methods have been developed to eliminate the common feature described above. In this direction, reinforcement learning methods have been developed in the literature; see, for instance,4,5 to cite only a few of them. They mainly deal with system identification, estimation of the model, but rarely address closed-loop guarantees,andonlyfewpapersprovidenon-conservativeconstructivestabilizationconditionsfortheclosed-loopsystem This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2023 The Authors. International Journal of Robust and Nonlinear Control published by John Wiley & Sons Ltd. Int J Robust Nonlinear Control. 2023;1–22. wileyonlinelibrary.com/journal/rnc 1 2SEURET et al. usingnoisydataoffinitelength,whichremainsanopenproblem,evenifthedataaregeneratedbyalineartime-invariant (LTI) system. Indeed,theautomaticcontrolcommunityhasrecentlyshownagrowinginterestintheproblemofdata-drivencontrol designforvariousclassicalcontrol problems(see, forinstance6-8),enhancingthe tools arisingfromrobustcontroltheory. Inthiscontext,theproblemcanbesummarizedasfollows:Howtotranslateamodel-basedstabilityorstabilizationcriteria into a data-driven ones, without introducing conservatism? A catalog of formulas has been provided recently in Reference 9, where several problems related to the design of state/output feedback controllers for linear systems have been considered. It is notably shown therein that in the case of exact data experiments arising from linear time-invariant systems, that is, without noise or uncertainties, equivalent formulations between model-based and data-based criteria have been obtained. More recently, another elegant solution to the data-driven design problem for linear systems has been provided in References 10 and 11. In this paper, an extension of the S-procedure has been proposed to eliminate in an LMI condition the uncertainties arising, for instance, from the model matrices. The underlying idea is to embed the uncertainties brought by the data experiments into an LMI constraint, which is considered an assumption in the design method. The authors have interestingly demonstrated that this assumption allows eliminating the model matrices at the price of introducing a single decisionvariabletotheinitialconditionsforstabilityorstabilization.AsimilarassumptionwasalsomadeinReferences6 and 12. To the best of our knowledge, this trend of research has mainly been considered for linear systems. For the nonlinear cases, the reader may look at the case of bilinear input-affine systems,13 referring to many relevant applications in engineering,medicine, orecology.14 Theauthorsdemonstratethereinhowtheirformulasprovidedin Reference9for the linear case can be efficiently adapted to this class of nonlinear systems. In this paper, the objective is to demonstrate that data-driven methods can also be applied to a particular class of nonlinear systems, namely switched affine systems. This class of systems represents a highly relevant theoretical and practical area of research. From a practical point of view, they have been employed to model numerous applications such as embedded systems, electronic power converters, mixing fluids, damping of vibrating structures, mobile sensor networks,15 characterizing these complex systems and not intuitive behaviors. From a theoretical point of view, they represent a particular class of hybrid dynamical systems.16,17 A particularity of switched affine systems arises from the fact that, in general, it is not possible to stabilize their solution to a single equilibrium point but rather to a(hybrid) limit cycle, as shown in Reference 18 and later refined in Reference 19. In the context of data-driven methods for switched systems (affine or not), the authors of References 20 and 21 provided the first attempts on switched systems (without affine terms). However, there are only a few works considering data-driven control design for switched affine systems, which is the objective of this paper. To the best of our knowledge, the only contribution in this direction has been presented in Reference 22, where the authors consider the stability analysis of this class of systems, when the switching signal is assumed to be an arbitrary exogenous input. The authors of Reference 23 use machine learning algorithms, especially regression trees and random forests to model a switched affine system using only historical data. The model is later used to establish a model predictive control or more precisely a Data Predictive Control (DPC) to obtain optimal control system trajectories. In this paper, the model under consideration is of the form x+=A𝜎x+B𝜎u+F𝜎,wherexis the state (and x+the forward value of x), uthe control law and 𝜎the switchingsignal.Thissystemreferstoadifferentclassofswitchedaffinesystemssinceitiscomposedofaswitchedlinear controlled systems (A𝜎x+B𝜎u)with affine switching terms (f𝜎)where 𝜎is an exogenous signal. In contrast to these works, here we consider a data-driven control design method for switched affine systems, where thecontrolvariableisthe switching signal.Following theframeworkprovided inReference24 dealing withanuncertain model-based method, this paper will present a robust data-driven control design for this class of systems. To do so the following objectives will be addressed in the paper •Provide a technical generic tool that allows transforming a model-based criteria for discrete-time systems into a data-driven one. •Solveaproblemofmodel-baseddesignofastabilizingswitchingstate-feedbackcontrollawforswitchedaffinesystems subject to an external disturbance. •Illustrate the potential of the preliminary technical tool though the data-driven design of switching control laws for switched affine systems. The structure of the paper is the following. Section 2introduces a novel matrix-constraint relaxation, its relationship with the existing literature and a short discussion about its potential on the data-driven stabilization of linear systems. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 3 Then, Section 3deals with the model-based stabilization of switched affine systems subject to a bounded external disturbance.Then,thankstotheresultsinthetwoprevioussections,themaincontributionofthepaperispresentedinSection4 addressing the data-driven stabilization of switched affine systems. These results are then illustrated in Section 5,where a numerical application of the theoretical contributions is treated. Notations: Throughout the article, Ndenotes the set of natural numbers, Rthe real numbers, Rnthe n-dimensional Euclidean space, Rn×mthe set of all real n×mmatrices and Snthe set of symmetric matrices in Rn×n.Forallscalars0< a<b,notation[a,b]Nstands for [a,b]⋂N, which represents the set of integers included in [a,b].Foranynand min N, matricesInand0n,m(0n=0n,n)denotetheidentitymatrixofRn×nandthenullmatrixofRn×m,respectively.Foranyinteger n,1nstands for the vector in Rn, whose entries are all equal to 1. When no confusion is possible, the subscripts of these matrices that precise the dimension will be omitted. For any matrix Mof Rn×n, the notation M≻0, (M≺0) means that Mis symmetric positive (negative) definite and det(M)represents its determinant. For any matrices A=A⊤,B,C=C⊤of appropriatedimensions,matrix[AB ∗C]denotesthesymmetricmatrix[AB B⊤C].||x||denotestheEuclideannormofx.For amatrixM∈Sn,M≻0 and a vector h∈Rn, we denote the shifted ellipsoid (M,h)={x∈Rn,(x−h)⊤M(x−h)≤1}. 2MATRIX-CONSTRAINED RELAXATION 2.1 New lemma for data-driven analysis The following lemma, which is the first contribution of the paper, presents a generic method to transform a problem of a particular matrix inequality depending on parameters that verify a quadratic constraint into a formulation that is independentoftheseparameters.Severalsolutionstothisproblem,alsoknownasQuadraticMatrixInequality,arestated below. Lemma 1. For given positive integers n,m,q, consider matrices (1,2,1,2)in Sq×Sn×Rq×n×Rq×m and a symmetric matrix Ψ=[Ψ1Ψ2 ∗Ψ 3]∈Sn+psuch that Ψ3≺0. Then, the following statements are equivalent. (i) Inequality ()=[11+2⊤ ∗2]≻0,∀∈ Σ(Ψ) (1) holds.Σ(Ψ) representsthenonemptysetof allowableuncertainmatricescharacterizedby aquadraticconstraint defined as follows, Σ(Ψ) ∶= ⎧ ⎪ ⎨ ⎪ ⎩ ∈Rn×ms.t. [In ⊤]⊤[Ψ1Ψ2 ∗Ψ 3][In ⊤]≽0⎫ ⎪ ⎬ ⎪ ⎭ .(2) (ii) There exist matrices (,)∈Sn×Rq×nand a positive scalar 𝜂>0such that ⎡⎢⎢⎢⎢⎢⎣ 1 1− 2 ∗2−00 ∗∗−𝜂Ψ1−𝜂Ψ2 ∗∗ ∗−𝜂Ψ3 ⎤⎥⎥⎥⎥⎥⎦ ≻0.(3) (iii) There exists a positive scalar 𝜂>0such that ⎡⎢⎢⎢⎣ 112 ∗2−𝜂Ψ1−𝜂Ψ2 ∗∗−𝜂Ψ3⎤⎥⎥⎥⎦ ≻0.(4) 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4SEURET et al. Proof. The proof is divided into three steps. (i)⇒(ii): The first step of the proof is to find an appropriate expression of matrices that belong to Σ(Ψ).For any matrix in Σ(Ψ), it holds 0⪯[In ⊤]⊤[Ψ1Ψ2 ∗Ψ 3][In ⊤]= −[In ⊤]⊤[ −Ψ 1−Ψ2 ∗−Ψ 3][In ⊤], where  is any matrix in Sn. In addition, condition Ψ3≺0 ensures that there exists a matrix  such that [ −Ψ 1−Ψ2 ∗−Ψ 3]is positive definite, which allows applying the Schur complement as follows ⎡⎢⎢⎢⎣ [ −Ψ 1−Ψ2 ∗−Ψ 3]−1[In ⊤] ∗ ⎤⎥⎥⎥⎦ ≻0. NotethatitisnottheusualwaytoapplytheSchurcomplement,butthisdualwayhasbeenconsideredtokeep block (1,2)(resp. (2,1))with[In ⊤](resp. its transpose). Next, pre- and post-multiply the previous inequality by[ 0 0I]anditstranspose,respectively,where  isanymatrixinRp×(n+m).Then,having∈ Σ(Ψ)implies ⎡⎢⎢⎢⎣  [ −Ψ 1−Ψ2 ∗−Ψ 3]−1  ⊤ [In ⊤] ∗ ⎤⎥⎥⎥⎦ ≻0.(5) The previous calculations ensure that inequality (1) can be rewritten as ()≻0, for all so that (5) holds. Using an S-procedure, this statement is equivalent to the existence of a positive scalar 𝜂>0 such that [11+2⊤ ∗3]−𝜂⎡⎢⎢⎢⎣  [ −Ψ 1−Ψ2 ∗−Ψ 3]−1  ⊤ [In ⊤] ∗ ⎤⎥⎥⎥⎦ ≻0, which, together with the Schur complement, writes ⎡⎢⎢⎢⎢⎢⎢⎢⎣ 1([12]−𝜂 )[In ⊤]𝜂  ∗2−𝜂 0 ∗∗ [𝜂 −𝜂Ψ1−𝜂Ψ2 ∗−𝜂Ψ3] ⎤⎥⎥⎥⎥⎥⎥⎥⎦ ≻0. Selecting 𝜂 =and 𝜂 =[1− 2]so that ([12]−𝜂 )[In ⊤]=[0][In ⊤]=, 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 5 makes disappear the dependence of the condition on the uncertain matrix and concludes the first part of the proof. (ii)⇒(iii): Pre- and post-multiplying inequality (3)by[In000 0InIn0 000In]and its transpose, respectively, leads to (4), where matrices and have been eliminated. (iii)⇒(i): Pre- and post-multiplying inequality (4)by[In00 0In]and its transpose, respectively, leads to 0≺[11+2⊤ ∗2]−𝜂⎡⎢⎢⎢⎣ 00 ∗[In ⊤]⊤[Ψ1Ψ2 ∗Ψ 3][In ⊤]⎤⎥⎥⎥⎦ . This inequality proves that inequality (1) holds for all matrices that belong to Σ(Ψ), since the second term is negative semi-definite. ▪ 2.2 Comparison with the S-Lemma Lemma1providesanalternativeformulationinrobustanalysisforuncertain matricessubjecttoquadraticconstraintsof the form (2) compared to the one presented in References 10 and 11. For the sake of consistency, the S-Lemma provided in Reference 10 is recalled in the following lemma. Lemma 2 (10, Th.9). Let s,Ψs∈R(ns+ms)×(ns+ms)be symmetric matrices and assume that there exists some matrix s∈Rns×mssuch that [Ins ⊤ s]⊤Ψs[Ins ⊤ s]≻0. Then the following statements are equivalent: (i) [Ins ⊤ s]⊤s[Ins ⊤ s]≻0,∀s∈Σ(Ψ s)⊂Rns×ms, where set Σ(Ψs)has the same definition as in (2) but replacing Ψ by Ψs. (ii) There exists a scalar 𝜂>0such that s−𝜂Ψs≻0. The main similarities and differences with respect to this formulation are described here. Let us first note that both lemmas address the same problem, consisting of the satisfaction of a matrix inequality subject to uncertain matrices characterized by a quadratic constraint. The main interest of both lemmas is to derive equivalent inequalities that are independentoftheuncertainmatrix(ors).Finally,bothlemmascanbeseenasapplicationoftheusualmanipulations on LMI such as Schur Complement, Finsler’s lemma and S-procedure. Apart from presenting these similarities, both lemmas have substantial differences. First, Lemma 2requires that matrix shas the same size as Ψs, that characterizes the quadratic constraint on . Lemma 1is more flexible in this sense, as there is no relationship between matrices 1and Ψ, which are independent. This flexibility has the benefit of reducing the initial manipulations to derive, from usual stability or control problems, the appropriate expression of and Ψto fit the framework of Lemma 2. In fact, the relationship between both lemmas can be seen by selecting ns=q+n,ms=m,⊤ s=[0m×q⊤],s=⎡⎢⎢⎢⎣ 112 ∗20 ∗∗0⎤⎥⎥⎥⎦ ,Ψs=⎡⎢⎢⎢⎣ 00 0 ∗Ψ 1Ψ2 ∗∗Ψ 3⎤⎥⎥⎥⎦ . From this selection, it is clear that item (ii) of Lemma 2is equivalent to item (iii) of Lemma 1, showing that Lemma 1is a particularcaseofLemma2.Therefore,itisimportanttounderstandtheadvantagesofLemma1withrespecttoLemma2, since they can be used for the same purpose. The first one is related to the structure of matrix (), which appears in manyconstructiveLMIproblemsforthestabilizationofdiscrete-timesystems.Asafirstillustration,considertheproblem 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6SEURET et al. ofstabilizationofa discrete-timelinear system x+=Ax +Buusingastate-feedback controlleru=Kx, whereweadopted the following notation x+=xk+1and x=xk. Using a quadratic Lyapunov function, with a positive definite matrix P,the condition for the stabilization of this system writes (A+BK)⊤P(A+BK)−P≤0. Then, as usual in this context, the design of the control gain is obtained by introducing W=P−1and applying the Schur complement to the first term of the previous inequality, leading to the equivalent problem 0≺(A,B)=[WWA ⊤+WK⊤B⊤ ∗W]=⎡⎢⎢⎢⎣ W[WWK ⊤][A⊤ B⊤] ∗W⎤⎥⎥⎥⎦ . This usual manipulation drives naturally to the formulation of item (i) in Lemma 1and, it is easy to identify that for this very simple example 1=2=W,1=0,2=[WWK ⊤],=[AB ]. Then Lemma 1leads to the following equivalent formulation 0≺⎡⎢⎢⎢⎣ W0[WWK ⊤] ∗W−𝜂Ψ1−𝜂Ψ2 ∗∗ −𝜂Ψ3⎤⎥⎥⎥⎦ , which can be rewritten as an LMI by introducing a new decision variable Y=KW. Keeping the same example and following the development presented in Reference 10 or Reference 6, the LMI stabilization problem cannot be treated directly. First, a manipulation is required to solve the problem, that is, to consider the dual or transpose problem that is (A+BK)P(A+BK)⊤−P≤0. Then, it is possible to apply the equivalence formulation proposed in Lemma 2. This manipulation is the key step in the developments provided in Reference 10. Note that this manipulation is correct when considering this simple linear time-invariantexample,sincebothLMIsarenecessary andsufficientconditionsformatrixA+BK tobeSchurstableand consequently for the transpose matrix (A+BK)⊤to be Schur stable as well. However, this manipulation is not permitted any more, or at least has to be studied carefully when other classes of systems are considered such as systems subject to nonlinearities, saturations, etc. In order to avoid working on the transpose matrix, one may apply the Schur complement and work on the same inequality s(A,B)=(A,B). However, by doing this, the problem fits exactly to the structure of the inequality presented in item (i) of Lemma 1, for which no additional manipulations are further needed. Even though this example may seem too simple, a more complicated stabilization problem will be considered in the next section on switched affine systems. In this situation, one can better understand the potential and simplicity of the proposed formulation. This is the main motivation to use Lemma 1. Another important issue is related to item (ii) in Lemma 1, which introduces two slack variables, namely  and . In light of item (iii), these slack variables are not needed, in general. However, there exist at least two situations in which these additional degrees of freedom could be useful. The first refers to the case where matrices in ()are subject to uncertainties. A second case of interest appears in the context of an optimization problem, where these slack variables may ease the search for the optimal solution, as we will show in the latter example section. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 7 3ROBUST HYBRID CYCLES FOR PERTURBED SWITCHED AFFINE SYSTEMS In this section, the objective is to present constructive stabilization conditions for switched affine systems subject to a boundedexternaldisturbance.Afterformulatingtheproblemunderconsideration,severalpreliminariesonrobusthybrid cycles and cycles for this class of systems will be documented. Then, the first contribution of this paper on the robust model-based stabilization of switched affine systems will be presented. 3.1 System data Consider the discrete-time switched affine system governed by the following dynamics. ⎧ ⎪ ⎨ ⎪ ⎩ x+=A𝜎x+B𝜎+w, 𝜎∈u(x)⊂K, x0∈Rn, (6) where x∈Rnis the state vector. At any time instant k∈N,x,andx+stand for x(k)and x(k+1), respectively. In (6), the time argument k∈Nis omitted for the sake of simplicity. Likewise, 𝜎∈K∶= {1,2,..,K}characterizes the active mode. The dynamics of the system is affected by an external disturbance vector w∈Rnthat verifies w⊤w≤𝜆2,(7) for some given positive real number 𝜆. Finally, Aj∈Rn×nand Bj∈Rn×1are the matrices of mode j∈K, which are not necessarilyconstantandknownbut it is assumed that theypossiblybelongto a polytopicsetofuncertainties. The particularity of this class of systems relies on its control action, which is only performed by selecting the active mode 𝜎,which requires particular attention. The objective here is to design a suitable set-valued map uin system (6) that ensures the convergence of the state trajectories to a set to be characterized accurately. Note that the property of uof being a set-value map comes from the factthatu∈K.Amodel-drivensolutiontothisproblemwasprovidedin Reference24 andwillbe summarizedhereafter. Interestingly, this paper provides a solution for robust stabilization of switched affine systems, which consists of proving that the solutions to the closed-loop system converge to a robust limit cycle which is composed of the union of shifted ellipsoid. This solution will be recalled in the next section. This formulation paves the way for the problem of the data-driven design of stabilizing controllers for switched affine systems. Let us first formulate the problem of data-driven control. 3.2 Cycles and robust limit cycles Following the ideas developed in Reference 24, the notion of limit cycles25,26 is adapted to the problem under consideration. Before presenting the concept of robust limit cycles, let us first introduce the following definitions. Definition 1 (Cycle). A cycle, denoted as 𝜈, is a periodic function from Nto K. More precisely, this means that there exists Nin N⧵{0},suchthat 𝜈(i+N)=𝜈(i),∀i∈N. For any cycle 𝜈,notationsN𝜈and D𝜈stand for the minimum period and the minimum domain of 𝜈, respectively. More formally, they are defined as follows N𝜈∶= minN∈N⧵{0}s.t. 𝜈(i+N)=𝜈(i)∈K,∀i∈N, D𝜈∶= {1,2,…,N𝜈}. Definition 2 (Set of cycles). Denote the set of cycles from Nto Kby ∶= {𝜈∶N→K,s.t. ∃N∈N⧵{0},∀i∈N,𝜈(i+N)=𝜈(i)}. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8SEURET et al. FIGURE 1 Illustration of a robust limit cycle composed of three subsets and associated to the cycle 𝜈={1,1,2}. Inclusions (8)impose the invariance of the trajectories inside the robust limit cycle. To ease readability, we introduce the following modulo notation: ⌊i⌋𝜈=((i−1)mod N𝜈)+1, for any i∈N⧵{0}.In particular,⌊i⌋𝜈=i,foranyi=1,…,N𝜈and⌊N𝜈+1⌋𝜈=1.Thenotionofrobustlimitcycles,whichextendsthedefinition of limit cycles in References 25 and 26, is adapted to discrete-time systems and is defined below. Definition 3 (Robust Limit Cycles). System (6) admits a robust limit cycle associated with a cycle 𝜈∈if there exist possibly disjoint subsets i⊂Rn, for i∈D𝜈such that A𝜈(i)i+B𝜈(i)⊂⌊i+1⌋𝜈,∀i∈D𝜈.(8) Ininclusion(8),theleft-handsideoftheinclusionmeans,withalightabuseofnotation,that,foranyi∈D𝜈andforall x∈i, vector A𝜈(i)x+B𝜈(i)belongs to ⌊i+1⌋𝜈. In the sequel, Figure 1illustrates this set of inclusions and the relationship withthecycle.If,insomecases,thesubsetscomposingtherobustlimitcyclesarereducedtosingletons,thatis,i={𝜌i}, for all i∈D𝜈, then inclusion (8) is rewritten as a set of equalities given by 𝜌⌊i+1⌋𝜈=A𝜈(i)𝜌i+B𝜈(i),∀i∈D𝜈,(9) illustrating then that inclusions (8) are the natural extension of (9). NotethattheideaofstudyinglimitcyclesforswitchedaffinesystemshasbeenconsideredinReference18forthecase of constant and known matrices Ajand Bj. In this case, necessary and sufficient constructive conditions for the existence of {𝜌i}i∈D𝜈, for a given cycle 𝜈, have been provided in Reference 24, while it was only an assumption in Reference 18.It is also worth noting that the structure of the control law provided in Reference 18 complicates the study of robust limit cycles since their control law requires exact knowledge of the system matrices. 3.3 Robust model-based stabilization of switched affine systems A solution to the stabilization of switched affine systems subject to external disturbances to a robust limit cycle is presented here following the idea and concept borrowed from Reference 24. The main difference with respect to Reference 24 is the addition of the bounded external disturbance in (6). The robust stabilization problem is formalized as follows. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 9 Theorem 1. For a given cycle 𝜈in and for a parameter 𝜇∈(0,1), consider the solution {(Wi,𝜁i,𝛿i)}i∈D𝜈in Sn×Rn×Rto the following problem. Φi(A𝜈(i),B𝜈(i))≻0,Wi≻0,𝛿 i>0,∀i∈D𝜈,(10) where matrices Φiare defined for all i in D𝜈as follows. Φi(A𝜈(i),B𝜈(i))=⎡⎢⎢⎢⎢⎢⎣ (1−𝜇)Wi00 WiA⊤ 𝜈(i) ∗𝜇−𝛿i𝜆20𝜁⊤ iA⊤ 𝜈(i)+B⊤ 𝜈(i)−𝜁⊤ ⌊i+1⌋𝜈 ∗∗𝛿iII ∗∗∗ W⌊i+1⌋𝜈 ⎤⎥⎥⎥⎥⎥⎦ .(11) Then, attractor 𝜈∶= ⋃ i∈D𝜈 (W−1 i,𝜁i)(12) with (W−1 i,𝜁i)={x∈Rn,i∈D𝜈|||(x−𝜁i)⊤W−1 i(x−𝜁i)≤1}, is robustly globally exponentially stable for system (6) with the disturbance signal (7) and with the switching control law u(x)={𝜈(𝜃),𝜃∈argmin i∈D𝜈 (x−𝜁i)⊤W−1 i(x−𝜁i)}⊂K.(13) Proof. The proof of this theorem is largely inspired by Serieye et al..24 This theorem is demonstrated thanks to the Lyapunov function built with matrices W−1 i≻0 and vectors 𝜁i’s that are the decision variables of (11) and given by V(x)=min i∈D𝜈 (x−𝜁i)⊤W−1 i(x−𝜁i).(14) The forward increment of the Lyapunov function writes ΔV(x)=min i∈D𝜈 (x+−𝜁i)⊤W−1 i(x+−𝜁i)−min 𝜄∈D𝜈 (x−𝜁𝜄)⊤W−1 𝜄(x−𝜁𝜄) =min i∈D𝜈 (x+−𝜁i)⊤W−1 i(x+−𝜁i)−(x−𝜁𝜃)⊤W−1 𝜃(x−𝜁𝜃).(15) The last equation holds since 𝜃is the value of D𝜈that minimizes (x−𝜁𝜄)⊤W−1 𝜄(x−𝜁𝜄)according to the control law (13). Furthermore, we have min i∈D𝜈 (x+−𝜁i)⊤W−1 i(x+−𝜁i)≤(x+−𝜁⌊𝜃+1⌋𝜈)⊤W−1 ⌊𝜃+1⌋𝜈(x+−𝜁⌊𝜃+1⌋𝜈) sincetheminimumisalwayslowerthanorequaltoanyotherterm.Thisexpressiononlyconsidersaparticular case,whichisi=⌊𝜃+1⌋𝜈.Alltogether,anupperboundoftheincrementoftheLyapunovfunctionisderived as follows ΔV(x)≤(x+−𝜁⌊𝜃+1⌋𝜈)⊤W−1 ⌊𝜃+1⌋𝜈(x+−𝜁⌊𝜃+1⌋𝜈)−(x−𝜁𝜃)⊤W−1 𝜃(x−𝜁𝜃).(16) Next, we note that x+−𝜁⌊𝜃+1⌋𝜈=A𝜈(𝜃)(x−𝜁𝜃)+A𝜈(𝜃)𝜁𝜃+B𝜈(𝜃)−𝜁⌊𝜃+1⌋𝜈 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝜈(𝜃) +w, 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 16 SEURET et al. FIGURE 3 Principles of the optimization of the attractor 𝜈thank to the minimization of the a decision variable 𝜀. Thisadditionalinequality canbeinterpretedas constrainingthe ellipsoids(Wi,𝜁i)tobeallincludedin theballof radius √𝜀, centered at the same location (see Figure 3). In other words, we impose (W−1 i,𝜁i)⊂(𝜀−1In,𝜁i),∀i∈D𝜈.(30) An additional feature can be added to reduce the “size” of this attractor by introducing an optimization problem, as described below. Optimization Problem 1. For a given 𝜇, the optimal solution to the condition of Theorem 1that verifies inclusion (30)withthesmallest𝜀is obtained by solving the following optimization problem 𝜀∗=min {(Wi,𝜁i,𝛿i)}i∈D𝜈 𝜀 s.t.Φi(A𝜈(i),B𝜈(i))≻0,𝜀I≻Wi≻0,𝛿 i>0,∀i∈D𝜈,(31) where Φi(A𝜈(i),B𝜈(i))is given in (11). Similarly, an optimization procedure can be added to the conditions of Theorem 2as provided below. Optimization Problem 2. For a given 𝜇, the optimal solution to the condition of Theorem 2, that verifies inclusion (30)withthesmallest𝜀is obtained by solving the following optimization problem 𝜀∗=min {(Wi,𝜁i,𝜂i,𝛿i)}i∈D𝜈 𝜀 s.t.Φi(Y𝜈(i))≻0,𝜀I≻Wi≻0,𝛿 i>0,𝜂 i>0,∀i∈D𝜈,(32) where Φi(Y𝜈(i))is given in (27). Introducing constraints 𝜀I≻Wiand minimizing 𝜀refer to a usual optimization to reduce the size of the components of the attractors as for Theorem 1. ApartfromtheusualcausesassociatedwithLyapunovandLMItechniques,tworeasonsmayexplainthisconservatism in the numerical result. The first is related to the fact that the optimization problem presented in Theorem 1requires the use of two successive S-procedures. It is well known that this introduces conservatism compared to the situation where only one is performed. Second, the optimal value 𝜀∗strongly depends on parameter 𝜇, which has been fixed a priori. It would be possible to refine the values obtained for 𝜀by tuning this parameter, using a “line-search” algorithm on 𝜇. 5.2 Optimization selection of the cycle FollowingtheideafirstpresentedinReference18,itispossibletointroduceacostfunctionthataimsatevaluatingtheperformance of a cycle. For instance, this cost function should reflect for each cycle the chattering effects, which means the 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 17 amplitudeofthetrajectorieswithinthe(robust)limitcycle,orthedistanceofthe(robust)limitcycletoadesiredfunctioningpointxdinRn,selectedbythedesigner.However,thesolutionproposedinReference18reliesontheexactknowledge of the limit cycle, that is, {𝜌i}i∈D𝜈generated by a given cycle 𝜈in the situation of system (6) without disturbances. When considering robust limit cycles and its outer estimation using the attractor 𝜈, such a procedure cannot be employed directly, since the estimation of the robust limit cycle is provided by the solution of the LMI optimization problems (31) and (32). Instead, the solution provided in Reference 24 allows formulating such a cost function based on the decision variables of the LMI optimization problem. For the sake of readability and to avoid an overlap with this contribution, such extension is not presented here but the reader can refer to Reference 24(Section 6). 6NUMERICAL APPLICATIONS 6.1 System data To illustrate the two results presented in this paper, here we will consider an example of switched affine systems (6) borrowed from Reference 27, where the matrices Aiand Biare defined as follows. Ai=eFiT,Bi= T ∫0 eFi𝜏d𝜏gi,∀i∈{1,2},(33) where T=1 is the sampling period of the associated continuous-time system  x=Fjx+gjdefined with matrices Fjand gj,withj∈K={1,2}given by F1=⎡⎢⎢⎢⎣ 010 001 −1−1−1⎤⎥⎥⎥⎦ ,F2=⎡⎢⎢⎢⎣ 010 001 0−1−1⎤⎥⎥⎥⎦ ,g1=⎡⎢⎢⎢⎣ 1 0 0⎤⎥⎥⎥⎦ ,g2=⎡⎢⎢⎢⎣ 0 1 0⎤⎥⎥⎥⎦ . 6.2 Model-based design: solution and simulations First, let us consider a model-based solution. To do so, we will consider three cycles 𝜈1={1,2},𝜈2={1,2,2,2},and 𝜈3={1,1,1,1,2,2}to illustrate the potential of the result. Solving the conditions of Theorem 1, for these three cycles and for three different values of 𝜆, the simulations of the closed-loopsystemsaredepictedin Figure4.First, the figureshowsthat the attractor𝜈is indeedcomposedofthe union ofellipsoids.Thenumber of ellipsoids intheattractorequals the length ofthecycle.The state trajectoriesconverge, inall cases, to the attractor. In addition, it can also be seen that the control input uconverges to a periodic trajectory, which is a shifted version of the desired cycle. This characteristic has been demonstrated in Reference 24, but not in this paper, as the proof is very similar. It is worth mentioning that this periodic behavior of the control input is only guaranteed when the intersection of any pair of ellipsoids composing the attractor is empty. This figure also illustrates that the size of the attractor increases with the amplitude of the disturbance 𝜆. Indeed, the effect of the disturbance on the state trajectory is visible in this figure. For the first and second cycles 𝜈={1,2}and 𝜈={1,2,2,2}, the values of 𝜀∗obtained by solving Theorem 1are given in Table 1. These solutions show that increasing 𝜆naturally leads to an increase in 𝜀∗. 6.3 Data-based design: Generation of the data In order to generate the data, the following procedure was employed. Matrix Xj0has been selected as the solution to the system with 𝜎=jand an arbitrary initial condition xj,0with magnitude less than 1. In other words, we have x+ j,𝓁=xj,𝓁+1=Ajxj,𝓁+Bj+wj,𝓁,∀(j,𝓁)∈K×[1,pmax]N. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 18 SEURET et al. FIGURE 4 State and input trajectories of the closed-loop system for three different cycles obtained by solving Theorem 1for 𝜆=0, 0.05 and 0.1 with 𝜇=0.1 with the initial condition x0=[2−50]⊤, for all the cycles and bounds 𝜆, considered here. TABLE 1 Optimal values of 𝜀∗obtained by solving problem (32) for various numbers of data pj=10, 20, and 30, for cycle 𝜈={1,2}. Cycle 𝝂={1,2}𝝂={1,2,2,2} 𝜀∗Data 𝜆j=0𝜆j=0.05 𝜆j=0.10 𝜆j=0𝜆j=0.01 𝜆j=0.02 Th.1Aj,Bj0.0005 1.1519 4.6078 0.0008 0.0709 0.2837 pj=30 0.0063 2.4167 23.1562 0.0081 0.5751 2.8390 Th.2pj=20 0.0065 2.8790 26.3315 0.0084 0.6905 3.1607 pj=10 0.0090 3.8825 43.3567 0.0107 0.6922 4.5794 with the corresponding matrices Xjand X+ jgiven in (19) and where the maximum number of data considered in this example is 30. Three cases will be used, corresponding to pj=10, 20, and 30 for all jin K. Note that these three cases use thesamesetof data. This meansthatthe10 first data fromcase pj=20 correspondto thedatausedwhen pj=10 and the 20 first data from case pj=30 are the same data as for case pj=20. In this setup, the measurement noise has been selected as an arbitrary vector, which corrupts the solution to the noiseless system. The noise wjis shown in Figure 5, only for mode j=1 but for three cases of 𝜆. The norm of the noise vectors is bounded by 𝜆,and it is also assumed that the overall noise matrices 𝜔jverify 𝜔j𝜔⊤ j≤𝜅pj𝜆2 jIn, for all j∈K,with 𝜅=0.3. Such requirements ensure that inequality [In 𝜔⊤ j]⊤[𝜅pj𝜆2 jIn0 ∗−Ipj] ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Ψj [In 𝜔⊤ j]≽0 (34) holds for all j∈K. Note that this assumption is related to the covariance matrix when 𝜔jis a random variable, as mentioned in Reference 10. 6.4 Data-based design: Solution and simulations Applying Theorem 2and solving optimization problem 31, we have obtained the results that are resumed in Table 1for 𝜈1={1,2}and the associated simulations are depicted in Figure 6. It can be seen from Table 1that augmenting 𝜆imposes an increase of 𝜀∗, and that increasing the number of data leads to a decrease of the optimal 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 19 FIGURE 5 Measurement noise 𝜔1=[𝜔11 𝜔12 𝜔13]⊤, affecting the data for mode 1. Similar noise affects the data of the other mode. value 𝜀∗. Compared to the model-based approach, the corresponding values of 𝜀∗suffers from a notable increase. For the case 𝜆=0, the large augmentation of the optimal value can be explained by the fact that Theorem 2 requires an additional S-procedure to obtain the stabilization conditions, which unavoidably introduces conservatism, as commented earlier. Nevertheless, the resulting values of 𝜀∗are still small and the attractors are already accurate asshowninFigure6. For the cases, where 𝜆≠0, the increase of the 𝜀∗compared to the model-based approach can also be explained using the previous argument but also by the fact that the noise in the data also leads to additional uncertainties to the system matrices, which need to be compensated by an increase of the size of the attractor. Finally, Figure 7presents the variation of 𝜀∗with respect to the number of data measurements for various values of 𝜆. First of all, this figure shows that optimization problem (32) has no solution when the number of data is less than 4=n+1. This is consistent with the assumption of informativity of the data matrices Yjin Definition 5.Moreover,as𝜆 increases, the same optimization problem requires more than the minimal necessary order of data (n+1=4) to obtain a solution.Theminimumnumberofdatatoobtainasolution,increaseswiththeamplitudeofthedisturbance.Thegeneral tendency is that the optimal solution decreases as the number of data increases. This figure also shows that the values of 𝜀∗tend to a constant as pincreases. The same example has been treated with a different cycle 𝜈2={1,2,2,2}. Applying again Theorem 1and 2 and solving the associated optimization problems, we have obtained the numerical results provided in Table 1.Note that the data-driven conditions deliver attractors composed of larger ellipsoids compared to the previous case with 𝜈1. This explains why the amplitudes of the disturbance noise have been reduced to 𝜆=0,0.01, and 0.02. Note that the optimal value of 𝜀obtained by solving the model-based optimization problem (31)with𝜆=0.1is7.0,whichis already a very large value. Figure 8presents the simulations resulting from the same initial condition as the one used in Figure 4. In this example, we have compared the numerical results obtained for two cycles, that is, one of short length, as 𝜈1 and another one with a larger length, as 𝜈2. Increasing the length of a cycle makes the attractor larger. Intuitively, this is because the uncertainties in the model matrices propagate from one element of the cycle to the next one, to guarantee the invariance of the attractor. As a final comment on the numerical application of the data-driven design, it is worth noting that the numerical solutions highly depend on the noise and are very sensitive to it. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 20 SEURET et al. FIGURE 6 Evolution of the state variables (blue dashed line) for cycle 𝜈1={1,2}with their associated robust limit cycles. The red crosses depict 𝜁i, which are the centers of the ellipsoids (W−1 i,𝜁i). From top to bottom, the plot show the attractors when the number of data available for the design are 30, 20, and 10, respectively. From left to right, the uncertainty bound in the data increases 𝜆=0, 0.05, and 0.1. FIGURE 7 Graphshowingtheoptimalvalue𝜀∗ofoptimizationproblem(32)forcycle𝜈={1,2}andforvariousvaluesof𝜆from0to0.1. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SEURET et al. 21 FIGURE 8 Evolution of the state variables (blue dashed line) for cycle 𝜈2={1,2,2,2}with their associated robust limit cycles. The red crosses represent 𝜁i, which are the centers of the ellipsoids (W−1 i,𝜁i). From top to bottom, the plot shows the attractors when the number of dataavailablefor thedesignare30, 20,and10,respectively.Fromlefttoright, theuncertaintyboundinthedataincreases𝜆=0,0.01,and0.02. 7CONCLUSION Inthispaper,afirstgeneralresultispresentedthatdealswiththeequivalenceofLMIs,whichcanbeseenasanalternative solution to the S-Lemma presented in Reference 10, dedicated to the stabilization problem of linear systems. Then, to demonstrate the potential of this alternative, the objective of the paper is to develop a data-based stabilization criterion for the class of switched affine systems, which corresponds to a relevant class of nonlinear systems. First, a model-based control design for switched affine systems subject to a bounded uncertainty is presented as an extension of Reference 24. Thismethod, basedon theselectionaprioriofaparticularsequenceofmodes, providesapurestate-feedbackcontrollaw that stabilizes the closed-loop systems to an attractor characterized by a level set of the Lyapunov function. The attractor is made up of the union of possibly disjoint ellipsoids resulting from an optimization problem. Then, this contribution is adapted to the data-driven control design, thanks to application of the first lemma. An academic example validates the main result, showing that the desiredattractor grows with larger noise and smaller amount of data. Future works would aim at applying this contribution to power converters modelled as switched affine systems. ACKNOWLEDGMENTS This work was supported in part by projects PID2019-105890RJ-I00 and PID2019-109071RB-I00, funded by MCIN/AEI /10.13039/501100011033/andFEDER A way of makingEurope,by the AgenceNationalede laRecherche(ANR)-France under Grant ANR-18-CE40-0022-01. CONFLICT OF INTEREST STATEMENT The authors declare that there is no conflict of interest for this article. DATA AVAILABILITY STATEMENT Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1. Ebihara Y, Peaucelle D, Arzelier D. S-Variable Approach to LMI-Based Robust Control. Vol 6. Springer; 2015. 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. 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Deaecto G, Geromel J. Stability analysis and control design of discrete-time switched affine systems. IEEE Trans Automat Control. 2016;62(8):4058-4065. How to cite this article: Seuret A, Albea C, Gordillo F. Linear matrix inequality relaxations and its application to data-driven control design for switched affine systems. Int J Robust Nonlinear Control. 2023;1-22. doi: 10.1002/rnc.6712 10991239, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/rnc.6712 by Universidad De Sevilla, Wiley Online Library on [22/05/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License