Stability analysis of discrete-time switched systems under arbitrary switching

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IFAC PapersOnLine 51-25 (2018) 371–376

Stability analysis of discrete-time switched systems Stability Stability analysis analysis of of discrete-time discrete-time switched switched systems systems under switching Stability analysis ofarbitrary discrete-time switched systems under arbitrary switching under arbitrary switching under arbitrary switching Thales S. Gomide ∗ M´arcio J. Lacerda ∗

∗ ∗ Thales Thales S. S. Gomide Gomide ∗∗ M´ M´aaarcio rcio J. J. Lacerda Lacerda ∗∗ Thales S. Gomide M´ rcio J. Lacerda ∗ M´ ∗ Thales S. Gomide arcio J. Lacerda ∗ Control Modelling Group (GCOM) ∗ Control and Modelling Group (GCOM) ∗∗ Control and and Group (GCOM) Department of Electrical Engineering and Modelling Modelling Group (GCOM) ∗ Control Department of Electrical Engineering Control and Modelling Group (GCOM) Department of Electrical Engineering Federal University of S˜ a o Jo˜ a o del-Rei (UFSJ) Department of Electrical Engineering Federal University of S˜ aaooEngineering del-Rei (UFSJ) Department ofdel-Rei, Electrical Federal University of S˜aaaooo Jo˜ Jo˜ del-Rei (UFSJ) S˜ a o Jo˜ a o MG, Brazil Federal University of S˜ Jo˜ a o del-Rei (UFSJ) S˜ a o Jo˜ a o del-Rei, MG, Brazil Federal University of S˜ a o Jo˜ a o del-Rei (UFSJ) S˜a ao o Jo˜ Jo˜aaoo del-Rei, del-Rei, MG, MG, Brazil Brazil e-mail:[email protected], [email protected] S˜ e-mail:[email protected], [email protected] S˜ a o Jo˜ a o del-Rei, MG, Brazil e-mail:[email protected], [email protected] e-mail:[email protected], [email protected] e-mail:[email protected], [email protected] Abstract: This work study the stability problem for discrete-time switched systems under arbitrary Abstract: This work study the stability problem for discrete-time switched systems under arbitrary Abstract: This work study the stability problem for discrete-time switched systems under arbitrary switching. The Lyapunov approach is employed to this end. The structure of the Lyapunov function Abstract: This work study the stability problem for discrete-time switched systems under arbitrary switching. The Lyapunov approach is employed to this end. The structure of the Lyapunov function Abstract: This work study the stability problem for discrete-time switched systems under arbitrary switching. The Lyapunov approach is employed employed to this this that end.contains The structure structure oforder the Lyapunov Lyapunov function is based on the existence of an augmented state vector higher shifted states. This switching. The Lyapunov approach is to end. The of the function is based on the existence of an augmented state vector that contains higher order shifted states. This switching. The Lyapunov approach is employed to this end. The structure of the Lyapunov function is based based on on the existence of an an augmented augmented state vector thatsystem contains higher order function, shifted states. states. This structure will bring the switched dynamic from the original to the Lyapunov providing is the existence of state vector that contains higher order shifted This structure will bring the switched dynamic from the original system to the Lyapunov function, providing based on the existence of anInaugmented state vector that contains higher order shifted states. This structure will bring the switched dynamic from the original system to the Lyapunov function, providing aais switched Lyapunov function. order to check stability, LMI (Linear Matrix Inequalities) conditions structure will bring the switched dynamic from the original system to the Lyapunov function, providing Lyapunov function. In order to check stability, LMI (Linear Matrix Inequalities) conditions structure will bring the switched dynamic from the original system to the Lyapunov function, providing aa switched switched Lyapunov function. In order to check stability, LMI (Linear Matrix Inequalities) conditions are derived to solve the problem. The Lyapunov function proposed in this work has proved stability using switched Lyapunov function. InThe order to check stability, LMI (Linear Matrix Inequalities) conditions are derived to solve the problem. Lyapunov function proposed in this work proved stability using aare InThe order to than check stability, LMIfrom (Linear Matrix Inequalities) conditions areswitched derived Lyapunov toless solve thefunction. problem. The Lyapunov function proposed inthe thisliterature. work has has proved stability using considerably scalar decision variables other methods A numerical example derived to solve the problem. Lyapunov function proposed in this work has proved stability using considerably less scalar decision variables than other methods from the literature. A numerical example are derived to solve the problem. The Lyapunov function proposed in this work has proved stability using considerably less scalar scalar decision variablesof than other methods from from the the literature. literature. A A numerical numerical example example has been explored to discuss the potential the method. considerably less decision variables than other methods has been explored to discuss the potential of the method. considerably less scalar decision variables than other methods from the literature. A numerical example has been explored to discuss the potential of the method. has been explored to discuss the potential of the method. © 2018, (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. has beenIFAC explored to discuss Federation the potential of the method. Keywords: Lyapunov function, hybrid systems, asymptotic stability Keywords: Lyapunov function, hybrid systems, asymptotic stability Keywords: Keywords: Lyapunov Lyapunov function, function, hybrid hybrid systems, systems, asymptotic asymptotic stability stability Keywords: Lyapunov function, hybrid systems, asymptotic stability In Daafouz et al. (2002), aa switched Lyapunov function was 1. INTRODUCTION In al. switched Lyapunov function was 1. INTRODUCTION 1. In Daafouz Daafouzforet etdiscrete-time al. (2002), (2002), aa switched switched systems Lyapunov function was proposed under arbitrary In Daafouz et al. (2002), switched Lyapunov function was 1. INTRODUCTION INTRODUCTION proposed for discrete-time switched systems under arbitrary In Daafouz et al. (2002), a switched Lyapunov function was 1. INTRODUCTION proposed for discrete-time switched systems under arbitrary switching. The stability could be checked using two different proposed for discrete-time switched systems under arbitrary Switched systems have been on the focus of aa great number switching. The stability could be checked using two different Switched systems have been on the focus of great number proposed for discrete-time switched systems under arbitrary switching. The stability could be checked using two different but equivalent in the of LMIs. addition, switching. The conditions stability could beform checked using In two differentaa Switched systems have been on focus of aa studies great of researches in the last years, one may cite about Switched systems have been on the the focus of great number number but equivalent conditions in the form of LMIs. addition, of researches in the last years, one may cite about switching. The stabilitycontrol could be checked using In two different but equivalent conditions in the form of LMIs. In addition, aa static output feedback has been designed. Later, (Sun Switched systems have been on the focus of a studies greatswitched number but equivalent conditions in the form of LMIs. In addition, of researches in the last years, one may cite studies about stability and control for continuous and discrete-time of researches in the last years, one may cite studies about static output feedback control has been designed. Later, (Sun stability and control for continuous and discrete-time switched but equivalent conditions in the form of LMIs. In addition, a static output feedback control has been designed. Later, (Sun and Ge, 2005) have developed control of switched linear syof researches in the last years, one may cite studies about static output feedback control has been designed. Later, (Sun stability and continuous and switched systems et al., 2006; Mason et al., 2007; Blanchini stability (Montagner and control control for for continuous and discrete-time discrete-time switched and Ge, 2005) have developed control of switched linear sysystems (Montagner et al., 2006; Mason et al., 2007; Blanchini static output feedback control has been designed. Later, (Sun and Ge, Ge,First, 2005) have developed and control of switched switched linear sysystems. the controllability observability problem for stability andYu control for continuous and discrete-time switched and 2005) have developed control of linear systems (Montagner et al., 2006; Mason et al., Blanchini et al., 2007; et al., Trofino et al., Deaecto et al., systems (Montagner et2007; al., 2006; Mason et2009; al., 2007; 2007; Blanchini stems. First, the controllability observability problem for et al., 2007; Yu et al., 2007; Trofino et al., 2009; Deaecto et al., and Ge, 2005) have developed and control of switched linear systems. First, the controllability and observability problem for both discrete and continuous-time cases have been considered. systems (Montagner et al., 2006; Mason et al., 2007; Blanchini stems. First, the controllability and observability problem for et al., 2007; Yu et al., 2007; Trofino et al., 2009; Deaecto et al., 2011; Souza et al., 2018). It is possible to say that a switched et al., 2007; Yu et al., 2007; Trofino et al., 2009; Deaecto et al., both discrete and continuous-time cases have been considered. 2011; Souza et al., 2018). It is possible to say that a switched stems. First, the controllability and observability problem for both discrete and continuous-time cases have been considered. Second, feedback control have been designed for two different et al., 2007; Yu et al., 2007; Trofino et al., 2009; Deaecto et al., both discrete and continuous-time cases have been considered. 2011; Souza et al., 2018). It is possible to say that aa switched system is formed by a number of subsystems and each one of 2011; Souza et al., 2018). It is possible to say that switched Second, feedback control have been designed for two different system is formed by a number of subsystems and each one of both discrete and continuous-time cases have been considered. Second, feedback control have been designed for two different scenarios: when the switching signal is a design variable and 2011; Souza et al., 2018). It is possible to say that a switched Second, feedback control have been designed for two different system is formed by of each of them can individually active accordingly to aaand switching rule. system is be formed by aa number number of subsystems subsystems and each one one of scenarios: when the switching signal is variable and them can individually active accordingly to switching rule. feedback control have been designed forand two different scenarios: when the switching signal is aaa design design variable and considering all possible switching signals. Sun Ge (2005) system is be formed by a number of subsystems each one of Second, scenarios: when the switching signal is design variable and them can be individually active accordingly to aaand switching rule. A crucial issue for this class of systems is stability. Even if all them can be individually active accordingly to switching rule. considering all possible switching signals. Sun and Ge (2005) A crucial issue for this class of systems is stability. Even if all scenarios: when the switching signal is for a design variable and considering all possible switching signals. Sun and Ge (2005) have also investigated optimal control switched systems. them can be individually active accordingly to a switching rule. considering all possible switching signals. Sun and Ge (2005) A crucial issue for this class of systems is stability. Even if all the subsystems are stable, the dynamics of the general system A crucial issue for this class ofdynamics systems isofstability. Even if all have also investigated optimal control for switched systems. the subsystems are stable, the the general system considering all possible switching signals. Sun and Ge (2005) have also investigated optimal control for switched systems. In (Lee and Dullerud, 2006) a condition to check the uniform A crucial issue for this class of systems is stability. Even if all also investigated optimal control for switched systems. the are the dynamics of may not be throughout time (Branicky, the subsystems subsystems are stable, stable, the dynamics1998). of the the general general system system have In and Dullerud, 2006) aa condition to the uniform may not be throughout time (Branicky, also optimal control for switched In (Lee (Lee andinvestigated Dullerud, 2006) condition to check check the systems. uniform stabilization for discrete-time switched systems was presenthe subsystems are stable, the dynamics1998). of the general system have In (Lee and Dullerud, 2006) a condition to check the uniform may not be throughout time (Branicky, 1998). may not be throughout time (Branicky, 1998). stabilization for discrete-time switched systems was presenIn (Lee and Dullerud, 2006) a condition to check the uniform Due certifying stability, many authors have stabilization for discrete-time switched systems was presented considering a path-dependent Lyapunov approach. Lee and may to notthe beimportance throughoutof time (Branicky, 1998). stabilization for discrete-time switched systems was presenDue to the importance of certifying stability, many authors have ted considering a path-dependent Lyapunov approach. Lee and Due to the importance of certifying stability, many authors have stabilization for discrete-time switched systems was presenstudied this class of hybrid systems under arbitrary switching, ted considering a path-dependent Lyapunov approach. Lee and Due to the importance of certifying stability, many authors have Dullerud (2006) also have studied stabilization of Markovian ted considering a path-dependent Lyapunov approach. Lee and studied this class of hybrid systems under arbitrary switching, Dullerud (2006) also have studied stabilization of Markovian Due to the importance of certifying stability, many authors have studied this class of hybrid systems under arbitrary switching, ted considering a also path-dependent Lyapunov approach. Lee and which means that the switching sequence is not known a priori. Dullerud (2006) also have studied stabilization of Markovian studied this class of hybrid systems under arbitrary switching, jump linear systems, which are a stochastic version of the linear Dullerud (2006) have studied stabilization of Markovian which means that the switching sequence is not known a priori. jump linear systems, which are a stochastic version of the linear studied this class of hybrid systems underis arbitrary switching, which means that the switching sequence not known aasystems priori. Dullerud (2006) alsowhich have studied stabilization of(2008) Markovian The classical approach to certify stability of switched jump linear systems, which are a stochastic version of the linear which means that the switching sequence is not known priori. systems with jumping parameters. Zhao and Hill have jump linear systems, are a stochastic version of the linear The classical approach to certifysequence stability switched systems with jumping parameters. Zhao and Hill (2008) have which means that the switching isof known asystems priori. The approach to stability of switched systems jump linear systems, which are aLyapunov stochastic version ofinthe linear under arbitrary switching is known as quadratic stability, that systems jumping parameters. Zhao Hill have The classical classical approach to certify certify stability ofnot switched systems showed aawith new approach for the function terms of systems with jumping parameters. Zhao and and Hill (2008) (2008) have under arbitrary switching is known as quadratic stability, that showed new approach for the Lyapunov function in terms of The classical approach to certify stability of switched systems under arbitrary switching is known as quadratic stability, that systems with jumping parameters. Zhao and Hill (2008) have makes use of a quadratic Lyapunov function to test asymptotic showed a new approach for the Lyapunov function in terms of under arbitrary switching is known as quadratic stability, that stabilization for continuous-time systems. They also studied the showed a new approach for the Lyapunov function in terms of makes use of a quadratic Lyapunov function to test asymptotic stabilization for continuous-time systems. They also studied the under arbitrary switching is known as quadratic stability, that makes use of a quadratic Lyapunov function to test asymptotic showed a new approach for the Lyapunov function in terms of stability of the system. It is important to recall that in this case, a stabilization for continuous-time systems. They also studied the makes use of a quadratic Lyapunov function to test asymptotic L gain and H control for this class of system. Hetel et al. stabilization for continuous-time systems. They also studied the 2 gain and H∞ control for this class of system. Hetel et al. stability of the system. It is important to recall that in this case, a L makes use of a quadratic Lyapunov function to test asymptotic ∞ 2 stability of the system. It is important to recall that in this case, a stabilization for continuous-time systems. They also studied the single Lyapunov matrix must assure stability for all subsystems L gaininvestigated and H H∞ control for this this class class of system. system. Hetel timeet al. al. stability of the system. Itmust is important to recallfor thatallinsubsystems this case, a L (2006) the stabilization problem considering 22 gain and for of Hetel et ∞ control single Lyapunov matrix assure stability (2006) investigated the stabilization problem considering timestability of thetime, system. Itthat is important to recall that this case, a L single Lyapunov matrix must assure stability for all and H∞which control for this class of system. Hetel etand al. at the same fact can bring some conservativeness 2 gaininvestigated (2006) the stabilization problem considering timesingle Lyapunov matrix must assure stability for allinsubsystems subsystems varying delays, has been further developed in (Zhang (2006) investigated the stabilization problem considering timeat the same time, fact that can bring some conservativeness varying delays, which has been further developed in (Zhang and single Lyapunov matrix must assure stability for allthe subsystems at the same time, fact that can bring some conservativeness (2006) investigated the stabilization problem considering timeto analysis. Thanks to the Lyapunov theory, stability varying delays, which has been further developed in (Zhang at the same time, fact that can bring some conservativeness Yu (2009); Zhao et al. (2012); Zong et al. (2015)). Zhang and delays, which has been further developed in (Zhang to the same analysis. Thanks to the theory, the stability varying Yu (2009); Zhao et (2012); Zong et al. Zhang and at time, factswitched that canLyapunov bring some conservativeness to Thanks to Lyapunov theory, the delays, been further developed in (Zhang and analysis problem for systems can be written as aa varying Yu (2009) (2009); Zhaowhich et al. al.has (2012); Zong et al. (2015)). (2015)). Zhang to the the analysis. analysis. Thanks to the the Lyapunov theory, the stability stability investigated time-delay for discrete systems consideYu (2009); Zhao et al. (2012); Zong et al. (2015)). Zhang and analysis problem for switched systems can be written as Yu (2009) investigated time-delay for discrete systems consideto the analysis. Thanks to the Lyapunov theory, the stability analysis problem for switched systems can be written as a Yu (2009); Zhao et al. (2012); Zong et al. (2015)). Zhang and set of LMIs (Linear Matrix Inequalities). Hybrid system is (2009) investigated time-delay for discretesignal. systems consideanalysis problem for Matrix switchedInequalities). systems canHybrid be written as isa Yu ring the average dwell time of the switching Zhao et al. (2009) investigated time-delay for discrete systems consideset of LMIs (Linear system ring the average dwell time of the switching signal. Zhao et al. analysis problem for switchedInequalities). systemsand canitHybrid be written as is of (Linear Matrix system Yu (2009) investigated time-delay for discrete systems consideaaset class of dynamic system is defined with the average dwell time of the switching signal. Zhao et al. setparticular of LMIs LMIs (Linear Matrix Inequalities). Hybrid system isa ring (2012) have investigated both continuous and discrete switched ring the average dwell time of the switching signal. Zhao et al. particular class of dynamic system and it is defined with (2012) have investigated both continuous and discrete switched set of LMIs (Linear Matrix Inequalities). Hybrid system is class of dynamic system and it is defined with ring the average dwell time of the switching signal. Zhao et al. aa particular continuous and discrete-time dynamics, and it also has a (2012) have investigated both continuous and discrete switched particular class of dynamic system and it is defined with systems and developed the concept of mode-dependent average (2012) have investigated both continuous and discrete switched continuous and discrete-time dynamics, and it also has a systems and developed the concept of mode-dependent average class of dynamic system and it is defined with aa particular continuous and discrete-time dynamics, and it also has aa (2012) have investigated both continuous and discrete switched discrete decision-making process. This process may change the systems and developed the concept of mode-dependent continuous and discrete-time dynamics, and it also has dwell time (MDADT) to deal with the restriction of average systems and developed the concept of mode-dependent discrete decision-making process. This process change time (MDADT) to with the restriction of average adiscrete continuous andsystem discrete-time dynamics, andmay it also hasthe a dwell discrete decision-making process. This may change the systems andof developed the concept mode-dependent average dynamic of the and sometimes it may depend on an dwell time (MDADT) to deal deal withof the restriction of decision-making process. This process process may change the the switching signal. Later, (Zong et al. (2015)) dwell (MDADT) to deal with the restriction of average dynamic of the system and sometimes it may depend on an dwell time of the switching signal. Later, (Zong et al. (2015)) discrete decision-making process. This process may change the dynamic of the system and sometimes it may depend on an (MDADT) to deal with the restriction of average outside factor. Many dynamic problems can be modeled as a dwell time of the switching signal. Later, (Zong et al. (2015)) dynamic of the system and sometimes it may depend on an have studied slowly switched systems, for continuous time, dwell time of the switching signal. Later, (Zong et al. (2015)) outside factor. Many dynamic problems can be modeled as a have studied slowly switched systems, for continuous time, dynamic of the system and sometimes it may depend on an outside factor. Many dynamic problems can be modeled as aa dwell time of the switching signal. Later, (Zong et al. (2015)) hybrid system, such as DC-DC converters, the motion of a car have studied slowly switched systems, for continuous time, outside factor. Many dynamic problems can be modeled as based on MDADT as well. Lin and Antsaklis (2009) have made have studied slowly switched systems, for continuous time, hybrid system, such as DC-DC converters, the motion of a car based on MDADT as well. Lin and Antsaklis (2009) have made outside factor. Many dynamic problems can be modeled as a hybrid system, such as DC-DC converters, the motion of a car have studied slowly switched systems, for continuous time, considering the gear box and air traffic control. based on MDADT as well. Lin and Antsaklis (2009) have made hybrid system, such as DC-DC converters, the motion of a car a very interesting study of a variety of switching stabilization based on MDADT as well. Lin and Antsaklis (2009) have made considering the gear box and air traffic control. a very interesting study of a variety of switching stabilization hybrid system, such as DC-DC converters, the motion of a car considering the gear box and air traffic control. based on MDADT as well. Lin and Antsaklis (2009) have made a very very interesting interesting study of of variety ofhave switching stabilization considering the gear box and air traffic control. Then, Jungers et (2017) described different amethods. study aaal. variety of switching stabilization methods. Then, Jungers et (2017) described different considering the gear box and air traffic control. amethods. very interesting study of aal. variety ofhave switching stabilization methods. Then, Jungers et al. (2017) have described different sets of LMIs that can be used to certify stability of discreteThen, Jungers et al. (2017) have described different  This work was supported by PPGEL-UFSJ/CEFET-MG and the Brazilian sets of LMIs that can be used to certify stability of discretemethods. Then, Jungers et al. (2017) have described different  This work was supported by PPGEL-UFSJ/CEFET-MG and the Brazilian sets of LMIs that can be used to certify stability of discretetime switched systems. Although some of these works have sets of LMIs that can be used to certify stability of discrete  time switched systems. some of these works have agencies Capes, FAPEMIG APQ-00692-17 This work work wasand supported by grant PPGEL-UFSJ/CEFET-MG and the the Brazilian Brazilian This was supported by PPGEL-UFSJ/CEFET-MG and sets of LMIs that can beAlthough used to certify stability of discretetime switched systems. Although some of these works have agencies Capes, and FAPEMIG grant APQ-00692-17  time switched systems. Although some of these works have This work wasand supported by grant PPGEL-UFSJ/CEFET-MG and the Brazilian agencies Capes, FAPEMIG APQ-00692-17 agencies Capes, and FAPEMIG grant APQ-00692-17 time switched systems. Although some of these works have

agencies Capes, and FAPEMIG grant APQ-00692-17 2405-8963 © 2018 2018, IFAC (International Federation of Automatic Control) Copyright © IFAC 519 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 519 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 519 Copyright © 2018 IFAC 519 10.1016/j.ifacol.2018.11.135 Copyright © 2018 IFAC 519

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provided studies about the arbitrary switching case for discretetime systems, there are plenty of room for less conservative and more efficient methods. This paper is concerned with the stability problem for discretetime switched systems under arbitrary switching. The stability condition is based on the existence of a Lyapunov function constructed with an augmented state vector that takes into account a generic number of higher order shifts states. This choice for the Lyapunov function will allow the dynamics of the switched system to be introduced in the Lyapunov function. In this way, the Lyapunov function presented in this paper is also a switched Lyapunov function. This approach is based on some recent ideas introduced for uncertain systems (Lacerda and Seiler, 2017). The main objective of this approach is to find a more efficient method to certify the stability of the switched system. First, a particular case is presented to make the reader familiar with the method. Then, the concept will be expanded to the generic case. A numerical example from the literature will be presented to evaluate the technique. The paper is organized as follows. Section 2 introduces the preliminary results with some classic conditions from the literature and the particular case of the proposed approach. The main results are presented in Section 3 with a generic representation of the Lyapunov function that have been considered. Section 4 presents the numerical experiments that illustrate the advantages of the proposed method when compared to other techniques from the literature and Section 5 concludes the paper.

One can say that the Lyapunov function is positive-definite, decrescent, and radially unbounded if V (k, 0) = 0, ∀k ≥ 0 and

β1 x(k)2  V (k, x(k))  β2 x(k)2 (6) for all x(k) ∈ Rn and k ≥ 0 with β1 and β2 positive scalars.

Before introduce the main idea of the proposed method, let us revisit some approaches presented in the literature to solve the same problem. In order to check the stability of system (1) under arbitrary switching, the authors from Daafouz et al. (2002) have introduced a switched Lyapunov function and two different LMI conditions to certify the stability of (1). The next Lemma states the conditions and the form of the Lyapunov function employed. Lemma 2. (Daafouz et al. (2002)). The following statements are equivalent. (1) There exists a Lyapunov function of the form  V (k, x(k)) = x(k) P(ξ (k))x(k) = x(k)

T

v

∑ ξi (k)Pi

i=1

2.1 System Description Consider the following discrete-time switched system (1) x(k + 1) = Aα x(k) where A ∈ Rn×n and x ∈ Rn is the state vector. The α parameter denotes the switching rule which can take a value in a finite set P = {1, . . . , v}. It means that matrix Aα may only assumes values in the finite set (2) {A1 , A2 , . . . , Av }. We are interested in studying the stability of the origin of a switched system represented by (1). Considering that Aα may assume values indicated in the set (2). It is also important to remember that only one mode of the switched system is active at a time, whereas the others do not have influence at the system. This situation can be represented by defining the indicator function T (3) ξ (k) = [ξ1 (k) ... ξv (k)] where  1, for Ai (the ith mode is active) ∀i = 1, . . . , v ξi (k) = 0, otherwise Therefore, system (1) can be written as v

• V is a positive-definite function, decrescent, and radially unbounded; • ∆V (k, x(k)) = V (k + 1, x(k + 1)) − V (k, x(k)) is negative definite along the solutions of (5).

T

2. PRELIMINARIES

x(k + 1) = ∑ ξi (k)Ai x(k)

Theorem 1. (Vidyasagar (1993)). The equilibrium 0 of (5) x(k + 1) = fk (x(k)) is globally uniformly asymptotically stable if there is a function V : Z+ × Rn → R such that:

(4)

i=1

2.2 Stability Analysis The next theorem introduces the conditions needed to certify the stability of the origin of a switched system considering a Lyapunov function. 520



x(k)

whose difference is negative definite, proving asymptotic stability of (1). (2) There exist v matrices P1 , ..., Pv satisfying   Pi ATi Pj > 0 ∀(i, j) ∈ P × P (7) Pj Ai Pj The Lyapunov function is given by  V (k, x(k)) = x(k)

T

v

∑ ξi (k)Pi

i=1



x(k)

(8)

(3) There exist v symmetric matrices S1 , ..., Sv and v matrices G1 , ...Gv which satisfy  Gi + GTi − Si GTi ATi >0 Ai Gi Sj



∀(i, j) ∈ PxP

(9)



(10)

The Lyapunov function is given by  V (k, x(k)) = x(k)

T

v

∑

i=1

ξi (k)Si−1

x(k)

Another approach to certify the exponential stability of system (1) was introduced in (Lee and Dullerud, 2006, Theorem 9). For the sake of completeness this results is stated in the next Lemma. Lemma 3. (Lee and Dullerud (2006)). Let (1) be a switched linear system. It is exponentially stable if and only if there is an integer M and symmetric positive definite matrices X( j1 , . . . , jM ) ∈ Rn×n , ( j1 , . . . , jM ) ∈ {1, . . . , v} such that ATiM Xi1 ...iM AiM − Xi0 ...iM−1 < 0

where (i0 , . . . , iM ) ∈ {1, . . . , switching path of length M.

v}M+1

(11)

represents an admissible

IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018Thales S. Gomide et al. / IFAC PapersOnLine 51-25 (2018) 371–376

Both Lemmas 2 and 3 propose conditions that rely on the existence of matrices that somehow copy the behavior of the switched system. For instance, in Lemma 2 the Lyapunov matrix has the same representation of the dynamic matrix from (1). In Lemma 3 the Lyapunov function is even more complex and depends on multiple indices that depend on the size of the path M chosen to test the stability of the system. Based on recent advances in the stability problem for polytopic uncertain systems (Lacerda and Seiler, 2017) the idea proposed in this paper takes into account a new class of Lyapunov function that depends on the dynamics of the switched system. Firstly, consider the Lyapunov function to be written as follows      P1 0 x(k) T T (12) V (k, x(k)) = x(k) x(k + 1) 0 P2 x(k + 1)

where P1 and P2 are symmetric matrices. Conditions from Theorem 1 must be satisfied in order to guarantee that (12) is a Lyapunov function. We do not require the matrices P1 or P2 to be positive-definite, in the sequel it will be clear that (12) can be written in such a way that avoid the imposition of more constraints on matrices P1 and P2 . The following lemma presents the results for this case Lemma 4. The following statements are equivalent (1) There exists a Lyapunov function in the form (12) proving asymptotic stability of (1) (2) There exist symmetric matrices P1 and P2 satisfying P1 + ATi1 P2 Ai1 > 0 ∀i1 ∈ P

(13)

ATi1 P1 Ai1 + ATi1 ATi2 P2 Ai2 Ai1 − (P1 + ATi1 P2 Ai1 ) < 0 ∀(i1 , i2 ) ∈ P × P Proof. Note that (12), can be written as V (k, x(k)) = x(k)T P1 x(k) + x(k + 1)T P2 x(k + 1) Moreover, based on (4), we can indicate x(k + 1) = A(ξ (k))x(k) Replacing (16) in (15) one has

(14)

(15) (16)

V (k, x(k)) = x(k)T P1 x(k) + x(k)T A(ξ (k))T P2 A(ξ (k))x(k) (17) that yields   (18) V (k, x(k)) = x(k)T P1 + A(ξ (k))T P2 A(ξ (k)) x(k) Clearly V (k, 0) = 0, ∀k ≥ 0. Theorem 1 states that V (k, x(k)) must be positive definite in order to be a Lyapunov function. Therefore, (19) P1 + A(ξ (k))T P2 A(ξ (k)) > 0 which may be verified by (13). To satisfy the  conditions from  Theorem 1, let us choose β1 = mini1 ∈P λmin P1 + ATi1 P2 Ai1   and β2 = maxi1 ∈P λmax P1 + ATi1 P2 Ai1 positive scalars such that β1 x(k)2 ≤ V (k, x(k)) ≤ β2 x(k)2 (20) for all x(k) ∈ Rn and k ≥ 0. Now we need to prove that V (k, x(k)) is descrescent along the trajectories of system (1). From (12) one may say that T

T

V (k + 1, x(k + 1)) = x(k + 1) P1 x(k + 1) + x(k + 2) P2 x(k + 2) (21) By using (4) it is possible to write x(k + 2) = A(ξ (k + 1))x(k + 1) (22) 521

373

or

x(k + 2) = A(ξ (k + 1))A(ξ (k))x(k) (23) Let us replace (23) and (16) in (21) and use (18) to write ∆V (k, x(k)) = V (k + 1, x(k + 1)) −V (k, x(k)) as ∆V (k, x(k)) = x(k)T [A(ξ (k))T P1 A(ξ (k))+

A(ξ (k))T A(ξ (k + 1))T P2 A(ξ (k + 1))A(ξ (k)) − (P1 + A(ξ (k))T P2 A(ξ (k)))]x(k) According to Theorem 1, ∆V (k, x(k)) must be a negative definite function, then one can write A(ξ (k))T P1 A(ξ (k)) + A(ξ (k))T A(ξ (k + 1))T P2 A(ξ (k + 1))A(ξ (k)) − (P1 + A(ξ (k))T P2 A(ξ (k))) < 0 which may be verified based on LMIs from (14). Furthermore ∆V (k, x(k)) ≤ −γ(x(k)) where γ=

λmin (ATi1 P1 Ai1

min

(i1 ,i2 )∈P×P

+ ATi1 ATi2 P2 Ai2 Ai1 − (P1 + ATi1 P2 Ai1 )) which concludes the proof. 3. MAIN RESULTS Based on the main idea proposed in Lemma 4 our goal in the present Section is to extend the approach to consider a Lyapunov function that can take into account a generic augmented state vector in the following form.  T   x(k) x(k)  x(k + 1)   x(k + 1)   Ψ  (24) V (k, x(k)) =  .. ..     . . x(k + (N − 1))

with

x(k + (N − 1))

  Ψ = blkdiag P1 , P2 , . . . , PN

(25)

where N depends on the size of the augmented state vector. In addition, P1 , . . . , PN ∈ Rn×n represent symmetric matrices. In this way the next theorem generalize the results presented in Lemma 4 Theorem 5. The following statements are equivalent (1) There exists a Lyapunov function in the form (24) proving asymptotic stability of (1) (2) There exist N symmetric matrices P1 , . . . , PN satisfying     N

m−1

m=1

β =0

∑

∏ ATiβ

m−1

∏ Aim−β −1

Pm

>0

(26)

β =0

∀(i1 , i2 , . . . , iN−1 ) ∈ P × P . . . P   N−1 times

N

∑

m=1



m−1

∏

β =0 N

−

∑

m=1

ATi1+β





m−1

∏

β =0

Pm

ATiβ



m−1

∏ Aim−β

β =0



Pm





m−1

∏ Aim−β −1

β =0



∀(i1 , i2 , . . . , iN ) ∈ P × P . . . P  

where Ai0 = I.

N times

<0

(27)

IFAC ROCOND 2018 374 Florianopolis, Brazil, September 3-5, 2018Thales S. Gomide et al. / IFAC PapersOnLine 51-25 (2018) 371–376

Proof. As stated in Theorem 1 the Lyapunov function (24) must be a positive definite function V (k, x(k)) > 0 (28) then, one can write T

T

V (k, x(k)) = x(k) P1 x(k) + x(k + 1) P2 x(k + 1) + . . . + x(k + N − 1)T PN x(k + N − 1) > 0 (29) According to (4) we may assume x(k + 1) = A(ξ (k))x(k) x(k + 2) = A(ξ (k + 1))A(ξ (k))x(k) (30) .. . x(k + N − 1) = A(ξ (k + N − 2)) · · · A(ξ (k))x(k) Replacing (30) in (29) we have a condition that need to be satisfied to guarantee a positive-definite Lyapunov function P1 + A(ξ (k))T P2 A(ξ (k)) + · · · N−1   N−1 + ∏ A(ξ (k + β − 1))T PN ∏ (A(ξ (k + N − β − 2))) > 0 β =0

β =0

(31)

with A(ξ (k − 1)) = I. Condition (31) can be written in a compact form as a sum of terms, from P1 to PN , which leads to (32). Condition (32) may be numerically checked using LMIs from (26). From Theorem 1 ∆V (k, x(k)) = V (k + 1, x(k + 1)) −V (k, x(k)) < 0 (33) Considering V (k, x(k)) in (24) and following the same lines from Lemma 4 we have V (k + 1, x(k + 1)) = x(k + 1)T P1 x(k + 1) + x(k + 2)T P2 x(k + 2) + . . . + x(k + N)T PN x(k + N)

(34)

By using (24), (30) and (34), condition (33), after some algebraic manipulation, can be satisfied by the following inequality A(ξ (k))T P1 A(ξ (k)) + A(ξ (k))T A(ξ (k + 1))T P2 A(ξ (k + 1))A(ξ (k)) · · · N−1   N−1 + ∏ A(ξ (k + β )T PN ∏ (Aξ (k + N − β − 1)) β =0

β =0

− (P1 + A(ξ (k))T P2 A(ξ (k)) + · · · N−1   N−1 + ∏ A(ξ (k + β − 1))T PN ∏ (A(ξ (k + N − 2 − β )))) < 0 β =0

β =0

(35) The terms of (35) may be expressed as in (36) and the inequality (36) may be numerically verified based on LMIs from (27). Remark 6. It is important to remember that, Theorem 5 with N = 1 is reduced to the classic quadratic stability condition (Liberzon, 2003). In other words, consider P2 = 0 in conditions obtained from Lemma 4 to get the quadratic stability constraints. 4. NUMERICAL EXPERIMENT The main goal of the numerical experiment is to compare the conditions proposed in this paper with other approaches from the literature. The routines were implemented in MATLAB, 522

version 8.3.03532 (R14a) utilizing Yalmip (L¨ofberg, 2004) and SeDuMi (Sturm, 1999). The computer used was an Intel Quad Core (2.4GHz), 4GB RAM, Windows 10. Consider the following discrete-time switched system borrowed from Lee and Dullerud (2006), which depends on the parameter θ x(k + 1) = Aα x(k) (37) The matrix Aα can assume values in a finite set P = {1, 2} with     θ θ −θ 0 A2 = A1 = 0 0 θ −θ Theorem 5 has been used to check stability of (37) under arbitrary switching, i.e., there is no switching rule between the two operation modes. The objective is to find the upper bound on θ such that system (37) is still stable. The adopted strategy was to consider different values of N and verify the different upper bounds for θ obtained with Theorem 5. Table 1 indicates the results for the upper bounds obtained for each value of N considered. One can see that increasing N in Theorem 5 provide less conservative results in terms of the upper bound obtained. Moreover, Table 1 shows the number of LMI rows NR5 and the number of scalar decision variables NV 5 that Theorem 5 has used in each case. Table 1 also shows the results, for the upper bounds, reported by Lee and Dullerud (2006) when applying Lemma 3. The size of the path is indicated by M and the number of LMI rows and scalar decision variables is given by NR3 and NV 3 respectively. It can be noted that by increasing M and N the upper bounds on θ also increase. For small values of M and N, the number of scalar decision variables when considering Theorem 5 and Lemma 3 is quite similar. However, when M and N increase, the number of scalar decision variables in the method proposed by Lee and Dullerud (2006) (NV 3 ) is considerably higher (approximately nine times bigger for the maximum θ ) when compared with the proposed approach (NV 5 ). It occurs because the number of decision variables used in Lemma 3 grows exponentially with M while the number of scalar decision variables of Theorem 5 grows linearly with N. It is also possible to verify that both approaches make use of the same number of LMI rows to solve the problem. The method proposed in Daafouz et al. (2002), presented in Lemma 2, also has been performed with the considered example. The stability was checked by applying condition (7) and then by applying condition (9). Both of them provided the same results, as expected since the conditions are equivalent. The method was only able to guarantee the stability of the system with a maximum upper bound θ = 0.75487. Both conditions have used 16 LMI rows, condition (7) has used 6 scalar decision variables while condition (9) has used 14. This result demonstrates the importance of consider more elaborate structures for Lyapunov functions. Furthermore, the proposed approach can be seen as a powerful tool to certify the stability of switched discrete-time systems under arbitrary switching. Figure 1 compares the number of scalar decision variables which were used by each method in this example. In the horizontal axis r means N = 1, . . . , 7 to Theorem 5 (red line) and M = 0, . . . , 6 for Lemma 3 (blue line). It is possible to observe a striking difference between both methods when r increases. Figure 2(a) shows the Lyapunov function for N = 7 and θ = 0.79370. The initial state is x(0)T = [−2 3] and the swit-

IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018Thales S. Gomide et al. / IFAC PapersOnLine 51-25 (2018) 371–376

N

∑

m=1 N

∑

m=1



m−1

∏ A(ξ (k + β )T

β =0



Pm





m−1

∏ A(ξ (k + β − 1))

T

β =0





m−1

∏ A(ξ (k + m − β − 1)

β =0

Pm



N

−

∑

m=1

m−1

∏ A(ξ (k + m − β − 2))

β =0



m−1

∏ A(ξ (k + β − 1)T

β =0





375

>0

Pm



(32) 

m−1

∏ Aξ (k + m − β − 2)

β =0

<0 (36)

Table 1. Upper bounds for θ , number of scalar decision variables and number of LMI rows obtained when employing the method from Lee and Dullerud (2006) and with Theorem 5. Lemma 3 from Lee and Dullerud (2006) M

0

1

2

3

4

5

6

θ

0.68125

0.75487

0.77943

0.78954

0.79296

0.79339

0.79370

NV 3

3

6

12

24

48

96

192

NR3

6

12

24

48

96

192

384

Theorem 5 N

1

2

3

4

5

6

7

θ

0.68125

0.75487

0.77943

0.78954

0.79296

0.79339

0.79370

NV 5

3

6

9

12

15

18

21

NR5

6

12

24

48

96

192

384

200

50

Lemma 3 Theorem 5

180

45 160

40 35

120

30

100

V(k,x(k))

Number of variables

140

80

25

60

20

40

15

20

10

0

1

2

3

4 r

5

6

5

7

0

Fig. 1. Number of scalar decision variables for Theorem 5 (red line) and Lemma 3 (blue line).

It is also possible to analytically calculate the number of LMI rows for each method. Lemma 3 builds LMIs supported by (11). The number of switched subsystems is also important, such as the dimension of matrices (A1 , A2 , . . . , Av ) ∈ Rn×n . The number of LMI rows from Lemma 3 can be computed as   NR3 = n vM + vM+1 (38)

The number of LMI rows from Theorem 5 is given by

523

5

10 k (a)

15

20

0

5

10 k (b)

15

20

2 Switching

ching was considered arbitrary and it occurs in every instant. Figure 2(b) presents the arbitrary switching signal that has been considered to illustrate the Lyapunov function behavior.

0

1.5

1

Fig. 2. Lyapunov function (a) for N = 7, θ = 0.79370 and x(0)T = [−2 3]. The switching (b) is arbitrary and occurs in every instant.

IFAC ROCOND 2018 376 Florianopolis, Brazil, September 3-5, 2018Thales S. Gomide et al. / IFAC PapersOnLine 51-25 (2018) 371–376

  NR5 = n vN−1 + vN (39) It is possible to see that for M = N − 1 both conditions make use of the same number of LMI rows. It is also possible to empirically demonstrate the number of scalar decision variables used to solved the optimization problem. They depend on the same parameters of (38) and (39). For Lemma 3 one has n(n + 1)vM (40) NV 3 = 2 while for Theorem 5 n(n + 1)N (41) NV 5 = 2 It is very important to observe that the number of variables in Theorem 5 does not increase geometrically with N. Therefore, this variable does not appear at any exponent of (41). 5. CONCLUSION This paper have discussed the stability problem for discretetime switched systems under arbitrary switching. Differently from existing approaches, this method makes use of the system dynamics in order to build a switched Lyapunov function. This new class of Lyapunov function can be constructed by considering an augmented state vector with higher shifted states. A numerical example from the literature have been used to demonstrate the efficacy of the proposed method. The proposed approach was capable to check the system stability requiring much less scalar decision variables than existing approaches. It is important to stress that this difference increases with the complexity of the system (number of states and number of operation modes). Further works should focus on building a state feedback controller by applying the proposed method. REFERENCES Blanchini, F., Miani, S., and Savorgnan, C. (2007). Stability results for linear parameter varying and switching systems. Automatica, 43(10), 1817–1823. Branicky, M.S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43(4), 475–482. Daafouz, J., Riedinger, P., and Iung, C. (2002). Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Transactions on Automatic Control, 47(11), 1883–1887. Deaecto, G.S., Geromel, J.C., and Daafouz, J. (2011). Switched state-feedback control for continuous time-varying polytopic systems. International Journal of Control, 84(9), 1500– 1508. Hetel, L., Daafouz, J., and Iung, C. (2006). Stabilization of arbitrary switched linear systems with unknown timevarying delays. IEEE Transactions on Automatic Control, 51(10), 1668–1674. Jungers, R.M., Ahmadi, A.A., Parrilo, P.A., and Roozbehani, M. (2017). A characterization of Lyapunov inequalities for stability of switched systems. IEEE Transactions on Automatic Control, 62(6), 3062–3067.

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