Stability of Switched T-S Fuzzy Systems with All Subsystems Unstable

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ScienceDirect IFAC PapersOnLine 52-24 (2019) 213–218

Stability of Switched T-S Fuzzy Systems Stability of Switched T-S Fuzzy Systems Stability of Switched T-S Fuzzy Systems with All Subsystems Unstable Stability of Switched T-S Fuzzy Systems with All Subsystems Unstable with All Subsystems Unstable with All Subsystems Unstable Can Liu ∗∗ Xiang Mao ∗∗ Hongbin Zhang ∗∗

∗ ∗ ∗ Can Can Liu Liu ∗∗ Xiang Xiang Mao Mao ∗∗ Hongbin Hongbin Zhang Zhang ∗∗ ∗ ∗ Can Liu Xiang Mao Hongbin Zhang ∗ ∗ ∗ University of Electronic Science and Technology of China, Chengdu University Science and of China, Chengdu ∗ ∗ University of of Electronic Electronic Science and Technology Technology 611731, Sichuan (e-mail: of China, Chengdu ∗ 611731, Sichuan (e-mail: University of Electronic Science and Technology [email protected],[email protected],[email protected]). 611731, Sichuan (e-mail: of China, Chengdu [email protected],[email protected],[email protected]). 611731, Sichuan (e-mail: [email protected],[email protected],[email protected]). [email protected],[email protected],[email protected]). Abstract: This This paper paper investigates investigates the the stability stability problem problem of of continuous-time continuous-time switched switched nonlinear nonlinear Abstract: systems without stable subsystems. The Takagi-Sugeno (T-S) fuzzy model is introduced to Abstract: This paper investigates the stability problem of continuous-time switched nonlinear systems without stable subsystems. The Takagi-Sugeno fuzzy is to Abstract: This paper stability problem of(T-S) continuous-time switched nonlinear systems without stableinvestigates subsystems.theBased The Takagi-Sugeno (T-S) fuzzy model model is introduced introduced to represent the nonlinear on a new bounded maximum average dwell time represent the nonlinear subsystems. Based on a new bounded maximum average dwell time systems without stableapproach, subsystems. The Takagi-Sugeno (T-S) fuzzyismodel is introduced to (BMADT) switching an exponential stability condition derived with a novel represent the nonlinear subsystems. Based on a new bounded maximum average dwell time (BMADT)the switching an exponential condition is derived with a novel represent nonlinearapproach, subsystems. Basedfunction on a stability new bounded maximum average dwell time time-varying piecewise multiple Lyapunov approach. Finally, the effectiveness of the (BMADT) switching approach, an exponential stability condition is derived with a novel time-varying piecewiseapproach, multiple Lyapunov function approach. Finally,isthe effectiveness the (BMADT) switching an exponential stability condition derived with a of established results results is illustrated illustrated with a numerical numerical example. time-varying piecewise multiple with Lyapunov function approach. Finally, the effectiveness ofnovel the established is a example. time-varying piecewise multiple with Lyapunov function approach. Finally, the effectiveness of the established results is illustrated a numerical example. Copyright © 2019. The is Authors. Published by aElsevier Ltd. All rights reserved. established results illustrated with numerical example. Keywords: switched switched nonlinear nonlinear systems, systems, Takagi-Sugeno Takagi-Sugeno (T-S) (T-S) fuzzy fuzzy model, model, time-varying time-varying Keywords: piecewise multiple Lyapunov function, bounded maximum average dwell time (BMADT) Keywords: switched nonlinear systems, Takagi-Sugeno (T-S) fuzzy model, time-varying piecewise maximum average dwell (BMADT) Keywords: switchedLyapunov nonlinearfunction, systems, bounded Takagi-Sugeno (T-S) fuzzy model, time-varying piecewise multiple multiple Lyapunov function, bounded maximum average dwell time time (BMADT) piecewise multiple Lyapunov function, bounded maximum average dwell time (BMADT) 1. INTRODUCTION systems without stable subsystems (Mao et al. (2017); 1. systems without stable subsystems (Mao et al. (2017); 1. INTRODUCTION INTRODUCTION Zhao et (2016a,b); Chiou et systems without stable subsystems (Mao et al. (2017); Zhao et al. al. (2016a,b); Chiou et al. al. (2010)). (2010)). 1. INTRODUCTION systems without stable subsystems (Mao et al. (2017); Zhao et al. (2016a,b); Chiou et al. (2010)). Switched system is a typical hybrid system and has been For the switched systems withetallal.subsystems unstable, the Switched system is a typical hybrid system and has been Zhao et al. (2016a,b); Chiou (2010)). the switched systems with all subsystems unstable, the Switched system a typicalautomotive hybrid system and has been For widely applied applied in isrobotics, robotics, industry, informaexisting stability results can mainly be divided into For the switched systems with all subsystems unstable, the widely in automotive industry, informaSwitched system aon. typical hybrid and has been existing stability systems results can mainly be divided into statestatewidely applied tion theory theory andin is so Thanks to system theindustry, numerous applirobotics, automotive informaFor the switched with all subsystems unstable, the dependent (Pettersson and Lennartson (2001); Pettersson existing stability results can mainly be divided into statetion and so on. Thanks to the numerous appliwidely applied robotics, automotive industry, informadependent (Pettersson and Lennartson (2001); Pettersson tion theory andin so on. Thanks to thesystem numerous applications, relevant theories of switched have been existing stability results can mainly be divided into statedependent (Pettersson and Lennartson (2001); Pettersson (2003)) and time-dependent switching strategies. Due to cations, relevant theories of switched have been tion theory and so on. Thanks to thesystem numerous and time-dependent switching strategies. Due cations, relevant theories of the switched system haveapplibeen (2003)) widely researched, especially stability problems. Most dependent and Lennartson Pettersson (2003)) and(Pettersson time-dependent switching strategies. Due to to the time-dependent switching strategies (2001); do not necessarily widely researched, especially the stability problems. Most cations, relevant theories of switched system have been the time-dependent switchingswitching strategies strategies. do not necessarily widely researched, especially thefocused stability problems. Most (2003)) efforts for switched systems are on the case where and time-dependent Due to need full plant state information and relatively be more the time-dependent switching strategies do not necessarily efforts for switched systems are focused on the case where widely researched, the stability problems. Most need full plant stateswitching information and relatively be more efforts for switchedespecially systems are focused all the subsystems are stable (Principle (1999); Allerhand on the case where the time-dependent strategies do not necessarily convenient for stability analysis, weand only discuss the full plant state information relatively be timemore all the for subsystems are stable (Principle (1999); Allerhand efforts switched systems focused Zhao on the where need convenient for analysis, we only discuss the timeall subsystems stable (Principle (1999); Allerhand andthe Shaked (2011);are Chesi et are al. (2012); et case al. (2012); need full plant state ininformation relatively be(2014) more convenient for stability stability analysis, weand only discuss the timedependent switching this paper. Xiang and Xiao and Shaked (2011); Chesi et al. (2012); Zhao et al. (2012); all the subsystems are stable (Principle (1999); Allerhand dependent switching in analysis, this paper. Xiang and Xiao (2014) and Shaked (2011);However, Chesi et equipment al. (2012); Zhao etsensor al. (2012); Li et al. (2019)). noise, fails convenient for stability we only discuss the timehas proposed a sufficient condition for switched linear dependent switching in this paper. Xiang and Xiao (2014) Li et al. However, noise, sensor fails and (2011); Chesi et equipment al.the (2012); Zhao al. (2012); has proposed a sufficient for and switched linear Li etShaked al. (2019)). (2019)). equipment noise,etunstable sensor fails or actuator actuator failureHowever, will make subsystems in dependent switching in this condition paper. Xiang Xiao (2014) systems by aa discretized Lyapunov function techhas proposed a sufficient condition for switched or failure will make the subsystems unstable in Li et al. (2019)). However, equipment noise, sensor fails systems by using using discretized Lyapunov function linear techor actuator failure will make the subsystems unstable practical application. At worst, all the subsystems may be in has proposed a sufficient condition for switched linear nique. Luo et al. (2018) has constructed a switching-timesystems by using a discretized Lyapunov function techpractical application. At worst, all the subsystems may be or actuator failure will make the subsystems unstable in nique. Luo et al. (2018) has constructed a switching-timepractical application. At worst, all the subsystems may be unstable. Therefore, Therefore, the the research research of of the the switched switched system system nique. systems bytime-varying a discretized Lyapunov function techLuo etusing al. (2018) has constructed a switching-timedependent Lyapunov function to investigate unstable. practical application. At worst, all the subsystems may be dependent time-varying Lyapunov function to investigate unstable. Therefore, the research of the switched system without stable subsystems also has important theoretical nique. Luo et al. (2018) has constructed a switching-timethe stability and stabilization problems of the switched dependent time-varying Lyapunov function to investigate without stable subsystems also has important theoretical unstable. Therefore, the research theGao switched system the stability and stabilization problems of to the switched without stable subsystems also hasof important theoretical and practical practical significance. Zhang and (2010); Zhai dependent time-varying Lyapunov function linear stochastic The of and Xiao the stability andsystems. stabilization problems of theinvestigate switched and significance. Zhang and Gao (2010); Zhai without stable subsystems also has important theoretical stochastic systems. The results results of Xiang Xiang and Xiao and practical significance. Zhang andZhang Gao (2010); Zhai linear et al. (2001); Xiang and Xiang (2009); et al. (2014); the stability and stabilization problems of the switched (2014) and Luo et al. (2018) are received under dwell linear stochastic systems. The results of Xiang and Xiao et al. (2001); Xiang and Xiang (2009); Zhang et al. (2014); and practical significance. Zhang and Gao (2010); Zhai (2014) and Luo et al. (2018) are received under dwell et al. (2001); Xiang and Xiang (2009); Zhang et al. (2014); Margaliot (2006); (2006); Zheng Zheng and and Zhang Zhang (2017) (2017) have have analyzed analyzed (2014) linear stochastic systems. Thetime results of Xiang and Xiao and Luoand et the al. dwell (2018) areisreceived under dwell time switching confined by a pair of Margaliot et al.switched (2001); Xiang andwith Xiang (2009); Zhanghave etsubsystems. al.analyzed (2014); time switching and the dwell time isreceived confined under by a pair of Margaliot (2006); Zheng and partly Zhang (2017) the systems unstable (2014) and Luo et al. (2018) are dwell upper and lower bounds. General speaking, average dwell time switching and the dwell time is confined by a pair of the switched systems with partly unstable subsystems. Margaliot (2006); Zheng and Zhang (2017) have analyzed upperswitching and lowerand bounds. General speaking, average dwell the switched systems with partly unstable subsystems. The key idea of these efforts is to activate the stable time the dwell time is confined by a pair of time (ADT) is more flexible and efficient than dwell time upper and lower bounds. General speaking, average dwell The key idea of these efforts is to activate the stable the switched systems with partly unstable subsystems. time (ADT) is more flexible and efficient than dwell dwell time The key idea subsystem for a sufficiently long time to absorb the sof these efforts is to activate the stable upper and lower bounds. General speaking, average in system stability Zhaoefficient et al. (2016a) studied time (ADT) is moreanalysis. flexible and than dwell time subsystem for a long time to absorb sThe key idea these efforts is to activate the the stable system stability analysis. Zhao et (2016a) studied subsystem for of aof sufficiently sufficiently long time to However, absorb the s- in tate divergence divergence the unstable subsystem. these time (ADT) is more flexibleofand efficient than dwell time in system stability analysis. Zhao et al. al. (2016a) studied the stabilization problems switched T-S fuzzy systems tate of the unstable subsystem. However, these subsystem for aofbe sufficiently long timethetosubsystems absorb the s- the stabilization problems of Zhao switched T-S(2016a) fuzzy systems tate divergence theapplied unstable subsystem. these methods cannot when all are However, in system stability analysis. et al. studied without stable subsystems by using ADT switching. Mao the stabilization problems of switched T-S fuzzy systems methods cannot be when all are tate divergence theapplied unstable subsystem. However, these without stable subsystems by switched using ADT Mao methods applied when all the the subsystems subsystems are the unstable. cannotofbe stabilization problems aof T-Sswitching. fuzzypiecewise systems et al. (2017) constructed “decreasing-jump” without stable subsystems by using ADT switching. Mao unstable. methods cannot be applied when all the subsystems are et al. (2017) constructed aaby“decreasing-jump” piecewise unstable. without stable subsystems using ADT switching. Mao et al. (2017) constructed “decreasing-jump” piecewise Lyapunov-like function to achieve exponential stabilizaOn the other hand, in recent years, the study of switched unstable. Lyapunov-like function to achieve exponential stabilizaOn the other hand, in recent years, the study of switched et al. (2017) constructed a “decreasing-jump” piecewise Lyapunov-like function to achieve exponential stabilization conditions for discrete-time switched nonlinear On the other hand,has in recent study ofattention. nonlinear systems also received received extensive years, the switched tion conditions for discrete-time switched nonlinear syssysnonlinear systems has also extensive attention. function toproposed achieve aexponential stabilizaOn the other hand, in recent years, the study of switched tems. Zhao (2016b) multion conditions discrete-time nonlinear sysnonlinear systems has also received extensive attention. The Takagi-Sugeno Takagi-Sugeno (T-S) (T-S) fuzzy fuzzy model model is is well-known well-known as aa Lyapunov-like tems. Zhao et et al. al.for (2016b) proposedswitched a time-scheduled time-scheduled mulThe as tion conditions for discrete-time switched nonlinear sysnonlinear systems has alsofuzzy received extensive attention. tiple Lyapunov solve the stabilization probZhao et al.function (2016b) to proposed a time-scheduled mulThe Takagi-Sugeno (T-S) model is well-known as a tems. very effective tool in analyzing and synthesizing complex tiple Lyapunov function to solve the stabilization probvery effective tool in analyzing and synthesizing complex tems. Zhao et al. (2016b) proposed a time-scheduled multiple Lyapunov function to solve the stabilization probThe Takagi-Sugeno (T-S) fuzzy model islinear well-known asdea lems of switched nonlinear systems under a new modevery effective tool inwhich analyzing and synthesizing complex nonlinear systems, utilizes local system lems of nonlinear systems aa new nonlinear systems, utilizes local linear system detiple function solve theunder stabilization problems Lyapunov of switched switched nonlinear systems under new modemodevery effective toolrule inwhich analyzing and synthesizing complex average dwell to time (MDADT) switching. It’s nonlinear systems, which utilizes local linear system de- dependent scription for each (Takagi and Sugeno (1985)). Recentdependent average dwell time (MDADT) switching. It’s scription for each rule (Takagi and Sugeno (1985)). Recentlems of switched nonlinear systems under a new modenonlinear systems, which utilizes local linear system deworth noting that the dwell time of each switching are still dependent average dwell time (MDADT) switching. It’s scription for each rule (Takagi and Sugeno (1985)). Recently, based based on on the the T-S T-S fuzzy fuzzy model, model, many many issues issues of of switched switched worth noting that the dwell time of each switching are still ly, dependent average dwell time (MDADT) switching. It’s scription eachT-S rule (Takagi and Sugeno (1985)). Recentlimitednoting by a lower dwell time boundaries in theseare works, worth that the dwell time of each switching still ly, based for onsystems the fuzzy model, many issues of as switched nonlinear have been researched such asynlimited by aa lower dwell time boundaries in these works, nonlinear systems have been researched such as asynworth noting that the dwell time of each switching are still limited by lower dwell time boundaries in these works, ly, based on the T-S fuzzy model, many issues of switched are somewhat conservative. Mao et al. (2018) first nonlinear have been researched such as asyn- which chronous H Hsystems problems (Zheng Zhang ∞ control which are conservative. Mao et in al. (2018) first chronous problems (Zheng and and such Zhangas(2016); (2016); byasomewhat anew lower dwell time boundaries works, ∞ control nonlinear been researched asynproposed BMADT approach to analysis switched which are somewhat conservative. Mao et al.these (2018) first chronous Hsystems Li et al. (2018)), Lhave performance analysis(Li et al. limited problems (Zheng and Zhang (2016); ∞ control 1 -gain ∞ proposed a new BMADT approach to analysis switched Li et al. (2018)), L -gain performance analysis(Li et al. which are somewhat conservative. Mao et al. (2018) 1 -gain chronous H control problems (Zheng and Zhang (2016); linear systems, which is more general than dwell time and proposed a new BMADT approach to analysis switched ∞ Li et al. (2018)), L performance analysis(Li et al. (2017)) and stabilization problems of switched nonlinear linear systems, which is more general than dwell time first 1 1 and (2017)) and stabilization of switched nonlinear a newwhich BMADT approach to analysis switched Li et al.and (2018)), L1 -gainproblems performance analysis(Li et al. proposed the traditional maximum average time linear systems, is more generaldwell than dwell(MADT). time and (2017)) stabilization problems of switched nonlinear the traditional maximum average dwell time (MADT). linear systems, which isconditions more general than dwell(MADT). time and traditional maximum average time (2017)) and stabilization problems of switched nonlinear the However, the obtained aredwell nonconvex and canHowever, the obtained conditions aredwell nonconvex and can This work was supported by the National Natural Science Founthe traditional maximum average timeTo(MADT). not be applied to nonlinear systems the canbest However, the obtained conditions aredirectly. nonconvex and  This work was supported by the National Natural Science Founnot be applied to nonlinear systems To and the best  However, the obtained conditions aredirectly. nonconvex dation China No. by 61374117). This of work was(Grants supported the National Natural Science Founnot be applied to nonlinear systems directly. To the canbest dation China No. by 61374117).  This of work was(Grants supported the National Natural Science Founnot be applied to nonlinear systems directly. To the best dation of China (Grants No. 61374117).

dation of China (Grants No.The 61374117). 2405-8963 Copyright © 2019. Authors. Published by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.410

Can Liu et al. / IFAC PapersOnLine 52-24 (2019) 213–218

214

of our knowledge, there is no literature researching on the continuous-time switched T-S fuzzy systems under BMADT switching. Firstly, based on the BMADT approach, we defined a compensation time span with a pair of fixed upper and lower bounds. According to whether the dwell time belongs to the compensation span, the corresponding switchings are divided into “stable-switching” and “unstable-switching”. The state divergence of the “unstable-switching” will be absorbed by the “stable-switching”. Based on that, we constructed a novel time-varying piecewise multiple Lyapunov function to analysis switched nonlinear systems and have derived out the convex stability conditions. The results have removed the minimum dwell time limitation relative to the ADT conditions. The rest of this paper is organized as follows. Section 2 gives the system descriptions and preliminaries. The stability conditions under BMADT switching are derived in Section 3. Section 4 provides a simulation example to demonstrate the feasibility and effectiveness of our results. Finally, the conclusion is included in Section 5. Notations: The notations used are fairly standard. P > 0(≥ 0) represents matrix P is positive definite (semipositive definite). N denotes non-negative integers set and N+ denotes positive integers set. The notation  ·  stands for the Euclidean norm. Rn denotes the n-dimensional Euclidean space. The superscript ”T” is matrix transpose. 2. SYSTEM DESCRIPTIONS AND PRELIMINARIES Consider the following class of continuous-time switched nonlinear system: (1) x(t) ˙ = fσ(t) (x(t)), where x(t) ∈ Rnx is the state vector and fσ(t) (·) is nonlinear function. σ(t) is switching signal which takes values in the finite set S = {1, 2, ..., N }, N ∈ N+ is the number of subsystems. For a switching sequence t0 < t1 < · · · < tp < · · · , while tp is the switching instant, ∆p = tp − tp−1 denotes the dwell time of p-th switching. The switched nonlinear systems are represented by T-S fuzzy model in this paper. The IF-THEN rules for i-th fuzzy subsystem are described as follows: Model Rule m for subsystem i: IF zi1 (t) is Mim1 and · · · and zil (t) is Miml , then (2) x(t) ˙ = Aim x(t), i ∈ S, m ∈ R = {1, 2, ..., r}, where r is the number of model rules. zi1 (t), · · · , zil (t) are the premise variables and Miml is the fuzzy set. Aim is a real matrix of the m-th local model. Through fuzzy blending, the global outputs of the i-th subsystem can be described as following form: r  x(t) ˙ = him (zi (t))Aim x(t), (3) m=1

l

where him (zi (t)) = r

m=1

Mimn (zi (t))

l

n=1

n=1

Mimn (zi (t))

are the nor-

malized r membership functions and satisfies him (zi (t)) ≥ 0, m=1 him (zi (t)) = 1. Definition 1. (Zhai et al. (2001)) The switched system (1) is globally exponentially stable if there exist scalars

λ > 0, µ < 0, such that x(t) ≤ λeµ(t−t0 ) x(t0 ), ∀t ≥ t0 holds for any initial condition x(t0 ). Definition 2. Given a switching sequence {tp } and compensation bounds t and t which satisfy 0 < t ≤ t. For any t ∈ [tp , tp+1 ), denote Tt,t (t) as the divergence time with the following form: Tt,t (t) =

p−1 

p s Tt,t (ts+1 ) + Tt,t (t),

(4)

s=0

where

  t − ts , s Tt,t (t) = 0,  t − ts − t,

t < ts + t, ts + t ≤ t ≤ ts + t, t > ts + t.

(5)

Remark 3. Only when the dwell time is less than t or larger than t can the total divergence time increase. The divergence time actually describes the time interval outside the compensation bounds [t, t]. Definition 4. (Mao et al. (2018)) For time interval [t0 , t], let N (t) denotes the number of switchings, if there exist positive constant τt,t satisfies N (t) ≥ Tt,t (t) ≤

t − t0 − N 0, τt,t + t τt,t · (t − t0 ) τt,t + t

+ T 0,

(6) (7)

where N 0 and T 0 are slack variables, then τt,t is called as BMADT. Remark 5. The BMADT describes the relationship between switching number and divergence time. If we choose Tt,t (tp ) = 0, ∀p ∈ N, i.e. each switching occurring inside the compensation boundary, the definition of BMADT will be reduced to dwell time in (Xiang and Xiao (2014); Luo et al. (2018)). When t = t = 0, then total divergence time T0,0 (t) = t − t0 is the total running time. Accordingly, τ0,0 is the traditional MADT defined in (Mao et al. (2017); Zhao et al. (2016a)). Therefore, the definition of BMADT is more general. Definition 6. For time interval [t0 , t] and t > 0, denote Nt (t) as the switching number whose dwell time is less than t. Definition 7. Given a switching signal σ(t) with switching number N (t), if there exist a constant εt ∈ [0, 1) satisfied the following condition: Nt (t) ≤ εt N (t) + N ∗ , t > 0, (8) then we said εt is the switching frequency of lower boundary switching. N ∗ > 0 is the corresponding slack variable. 3. MAIN RESULTS In this section, we will analyse the stability problem of continuous-time switched nonlinear systems without stable subsystems under BMADT switching. Lemma 8. Consider the continuous-time switched nonlinear system (1). Let α > 0, 0 < β1 < 1, β2 > 1 and 0 < t ≤ t, if there exist function Vσ(t) (x(t)) : RN → R, 0 ≤ εt < 1, τk,k ≥ 0 and two positive scalars γ1 and γ2 such that



Can Liu et al. / IFAC PapersOnLine 52-24 (2019) 213–218

γ1 (x(t))2 ≤ Vσ(t) (x(t)) ≤ γ2 (x(t))2 , (9)  t ∈ [t , t ), ∆ < t p−1 p p V˙ σ(t) (x(t)) ≤ αVσ(t) (x(t)), t ∈ [tp−1 + t, tp ), ∆p > t (10) V˙ σ(t) (x(t)) ≤ 0, t ∈ [tp−1 , tp−1 + t), ∆p ≥ t, (11) Vσ(tp ) (x(tp )) ≤ β1 Vσ(tp−1 ) (x(tp )), ∆p ≥ t, Vσ(tp ) (x(tp )) ≤ β2 Vσ(tp−1 ) (x(tp )), ∆p < t.

(12) (13)

Then the switched system (1) is globally exponentially stable under the BMADT switching satisfies ln β1 , ln β1 − ln β2 εt ln β2 + (1 − εt ) ln β1 . <− α

εt <

(14)

τt,t

(15)

Proof. Firstly, combined with (10)(11)(12)(13) and the definition of divergence time (7), it can be easily obtained that Vσ(tp ) (x(tp )) ≤ β1 e Vσ(tp ) (x(tp )) ≤ β2 e

αT p−1 (tp ) t,t

αT p−1 (tp ) t,t

Vσ(tp−1 ) (x(tp−1 )), ∆p ≥ t, (16) Vσ(tp−1 ) (x(tp−1 )), ∆p < t. (17)

Without loss of generality, we suppose t > tp + t, ∆p ≥ t, ∆p−1 < t, ∀t ∈ [tp , tp+1 ), p ∈ N+ , we have Vσ(t) (x(t)) ≤ eα(t−tp −t) Vσ(tp ) (x(tp + t)) ≤e

α(T p (t)) t,t

≤ β1 e

Vσ(tp ) (x(tp ))

α(T p (t)+T p−1 (tp )) t,t

≤ β1 β2 e ...

α(T

t,t

p t,t

(t)+T

p−1 t,t

Vσ(tp−1 ) (x(tp−1 ))

(tp )+T p−2 (tp−1 )) t,t

p−1

p T s (ts+1 )) N (t)−Nt (t) Nt (t) α(Tt,t (t)+ s=0 t,t β2 e Vσ(t0 ) (x(t0 )) N (t)−Nt (t) Nt (t) α(Tt,t (t)) β2 e β1 Vσ(t0 ) (x(t0 )).

≤ β1 ≤

Vσ(tp−2 ) (x(tp−2 ))

(18)

Let φ = obtain

∗ ( ββ21 )N ,

with (14) and Definition 4,7, we can

Vσ(t) (x(t)) β2 ∗ 1−εt εt α(T (t)) ≤ ( )N (β1 β2 )N (t) e t,t Vσ(t0 ) (x(t0 )) β1 t−t0

≤ ≤

1−εt εt τ +t φ(β1 β2 ) t,t

φe

αT

1−εt

(β1

Let λ =

γ2 γ1

τ

t,t

e

τ

·(t−t0 )

t,t

+t

+T 0 )

(1−εt ) ln β1 +εt ln β2 +ατ

0

τ

e

εt

β2 ) N 0 ·

−N 0 α(

(t−t0 )

Vσ(t0 ) (x(t0 )). (19)

αT 0

φe

1−εt

(β1

t,t

+t

Vσ(t0 ) (x(t0 )) t,t

εt

β 2 )N 0

,µ=

(1−εt ) ln β1 +εt ln β2 +ατt,t

with (9)(15) we have λ > 0, µ < 0,

τt,t +t

, then

1 Vσ(t) (x(t)) γ1 1 ≤ λeµ(t−t0 ) Vσ(t0 ) (x(t0 )) γ2 ≤ λeµ(t−t0 ) x(t0 )2 .

215

x(t)2 ≤

(20)

√ µ (t−t ) 0 so x(k) ≤ λe 2 x(k0 ) satisfied Definition 1. Therefore, the switched system (1) is globally exponentially stable under the BMADT switching. This completes the proof. We define the time span [tp−1 + t, tp−1 + t] with compensation bounds [t, t] for p-th switching. The switching is called “stable-switching” when the corresponding dwell time belonging to this time span, otherwise it is “unstableswitching”. For the “stable-switching”, the value of Lyapunov function falls exponentially. For the “unstableswitching”, the value of Lyapunov function can be either increased or decreased. Therefore, the state divergence can be compensated by the “stable-switching”, which is similar to the idea of handling the switched system with partly unstable subsystems. Remark 9. The dwell time of the stability conditions under ADT switching should not be less than a lower bound τmin in (Mao et al. (2017); Zhao et al. (2016a,b)). The new BMADT method eliminates this limitation by introducing the switching frequency εt . For switched T-S fuzzy system, we first construct a timevarying piecewise multiple Lyapunov function (3) as follows: (21) Vi (t) = xT (t)Pi (t)x(t), ∀i ∈ S. If ∆p < t, ∀p ∈ N+ , we fix Pi (t) = Pi,0 . If ∆p ∈ [t, t], then we divide the interval [tp−1 , tp ] into M subintervals. M is a positive integer. The length of each subinterval is h = ∆p /M . Therefore, we can redescribe the interval [tp−1 , tp ] as Di,f = [tp−1 + lf , tp−1 + lf +1 ], f = 0, 1, ..., M − 1, where lf = f ∆p /M . Then, for any t ∈ Di,f , (22) Pi (t) = (1 − ω)Pi,f + ωPi,f +1 , where ω = (t − tp−1 − lf )/h. If ∆p > t, we divide the interval [tp−1 , tp−1 + t] into M subintervals. In this case, h = t/M, lf = f t/M . For any t ∈ Di,f , Pi (t) is also satisfied (22). When t ∈ [tp−1 + t, tp ], we fix Pi (t) = Pi,M . Then, we can obtain the following stability result for continuous-time switched T-S fuzzy system (3). Theorem 10. Consider switched T-S fuzzy system (3). Let α > 0, 0 < β1 < 1, β2 > 1, 0 < t ≤ t, m ∈ R and a positive integer M , if there exist matrices Pi,g > 0, g ∈ {0, ..., M }, such that ∀f ∈ {0, ..., M − 1}, d ∈ {0, 1}, (i, j) ∈ S × S, i = j (23) ATim Pi,0 + Pi,0 Aim − αPi,0 ≤ 0, ATim Pi,M + Pi,M Aim − αPi,M ≤ 0, M (Pi,f +1 − Pi,f ) ATim Pi,f +d + Pi,f +d Aim + ≤ 0, t M (Pi,f +1 − Pi,f ) ≤ 0, ATim Pi,f +d + Pi,f +d Aim + t Pi,0 ≤ β1 Pj,M , Pi,0 ≤ β2 Pj,0 ,

(24)

(25) (26) (27) (28)

Can Liu et al. / IFAC PapersOnLine 52-24 (2019) 213–218

216

then the switched T-S fuzzy system (3) is globally exponentially stable under the BMADT switching satisfies ln β1 εt < , (29) ln β1 − ln β2 εt ln β2 + (1 − εt ) ln β1 τt,t < − . (30) α

0, t ∈ [tp−1 , tp−1 +t] hold with (25). When t ∈ [tp−1 +t, tp ], with (24), one can obtain V˙ i (x(t)) − αVi (x(t))

Proof. Case 1: When ∆p < t, with (23), we can obtain V˙ i (x(t)) − αV˙ i (x(t))

(35) On the other hand, Vσ(tp ) (x(tp )) ≤ β1 Vσ(tp−1 ) (x(tp )) hold, which is same as Case 2.

him (zi (t))xT (t)[ATim Pi (t) + Pi (t)Aim

Based on the above discussion, the conditions of Lemma 8 hold. Therefore, the switched T-S fuzzy system (3) is globally exponentially stable under the BMADT switching satisfies (29)(30). This completes the proof. Remark 11. The time-varing Lyapunov function makes full use of the dwell time and can ensure enough interpolation points. Larger M will lead to less conservatism but increase the computational burden.

m=1

+ P˙ i (t) − αPi (t)]x(t) r  = him (zi (t))xT (t)[ATim Pi,0 + Pi,0 Aim − αPi,0 ]x(t) m=1

≤ 0.

(31) By using (26), let σtp = i, σtp−1 = j, ∀p ∈ N+ , we can proof Vi (x(tp )) − β2 Vj (x(tp ))

= xT (t)(Pi,0 − β2 Pj,0 )x(t) ≤ 0,

(32)

which satisfied condition (13).

Case 2: When ∆p ∈ [t, t], for any t ∈ Di,f , f = 0, 1, ..., M − 1, ˙ i,f + ωP ˙ i,f +1 P˙ i (t) = −ωP

M (Pi,f +1 − Pi,f ) ∆p M (Pi,f +1 − Pi,f ) M (Pi,f +1 − Pi,f ) ≤ max{ , }. t t (33) If (25)(26) hold, then V˙ i (x(t)) =

= =

r 

m=1 r 

m=1

m=1

≤ 0.

4. NUMERICAL EXAMPLE A simulation example is given to illustrate our results in this section. Consider the following switched nonlinear systems composed of two subsystems  x˙ 1 (t) = 0.5x1 (t) − 0.21 sin2 (x1 (t))x2 (t)     − 0.8x2 (t) + 0.02 sin2 (x1 (t))x1 (t) Ω1 =  x˙ 2 (t) = 1.2x1 (t) − 0.7 sin2 (x1 (t))x2 (t)    − 2.3x2 (t) + 0.31 sin2 (x1 (t))x1 (t)  2   x˙ 1 (t) = −1.7x1 (t) + 0.3 sin (x2 (t))x2 (t)   + 0.6x2 (t) − 0.95 sin2 (x2 (t))x1 (t) Ω2 =  x˙ 2 (t) = −0.9x1 (t) + 0.1 sin2 (x2 (t))x2 (t)    + 0.4x2 (t) − 0.45 sin2 (x2 (t))x1 (t). Subsystem 1

+ [(1 − ω)Pi,f + ωPi,f +1 ]Aim M (Pi,f +1 − Pi,f ) + }x(t) ∆p r  = him (zi (t))xT (t){(1 − ω)[ATim Pi,f + Pi,f Aim m=1

M (Pi,f +1 − Pi,f ) + ] + ω[ATim Pi,f +1 ∆p M (Pi,f +1 − Pi,f ) + Pi,f +1 Aim + ]}x(t) ∆p ≤ 0.

(34) Similar to Case 1, it is easy to obtain Vσ(tp ) (x(tp )) ≤ β1 Vσ(tp−1 ) (x(tp )) by using (25). M (Pi,f +1 −Pi,f ) ,t t

Case 3: When ∆p > t, P˙ i (t) = ∈ Di,f , f = 0, 1, ..., M − 1. Similar to Case 2, we can proof V˙ i (x(t)) ≤

Subsystem 2

250

700 x1(t)

him (zi (t))xT (t)[ATim Pi (t) + Pi (t)Aim + P˙ i (t)]x(t) him (zi (t))xT (t){ATim [(1 − ω)Pi,f + ωPi,f +1 ]

him (zi (t))xT (t)[ATim Pi,M + Pi,M Aim − αPi,M ]x(t)

200

600

x2(t) State response

r 

r 

State response

=

=

150 100 50 0 0

x1(t) x2(t)

500 400 300 200 100

50 t

100

0 0

50 t

100

Fig. 1. States of subsystems. the state response of subsystems is shown in Fig. 1 with the initial state condition x(0) = [5, 3]T . As shown in Fig. 1 that both the subsystems are unstable. Using the T-S fuzzy modeling method, set z1 (t) = sin2 (x1 (t)), z2 (t) = sin2 (x2 (t)) with fuzzy rules as follows: Rule 1 for subsystem 1: If z1 (t) is 0, then x(t) ˙ = A11 x(t), Rule 2 for subsystem 1: If z1 (t) is 1, then x(t) ˙ = A12 x(t). ˙ = A21 x(t), Rule 1 for subsystem 2: If z2 (t) is 0, then x(t) Rule 2 for subsystem 2: If z2 (t) is 1, then x(t) ˙ = A22 x(t).

Can Liu et al. / IFAC PapersOnLine 52-24 (2019) 213–218

   0.52 −1.01 0.50 −0.80 , = , A12 = 1.51 −3.00 1.20 −2.30     −1.70 0.60 −2.65 0.90 = , A22 = . −0.90 0.40 −1.35 0.50 

However, no feasible solutions can be obtained by using the methods in Mao et al. (2017); Zhao et al. (2016a). Through Theorem 10 in this paper, we choose t = 1, t = 7, M = 5, α = 0.21, β1 = 0.75, β2 = 6.9 and εt = 0.1 < ln β1 ln β1 −ln β2 = 0.1296. By using the Matlab LMI toolbox, one can obtain feasible solutions as follows:    4.5262 −1.9271 6.1056 −2.3126 P1,0 = , , P1,1 = −1.9271 1.4809 −2.3126 1.3275     3.4455 −1.7620 2.7784 −1.8651 , P1,3 = , P1,2 = −1.7620 1.8162 −1.8651 2.5017     2.5174 −2.3777 2.7218 −3.5315 , P1,5 = , P1,4 = −2.3777 3.8717 −3.5315 6.4281     2.0124 −2.5910 2.1705 −2.1054 P2,0 = , P2,1 = , −2.5910 4.7058 −2.1054 3.4675     2.6880 −1.8904 3.6898 −1.9453 , P2,3 = P2,2 = , −1.8904 2.6209 −1.9453 2.0814     5.3919 −2.3118 8.1465 −3.0854 , P2,5 = . P2,4 = −2.3118 1.8020 −3.0854 1.7708 The divergence time Tt,t (t) and state responses of the switched T-S fuzzy system are shown in Fig. 2. From Fig. 2 it can be seen that only when the dwell time is less than t or larger than t can the total divergence time increase. There also have no minimum dwell time limitation compared with the work in Zhao et al. (2016b). The system state will finally converge to zero under the BMADT switching signal which satisfies τ1,7 = 0.3063 < εt ln β2 +(1−εt ) ln β1

τ∗ = − = 0.3132. Therefore, the results α in Theorem 10 are verified. The curve of the Lyapunov function is displayed in Fig. 3. When the dwell time ∆2 = 0.5 < t, the value of the Lyapunov function in switching instant goes up from V2 (x(3.5)) = 1.979 to V1 (x(3.5)) = 13.65 and satisfies V1 (x(3.5)) V2 (x(12.5)) 3.701 V2 (x(3.5)) = 6.897 < β2 . Similarly, V1 (x(12.5)) = 4.936 = 0.749 < β1 when the corresponding dwell time ∆3 = 9 > t. In addition, from the third switching ∆3 , it can be seen that the Lyapunov function must be non-increase before t, which satisfies the inequalities in Lemma 8. Finally, the boundary τ ∗ of BMADT under different M is shown in Table 1. The boundary of BMADT will expand as the number of interpolations increasing, thus reducing the conservatism. 5. CONCLUSION The stability problems of continuous-time switched nonlinear systems with all subsystems unstable are studied in

σ(t)

2

10

1 0

0

10

20

30 t

40

0 60

50

6 x1 (t)

x2 (t)

4 2 0

0

10

20

30 t

40

50

60

Fig. 2. Divergence time and state trajectories. 100

15

80

V (x (t)) σ(t)

10 3 5 4

60

6

8 t

10

12 2

σ(t)

A21

σ(t)

x(t)

A11

T1,7 (t)

V (x (t))

and

h21 (t) = 1 − sin2 (x2 (t)), h22 (t) = sin2 (x2 (t)).

217

20

V (x (t))

The normalized membership functions are calculated as follows: h11 (t) = 1 − sin2 (x1 (t)), h12 (t) = sin2 (x1 (t)),

Tt,t (t)



40

20

0

1

0

10

20

30 t

40

50

60

Fig. 3. The proposed Lyapunov function. Table 1. Relationship between M and τ ∗ (t = 1, t = 7, εt = 0.1). M 5 10 15 20

α 0.21 0.21 0.21 0.21

β1 0.75 0.62 0.58 0.56

β2 6.9 7.3 7.5 7.7

τ∗ 0.3132 1.1021 1.3751 1.5129

this paper. A more general BMADT approach is first introduced to analyze switched nonlinear systems. Utilizing the information of dwell time, we construct a time-varying piecewise multiple Lyapunov function to achieve convex stability conditions. In addition, combining with T-S fuzzy modeling technique, less conservative stability results are obtained in the form of linear matrix inequalities. Finally, a numerical example is given to verify the conservatism and effectiveness of our results. REFERENCES Allerhand, L.I. and Shaked, U. (2011). Robust stability and stabilization of linear switched systems with dwell time. IEEE Transactions on Automatic Control, 56(2), 381–386. doi:10.1109/TAC.2010.2097351.

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