Stability Analysis and Synthesis for Nonlinear Networked Control Systems

6th IFAC Workshop on Distributed Estimation and Control in 6th IFAC 6th IFAC Workshop Workshop on Distributed Distributed Estimation Estimation and and Control Control in in Networked Systemson 6th IFAC Workshop on Distributed Estimation and Control in Networked Networked Systems Systems Available online at www.sciencedirect.com September 8-9, 2016. Tokyo, Japan Networked Systems September September 8-9, 8-9, 2016. 2016. Tokyo, Tokyo, Japan Japan September 8-9, 2016. Tokyo, Japan

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IFAC-PapersOnLine 49-22 (2016) 297–302

Stability Analysis and Synthesis for Stability Analysis and Synthesis for Stability Networked Analysis and Synthesis for Nonlinear Control Systems Nonlinear Networked Control Systems Nonlinear Networked Control Systems

Jun Yoneyama ∗∗∗ Kenta Hoshino ∗∗ ∗∗ Jun Yoneyama ∗ Kenta Hoshino ∗∗ Jun Jun Yoneyama Yoneyama Kenta Kenta Hoshino Hoshino ∗∗ ∗ Aoyama Gakuin University, Sagamihara, Kanagawa, 252-5258 Japan ∗ ∗ Aoyama Gakuin University, Sagamihara, Kanagawa, 252-5258 Japan University, Sagamihara, ∗ Aoyama Gakuin (e-mail: [email protected]). Aoyama Gakuin University, Sagamihara, Kanagawa, Kanagawa, 252-5258 252-5258 Japan Japan (e-mail: [email protected]). ∗∗ (e-mail: [email protected]). Aoyama Gakuin University, Sagamihara, Kanagawa, 252-5258 (e-mail: [email protected]). ∗∗ ∗∗ Aoyama Gakuin University, Sagamihara, Kanagawa, Kanagawa, 252-5258 Gakuin University, Sagamihara, ∗∗ Aoyama Japan [email protected]). Aoyama Japan Gakuin(e-mail: University, Sagamihara, Kanagawa, 252-5258 252-5258 (e-mail: [email protected]). Japan (e-mail: [email protected]). Japan (e-mail: [email protected]). Abstract: The paper is concerned with stabilization of nonlinear networked systems which are Abstract: is concerned with stabilization of nonlinear networked systems are Abstract: The paper is with stabilization of nonlinear which are described byThe the paper Takagi-Sugeno fuzzy systems. Based on stabilitywhich theorem, Abstract: The paper is concerned concerned with stabilization of Lyapunov-Krasovskii nonlinear networked networked systems systems which are described by the Takagi-Sugeno fuzzy systems. Based on Lyapunov-Krasovskii stability theorem, described by the Takagi-Sugeno fuzzy systems. Based on Lyapunov-Krasovskii stability theorem, stability analysis and control design problems are considered. Novel Lyapunov-Krasovskii described by the Takagi-Sugeno fuzzy systems. Based on Lyapunov-Krasovskii stability theorem, stability analysis and control design problems are considered. Lyapunov-Krasovskii stability analysis and design problems are Novel Lyapunov-Krasovskii function and corresponding generalized to obtain lessNovel conservative conditions for stability analysis and control control design controller problems allow are considered. considered. Novel Lyapunov-Krasovskii function and corresponding generalized controller allow to obtain less conservative conditions for function and corresponding generalized controller allow to obtain less conservative conditions for the networked control system to be asymptotically stable. Furthermore, free matrix parameter function and corresponding generalized controller allow to obtain less conservative conditions for the networked control system to be asymptotically stable. Furthermore, free matrix parameter the networked control system to be asymptotically stable. Furthermore, free matrix parameter method is introduced. This method is known to stable. reduce Furthermore, the conservatism in control design the networked control system to be asymptotically free matrix parameter method is introduced. This method is known to reduce the conservatism control design method is This method is to in design conditions. With such novel novel Lyapunov-Krasovskii function,in new controller method is introduced. introduced. Thiscontroller method and is known known to reduce reduce the the conservatism conservatism inaa control control design conditions. With such novel controller and novel Lyapunov-Krasovskii function, new controller conditions. With such novel controller and novel Lyapunov-Krasovskii function, a new controller design method is proposed. Resulting conditions for a controller to stabilize the networked conditions. With such novel controller and novel Lyapunov-Krasovskii function, a new controller design method method is proposed. proposed. Resulting conditions for aaforcontroller controller to stabilize stabilize the networked networked design is Resulting for to the control system are less conservative andconditions are applicable a wider class of nonlinear systems. design method is proposed. Resulting conditions for aforcontroller to stabilize the networked control system are less conservative and are applicable aa wider class of nonlinear systems. control system are less conservative and are applicable for wider class of nonlinear systems. control system are less conservative and are applicable for a wider class of nonlinear © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rightssystems. reserved. Keywords: Fuzzy systems, Intelligent control, Networks, Nonlinear control systems, Keywords: Fuzzy Fuzzy systems, systems, Intelligent Intelligent control, control, Networks, Networks, Nonlinear Nonlinear control control systems, systems, Keywords: Time-Delay. Keywords: Time-Delay.Fuzzy systems, Intelligent control, Networks, Nonlinear control systems, Time-Delay. Time-Delay. 1. INTRODUCTION iterative technique but still needs much computational 1. INTRODUCTION INTRODUCTION iterative technique but still still needs much computational 1. iterative but much computational load. The technique paper Tanaka(2007) adopted a descriptor system 1. INTRODUCTION iterative technique but still needs needs much computational load. The paper Tanaka(2007) adopted a descriptor system The paper Tanaka(2007) adopted a descriptor system Most physical systems are nonlinear and they appear in load. approach, whichTanaka(2007) reduces control designa conditions for the load. The paper adopted descriptor system Most physical systems are nonlinear and they appear in approach, which reduces control design conditions for the the Most physical systems are nonlinear and they appear in approach, which reduces control for many engineering fields.are In the past three decades, Takagistate feedback control design. Thedesign papersconditions by Bourar(2013) Most physical systems nonlinear and they appear in approach, which reduces control design conditions for the many engineering fields. In the past three decades, Takagistate feedback control design. The papers by Bourar(2013) many engineering fields. In the past three decades, Takagistate feedback control design. The papers by Bourar(2013) Sugeno fuzzy model hasInbeen widely used for nonlinear Guelton(2009) the papers descriptor system apmany fields. the past three decades, Takagi- and state feedback controlextended design. The by Bourar(2013) Sugenoengineering fuzzy model model has been widely used for nonlinear nonlinear and Guelton(2009) Guelton(2009) extended the descriptor system apSugeno fuzzy has widely used for descriptor system control systems since it canbeen universally approximate or can and proach to the case of extended the outputthe feedback control design.apIt Sugeno fuzzy model has been widely used for nonlinear and Guelton(2009) extended the descriptor system apcontrol systems systems since it can can nonlinear universallysystems(Lam(2000), approximate or or can can proach proach to the case of the output feedback control design. It control since it universally approximate to the case of the output feedback control design. It exactly describe general is well-known thatofalthough descriptor systems are more control systems since it can universally approximate or can proach to the case the output feedback control design. It exactly describe general nonlinear systems(Lam(2000), is well-known that although descriptor systems are more exactly describe general nonlinear systems(Lam(2000), well-known that although descriptor are more Takagi(1985) and general Tanaka(1992)). In systems(Lam(2000), fact, Takagi-Sugeno is complicated than state-space systems, systems they have richer exactly describe nonlinear is well-known that although descriptor systems are more Takagi(1985) and Tanaka(1992)). Inrole fact, Takagi-Sugeno complicatedwhich than produces state-space systems, they have have richer Takagi(1985) and Tanaka(1992)). In fact, than state-space they fuzzy model has played an important forTakagi-Sugeno nonlinear sys- complicated structures, lesssystems, conservatism in thericher conTakagi(1985) and Tanaka(1992)). Inrole fact, Takagi-Sugeno complicated than produces state-space systems, they have richer fuzzyanalysis model has has played an important important for nonlinear sys- structures, structures, which less conservatism in the confuzzy model played an role for nonlinear syswhich produces less conservatism in the contem and its control design. The stability of fuzzy trol design conditions. Marouf(2016) and Shao(2015) confuzzy model has played an important role for nonlinear sysstructures, which produces less conservatism in the tem analysis analysis and its control control in design. The stability stability of fuzzy fuzzy trol trol design design conditions. Marouf(2016) and Shao(2015) contem and its design. The of conditions. Marouf(2016) and Shao(2015) consystems was investigated Lee(2014), Tanaka(1992), sidered control design problems for nonlinear networked tem analysis its control in design. The stability of fuzzy trol design conditions. Marouf(2016) and Shao(2015) consystems was and investigated Lee(2014), Tanaka(1992), sidered systems. control design design problems for for nonlinear networked networked systems was investigated in Lee(2014), Tanaka(1992), control Tanaka(2000), Teixiera(2003), Tuan(2002). The recent pa- sidered control systems was investigated in Lee(2014), Tanaka(1992), sidered control design problems problems for nonlinear nonlinear networked Tanaka(2000), Teixiera(2003), Tuan(2002). The recent pacontrol systems. Tanaka(2000), Teixiera(2003), Tuan(2002). The recent pacontrol systems. per Lee(2014) gave novel stability conditions but no conTanaka(2000), Teixiera(2003), Tuan(2002). The recent pa- control systems. per Lee(2014) Lee(2014) gave novel stability conditions but nofuzzy conIn this paper, we consider a nonlinear networked control per gave novel stability conditions but no control design method was provided there. Recently, per Lee(2014) gave novel stability conditions but no conIn this this design paper, based we consider consider nonlinearfuzzy networked control In paper, we aaa nonlinear networked trol design method was provided there. Recently, fuzzy system on Takagi-Sugeno models.control First, trol design method was provided fuzzy In this design paper, based we consider nonlinearfuzzy networked control control theory has been extended a there. class ofRecently, fuzzy systems trol design method was provided there. Recently, fuzzy system on Takagi-Sugeno models. First, system design based on Takagi-Sugeno fuzzy models. First, control theory has been extended a class of fuzzy systems we assume a special form of fuzzy feedback controller control theory has been extended a class of fuzzy systems system design based on Takagi-Sugeno fuzzy models. First, with local nonlinear systems(Yoneyama(2014)). Since lo- we control theory has been extended a class of fuzzy systems we assume a special form of fuzzy feedback controller assume a special form of fuzzy feedback controller with local nonlinear systems(Yoneyama(2014)). Since loand consider the closed-loop system with such a feedback with local nonlinear systems(Yoneyama(2014)). Since lowe assume a special form of fuzzy feedback controller cal subsystems are nonlinear, a fewer number Since of local with local nonlinear systems(Yoneyama(2014)). lo- and and consider consider the closed-loop system with such such a feedback closed-loop system with cal subsystems subsystems are nonlinear, nonlinear, fewernonlinear number systems, of local local controller. Inthe order to obtain less conservative stability cal are aa fewer number of and consider the closed-loop system with such aa feedback feedback subsystems are required to represent cal subsystems are nonlinear, a fewer number of local controller. In order to obtain less conservative stability controller. In order to obtain less conservative stability subsystems are required to represent nonlinear systems, conditions, we introduce a new type of multiple Lyapunovsubsystems are required to represent systems, controller. In order to obtain less of conservative stability compared with that of fuzzy systemsnonlinear with local linear conditions, subsystems are required to represent nonlinear systems, conditions, we introduce a new type multiple Lyapunovwe introduce a new type of multiple Lyapunovcomparedwhich with in that of fuzzy fuzzy systems systems with local linear linear Krasovskii function, which reduces conservatism in compared with that of local conditions, we introduce a new type ofthe multiple Lyapunovsystems, turn greatly reduceswith conservatism in Krasovskii compared with that of fuzzy systems with local linear Krasovskii function, which reduces the conservatism in function, which reduces the conservatism in systems, which in turn greatly reduces conservatism in our stability condition. Such a new multiple Lyapunovsystems, which in turn greatly reduces conservatism in Krasovskii function, which reduces the conservatism in control design conditions. Such areduces generalized fuzzy syssystems, which in turn greatly conservatism in our stability condition. Such a new multiple Lyapunovstability condition. Such aa new multiple control design conditions.and Such a generalized generalized fuzzy syssys- our Krasovskii function employs an integral of theLyapunovmembercontrol design conditions. Such a fuzzy our stability condition. Such new multiple Lyapunovtem has been considered, stability and stabilizability control conditions.and Such a generalized fuzzy sys- Krasovskii Krasovskii function employs an integral integral of the the membermemberfunction employs an of tem has has design been considered, stability and stabilizability ship functions of fuzzy systems. A multiple tem been considered, and stability stabilizability Krasovskii function employs an integral of theLyapunovmemberconditions have been obtained. However,and those conditions ship tem has been considered, and stability and stabilizability ship functions of fuzzy systems. A multiple Lyapunovfunctions of fuzzy systems. A multiple Lyapunovconditions have been been obtained. obtained. However, those conditions Krasovskii function is a systems. natural extension of aLyapunovcommon conditions have conditions ship functions of fuzzy A multiple are still conservative, and there However, is room tothose improve. conditions have been obtained. However, those conditions Krasovskii function is a natural extension of a common Krasovskii function is aa natural of aa common are still still conservative, conservative, and and there there is is room room to to improve. improve. Lyapunov function. However, sinceextension a conventional multiple are Krasovskii function is natural extension of common are still conservative, and there is room to improve. Lyapunov function. However, since a conventional multiple function. However, aa conventional multiple The state feedback control is simple and is widely applied Lyapunov Lyapunov-Krasovskii functionsince contains the membership Lyapunov function. However, since conventional multiple The state feedback control is simple and is widely applied Lyapunov-Krasovskii function contains the membership The state feedback control is simple and is widely applied Lyapunov-Krasovskii function contains the membership to controlled systems, and many results the state feed- function, a resulting stability condition depends on the The state feedback control is simple andon is widely applied Lyapunov-Krasovskii function contains the membership to controlled systems, and many results on the state feedfunction, aa ofresulting resulting stability function conditionthat depends on the the to controlled systems, and many results on the feedfunction, stability condition depends on back controller design method have appeared in state the literaderivatives the membership is a function to controlled systems, and many results on the state feedfunction, a resulting stability condition depends on the back controller controller design method Precup(2012), have appeared appeared anaka(1992), in the the literalitera- derivatives derivatives of the membership function that is a function back design method have in of the membership function that is a function ture(for example, Lam(2000), the premise variables. In fact, the paper back controller design method Precup(2012), have appeared anaka(1992), in the litera- of derivatives of the membership function thatTanaka(2013) is a function ture(for example, Lam(2000), of the the premise premise variables. In fact, the paper paper Tanaka(2013) ture(for example, Lam(2000), Precup(2012), anaka(1992), variables. the Tanaka(2000), Wang(1996), and references therein.). The of assumes the upper bound In of fact, the derivative ofTanaka(2013) the memberture(for example, Lam(2000), Precup(2012), anaka(1992), of the premise variables. In fact, the paper Tanaka(2013) Tanaka(2000), Wang(1996), and references therein.). The assumes assumes the upper upper boundit of of the derivative of the the membermemberTanaka(2000), Wang(1996), references therein.). The the bound derivative of paper Lam(2000) uses a stateand feedback parallel distributed function. However, is the in general difficult know it Tanaka(2000), Wang(1996), and references therein.). The ship assumes the upper boundit of the derivative of theto memberpaper Lam(2000) uses a state feedback parallel distributed ship function. However, is in general difficult to know it paper Lam(2000) uses a state feedback parallel distributed ship function. However, it is in general difficult to know it compansator(PDC), whose membership functions are mis- aship priori because the membership function depends on the paper Lam(2000) uses a state feedback parallel distributed function. However, it is in general difficult to know it compansator(PDC), whose membership membership functions are mismisa priori because the membership function depends on the compansator(PDC), whose functions are a priori because the membership function depends on the matched with local controlled systems, and gives less conpremise variables,the which are actually the state and/or the compansator(PDC), whose membership functions are misa priori because membership function depends on matched with with localconditions controlledthan systems, and gives less less con- premise premiseof variables, which areare actually the state state and/or the matched local controlled systems, gives variables, which actually the the servative stability thoseand of Tanaka(1992), the system andare to be controlled. Our new matched with localconditions controlledthan systems, and gives less concon- output premise variables, which areare actually the state and/or and/or the servative stability those of Tanaka(1992), output of the system and to be controlled. Our new servative stability conditions than those of Tanaka(1992), output of the system and are to be controlled. Our new Tanaka(2000) and conditions Wang(1996). The paper Precup(2012) multiple Lyapunov-Krasovskii function eventually avoids servative stability than those of Tanaka(1992), output of the system and are to be controlled. Our new Tanaka(2000) and Wang(1996). The paper Precup(2012) multiple Lyapunov-Krasovskii Lyapunov-Krasovskii function function eventually eventually avoids avoids Tanaka(2000) and The recently provided a state feedback controlPrecup(2012) method by multiple Tanaka(2000) and Wang(1996). Wang(1996). The paper paper recently provided provided a state state feedback feedback controlPrecup(2012) method by by multiple Lyapunov-Krasovskii function eventually avoids recently a control method recently provided a state feedback control method by Copyright © 2016, 2016 IFAC 297Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2016 297 Copyright ©under 2016 IFAC IFAC 297Control. Peer review© of International Federation of Automatic Copyright © 2016 responsibility IFAC 297 10.1016/j.ifacol.2016.10.413

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such a difficulty and does not require the upper bound of the derivative of the membership function. Furthermore, our stability condition is given in terms of linear matrix inequalities (LMIs), which are easily solved by the standard numerical software. Based on an obtained stability condition, we propose a controller design method. Finally, we show a numerical example to illustrative our design procedures and to show the effectiveness of our design approach. 2. FUZZY MODEL OF NETWORKED CONTROL SYSTEMS In this section, we introduce Takagi-Sugeno fuzzy systems, which describes nonlinear systems. Consider the TakagiSugeno fuzzy model, described by the following IF-THEN rules: IF ξ1 is Mi1 and · · · and ξp is Mip , T HEN x(t) ˙ = Ai x(t) + Bi u(t). where x(t) ∈ n is the state and u(t) ∈  is the control input. The matrices Ai and Bi are constant matrices of appropriate dimensions. The time-delay τ may be unknown but is assumed to be finite. r is the number of IFTHEN rules. Mij are fuzzy sets and ξ1 , · · · , ξp are premise variables. We set ξ = [ξ1 · · · ξp ]T . The premise variable ξ(t) is assumed to be measurable. Then, the state equation is described by x(t) ˙ =

r 

λi (ξ){Ai x(t) + Bi u(t)}

i=1

∆

= Aλ x(t) + Bλ u(t)}.

(1)

where βi (ξ) λi (ξ) = r ,  βi (ξ)

βi (ξ) =

p 

where



   −2 0.7α −2 1.3α , A2 = , A1 = −0.2α 2 0.2α 2     −0.5α 0.5α B1 = , B2 = , 1 1 −x1 + α x1 + α , λ2 (x1 ) = . λ1 (x1 ) = 2 2

(5) is an exact representation of the nonlinear system (3)(4). If α < ∞, we can discuss local system analysis, and if α → ∞, we can investigate the system globally. In the networked control system, the information is exchanged with packets through a network where the data packets encounter delays. In what follows, considering the effects of network-induced delays, the closed-loop system is modeled as a fuzzy system with bounded delays. In the considered networked control system, the controller and the actuator are event-driven and sampler is clock-driven. The actual input of the system (1) is realized via a zeroorder hold device. The sampling period is assumed to be a positive constant T and the information of the zeroorder hold may be updated between sampling instants. The updating instants of the zero-order hold are denoted by tk , and τ1 and τ2 are the time-delays from the sampler to the controller and from the controller to the zero-order hold at the updating instant tk , respectively. So, the successfully transmitted data in the networked control system at the instant tk experience round trip delay τ = τ1 + τ2 which does not need to be restricted inside one sampling period. Regarding the role of the zero-order hold, for a state sample data tk − τ , the corresponding control signal would act on the plant from tk unto tk+1 . Therefore, the fuzzy control input for tk ≤ t ≤ tk+1 , is written as the following: For a fuzzy system (1), the parallel distributed compensator(PDC) has been used. Its rules are given as follows:

Mij (ξj )

j=1

i=1

and Mij (·) is the grade of the membership function of Mij . We assme r  β(ξ(t)) ≥ 0, i = 1, · · · , r, βi (ξ(t)) > 0

IF ξ1 is Mi1 and · · · and ξp is Mip , T HEN u(t) = Ki x(t − τ (t)), i = 1, · · · , r. where Ki , i = 1, · · · , r are a constant matrices. Then, an overall controller is given by

i=1

for any ξ(t). Hence λi (ξ(t)) satisfy r  λi (ξ(t)) ≥ 0, i = 1, · · · , r, λi (ξ(t)) = 1

µi (ξ(t − τ (t)))Ki x(t i=1 = Kµτ x(t − τ (t))

u(t) =

(2) where

i=1

1 µi (ξ(t)) = τ

for any ξ(t). As we can see below, the fuzzy model representation has a rich structure and can describe a wide class of nonlinear systems. Example 1. We consider a nonlinear system described by x˙ 1 = −2x1 + 0.3x1 x2 + x2 + 0.5x1 u,

x˙ 2 = −0.2x21 + 2x2 + x1 u. Assuming x1 = [−α. α] for some α > 0, we have 2  λi (ξ){Ai x(t) + Bi u(t)} x(t) ˙ =

(3) (4)

(5)

i=1

298

r 

t

− τ (t)) (6)

λi (ξ(s))ds,

t−τ

and τ > 0 is a known upper bound of the time-delay τ (t). The closed-loop system (1) with (6) is given by x(t) ˙ =

r  r  i=1 l=1

λi (ξ(t))µl (ξ(t − τ (t))) ×{Ai x(t) + Bi Kl x(t − τ (t))}

= Aλ x(t) + Bλ Kµτ x(t − τ (t)).

We note that µi (ξ(t)) ≥ 0, i = 1, · · · , r and

(7)

2016 IFAC NECSYS September 8-9, 2016. Tokyo, Japan

r 

1 µi (ξ(t)) = τ i=1

Jun Yoneyama et al. / IFAC-PapersOnLine 49-22 (2016) 297–302

t  r

λi (ξ(s))ds

t−τ i=1

1 = τ

t

1ds

t−τ

= 1, which imply that µi (ξ(t)) and λi (ξ(t)) share the same properties as seen in (2). Our problem is to find a state feedback controller (6) such that the closed-loop system (7) is asymptotically stable.

where Φ11ijkl = Φ1ijkl + Φ2il + ΦT2il + Φ3il + ΦT3il ,   1 Rj + (Pi − Pk ) 0 0 Pj   h , 0 0 0 0 Φ1ijkl =    0 0 −Rk 0 Pj 0 0 τ (Z1 + Z2 ) Φ2il = [ Nil + Mil −Nil + Sil −Mil − Sil 0 ] , Φ3il = [ −T Ai −T Bi Kl 0 T ] . Proof. First, it follows from the Leibniz-Newton formula that the following equations hold for any matrices Nij , Sij and Mij , the forms of which are given in Theorem 2.

3. STABILITY ANALYSIS

2

Let us first assume that all the controller gain matrices Ki , i = 1, · · · , r are given.

Stability of fuzzy time-delay system has been investigated in the literature. Stability conditions are obtained by Lyapunov-Krasovskii stability theorem. Importance on stability conditions lies on how to choose an appropriate Lyapunov-Krasovskii function. So far, LyapunovKrasovskii function with common matrices has been employed for the analysis. In this paper, we introduce a new Lyapunov-Krasovskii function. To begin with, let us consider a polytopic matrix: r  µi (ξ(t))Pi . Pµ = i=1

It is easy to calculate that the time-derivative of Pµ is given by P˙ µ =

r 

µ˙ i (ξ(t))Pi

1 = (Pλ − Pλτ ). τ ∆

(8)

Now, we are ready to give our first main result. Theorem 2. Given control gain matrices Kl , l = 1, · · · , r. A controller (6) stabilizes the fuzzy system (1) if there exist matrices Pi > 0, Ri ≥ 0, i = 1, · · · , r, Z1 > 0, Z2 > 0,       N1il S1il M1il N  S  M  Nil =  2il  , Sil =  2il  , Mil =  2il  , N3il S3il M3il N4il S4il M4il   T1  T2  i, l = 1, · · · , r and T =   T3 T4

Φijkl

Φ11ijkl  τNT il =   τ ST il τ MilT

τ Nil −τ Z1 0 0

i=1 l=1



λi (ξ(t))µl (ξ(t − τ (t)))ζ T (t)Nil

 × x(t) − x(t − τ (t)) − 2

r  r  i=1 l=1



2

r  r  i=1 l=1

τ Sil 0 −τ Z1 0



τ Mil 0   < 0, 0  −τ Z2 i, j, k, l = 1, · · · , r

 x(s)ds ˙  = 0,

t−τ (t)

(10)

λi (ξ(t))µl (ξ(t − τ (t)))ζ T (t)Sil 

t−τ  (t) t−τ

 x(s)ds ˙  = 0, (11)

λi (ξ(t))µl (ξ(t − τ (t)))ζ T (t)Mil



xT (t − τ (t))

t

t−τ



=0 x(s)ds ˙

xT (t − τ )

(12)

T

x˙ T (t) ] .

It is also clear from the nominal closed-loop system (7) that the following is true for any T . 2

r  r  i=1 l=1

λi (ξ(t))µl (ξ(t − τ (t)))ζ T (t)T

× [x(t) ˙ − Ai x(t) − Bi Kl x(t − τ (t))] = 0

(13)

Now, we consider the following Lyapunov-Krasovskii function: V (xt ) = V1 (x) + V2 (xt ) + V3 (xt ) where xt = x(t + θ), −h ≤ θ ≤ 0,

V1 (x) = xT (t)Pµ x(t), t xT (s)Rµ x(s)ds, V2 (xt ) = V3 (xt ) =

t−τ 0

t

x˙ T (s)(Z1 + Z2 )x(s)dsdθ, ˙

−τ t+θ

(9) Pµ =

r  i=1

299



t

 × x(t − τ (t)) − x(t − τ ) −

where ζ(t) = [ xT (t)

r 1 = (λi (ξ(t)) − λi (ξ(t − τ )))Pi τ i=1



r  r 

× x(t) − x(t − τ ) −

i=1

such that

299

µi (ξ)Pi , Rµ =

r  i=1

µi (ξ)Ri ,

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Jun Yoneyama et al. / IFAC-PapersOnLine 49-22 (2016) 297–302

and Pi > 0, Ri ≥ 0, i = 1, · · · , r, Z1 > 0, Z2 > 0 are to be determined. Then, V (x, t) > 0, ∀x = 0. Now, we take the derivative of V (xt ) with respect to t along the solution of the system (7) and add (10)-(13): d V (xt ) dt T ≤ 2x˙ (t)Pµ x(t) + xT (t)P˙µ x(t) + xT (t)Rµ x(t) −xT (t − τ )Rµτ x(t − τ ) + τ x˙ T (t)(Z1 + Z2 )x(t) ˙ t−τ (t) t   x˙ T (s)Z1 x(s)ds ˙ − x˙ T (s)Z1 x(s)ds ˙ − −

t−τ

x˙ T (s)Z2 x(s)ds ˙ + (10) + · · · + (13)

r r  r  r  

≤

i=1 j=1 k=1 l=1

t

[ζ T (t)Nil + x˙ T (s)Z1 ]Z1−1 [NilT ζ(t) + Z1 x(s)]ds ˙

−

 τ Mil 0  < 0, 0  −τ Z2 i, l = 1, · · · , r

(16)

0],

4. CONTROL DESIGN

[ζ T (t)Sil + x˙ T (s)Z1 ]Z1−1 [SilT ζ(t) + Z1 x(s)]ds ˙

t−τ

t

τ Sil 0 −τ Z1 0

As in Corollary 3, conditions (9) recovers (16) which are already given in the literature. This implies that common Lyapunov-Krasovskii matrix approach is a special case of our result. In other words, our result is generalized one from the literature.

t−τ (t) t−τ  (t)

−

τ Nil −τ Z1 0 0

Proof. If all the matrices Pi and Qi are the same;Pi = P and Qi = Q, i = 1, · · · , r, conditions (9) are reduced to λi (ξ(t))µj (ξ(t))λk (ξ(t − τ ))µl (ξ(t − τ (t)) (16).

×{ζ T (t)Ψijkl ζ(t) −

Φ11il  τ NilT Φil =  τ SilT τ MilT

where T T Φ11il = Φ  1 + Φ2il + Φ2il + Φ3il + Φ3il , R 0 0 P 0 0 0 0  Φ1 =  , 0 0 −R 0 P 0 0 τ (Z1 + Z2 ) Φ2il = [ Nil + Mil −Nil + Sil −Mil − Sil Φ3il = [ −T Ai −T Bi Kl 0 T ] .

t−τ

t−τ (t) t



[ζ T (t)Mil + x˙ T (s)Z2 ]Z2−1 [MilT ζ(t) + Z2 x(s)]ds} ˙ (14)

t−τ

where Ψijkl = Φ11ijkl + τ Nil Z1−1 NilT + τ Sil Z1−1 SilT +τ Mil Z2−1 MilT . In the above derivation, we have used (8). If (9) is satisfied, then by the Schur complement formula we have Ψijkl < 0, i, j, k, l = 1, · · · , r.

(15)

r  r

If (15) holds, we have i j λi (ξ(t))λj (ξ(tk ))ζ T (t)Ψij ζ(t) < 0, which implies that V˙ (xt ) < 0 because the last three terms in (14) are all less than 0. This proves that the conditions (9) suffice to achieve the asymptotic stability of the system (7). The following corollary is a special case of Theorem 2. Corollary 3. Given control gain matrices Kl , l = 1, · · · , r. A controller (6) stabilizes the fuzzy system (1) if there exist matrices P > 0, R ≥ 0, Z1 > 0, Z2 > 0,       N1il S1il M1il N  S  M  Nil =  2il  , Sil =  2il  , Mil =  2il  , N3il S3il M3il N4il S4il M4il   T1  T2  i, l = 1, · · · , r and T =   T3 T4 such that 300

We shall propose a control design method based on Theorem 2. Theorem 4. Given scalars ti , i = 1, · · · , 4, a controller (6) stabilizes the fuzzy system (1) if there exist matrices ¯ i ≥ 0, Z¯1 > 0, Z¯2 > 0, L, Yl , i, l = 1, · · · , r, P¯i > 0, R  ¯  ¯   ¯  N1il M1il S1il ¯2il  ¯  N  S¯2il  M ¯ ¯ ¯ Nil =  ¯  , Sil =  ¯  and Mil =  ¯ 2il  , N3il S3il M3il ¯4il ¯ 4il N S¯4il M i, l = 1, · · · , r such that  Θijkl

Θ11ijkl ¯T  hN il =   hS¯T il ¯T hM il

¯il hN −hZ¯1 0 0

hS¯il 0 −hZ¯1 0

 ¯ il hM 0   < 0, (17) 0  −hZ¯2 i, j, k, l = 1, · · · , r

where Θ11ijkl = Θ1ijkl + Θ2il + ΘT2il + Θ3il + ΘT3il ,   ¯ j + 1 (P¯i − P¯k ) 0 R 0 P¯j   h , 0 0 0 0 Θ1ijkl =    ¯k 0 0 −R 0 ¯ ¯ ¯ Pj 0 0 h(Z1 + Z2 ) ¯il + M ¯ il −N ¯il + S¯il −M ¯ il − S¯il 0 ] , Θ2il = [N  −t1 Ai LT −t1 Bi Yl 0 t1 LT T T −t2 Bi Yl 0 t2 L   −t A L . Θ3il =  2 i T −t3 Ai L −t3 Bi Yl 0 t3 LT  −t4 Ai LT −t4 Bi Yl 0 t4 LT

In this case, state feedback control gains in (6) are given by Ki = Yi L−T . (18) ¯ i = 1, · · · , 4 where ti are scalars, Proof. We let Ti = ti L, and substitute them into the conditions (9). If (9) holds,

2016 IFAC NECSYS September 8-9, 2016. Tokyo, Japan

Jun Yoneyama et al. / IFAC-PapersOnLine 49-22 (2016) 297–302

it follows that the (4, 4)-block of Φ11ijkl must be negative ¯+L ¯ T ) < 0, which definite. It follows that T4 + T4T = t4 (L ¯ is nonsingular if t4 = 0. Then we calculate implies that L Θijkl = QΦijkl QT with Q = diag [ L L L L L L L ] ¯ −1 . Defining where L = L ¯ i = LRi LT , Z¯1 = LZ1 LT , Z¯2 = LZ2 LT , P¯i = LPi LT , R ¯ pij = LMpij LT , ¯pij = LNpij LT , S¯pij = LSpij LT , M N p = 1, · · · , 4,

We can readily obtain the following corollary from Theorem 4. Corollary 5. Given scalars ti , i = 1, · · · , 4, a controller (6) stabilizes the fuzzy system (1) if there exist matrices ¯ ≥ 0, Z¯1 > 0, Z¯2 > 0, L, Yl , l = 1, · · · , r, P¯ > 0, R  ¯  ¯   ¯  N1il S1il M1il ¯2il  ¯2il  ¯    N S M ¯ il =  2il  ¯il =  , S¯il =  ¯  and M N  ¯ ¯ 3il  , N3il S3il M ¯4il ¯ 4il N S¯4il M i, l = 1, · · · , r



Θ11il ¯T  hN Θil =  ¯Til hSil ¯T hM il

¯il hN −hZ¯1 0 0

hS¯il 0 −hZ¯1 0

¯ il  hM 0  < 0, 0  ¯ −hZ2 i, l = 1, · · · , r

where T T Θ11il = Θ 3il ,  1¯+ Θ2il + Θ2il + Θ¯3il + Θ R 0 0 P  0 0 0 0 Θ1 =  , ¯ 0 0 −R 0 P¯ 0 0 h(Z¯1 + Z¯2 ) ¯il + M ¯ il −N ¯il + S¯il −M ¯ il − S¯il Θ2il =  [N  −t1 Ai LT −t1 Bi Yl t1 LT T T  −t A L −t2 Bi Yl t2 L  . Θ3il =  2 i T −t3 Ai L −t3 Bi Yl t3 LT  T T −t4 Ai L −t4 Bi Yl t4 L

(19)

0],

In this case, state feedback control gains in (6) are given as in (18). 5. NUMERICAL EXAMPLE We consider the system x(t) ˙ =

2 

λi (ξ){Ai x(t) + Bi u(t)}

(20)

i=1

where x1 is supposed to be in [−1, 1], and     −1 0.7 −1 1.3 , A2 = , A1 = −0.2 2 0.2 2     −0.5 0.5 , B2 = , B1 = 1 1 301

−x1 + 1 x1 + 1 , λ2 (x1 ) = . 2 2

Given the scaling parameters t1 = 2.0, t2 = −1.8, t3 = 0.2, t4 = 0.85, Theorem 4 gives a stabilizing controller for the system with the maximum time-delay τ = 0.3077. In this case, the control gain matrices are given by K1 = [ 1.9296

and applying the Schur complement formula, we obtain Θijkl in (17), where we define Yj = Kj LT , and Θ11ijkl , Θ1ijkl , Θ2il and Θ3il are given as in Theorem 4.

such that

λ1 (x1 ) =

301

−21.7549 ] , K2 = [ −1.8841

−21.7825 ] .

On the other hand, Corollary 5 only guarantees the stabilization with maximum time-delay τ = 0.2458. This obviously show that our new multiple Lyapunov-Krasovskii function method is better than the existing common Lyapunov-Krasovskii function method. 6. CONCLUSIONS State feedback stabilization of nonlinear networked control systems described by Takagi-Sugeno fuzzy systems has been considered. A new multiple Lyapunov-Krasovskii function was introduced to show new stability conditions for the closed-loop system. Based on such stability conditions, a control design method with a descriptor system approach was proposed. Our stability conditions do not require any information on the derivative of the membership function for our fuzzy systems. A generalized nonlinear controller has been used. These advantages widen a class of nonlinear systems for its stability analysis and control design. REFERENCES T. Bouarar, K. Guelton, and N. Manamanni, Robust non-quadratic static output feedbak controller design for Takagi-Sugeno systems using descriptor redundancy, Engineering Applications of Artificial Intelligence,Vol.26, No.2, pp.739-756, 2013. H. Du, J. Lam, and K. Y. Sze, Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm, Engineering Applications of Artificial Intelligence, Vol.16, pp.667-680, 2003. K. Guelton, T. Bouarar, and N. Manamanni, Robust dynamic output feedback Lyapunov stabilization of Takagi-Sugeno systems - a descriptor redundancy approach, Fuzzy Sets and Systems, Vol.160, pp.2796-2811, 2009. H.K. Lam, F.H.F. Leung, and P.K.S. Tam, Stable and robust fuzzy control for uncertain nonlinear systems, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, Vol.30, No.6, pp.825-840, 2000. D.H. Lee, Y.H. Joo, M.H. Tak, Local stability analysis of continuous-time Takagi-Sugeno fuzzy systems: A fuzzy Lyapunov function approach, Information Sciences, Vol.257, pp.163-175, 2014. C. Lien, H∞ non-fragile observer-based controls of dynamical systems via LMI optimization approach, Chaos, Solitons and Fractals, Vol.34 pp.428-436, 2007. S. Marouf, R.M. Esfanjani, A. Akbari, M. Barforooshan, T-S fuzzy controller design for stabilization of nonlinear networked control system, Engineering Applications of Artificial Intelligence, Vol.50, pp.135-140, 2016.

2016 IFAC NECSYS 302 September 8-9, 2016. Tokyo, Japan

Jun Yoneyama et al. / IFAC-PapersOnLine 49-22 (2016) 297–302

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