# Switching Fuzzy Guaranteed Cost Control for Nonlinear Networked Control Systems

Authors: Linqin Cai; Zhuo Yang; Jimin Yu; Zhenhua Zhang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper deals with the problem of guaranteed cost control for a class of nonlinear networked control systems (NCSs) with time-varying delay. A guaranteed cost controller design method is proposed to achieve the desired control performance based on the switched T-S fuzzy model. The switching mechanism is introduced to handle the uncertainties of NCSs. Based on Lyapunov functional approach, some sufficient conditions for the existence of state feedback robust guaranteed cost controller are presented. Simulation results show that the proposed method is effective to guarantee system’s global asymptotic stability and quality of service (QoS).


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## Body

## 1. Introduction

As network technology advanced in the last decade, networked control system (NCS) has increasingly become a research focus. Considerable attention on the modeling and controller design of NCSs has been paid in [1–5]. There are many advantages to NCSs, such as reduced system wiring, facilitated system maintenance, and increased systems flexibility.However, due to the insertion of communication channels, this brings many challenging problems such as network-induced delay and data packet dropout. Regardless of the type of network used, these special issues degrade the system dynamic performance and are a source of potential instability. There are a number of design methods that have been proposed to deal with these problems. One of the most general methods is to model the NCS as a system with time-varying delays. So the stability of an NCS is equivalent to the stability of a system with time-varying delays [2]. Moreover, the sampling behavior has also an important impact on the design of the NCS controller because the states of the feedback controller are not continuous as a result of the existence of zeroth-order hold (ZOH). In [1], a model of NCS was provided considering network-induced delay and packet dropout in the transmission. In [3], an observer-based stabilizing controller was designed for the NCSs involving both random measurement and actuation delays. Robust controllers for uncertain NCSs were also obtained in [4, 5]. However, how to analyze the stability of nonlinear NCSs has increasingly become a challenging topic. Some results about the stability of nonlinear NCSs were obtained in [6, 7]. In [8], a stochastic optimal controller design for nonlinear NCSs with uncertain dynamics via neurodynamic programming was proposed. The closed-loop stability of the nonlinear NCSs was demonstrated by selecting neural network (NN) update laws. However, these methods often require some strict assumptions for a system model, so it is difficult for practical applications.In the last few years, the fuzzy control is a useful approach to solve the control problems of nonlinear systems. The Takagi-Sugeno (T-S) fuzzy system proposed in [9] is widely applied to industrial control fields because of its simple structure with local dynamics. In the T-S fuzzy model, local dynamics in different state-space regions are represented by many linear models so that linear system theory can conveniently be employed to analyze the stability of overall closed-loop system and to design the feedback controller. The typical design approaches are carried out based on fuzzy model via the so-called parallel distributed compensation (PDC) method [10]. Considering the time-delay characteristic of nonlinear systems, many results about the T-S fuzzy model with a time-delay term are obtained to deal with stability and stabilization problems of nonlinear systems with time delays [2, 11–13]. The guaranteed cost control for a T-S fuzzy system with time delays was presented in [2, 11, 12]. In [13], robust control problem was studied for a class of large-scale NCSs, and the decentralized design was presented using T-S fuzzy approach.Nevertheless, an inherent drawback remains since the number of fuzzy rules of a T-S model increases exponentially with the number of nonlinearities constituting the matched nonlinear system [14]. This makes fuzzy controller design and implementation difficult as the complexity of the nonlinear system to be controlled increases [15]. To outline the problem of rules explosion in fuzzy T-S modeling, some authors have proposed to combine the merit of switched systems with T-S ones to deal with nonlinear control problems [16–18]. To do so, partitioning the state space of a nonlinear system allows defining a switched nonlinear system. Then, inside each partition, a T-S model can be obtained. As stated in [18], the resulting switched T-S system inherits some essential features of hybrid systems and maintains all the information and knowledge representation capacity of fuzzy systems. However, few papers have studied stabilization issues of switched fuzzy systems based on switching Lyapunov functions [16] or quadratic approaches [17]. In [19], based on barrier Lyapunov functions (BLFs), a new control design for constrained nonlinear switched systems was investigated to achieve output tracking. By ensuring the boundedness of the BLFs in the closed loop, the proposed approaches can guarantee that all states in the switched systems do not violate the desired constraints and that all closed-loop signals are bounded. For many nonlinear systems, some of the premise variables of the corresponding T-S fuzzy models are measurable when they are modeled as T-S fuzzy models, while the parts with unmeasurable premise variables can be modeled as uncertainties. Thus, the overall systems can be described by T-S fuzzy parts with measurable premise variables and uncertainty parts. In [20], a switching stabilizing controller is designed for a nonlinear system with unknown parameters or unmeasurable premise variables. However, the uncertainties cannot be well handled. In [21], a switching fuzzy dynamic output feedback control scheme is proposed, where the switching mechanism is introduced to handle the unknown parameters.In designing a controller for a real plant, it is invariably necessary to design a control system which not only is stable but also possesses a strong robust performance. One way to deal with this is the so-called guaranteed cost control approach [2]. This addresses the robust performance problem and has the advantage of providing an upper bound on a given performance index guaranteeing that system performance degradation incurred by uncertainty is less than this bound. For the NCSs, the quality of service (QoS) is one of the most important performance indexes. Therefore, it is vital to design a guaranteed cost controller such that the NCSs are stable as well as satisfactory with the required QoS.In this paper, we aim at the problem of guaranteed cost control for a class of uncertain nonlinear NCSs with time delays. Considering the QoS of NCSs, we propose a guaranteed cost control scheme to achieve the desired control performance based on switched T-S fuzzy control method, where the switching mechanism is introduced to handle the uncertainties. Moreover, the sufficient condition for the existence of the robust guaranteed cost controller and the design method of the corresponding switching control law are obtained via Lyapunov functions. Comparing with [2, 11, 12], the proposed switching fuzzy approach inherits some essential features of hybrid systems to deal with the uncertain nonlinear NCSs and avoids the inherent drawback of rules explosion in modeling a fuzzy T-S model. In addition, we consider the stabilization problem of the switched fuzzy T-S system with time delays in NCSs.The innovations of this paper are as follows: (1) the guaranteed cost controller is proposed for nonlinear NCSs with time-varying delay to achieve the desired control performance based on the switched T-S fuzzy model with uncertain parameters, and (2) the sufficient condition for the robust guaranteed cost control law is presented to uncertain nonlinear NCSs.The paper is organized as follows. The basic problem formulation of the nonlinear networked control system is given in Section2. The switching fuzzy controller via state feedback is analyzed in Section 3. In Section 4, the sufficient stability conditions and guaranteed cost control law are discussed by Lyapunov functions. Section 5 provides simulation results to demonstrate the effectiveness of the proposed method. Finally, concluding remarks are given in Section 6.


## 2. Problem Formulation

A general NCS configuration is illustrated in Figure1, which is composed of a controller and a remote system containing a physical plant, sensors, and actuators. The controller and the plant are physically located at different locations and are directly linked by a data network in order to perform remote closed-loop control.Figure 1
A general NCS.In Figure1, the nonlinear networked control system can be described as follows:
(1)x˙(t)=f(x(t),u(t)),y(t)=g(x(t),u(t)),
where x(t)∈Rn is the state variation, u(t)∈Rm is the control input, y(t)∈Rs is the regulated output, f(x,u) and g(x,u) are the nonlinear functions. Currently, it has been proved that the T-S fuzzy models are universal approximations of many nonlinear dynamic systems. So we introduce a T-S fuzzy system to model a class of nonlinear NCSs. Without considering the disturbance input, we use the following T-S fuzzy model to approximate f(x(t),u(t)) and g(x(t),u(t)):
(2)Ri:IFθ1(t)isNi1,…,andθg(t)isNig,THEN{x˙(t)=Aix(t)+Biu(t),y(t)=Cix(t),
where i=1,2,…,r is the index number of fuzzy rules, θ1(t),θ2(t),…,θg(t) are the known premise variables, Nik is the fuzzy sets (k=1,2,…,g), x(t)∈Rn is the state vector, y(t)∈Rs is the output vector, u(t)∈Rm is the control input, Ai∈Rn×n is known system matrices, Bi∈Rn×m is the input matrix, and Ci∈Rn×s is the output matrix.Before designing the controller, we make the following reasonable assumptions.Assumption 1.
The sensor is clock-driven. The controller and actuator are event-driven. The clocks among them are synchronized.Assumption 2.
Time-varying network-induced delay is less than one sampling period.Assumption 3.
The computational delay is negligible.Assumption 4.
The signal is single-packet transmission without packet drop.Assuming the node sampling period isT, so we can obtain the discrete T-S fuzzy model of nonlinear networked control systems with time-varying delay:
(3)Ri:IFθ1(k)isNi1,…,andθg(k)isNig,THEN{x(k+1)=Φix(k)+Γi0(τk)u(k)+Γi1(τk)u(k-1),y(k)=Cix(k),
where Φi=eAiT, Γi0(τk)=∫0T-τkeAiTdt·Bi, Γi1(τk)=∫T-τkTeAiTdt·Bi, and τk is the network-induced delay, i=0,1,2,…,r.Furthermore, (3) can be transformed into the following form with uncertain parameters:
(4)x(k+1)=Φix(k)+(Γi0+DiFi(τk)Ei)u(k)+(Γi1-DiFi(τk)Ei)u(k-1),y(k)=Cix(k),
where Di, Ei are known constant matrices of appropriate dimensions and Fi  is an unknown matrix function satisfying
(5)FTF≤I,Γi0(τk)=Γi0+DiFi(τk)Ei,Γi1(τk)=Γi1-DiFi(τk)Ei.
For any given x(k) and u(k), by using a weighted-average defuzzifier, product inference, and singleton fuzzifier, the local models can be integrated into a global nonlinear model:
(6)x(k+1)=∑i=1rμi(θ(k))[Φix(k)+Γi0(τk)u(k)∑i=1rμi(θ(k))+Γi1(τk)u(k-1)],y(k)=Cix(k),
where
(7)θ(k)=[θ1T(k),θ2T(k),…,θgT(k)]T,μi(θ(k))=∏l=1gNil(ql(k))∑i=1r∏l=1gNil(θl(k)).Nil(θl(k)) is the grade of membership of θl(k) in the fuzzy set Nil. Notice the following facts:
(8)μi(θ(k))≥0,∑i=1rμi(θ(k))=1.The guaranteed cost function associated with system (3) is given by
(9)J=∑k=0∞[xT(k)Qx(k)+uσT(k)Ruσ(k)].Definition 5.
Consider the uncertain system (3) and cost function (9). If there exists a control law uσ*(k) and a positive scalar J* such that for all admissible uncertainties, the closed-loop system is asymptotically stable and the value of the cost function (9) satisfies J≤J*, then J* is said to be a guaranteed cost and uσ*(k) is said to be a guaranteed cost control law.


## 3. Controller Design

We assume that switched fuzzy controller is constituted withN switching rules. The σth subfuzzy controller is
(10)Rσi:IFθ1(t)isNi1,…,andθg(t)isNig,THENuσ(k)=Kσix(k),
where σ=1,2,…,N is a piecewise constant function representing the switching signal and Riσ represents ith fuzzy rule of the σth subfuzzy controller. The switching control law is constituted by the following fuzzy controller:
(11)u1(k)=∑i=1rμi(θ(k))K1ix(k),u2(k)=∑i=1rμi(θ(k))K2ix(k),⋮uN(k)=∑i=1rμi(θ(k))KNix(k).When the controlled system is inσth switching subsystem, the global fuzzy equation is as follows
(12)x(k+1)=∑i=1rμi[Φix(k)+Γi0(τk)u(k)+Γi1(τk)u(k-1)]=∑i=1rμiΦix(k)+∑i=1r∑j=1rμiμj[Γi0(τk)Kσjx(k)+Γi1(τk)Kσjx(k-1)]=∑i=1r∑j=1rμiμj×[Φix(k)+Γi0(τk)Kσjx(k)h+Γi1(τk)Kσjx(k-1)]=∑i=1r∑j=1rμiμj×{[Φi+Γi0(τk)Kσj]x(k)h+Γi1(τk)Kσjx(k-1)}=∑i=1r∑j=1rμiμj{[Φi+(Γi1-DiFi(τk)Ei)Kσj]x(k)=∑i=1r∑j=1rμiμj+(Γi1-DiFi(τk)Ei)Kσjx(k-1)}.
Let Ω1,Ω2,…,ΩN be a partition of the set Rn; then ⋃i=1NΩi=Rn∖{0}, Ωi⋂Ωj=Φ, i=j. The switching law that is determined by Ω1,Ω2,…, and ΩN is σ=σ(x(t))=i, when x(t)∈Ωi. This switching law can be completely described by the following function:
(13)vσ(x(t))={1,x(t)∈Ωσ,0,x(t)∉Ωσ,σ=1,2,…,N.
Thus, we have
(14)x(k+1)=∑σ=1N∑i=1r∑j=1rvσμiμj×{[Φi+(Γi1-DiFi(τk)Ei)Kσj]x(k)+(Γi1-DiFi(τk)Ei)Kσjx(k-1)},
where τk is the network-induced delay.


## 4. Sufficient Condition for Guaranteed Cost Control

Lemma 6 (see [22] (Schur complement)).
For a given symmetric matrix,(15)S=[S11S12S21S22].
Then, the following three conditions are mutually equivalent:(1)
S<0,(2)
S11<0, S22-S12TS11-1S12<0,(3)
S22<0, S11-S12S22-1S12T<0.Lemma 7 (see [23]).
Given matricesY, D, E, and I of appropriate dimensions and with Y and I symmetrical and I>0, then
(16)Y+DFE+ETFTDT<0
for all F satisfying FTF≤I, if and only if there exists some ε>0 such that
(17)Y+ε2DDT+ε-2ETE<0.Theorem 8.
Consider the uncertain nonlinear networked control systems (3) and the cost function (9). If there exist some constants λσ∈[0,1], ∑σ=1lλσ=1, a group of positive constants εσ>0, and positive definite matrices X, Z, Yσi, σ=1,…,l, i=1,…,r, such that the following matrix inequalities (18) hold:(18)∑σ=1lλσ[LUV0000*-2X+2Z02X(Yσi+Yσj)T(EiYσj)T(EjYσi)T**-2Z00-(EiYσj)T-(EjYσi)T***-2Q-1000****-2R-100*****-εσI0******-εσI]<0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1≤i≤j≤r,where
(19)X=P-1,Yσi=kσiX,Z=XSX,L=-2X+εσ(DiDiT+DjDjT),U=Φi+Φj+Γi0Yσj+Γj0Yσi,V=Γi1Yσj+Γj1Yσi,
then close-loop system (14) with the guaranteed cost controller (11) and the switching law σ=σ(x(k)) is globally asymptotically stable. The guaranteed cost function (9) satisfies the following bound:
(20)J≤xT(0)Px(0)+xT(-1)Sx(-1).Proof.
For the networked control system (3), we lead the performance index as follows:
(21)J=∑k=0∞[xT(k)Qx(k)+uσT(k)Ruσ(k)]=∑k=0∞[∑i=1r∑j=1rμiμjxT(k)(Q+KσiTRKσj)x(k)]=∑k=0∞[(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)(Q+KσiTRKσj)x(k)]=∑k=0∞[(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)×(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)].
Consider the Lyapunov function as
(22)V(x(k))=xT(k)Px(k)+xT(k-1)Sx(k-1).
Let
(23)Θ=[GσijTPGσnm-P+SGσijTPHσnmHσijTPGσnmHσijTPHσnm-S],Gσij=Φi+Γi0Kσj+DiFi(τk)EiKσj,ασij=Gσij+Gσji,Hσij=Γi1Kσj-DiFi(τk)EiKσj,βσij=Hσij+Hσji.
Along any trajectory of the closed-loop system (14), the forward difference of V(k) is(24)ΔV(x(k))=V(x(k+1))-V(x(k))=xT(k+1)Px(k+1)+xT(k)Sx(k)-xT(k)Px(k)-xT(k-1)Sx(k-1)=∑σ=1N∑i=1r∑j=1r∑n=1r∑m=1rvσμiμjμnμm[x(k)x(k-1)]TΘ[x(k)x(k-1)]≤∑σ=1Nvσ{14∑i=1r∑j=1rμiμj[x(k)x(k-1)]T[ασijTPασij-4P+4SασijTPβσijβσijTPασijβσijTPβσij-4S][x(k)x(k-1)]}=∑σ=1Nvσ{[[ασij2]TPασij2-P+S[ασij2]TPβσij2[βσij2]TPασij2[βσij2]TPβσij2-S]∑i=1rμi2[x(k)x(k-1)]T[GσiiTPGσii-P+SGσiiTPHσiiHσiiTPGσiiHσiiTPHσii-S][x(k)x(k-1)]+2∑i=1r∑j>irμiμj[x(k)x(k-1)]T[[ασij2]TPασij2-P+S[ασij2]TPβσij2[βσij2]TPασij2[βσij2]TPβσij2-S][x(k)x(k-1)]}.
Thus(25)ΔV(x(k))+xT(k)Qx(k)+uσT(k)Ruσ(k)≤∑σ=1Nvσ{∑i=1r∑j>irμiμjxT(k)(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)∑i=1rμi2[x(k)x(k-1)]T×[GσiiTPGσii-P+SGσiiTPHσiiHσiiTPGσiiHσiiTPHσii-S][x(k)x(k-1)]+2∑i=1r∑j>irμiμj[x(k)x(k-1)]T[[ασij2]TPασij2-P+S[ασij2]TPβσij2[βσij2]TPασij2[βσij2]TPβσij2-S][x(k)x(k-1)]+∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)∑i=1r∑j>irμiμjxT(k)(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)∑i=1rμi2[x(k)x(k-1)]T}=∑σ=1Nvσ{∑i=1rμi2[x(k)x(k-1)]T×[GσiiTPGσii-P+S+Q+KσiTRKσiGσiiTPHσiiHσiiTPGσiiHσiiTPHσii-S][x(k)x(k-1)]+2∑i=1r∑j>irμiμj[x(k)x(k-1)]T×[[ασij2]TPασij2-P+S+Q+(Kσi+Kσj)T2R(Kσi+Kσj)2[ασij2]TPβσij2[βσij2]TPασij2[βσij2]TPβσij2-S]×[x(k)x(k-1)]∑i=1rμi2[x(k)x(k-1)]T∑i=1rμi2}.
Thus, if the matrix inequalities (26) and (27) hold(26)[GσiiTPGσii-P+S+Q+KσiTRKσiGσiiTPHσiiHσiiTPGσiiHσiiTPHσii-S]<0,1≤i≤r,(27)[[ασij2]TPασij2-P+S+Q+(Kσi+Kσj)T2R(Kσi+Kσj)2[ασij2]TPβσij2[βσij2]TPασij2[βσij2]TPβσij2-S]<0,1≤i<j≤r,then, the following inequality can hold:
(28)ΔV(x(k))+xT(k)Qx(k)+uσT(k)Ruσ(k)<0,
and (26) is the special case of (27) when i=j. Thus, when (27) holds, the inequality (28) can hold. Consider(29)θσij=[[ασij2]TPασij2-P+S+Q+(Kσi+Kσj)T2R(Kσi+Kσj)2[ασij2]TPβσij2[βσij2]TPασij2[βσij2]TPβσij2-S]<0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1≤i≤j≤r.
Let(30)Ωσij=-4P+4S+4Q+(Kσi+Kσj)TR(Kσi+Kσj),θσij=[Ωσij+ασiiTPασijασiiTPβσijβσiiTPασijβσiiTPβσij-4S]<0,ffffffffffffffffffffffffffffffffff1≤i≤j≤r.
Therefore, according to the fact of μi(θ(k))≥0 in (8), the inequality (30) and the equality (24), we can obtain ΔV<0.
Define setsΩσ={y∈R2n:yTθσijy<0}; thus ∪Ωi=R2n∖{0}. Construct the sets Ω~1=Ω1,…,Ω~σ=Ωσ-⋃i=1σ-1Ω~i,….
Obviously, we have⋃i=1lΩ~i=R2n∖{0}, Ω~i∩Ω~j=Φ, i≠j.
Construct a switching law as follows:(31)σ(x(k))=i,whenx(k)∈Ω~i,σ∈M.
Thus, ΔV(x(k))+xT(k)Qx(k)+uσT(k)Ruσ(k)≤ξT(k)θσijξ(k)<0, ∀ξ(k)≠0, where ξ(k)=[xT(k)xT(k-1)]T.
Notice thatFTF≤I. Applying Lemmas 6 and 7 to the inequality (29), we have(32)[ΨΛΠ0000*-2P+2S02I(kσi+kσj)T(Eikσj)T(Ejkσi)T**-2S00-(Eikσj)T(Ejkσi)T***-2Q-1000****-2R-100*****-εσI0******-εσI]<0,where
(33)Ψ=-2P-1+εσ(DiDiT+DjDjT),Λ=Φi+Φj+Γi0kσj+Γj0kσi,Π=Γi1kσj+Γj1kσi.
By inequality (32) left-multiplied and right-multiplied by diag⁡(IP-1P-1IIII) and defining new variables X=P-1, Yσi=kσiX, and Z=XSX, we have(34)[LUV0000*-2X+2Z02X(Yσi+Yσj)T(EiYσj)T(EjYσi)T**-2Z00-(EiYσj)T(EjYσi)T***-2Q-1000****-2R-100*****-εσI0******-εσI]<0,where
(35)L=-2X+εσ(DiDiT+DjDjT),U=Φi+Φj+Γi0Yσj+Γj0Yσi,V=Γi1Yσj+Γj1Yσi.
Taking (28) into account, for all admissible uncertainties, we can infer that as follows:
(36)ΔV(x(k))≤-xT(k)Qx(k)-uσT(k)Ruσ(k).
According to (18), we can infer that there exists at least one σ such that (29) is established. Therefore, the closed-loop system (14) is asymptotically stable under the controller (11) and the switching law (31). Moreover, we have
(37)ΔV<∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)dd×(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k),
thus
(38)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)×(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)<-ΔV.
The inequalities (38) are added up together in the case that k is 0,1,2,…,∞; we have
(39)J=∑k=0∞[(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)×(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)]≤xT(0)Px(0)+xT(-1)Sx(-1).
Therefore, Theorem 8 is proved.Theorem 9.
Consider the uncertain nonlinear networked control systems (3) and the cost function (9). If there exist some nonpositive or nonnegative constants δσλ(σ,λ=1,2,…,l), a group of positive constants εσ>0, and positive-definite matrices Xσ, Zσ, matrix Yσi,  σ=1,…,l,  i=1,…,r, such that the following matrix inequalities:(40)[LUV0000*Ξ+∑λ=1,λ≠σlδσλ(Xλ-Xσ)02X(Yσi+Yσj)T(EiYσj)T(EjYσi)T**-2Z00-(EiYσj)T-(EjYσi)T***-2Q-1000****-2R-100*****-εσI0******-εσI]<0,1≤i≤j≤r,hold, where
(41)Ξ=-2Xσ+2Zσ,Xσ=Pσ-1,Yσi=kσiXσ,Zσ=XσSσXσ,L=-2Xσ+εσ(DiDiT+DjDjT),U=Φi+Φj+Γi0Yσj+Γj0Yσi,V=Γi1Yσj+Γj1Yσi,
then system (14) with the guaranteed cost controller (11) and the switching law σ=σ(x(k)) is globally asymptotically stable. The guaranteed cost function (9) satisfies the following bound:
(42)J≤xT(0)Pσx(0)+xT(-1)Sσx(-1).Proof.
By the Lemmas6 and 7, and Xσ=Pσ-1, Yσi=kσiXσ, Zσ=XσSσXσ, from (40), we have(43)[[ασij2]TPσασij2-Pσ+Sσ+Q+(Kσi+Kσj)T2R(Kσi+Kσj)2+∑λ=1,λ≠σlδσλ(Pλ-Pσ)[ασij2]TPσβσij2[βσij2]TPσασij2[βσij2]TPσβσij2-Sσ]<0.
Without loss of generality, we assume thatδσλ≥0.
Obviously, there exists at least oneσ∈M when x(k)∈Rn∖{0}, such that xT(k)(Pλ-Pσ)x(k)≥0, ∀λ∈M.
LetΩσ={x(k)∈Rn∣xT(k)(Pλ-Pσ)x(k)≥0,∀λ∈M,λ≠σ,x(k)≠0}; thus ∪Ωi=Rn∖{0}.
Construct the setsΩ~1=Ω1,…,Ω~σ=Ωσ-⋃i=1σ-1Ω~i,….
Obviously, we have⋃i=1lΩ~i=Rn∖{0}, Ω~i∩Ω~j=Φ, i≠j.
Construct a switched law by(44)σ(x(k))=i,whenx(k)∈Ω~i,σ∈M.
Thus, from (43), we have(45)[x(k)x(k-1)]T[[ασij2]TPσασij2-Pσ+Sσ+Q+(Kσi+Kσj)T2R(Kσi+Kσj)2[ασij2]TPσβσij2[βσij2]TPσασij2[βσij2]TPσβσij2-S][x(k)x(k-1)]<0,1≤i≤j≤r.
Following the similar lines as in the proof of Theorem8, we have(46)θσij=[[ασij2]TPσασij2-Pσ+Sσ+Q+(Kσi+Kσj)T2R(Kσi+Kσj)2[ασij2]TPσβσij2[βσij2]TPσασij2[βσij2]TPσβσij2-S]<0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1≤i≤j≤r,then, ΔVσ(x(k))+xT(k)Qx(k)+uσT(k)Ruσ(k)<0. Thus,
(47)∑i=1rμi2xT(k)(Q+KσiTRKσi)x(k)+2∑i=1r∑j>irμiμjxT(k)(Q+(Kσi+Kσj)T2R(Kσi+Kσj)2)x(k)<-ΔVσ.
The above inequality (47) is added up together in the case that k is 0,1,2,…,∞, according to the stability of the system, then
(48)J≤xT(0)Pσx(0)+xT(-1)Sσx(-1).
Theorem 9 is proved.


## 5. Simulation Example

Consider the nonlinear system with the following differential equation [24]:
(49)s¨(t)+f(s(t),s˙(t))-0.1s(t)=F(t),
where f(s(t),s˙(t))=0.5s(t)+0.75sin(s˙(t)/0.5).Choose the state variable and the input variable asx(t)=[s(t),s˙(t)]T, u(t)=F(t), respectively. It can be represented by the following fuzzy model consisting of two rules:
(50)R1:IFx2(t)isM1,THENx˙(t)=A1x(t)+B1u(t),R2:IFx2(t)isM2,THENx˙(t)=A2x(t)+B2u(t),
where x(t)=[x1(t),x2(t)]T(51)A1=[010.1-2],A2=[010.1-0.5-1.5β],B1=B2=[-0.1-0.2]T,β=0.01π,
and β is used to avoid system matrices being singular.The sampling periodT=0.3s, then the discrete model of the system is as follows:
(52)R1:IFx2(t)0.5isabout0,THENx(k+1)=Φ1x(k)+(Γ10+D1F1(τk)E1)u(k)THENx(k+1)=+(Γ11-D1F1(τk)E1)u(k-1),R2:IFx2(t)0.5isaboutπor-π,THENx(k+1)=Φ2x(k)+(Γ20+D2F2(τk)E2)u(k)THENx(k+1)=+(Γ21-D2F2(τk)E2)u(k-1),
where
(53)Φ1=[1.00370.22590.02260.5519],Φ2=[1.00430.27880.02790.8635],Γ10=[-10.0018-0.0001],Γ11=[-10.0390-0.2260],Γ20=[-100],Γ21=[-10.0432-0.2788],D1=[20.76930.21411.0137-0.4386],D2=[6.86910.66760.3353-1.3678],E1=[0.47731.0869],E2=[1.25001.4786],F1(τk)=[e-0.0488τk00e-2.0488(0.3-τk)],F2(τk)=[e-0.1522τk00e-0.6570(0.3-τk)].
Obviously, F1(τk) and F2(τk) satisfy the uncertain matching conditions:
(54)F1T(τk)F1(τk)≤I,F2T(τk)F2(τk)≤I.
The membership functions of “about 0” and “about π or -π” are selected as in Figure 2.Figure 2
The membership function of inputx2.Suppose the switched fuzzy feed-back controllers are the following fuzzy controllers:(55)u1(k)=∑i=12μi(x2(k))k1ixi(k),u2(k)=∑i=12μi(x2(k))k2ixi(k).ChooseQ=0.1I2×2, R=1. Carrying out computations for matrices inequality (40), we obtain
(56)P1=[8.01775.23625.23628.5369],P2=[7.89265.40265.40269.1487].
The controller gain:
(57)k11=[-0.6138-0.4204],k12=[-0.2936,-0.3036],k21=[-0.4156-0.3852],k22=[-0.0893,-0.1025].
Let
(58)Ω1={x(k)∈R2∣xT(k)(P2-P1)x(k)≥0,x(k)≠0},Ω2={x(k)∈R2∣xT(k)(P1-P2)x(k)≥0,x(k)≠0}.
Then Ω1∪Ω2=R2∖{0}. We design a switched law as follows:
(59)σ(x(k))={1,x(k)∈Ω1,2,x(k)∈Ω2∖Ω1.
The initial condition is [1.80.5]T, guaranteed cost bound J*=36.53.Figures3 and 4 show the system state trajectories that use fuzzy controller 1 and fuzzy controller 2, respectively. Figure 5 shows the simulating results for the proposed switched fuzzy controller method. In Figure 5, the system state trajectories indicate that nonlinear networked control system is asymptotically stable and satisfies the performance index via the designed guaranteed cost controller and the switching law. From the simulating results, we can confirm that the guaranteed cost controller in the switched fuzzy model is able to stabilize the nonlinear delay system via switching T-S fuzzy method. In addition, the performance of switching fuzzy controller is better than that of the fuzzy controllers.Figure 3
The state trajectory using fuzzy controller 1.Figure 4
The state trajectory using fuzzy controller 2.Figure 5
The state trajectory using switched fuzzy controller.


## 6. Conclusions

In this paper, we have presented a novel controller design methodology for a class of nonlinear NCSs based on switched T-S fuzzy model. By introducing the switching mechanism into the fuzzy T-S systems, the proposed methods can deal with the uncertainties of nonlinear NCSs with time delays and furthermore avoid the inherent drawback of a fuzzy T-S model in controller design and implementation of nonlinear systems. In addition, considering QoS of nonlinear NCSs, some sufficient conditions for the existence of the robust guaranteed cost control law have been built via Lyapunov functional approach. Simulation results have verified and confirmed the effectiveness of the guaranteed cost controller based on the switched T-S fuzzy model for nonlinear NCSs.At present, this paper only presents a numerical example to show the validity of our control scheme on the nonlinear NCS with time delays. In next step, we plan to further verify this control scheme via practical NCSs and investigate the stability analysis and controller design with multiple-packet transmission in nonlinear NCSs. Moreover, the boundedness of the parameter constraints for NCSs will be studied. The switched dynamics of nonlinear NCSs will also be considered in future investigation.



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*Source: 539617-2014-02-09.xml*