# Robust Constrained Model Predictive Control for T-S Fuzzy Uncertain System with Data Loss and Data Quantization

Authors: Hongchun Qu; Yu Li; Wei Liu
Journal: Complexity
 (2021)

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## Abstract

This paper addresses the robust constrained model predictive control (MPC) for Takagi-Sugeno (T-S) fuzzy uncertain quantized system with random data loss. To deal with the quantization error and the data loss over the networks, the sector bound approach and the Bernoulli process are introduced, respectively. The fuzzy controller and new conditions for stability, which are written as the form of linear matrix inequality (LMI), are presented based on nonparallel distributed compensation (non-PDC) control law and an extended nonquadratic Lyapunov function, respectively. In addition, slack and collection matrices are provided for reducing the conservativeness. Based on the obtained stability results, a model predictive controller which explicitly considers the input and state constraints is synthesized by minimizing an upper bound of the worst-case infinite horizon quadratic cost function. The developed MPC algorithm can guarantee the recursive feasibility of the optimization problem and the stability of closed-loop system simultaneously. Finally, the simulation example is given to illustrate the effectiveness of the proposed technique.


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## Body

## 1. Introduction

Model predictive control (MPC) is an effective advanced control method that has aroused extensive attention among the academic and industrial communities in the past decades [1–3]. The main idea of MPC is online solution of an optimization problem to obtain a sequence of optimal control inputs; then, only the first one is implemented and it repeats this procedure with new measurements at the next sampling time [4]. The defining feature of MPC comes from its ability to handle the multivariable systems with hard constraints. A large amount of results on MPC of ideal systems without model uncertainties were reported in [5–7]. However, model uncertainty exists in practical industrials, and hence robust MPC is of theoretical and practical significance and need to be considered. To solve this problem, the authors in [8] firstly proposed a robust MPC synthesis approach for the uncertain system by solving a linear matrix inequalities (LMIs) optimization problem which explicitly considered the input and state constraints. The key point of [8] was to utilize the concept of robust invariant set to guarantee the recursive feasibility of the LMI optimization problem. Since then, much progress has been made in the research of MPC synthesis approach. There are mainly two aspects: one is to reduce the computational burden (see [9, 10]) and the other is to improve the control performance (see [11]).Meanwhile, networked control systems (NCSs) have been a hot research topic in the last decade which combines sensors, controllers, and actuators via a shared communication network. Compared with traditional point-to-point control systems, NCSs bring incomparable advantages, such as high reliability, easy installation and maintenance, and excellent flexibility [12, 13]. However, some phenomenons, such as data loss and quantization, inevitably exist in the communication channels which would deteriorate the performance or even destroy the stability of the systems. There have emerged many works about the controller design and stability analysis for the NCSs with packet loss and/or quantization (see, e.g., [14–16]). Works on the stabilization problem for NCSs with packet loss can be found in [17–20]. Some nice results about the quantized feedback control problem for NCSs were reported in [21, 22]. Further, most recently, some researchers considered more complex network environment where packet loss and quantization were coexisted in NCSs [23–27].Some interesting results are presented in the aforementioned documents; however, all of the results are under the assumption that the controlled plant is a linear system without system uncertainty. Unfortunately, almost all the systems in industry are nonlinear with uncertainty. The research studies on the uncertain system have been made by many researchers, and several common descriptions of uncertain systems can be found in [8, 28, 29], such as polytopic paradigm and structured uncertainty. In addition, Takagi–Sugeno (T-S) fuzzy model is employed in this paper to describe the nonlinear system. As we all know that the T-S fuzzy model can transform the nonlinear system into a set of linear submodels by IF-THEN rules instead of handling it directly. In the analysis of T-S fuzzy systems, the controller is designed based on the linear submodels, while the implementation of the controller clearly depends on the membership functions which explicitly consider the nonlinearity of the system. The local linearities of the T-S fuzzy system provide a bridge between the analysis of nonlinear systems and the fruitful linear control theory results [30–35]. For the T-S fuzzy system, some interesting technologies are introduced for the purpose of reducing conservatism, such as nonquadratic Lyapunov function [36, 37], extended nonquadratic Lyapunov function [38, 39] with nonparallel distributed compensation (non-PDC) law, and slack matrices [40]. Recently, some researchers begin applying the T-S fuzzy model to describe complex nonlinear plant in NCSs, and more investigations have been carried out. The controller design and stability analysis for T-S fuzzy model-based NCSs with data losses were considered in [41, 42], and for NCSs with quantization, the same problems were addressed in [44, 45].There also have been some nice works that introduce MPC to deal with the problem for NCSs represented by the T-S fuzzy system (see, e.g., [43, 46, 47]). The authors in [43] used Bernoulli random binary distribution to model data loss occurring intermittently between the controller and the physical plant. Meanwhile, piecewise approach is more effective for fuzzy systems with trapezoidal membership functions. Consequently, the fuzzy predictive control problem of nonlinear NCSs subject to parameter uncertainties and data loss was considered in [46]. The difference between [43, 46] is that the authors in [46] employed slack matrices to develop less conservative conditions for stability analysis of the closed-loop system. Yu et al. [47] considered more intricate network where data loss and quantization were existed simultaneously. In [48], a fuzzy predictive controller which guaranteed the stability of the closed-loop system was designed in terms of sector bound approach. Tang et al. [49] investigated the output feedback model predictive control for networked control systems with packet loss and data quantization. Although some remarkable results are presented in the foregoing documents, most of the T-S fuzzy systems are described without uncertainty. In addition, the same problem for NCSs, packet loss, data quantization, and the model predictive control strategy, were considered in both of this paper and [49]. However, this paper is different from [49]. First, the interval-type 2 T-S fuzzy model is introduced to describe the nonlinear NCSs, while the type 1 T-S fuzzy model is applied in this paper and the system uncertainty is also considered which is not involved in [49]; second, the result for [49] was obtained based on the assumption that the system states were not measurable and the output feedback MPC strategy was addressed, while this paper assumed that the system states can be sampled and state feedback MPC approach was investigated. It is noted that different assumptions resulted in completely different solutions. Indeed, few works are focused on the problem of reducing the conservatism for fuzzy MPC for NCSs which motivates our research. To illustrate this research more clearly, we provide Table 1 to compare published works with this paper.Table 1
Comparisons of published papers and this paper.
[13][21][22][25][31][33][35][40][45]This paperData lossYesNoNoYesNoNoNoYesNoYesData quantizationNoYesYesYesNoNoNoNoYesYesUncertaintiesNoNoYesNoNoYesYesNoYesYesT-S fuzzy modelNoNoNoNoYesYesYesYesYesYesNon-PDC control lawNoNoNoNoNoNoYesNoNoYesThis paper focuses on the synthesis approach of the robust constrained MPC with quantization and data loss for the uncertain fuzzy system. The quantization error is regarded as sector bound uncertainties by using the sector bound approach, and the data loss process is described by Bernoulli distributed sequence. A new stability condition for NCSs is achieved based on extended nonquadratic Lyapunov function and additional slack and collection matrices. By optimizing an objective function in the infinite time horizon at each sampling time, a robust MPC optimization problem which explicitly considers the data loss and quantization is presented. Furthermore, the optimization problem is proved to be recursive feasible, and the networked control system turns out to be asymptotically stable.The remainder of this paper is summarized as follows. The problem formulation is given in Section2. Section 3 provides the new stability results for fuzzy NCSs. Online synthesis of MPC for fuzzy NCSs is presented in Section 4. Section 5 gives the analysis of feasibility and stability. Section 6 presents a simulation example, and Section 7 draws the conclusion.Notations. For any vector x and positive-definite matrix W, xW2=xTWx. Rn and Rn×m denote n-dimensional Euclidean space and the set of n×m real matrices, respectively. I is the identity matrix with appropriate dimension. For any vector, xk+s|k is the value of x at sampling instant k+s. The superscript T denotes the transpose of matrix or vector. The symbol ∗ stands for a symmetric element or submatrix. For simplicity, μt,t∈i,j and μz+,z∈l,m are used in place of μtfk, μzfk+1 and μtfk+s|k, μzfk+s+1|k, respectively. χh=∑i=1rμi⋅χi, χh+=∑l=1rμi⋅χl, χhh=∑i=1r∑j=1rμi⋅μj⋅χij, and χhh+=∑l=1r∑m=1rμl⋅μm⋅χlm, with χ∈A,B,E,K,Q,S.


## 2. Problem Formulation

Consider a discrete-time T-S fuzzy model withr rules and describe its ith rule as(1)Rulei:IFf1kisξ1i,f2kisξ2i,…,andfθkisξθi,THENxk+1=Ai+ΔAixk+Bi+ΔBiuk,i=1,2,…,r,where ξgig=1,2,…,θ stands for the fuzzy set of rule i and fk=f1k,…,fgk is the corresponding premise variable which depends on the states of the system; xk∈ℜn and uk∈ℜm denote the system state and control input, respectively; Ai∈ℜn×n and Bi∈ℜn×m are known matrices; and ΔAi and ΔBi are introduced to describe the system uncertainty, which satisfy(2)ΔAiΔBi∈Ω≔DFkE1iDFkE2iFkTFk≤I,i=1,2,…,r,k≥0,where Fk is an uncertain matrix and D, E1i, and E2i are known matrices.According to the above discussion, the definition of the inferred T-S fuzzy model can be represented as follows:(3)xk+1=∑i=1rμifkAi+ΔAixk+Bi+ΔBiuk,where(4)μifk=ϖifk∑i=1rϖifk,ϖifk=∏g=1θξgifgk,i∈1,2,3…,r,in which ϖixk represents the weight of the ith rule and μifk is the membership grade of fk in ξji and satisfies(5)μifk≥0,∑i=1rμifk=1,i∈1,2,3…,r.In this paper, we assume that the communication network exists in sensor-to-controller (S/C) and controller-to-actuator (C/A) links, and the configuration of quantized NCSs with data loss is shown in Figure1. At time k, the output of controller is quantized as(6)uck=φvk,where φ⋅ denotes a logarithmic quantizer which is defined as(7)φv=ɛi,if11+τɛi<v≤11−τɛi,v>0,0,ifv=0,−φ−v,ifv<0,in which τ=1−ρ/1+ρ and ρ is the quantization density. The set of quantization levels is presented as follows: ν=±ɛi,ɛi=ρiɛ0,i=±1,±2,…∪±ɛ0∪0, 0<ρ<1, ɛ0>0. Each quantization level corresponds to a segment such that the quantizer maps the whole segment. Based on the sector bound approach in [48], the control input can be expressed as follows:(8)uck=φvk=I+ςkvk,ςk∈−τ,τ,where ςk=diagς1k,ς2k,…,ςmk,ςik≤τi.Figure 1
Configuration of quantized NCSs with data loss.Whenxk and uck are transmitted in the communication channel, it may be lost due to the poor network traffic. Hence, two Bernoulli processes are introduced to model the data loss in NCSs.(9)xck=αkxk,uk=βkuck,where xck is input of controller. αk and βk are two mutually independent Bernoulli processes. αk,βk∈0,1 (1 means that the data are transmitted successfully and 0 indicates that data packet is lost).Then, we assume thatαk and βk satisfy the following conditions:(10)Prαk=1=Eαk=α¯,Prαk=0=1−α¯,Prβk=1=Eβk=β¯,Prβk=0=1−β¯.In the following, the fuzzy controller is presented.(11)RuleRi:IFf1xkisξ1i,f2xkisξ2i,…,andfθxkisξθi,THENvk=KiQi−1xck,i∈1,2,…,r,where Ki∈ℜm×nin whichi∈1,2,…,r and Qi∈ℜn×nin whichi∈1,2,…,r are the controller gains. Then, the state feedback fuzzy controller is defined by(12)vk=∑i=1rμifkKi∑i=1rμifkQi−1xck=KhQh−1xck,i∈1,2,…,r.Based on (9) and (12), the following state feedback fuzzy controller is achieved:(13)uk=I+ςkαkβkKhQh−1xk,i∈1,2,…,r.Letωk=αkβk. It is obvious that if αk=1 and βk=1, ωk=1; otherwise, ωk=0. Thus, we have(14)Prωk=1=Eωk=ω¯,Prωk=0=1−ω¯.According to (3), (8), and (13), the overall uncertain closed-loop fuzzy system is described as follows:(15)xk+1=∑i=1rμifkAi+ΔAi+Bi+ΔBiI+ςkωkKhQh−1xk=A˜h+ω¯B˜hI+ςkKjQj−1+ω˜kB˜hI+ςkKjQj−1xk,where Ah=∑i=1rμifkAi, ΔAh=∑i=1rμifkΔAi, Bh=∑i=1rμifkBi, ΔBh=∑i=1rμifkΔBi, Kj=∑i=1rμifkKh, and Qj=∑i=1rμifkQh. A˜h=Ah+ΔAh, B˜h=Bh+ΔBh, and ω˜k=ωk−ω¯. Obviously, Eω˜k=0 and Eω˜kω˜k=ω¯1−ω¯.


## 3. New Stability Results for Fuzzy NCSs

Compared with the conditions of stability for traditional control fuzzy systems, a new stability condition which can reduce the conservatism is presented in this section by introducing the notation of slack matrices. This new stability condition is the foundation of the MPC for fuzzy NCSs which will be provided in the next section.First, the following two lemmas that play a significant role in the proof of Theorem1 are introduced.Lemma 1 (see [50]).
For vectorsa and b and matrices N, X, Y, and Z of any appropriate dimensions, if XYYTZ≥0, then the following inequality holds:(16)−2aTNb≤abTXY−NYT−NTZab.
(17)−2aTNb≤abT0−N−NT0ab≤abT0−N−NT0ab+abTXYYTZab≤abTXY−NYT−NTZab.This inequality is generally called Moon inequality. It can be seen that ifN=Y=I and Z=X−1, then −2aTb≤aTXa+bTX−1b. For any w,z∈Rn and any positive-definite matrix c∈Rn×n, the following inequality holds:(18)2wTz≤wTc−1w+zTcz.Lemma 2 (see [51]).
Symmetric matrixL1 and matrices L2andL3 satisfy the following inequality:(19)L1+L2FkL3+L3TFkTL2T<0,for FkTFk≤I, if there exists a scalar η>0 such that(20)L1+ηL2L2T+η−1L3TL3<0.Next, the following theorem is given to obtain the conditions of the stability for NCSs.Theorem 1.
Considering the T-S fuzzy system (3), the asymptotic stability is guaranteed if there exist scalars ηijandη¯ij, matrices Kj, Qj, and Y, symmetric matrices Giil and Gijl, and Si>0,Sj>0 such that(21)Ψiil≤−Giil,i,l∈1,2,…,r,(22)Ψijl+Ψjil≤−Gijl,j>i,i,j,l∈1,2,…,r,(23)2G11l∗⋯∗G12l2G22l⋯∗⋮⋮⋱∗G1rl⋯Gr−1rl2Grrl≥0,l∈1,2,…,r,where(24)Ψijl=d¯1∗∗d¯2d¯3∗d¯4d¯5d¯6,i,j,l∈1,2,…,r,d¯1=Si−Qj−QjT∗∗AiQj+ω¯BiKj−Sl++ω¯2BiτYτBiT∗κBiKjκω¯BiτYτBiT−Sl++κ2BiτYτBiT,d¯2=E1iQj+ω¯E2iKjω¯2E2iτYτBiTω¯κE2iτYτBiT0ηijDT0κE2iQjKjω¯κE2iτYτBiTκ2E2iτYτBiT,d¯3=−ηijI+ω¯2E2iτYτE2iT∗∗0−ηijI∗ω¯κE2iτYτE2iT0−η¯ijI+κ2E2iτYτE2iT,d¯4=00−η¯ijDTKj00,d¯5=000000,d¯6=−η¯ijI∗0−Y.Proof.
See AppendixA for details.


## 4. Online Synthesis of MPC for Fuzzy NCSs

Based on the stability condition achieved in the above section, the online synthesis of MPC is derived for fuzzy NCSs with the quantization and data loss. Furthermore, the model uncertainty and physical constraints are considered.The following non-PDC fuzzy predictive controller is used for the T-S fuzzy system:(25)uk+s|k=αkβkI+ςk∑i=1rμifk+s|kKi×∑i=1rμifk+s|kQi−1xk+s|k=ωkI+ςkKhQh−1xk+s|k.In view of (25), we can get the predictive model of uncertain T-S fuzzy NCSs:(26)xk+s+1|k=A˜h+ω¯B˜hI+ςkKhQh−1+ω˜kB˜hI+ςkKhQh−1xk+s|k,where A˜h=Ah+ΔAh,B˜h=Bh+ΔBh, and ω˜k=ωk−ω¯.The following is the optimization problem for the online constrained robust fuzzy MPC, which is optimized in the infinite time horizon at each time:(27)minuk+s|k,s≥0maxςk,ΔAik+sΔBik+s∈Ω,i∈1,2,…,r,s≥0J0∞k,where J0∞k=∑s=0∞Exk+s|kTWxk+s|k+uk+s|kTRuk+s|k, with W>0 and R>0 the weighting matrices. It is obvious that this objective function is related to the model uncertainty, quantization density, and expected values of stochastic variables.The following extended nonquadratic Lyapunov function is considered:(28)Vxk+s|k=xk+s|kT∑i=1r∑j=1rμifk+s|kμjfk+s|kSij−1×xk+s|k=xk+s|kTShh−1xk+s|k,Sij>0,s≥0.Then, imposing the following stability constraint at each time,(29)EVxk+s+1|k−Vxk+s|k≤−Exk+s|kTWxk+s|k+uk+s|kTRuk+s|k.Summing up both sides of (29) from s=0 to s=∞, we have(30)EVxk+∞|k−Vxk|k≤−E∑s=0∞xk+s|kTWxk+s|k+uk+s|kTRuk+s|k.As we all know, whens⟶∞, Exk+s|k=0. Then, we can obtain lims⟶∞EVxk+s|k=0. Thus, J0∞k≤EVxk|k, which is equal to(31)maxΔAik+sΔBik+s∈Ω,i∈1,2,…,rJ0∞k≤∑i=1r∑j=1rμiξk|kμjξk|kxk|kSij−12,where s≥0.Hence, the problem of minimizingJ0∞ is transformed into minimizing a scalar γ satisfying(32)xk|kT∑i=1r∑j=1rμiξk|kμjξk|kSij−1xk|k≤γ.Moreover, input and state constraints are(33)Euak+s|k2≤u¯,a=1,2,…,nu,(34)Eζxk+s+1|ka2≤x¯,a=1,2,…,nx,where uak+s|k and xak+s+1|k are the ath component of the vector uk+s|k and xk+s+1|k, respectively.Can be solved by the following theorem.Theorem 2.
Consider the uncertain fuzzy NCS (26) and assume that the communication link parameter ω¯ and quantization density ρ are known. The non-PDC state feedback control law (25) can guarantee the robust asymptotically stability of the closed-loop fuzzy system if there exist scalars γ, ηij, η¯ij, ηijlm, and η¯ijlm, positive-definite matrices S˜ij, S˜lm, and Gijlm, and any matrices Kj, Qj, Gijlm=GjilmT, and Ulm=UmlTm>l such that the following minimization problem is feasible:(35)minγ,ηij,ηijlm,η¯ij,η¯ijlm,S˜ij,S˜lm,Giilm,Gijlm,Qj,Kj,Ull,Ulm,N,O,Tγ,subject to(36)1∗xk|kS˜ij≥0,i,j∈1,2,…,r,(37)Ψiilm≤−Giilm,i,l,m∈1,2,…,r,(38)Ψijlm+Ψjilm≤−Gijlm−Gjilm,j>i,i,j,l,m∈1,2,…,r,(39)ϕll≥0,l∈1,2,…,r,(40)ϕlm+ϕml≥0,m>l,l,m∈1,2,…,r,where(41)Ψijlm=m1∗∗m2m3∗m4m5m6,(42)ϕlm=G11lm−UlmG12lm−Ulm⋯G1rlm−UlmG21lm−UlmG22lm−Ulm⋯G2rlm−Ulm⋮⋮⋱⋮Gr1lm−Ulm⋯⋯Grrlm−Ulm,l,m∈1,2,…,r,(43)m1=m¯1∗∗AiQj+ω¯BiKjm¯2∗κBiKjω¯κBiτNτBiTm¯3,m2=E1iQj+ω¯E2iKjω¯2E2iτNτBiTω¯κE2iτNτBiT0ηijlmDT0κE2iKjω¯κE2iτNτBiTκ2E2iτNτBiT,m3=m¯4∗∗0−ηijlmI∗ω¯κE2iτNτE2iT0m¯5,m4=00η¯ijlmDTW1/2Qj00ϱ¯Kjω¯ϱ¯τNτBiTκϱ¯τNτBiTKj00,m5=000000ω¯ϱ¯τNτE2iT0κϱ¯τNτE2iT000,m6=−η¯ijlmI∗∗∗0−γI∗∗00−γI+ϱ¯2τNτ∗000−N,m¯1=S˜ij−Qj−QjT,m¯2=−S˜lm++ω¯2BiτNτBiT,m¯3=−S˜lm++κ2BiτNτBiT,m¯4=−ηijlmI+ω¯2E2iτNτE2iT,m¯5=−η¯ijlmI+κ2E2iτNτE2iT,U11U12⋯U1rU21U22⋯U2r⋮⋮⋱⋮Ur1Ur2⋯Urr≥0.Proof.
See AppendixB for details.Then, we handle the input constraint (33). For any ΔAiΔBi∈Ω,i∈1,2,…,r, we have(44)maxs≥0Euak+s|k2=maxs≥0Eω¯+ω−ω¯I+ςkKhQh−1xk+s|ka2≤maxxk+s|k∈εS˜hhω¯I+ςkKhQh−1S˜hh1/2a22S˜hh−1/2xk+s|ka22≤ω¯I+ςkKhQh−1S˜hh1/2a22,a∈1,2,…,nu.If there exists a symmetric matrixX that satisfies(45)X−ω¯2I+ςkKhQh−1S˜hhI+ςkKhQh−1T≥0Xaa≤ua,max2,a∈1,2,…,nu.By applying the Schur complement andQhTSh−1Qh≥Qh+QhT−Sh [52], (45) becomes(46)S˜ij−Qj−QjT∗ω¯I+ςkKj−X≤0,Xaa≤ua,max2,a∈1,2,…,nu,which equals to(47)S˜ij−Qj−QjT∗ω¯Kj−X+0ω¯ςkKj0+KjT0ςk0ω¯≤0.Using Lemma2 and Schur complement, it yields(48)S˜ij−Qj−QjT∗∗ω¯Kj−X+ω¯τOτω¯∗Kj0−O≤0,where O is a positive-definite matrix. In consequence, (33) is guaranteed by (49) through the convex feature of μi and μj.In addition, we deal with the state constraint (34). For any ΔAiΔBi∈Ω,i∈1,2,…,r, we can achieve(49)maxs≥0Eζxk+s+1|ka2=maxs≥0EζaA˜h+ω¯B˜hI+ςkKhQh−1+ω˜k+s|k×B˜hI+ςkKhQh−1xk+s|ka2≤maxxk+s|k∈εS˜hhζaA˜h+ω¯B˜hI+ςkKhQh−1S˜hh1/2a22S˜hh−1/2xk+s|ka22≤ζaA˜h+ω¯B˜hI+ςkKhQh−1S˜hh1/2a22.If there exists a symmetric matrixZ that satisfies(50)Z−ζA˜hQh+ω¯B˜hI+ςkKhQh−1S˜hhQh−TA˜hQh+ω¯B˜hI+ςkKhTζT≥0Zaa≤xa,max2,a∈1,2,…,nx.Using Lemmas1 and 2 and Schur complement, (50) holds if(51)S˜ij−Qj−QjT∗∗∗ζAiQj+ω¯BiI+ςkKj−Z∗∗E1iQj+ω¯E2iI+ςkKj0−ηijI∗0ηijDTζT0−ηijI≤0,Zaa≤xa,max2,a∈1,2,…,nx,for all Fk+s|k satisfying Fk+s|kTFk+s|k≤I. Equation (52) can be transformed into(52)S˜ij−Qj−QjT∗∗∗ζAiQj+ω¯BiKj−Z∗∗E1iQj+ω¯E2iKj0−ηijI∗0ηijDTζT0−ηijI+0ζω¯Biω¯E2i0ςkKjT000T+Kj000Tςk0ζω¯BiTω¯E2iT0≤0.According to Lemma1 and utilizing Schur complement, the following inequality is obtained:(53)S˜ij−Qj−QjT∗∗∗∗ζAiQj+ω¯BiKjm18∗∗∗E1iQj+ω¯E2iKjω¯E2iτTτζω¯BiT−ηijI+ω¯E2iτTτω¯E2iT∗∗0ηijDTζT0−ηijI∗Kj000−T≤0,where m18=−Z+ζω¯BiτTτζω¯BiT and T is a positive-definite matrix. We can conclude that (34) is guaranteed by (53). The proof is completed.Based on the above discussions, the solution of (35) can be obtained by solving the following optimization problem:(54)minγ,Ωγsubject to27−34and44,where Ω=ηij,η¯ij,ηijlm,η¯ijlm,S˜ij,S˜lm,Giilm,Gijlm,Qj,Kj,Ull,Ulm,X,Z,N,O,T. At time instant k≥0, optimization problem (54) is solved online and we can get optimal control sequence uk+s|k,s≥0. Then, only the first control input uk|k is implemented.The implementation steps of designing RMPC are summarized in Algorithm1.Algorithm 1: Robust constrained model predictive control.
(1)
Set the communication link parameterω¯ and quantization density ρ(2)
At timek≥0, solve the optimization problem (54) and get γ,ηij,η¯ij,ηijlm, η¯ijlm,S˜ij,S˜lm,Giilm,Gijlm,Qj,Kj,Ull,Ulm,X,Z,N,O,T(3)
Using (25), calculate the optimal control sequence uk+s|k,s≥0(4)
Quantize and transmit the first control inputuk|k into the communication network(5)
Update time tok+1 and then return to step (2)The complexity of solving the LMI optimization problem is polynomial time, which (regarding the fastest interior-point algorithms) is proportional toK3L, where K is the total number of scalar variables and L is the total row size of the LMI system. For the optimization problem which is described by (54), the total number of scalar variables isK=1+2r2+2r4+nxnx+1r2+rnx2+rnxnu+1/2nxnx+1+2nunu+1+8nx+2nu8nx+2nu+1r4+r2 and the total row size of the LMI system is L=8nx+2nu3/4r4+8nx+2nur3+6nx+3nu+1r2, where r is the number of inference rules. It can be seen that the increase in r,nx, and nu will increase the computational complexity exponentially. Let r=nx=nu and take the most influential parts of K and L, respectively. Then, the computational complexity is 15/2nx24.


## 5. The Analysis of Feasibility and Stability

### 5.1. The Analysis of Feasibility

Theorem 3.
Consider the uncertain fuzzy NCS (26), if any feasible solution is achieved from the optimization problem (55) at time k, then the optimization problem (55) is feasible for all time t>k.Proof.
See AppendixC for details.


### 5.2. The Analysis of Stability

Theorem 4.
Consider the uncertain fuzzy NCS (26). The control input uk, which is applied in a receding horizon way by solving optimization problem (55), robustly asymptotically stabilizes the closed-loop fuzzy NCSs.Proof.
See AppendixD for details.


## 5.1. The Analysis of Feasibility

Theorem 3.
Consider the uncertain fuzzy NCS (26), if any feasible solution is achieved from the optimization problem (55) at time k, then the optimization problem (55) is feasible for all time t>k.Proof.
See AppendixC for details.


## 5.2. The Analysis of Stability

Theorem 4.
Consider the uncertain fuzzy NCS (26). The control input uk, which is applied in a receding horizon way by solving optimization problem (55), robustly asymptotically stabilizes the closed-loop fuzzy NCSs.Proof.
See AppendixD for details.


## 6. Simulation Example

In this section, the following example of a continuous stirred tank reactor (CSTR) is given to illustrate the effectiveness of the proposed method.Example 1.
As shown in Figure2, consider the nonlinear model of CSTR which has been described in [11].
Assuming constant liquid volume, the CSTR liquid volume, the CSTR for an exothermic, irreversible reaction, i.e.,A⟶B, is presented by the following dynamic model based on a component for reactant A and an energy balance:(55)C˙A=qVCAf−CA−k0exp−E/RTCA.T˙=qVTf−T−−ΔHρCpk0exp−E/RTCA+UAVρCpTc−T,where CA, T, and Tc denote the concentration of A in the reactor, the reactor temperature, and temperature of the coolant stream, respectively.
The aim of using CSTR is to regulateT by manipulating Tc, satisfying the constraint Tcl≤Tc≤Tcu. The system state and input variables are set as x=CA−CAeqT−TeqT, u=Tc−Tceq. In addition, let y=x2. In this example, we choose x2 as the premise variable of the fuzzy system and x¯2≤x2≤x¯2, where x¯2=Tl−Teq and x¯2=Tu−Teq. Then, the following equations are defined:(56)g1x2=k0exp−E/Rx2+Teq,g10=g1x¯2+g1x¯22,g2x2=k0exp−E/Rx2+Teq−exp−E/RTeqCAeq1x2,g20=g2x¯2+g2x¯22,ϖ1x2=g1x2−g10,ϖ2x2=g2x2−g20.
The membership functions are described as(57)μ¯1x2=12ϖ1x2−ϖ1−x¯2ϖ1x¯2−ϖ1x¯2,μ¯2x2=12ϖ1x¯2−ϖ1x2ϖ1x¯2−ϖ1x¯2,μ¯3x2=12ϖ2x2−ϖ2x¯2ϖ2x¯2−ϖ2x¯2,μ¯4x2=12ϖ2x¯2−ϖ2x2ϖ2x¯2−ϖ2x¯2.
Thus, (55) can be transformed into the following T-S fuzzy model:(58)A1c=−qV−g10−2ϖ1x¯2−g20−ΔHρCpg10+2−ΔHρCpϖ1x¯2−qV−UAVρCp+−ΔHρCpg20,A2c=−qV−g10−2ϖ1x¯2−g20−ΔHρCpg10+2−ΔHρCpϖ1x¯2−qV−UAVρCp+−ΔHρCpg20,A3c=−qV−g10−g20−2ϖ2x¯2−ΔHρCpg10−qV−UAVρCp+−ΔHρCpg20+2−ΔHρCpϖ2x¯2,A4c=−qV−g10−g20−2ϖ2x¯2−ΔHρCpg10−qV−UAVρCp+−ΔHρCpg20+2−ΔHρCpϖ2x¯2,B1c=B2c=B3c=B4c=0UAVρCp.
Choose sampling periodTs=0.06min. The values and meaning of other parameters are given in Table 2.(59)A1d=0.7899−0.00207.16380.9232,A2d=0.9588−0.0022−0.80350.9326,A3d=0.8678−0.00353.49320.9951,A4d=0.8728−0.00073.25190.8650,B1d=−0.00010.1208T,B2d=−0.00010.1212T,B3d=−0.00020.1254T,B4d=−0.0000480.1169T.
In this section, for simplicity, we adopt the following T-S fuzzy model:(60)xk+1=∑i=12μix2kAixk+Biuk,with(61)μ1x2k=μ¯1x2k+μ¯3x2k,μ2x2k=μ¯2x2k+μ¯4x2k,A1=12A1d+A3d,A2=12A2d+A4d,B1=12B1d+B3d,B2=12B2d+B4d.
According to the above discussion, the T-S fuzzy model can be summarized as(62)Rule1:IFx2kisΓ11,THENxk+1=A1+ΔA1kxkB1+ΔB1kuk,Rule2:IFx2kisΓ12,THENxk+1=A2+ΔA2kxkB2+ΔB2kuk,with(63)A1=0.8288−0.00275.32850.9592,A2=0.9158−0.00141.22420.8988,B1=−0.00020.1231T,B2=−0.00010.1191T.
Considering the uncertainty of the T-S fuzzy system, the following conditions must be satisfied:(64)ΔA1k=DFkE11,ΔA2k=DFkE12,ΔB1k=DFkE21,ΔB2k=DFkE22,with D=I, E11=diag0.01;0.001A1,E12=diag0.01;0.001A2, E21=diag0.05;0.005B1,andE22=diag0.01;0.001B2. For more rational analysis, we assume that Fk is norm-bounded and satisfies Fk=sink, FkTFk≤I.
The control input and state constraints must satisfyuk≤3.5K,x2k≤10K,s≥0. We choose the initial conditions as x0=−0.01425;−0.60872 and u0=−0.14. Let W=diag0.5;0.5 and R=1I. Assume ω¯=0.3. The comparison in this example is made in the view of Theorem 2 with quantization density τ=0.6567andτ=0.0191 and the method in [43], which provided a designing approach of MPC for fuzzy NCSs with data loss. The state feedback gains at k=1, and the performance costs for different methods are shown in Table 3. As we can see from Figures 3–5 and Figures 6–8, the state, upper bound γ of the objective function and control input all converge to zero as k⟶∞. However, some differences can be found from these figures. The convergence speed of Theorem 2 in this paper is faster than that of the method in [43]. Further, the control performance of the system in this paper is better than that in [43]. Figures 9 and 10 show the state trajectories and invariant sets of these two methods which show the superiority of the presented approach. It is obvious that the recursive feasibility and asymptotic stability of the closed-loop T-S fuzzy system can be guaranteed, which illustrates the effectiveness of our method.Figure 2
Structure of the continuous stirred tank reactor.Table 2
The operating parameters of CSTR.
ParameterDescriptionValueUnitqProcessflowrate100L/minTfActualfeedtemperature350KCAfFeedconcentration1mol/LVReactorvolume100LρLiquiddensity1000g/LCpHeatcapacity0.239J/g KΔHHeatofreaction−120,000J/molE/RActivationenergy8750Kk0Reactionrateconstant7.2×1010min−1UAHeattransfercoefficient50,000J/min KTable 3
The comparison of different methods.
MethodsSate feedback gain atk=1Performance costsJTheorem2, δ=0.0191[–6.90906, –0.47676]0.6686Theorem2, δ=0.6567[–6.94362, –0.48018]0.6575Corollary 7 in [43][–3.2213, –0.169]0.7075Figure 3
State trajectory.Figure 4
Control input.Figure 5
γk trajectory.Figure 6
State trajectory.Figure 7
Control input.Figure 8
γk trajectory.Figure 9
Invariant sets.Figure 10
Invariant sets.


## 7. Conclusion

In this paper, the robust constrained MPC scheme with quantization and data loss for the uncertain fuzzy system has been studied. Compared with other documents, the key idea of our approach is that the non-PDC control law and nonquadratic Lyapunov function are employed in the case of taking the problem of NCSs into consideration. Based on the nonquadratic Lyapunov function, the new conditions for stability related to slack and collection matrices are obtained. Further, the fuzzy controller is designed, and online synthesis of MPC for fuzzy NCSs is presented by solving the optimization problem in view of non-PDC control law and an extended nonquadratic Lyapunov function, respectively. In future work, we will focus on reducing the impact of network-induced delay through the MPC method or study the problem of reducing computational burden of the networked MPC method.



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