# Robust Adaptive Fuzzy Control via State-Dependent Function for Nonlinear Stochastic Large-Scale Systems Subject to Dead Zones

Authors: Chang-Qi Zhu; Lei Liu
Journal: Complexity
 (2021)

---

## Abstract

This paper concentrates on the adaptive fuzzy control problem for stochastic nonlinear large-scale systems with constraints and unknown dead zones. By introducing the state-dependent function, the constrained closed-loop system is transformed into a brand-new system without constraints, which can realize the same control objective. Then, fuzzy logic systems (FLSs) are used to identify the unknown nonlinear functions, the dead zone inverse technique is utilized to compensate for the dead zone effect, and a robust adaptive fuzzy control scheme is developed under the backstepping frame. Based on the Lyapunov stability theory, it is proved ultimately that all signals in the closed-loop system are bounded and the tracking errors converge to a small neighborhood of the origin. Finally, an example based on an actual system is given to verify the effectiveness of the proposed control scheme.


---

## Body

## 1. Introduction

The nonlinear large-scale systems are a common and significant class of nonlinear systems. As its name implies, it is composed of multiple subsystems, and each subsystem is connected by the terms. Numerous actual systems can be described as nonlinear large-scale systems, such as instance power supply systems, ecological systems, water resources systems, and social-economic systems. The decentralized control study for nonlinear large-scale systems has achieved a lot over the years. In [1, 2], the authors studied the adaptive state feedback control problems of nonlinear large-scale systems. Then, the authors in [3, 4] have extended the works to the output feedback control problems of nonlinear large-scale systems by designing a state observer. To remove the restriction that the system dynamics must be known for the control design exists in [1–4], adaptive fuzzy or neural network (NN) control schemes have been developed [5–12]. In order to achieve the objective in a finite time, the authors in [5, 6, 13–15] presented the finite-time stabilities for nonlinear large-scale systems with measured and unmeasured states, respectively. To improve the robustness of the system, Tong et al. [7, 8] studied the adaptive fuzzy robust control designs for nonlinear large-scale systems with actuator faults and dead zones, respectively. To spare the resources of network communication, the authors in [9, 10] studied the event-trigger control problems of nonlinear large-scale systems. However, the results [9–11] will no longer be applicable when stochastic disturbances exist. To deal with this problem, the authors in [16] first presented the method using a quartic Lyapunov function. Based on [16], the authors in [17, 18] studied the adaptive fuzzy decentralized control designs for stochastic nonlinear large-scale systems with dynamic uncertainties, the authors in [19] investigated the adaptive fuzzy decentralized control problem for stochastic nonlinear large-scale systems with unknown control directions, and Tong et al. [20] proposed a control algorithm for stochastic nonlinear large-scale systems via the dynamic surface control (DSC) technique.Note that the above results do not have any requirements for the states of the plants. However, due to the restriction of some physical conditions, the constraints are inevitable in practical engineerings, such as the chemical industry, boiler industry, and robot industry. Therefore, the control study for nonlinear constrained systems becomes necessary, and some meaningful results have been achieved in [21–37]. In [21, 22], the authors proposed two approaches for uncertain nonlinear systems with output constraints and time-varying output constraints, respectively. To investigate the more complex constrained problem, Liu et al. [23–25] studied adaptive control designs for nonlinear systems with full state constraints. Subsequently, by utilizing the fuzzy logic systems (FLSs) and radial basis function neural networks (RBFNNs) [26, 27], the authors in [30–32] studied adaptive fuzzy or NN control problems of uncertain state-constrained nonlinear systems. Furthermore, the authors in [33] investigated the adaptive fuzzy control design for stochastic high-order nonlinear systems with asymmetric output constraints by designing a novel barrier Lyapunov function (BLF). However, the results above all depend on the feasibility condition, which means that the states of the controlled system cannot be constrained directly. To handle this problem, Zhao et al. [34, 35] studied the control problems for single-input and single-output (SISO) nonlinear systems and multiple-input and multiple-output (MIMO) nonlinear systems with full state constraints by utilizing the nonlinear state-dependent functions, respectively. Then, the authors in [36–38] extended the results in [34, 35] to the time-varying state-constrained problems of SISO nonlinear systems and MIMO nonlinear systems, respectively. As another unavoidable factor in the actual system, the dead zone input nonlinearity has been also paid much attention and some valuable results have been achieved in [7, 39–47]. In [39, 40], the authors have investigated the adaptive control problems for nonlinear systems with the symmetric dead zone and asymmetric dead zone, respectively. To deal with the nonlinear dead zone problem, an adaptive NN control approach has been proposed for uncertain nonlinear systems via the Lagrange mean value theorem in [42]. Furthermore, by using the adaptive estimation mechanism, the dead zone inverse technique has been proposed to compensate the dead zone effect in [7].It is worth mentioning that most of the results mentioned are for the strict-feedback nonlinear systems instead of nonstrict-feedback nonlinear systems. Different from the strict-feedback nonlinear systems, the unknown nonlinear functions in the nonstrict-feedback nonlinear systems are composed of the whole states. If the traditional control schemes for strict-feedback nonlinear systems are adopted, the algebraic loop problem will be ineluctable. To find a way out of the dilemma, the authors in [46–48] proposed the control algorithms for nonstrict-feedback nonlinear systems via the variable separation technique. Since the variable separation technique requires the nonlinear functions to be strictly monotonic increasing, the authors in [49, 50] proposed the novel control algorithms for nonstrict-feedback nonlinear systems by using the property of radial basis function, which do not have any restriction for the nonlinear functions. Motivated by all the mentioned works, an adaptive fuzzy robust control scheme is developed for stochastic state-constrained nonlinear large-scale systems with unknown dead zones. Its main contributions can be summarized as follows.(1)
This paper studied the constrained problem for nonlinear large-scale systems with unknown dead zones. Note that stochastic disturbance is inevitable in engineering practice and the stochastic system is always a research hot spot [51–55]. A novel adaptive law is used to overcome the algebraic loop problem, a variable transformation method is utilized to solve the constrained problem, and a dead zone inverse technique is used to compensate for the dead zone effect.(2)
Although the results in references, like [21–24], are also for the constrained study, they all depend on the feasibility condition, which cannot constrain the states of the system directly, and the developed control scheme in this paper can remove the restriction. On the other hand, the controlled plants in [21–24] are all SISO or MIMO nonlinear systems instead of large-scale nonlinear systems.


## 2. Preliminaries

Consider a class of stochastic nonlinear large-scale systems with unknown dead zones which are composed ofN subsystems connected by outputs. The ith subsystem can be expressed as(1)∑idxi,1=xi,2+fi,1xi+Δi,1y¯dt+gi,1xidwdxi,2=xi,3+fi,2xi+Δi,2y¯dt+gi,2xidw⋮dxi,ni−1=xi,ni+fi,ni−1xi+Δi,ni−1y¯dt+gi,ni−1xidwdxi,ni=Diui+fi,nixi+Δi,niy¯dt+gi,nixidwyi=xi,1,where the state vectors of the system are expressed as xi=xi,1,xi,2,…,xi,niT, x¯i,j=xi,1,xi,2,…,xi,jT, 1≤i≤N and 1≤j≤ni, ui∈R and yi∈R denote the actuator input and sensor output of the system, respectively, Diui∈R denotes the output of the dead zone, fi,jxi and gi,jxi are the unknown nonlinear functions, Δi,jy¯ is the interconnected term which connects each subsystem, and w is an independent r- dimensional Wiener process, and we assume that the states of the system can be measured directly.Remark 1.
Different from the research results in references, such as [1–3, 6–8], the control study in this paper is to design a robust adaptive fuzzy control method for nonstrict-feedback nonlinear systems. If the traditional control schemes are adopted, the algebraic loop problem will not be able to be avoided. The algebraic loop problem means that if the control design method in strict feedback systems is adopted, the virtual control signal αi in nonstrict-feedback nonlinear systems for the ith subsystem will be a function which contains the entire states x=x1,x2,…,xnT, but the states xi+1,xi+2,…,xn are not available at this time. Also, the time-varying state constrained problem is considered in this paper.
Similar to [7, 40], the output of the dead zone is defined as(2)Diui≜mi,rui−di,rifui≥di,r,0,if−di,l<ui<di,r,mi,lui+di,l,ifui≤−di,l.
In (2), ui is the input to the dead zone, mi,r and mi,l are the slopes of the dead zone, and di,r and di,l denote the dead zone width parameters. In this paper, we assume that the output of the dead zone is unmeasurable, and the dead zone parameters mi,r, mi,l, di,r, and di,l are not available, but their signs are available (mi,r>0, mi,l>0, di,r≥0, and di,l≥0, respectively). Dead-zone slopes are bounded by known constants mi,rmax, mi,rmin, mi,lmax and mi,lmin such that 0<mi,rmin≤mi,r≤mi,rmaxand 0<mi,lmin≤mi,l≤mi,lmax. The dead-zone inverse technique is used to compensate the dead-zone effect [7]. Setting ui∗ as the control input which is free of a dead zone, the control signal ui can be expressed as follows:(3)ui=Diui∗−1=ui∗+d^i,r,mm^i,rδi+ui∗+d^i,l,mm^i,l1−δi,where m^i,r, m^i,l, d^i,r,m, and d^i,r,l are the estimations of mi,r¸ mi,l, mi,rdi,r, and mi,ldi,l, respectively. And,(4)δi=1,ifui∗≥0,0,ifui∗≤0.
The resulting error betweenui∗ and ui is given by(5)Diui−ui∗=d˜i,r,m−ui∗+d^i,r,mm^i,rm˜i,rδi+d˜i,l,m−ui∗+d^i,l,mm^i,lm˜i,l1−δi+εiui∗,where parameter errors are defined as m˜i,r=mi,r−m^i,r, m˜i,l=mi,l−m^i,l, d˜i,r,m=di,r,m−d^i,r,m, and d˜i,r,l=di,l,m−d^i,l,m.
The boundεiui∗ is expressed as(6)εiui∗=−mi,rκi,rui−di,r−mi,lκi,lui−di,l,where(7)κi,r=1,if0≤ui<di,r0,otherwise,κi,l=1,ifdi,l≤ui<0,0,otherwise.
To solve the state-constrained problem as well as removing the restriction of the feasibility condition, a nonlinear state-dependent function is introduced, which has form below:(8)χi,j=logλi,j,1+xi,jλi,j,2−xi,j,where λi,j,1 and λi,j,2 are time-varying bounded functions and χi,j is the state which is constrained.
From (8), χi,j holds the following property:(9)χi,j⟶±∞,only ifxi,j⟶λi,j,2orxi,j⟶−λi,j,1.
The property means that the statexi,j will be constrained in the region −λi,j,1,λi,j,2 only when χi,j is bounded.
In order to complete the constrained control design without the feasibility condition, system (1) will be transformed into a brand-new system without any constraint. From (6), we can obtain(10)xi,j=eχi,jλi,j,2−λi,j,1eχi,j+1,(11)dxi,j=eχi,jλ˙i,j,2−λ˙i,j,1eχi,j+1+eχi,jλi,j,2+λi,j,1eχi,j+12dχi,j.
Substituting (10) and (11) into system (1), a new system dynamic is obtained as(12)∑idχi,1=hi,1χi,2+fi,1χi+Δi,1Y¯+li,1dt+g¯i,1χidwdχi,2=hi,2χi,3+fi,2χi+Δi,2Y¯+li,2dt+g¯i,2χidw⋮dχi,ni−1=hi,ni−1χi,ni+fi,ni−1χi+Δi,ni−1Y¯+li,ni−1dt+g¯i,ni−1χidwdχi,ni=hi,niDiui+fi,niχi+Δi,niY¯+li,nidt+g¯i,niχidwYi=χi,1,where(13)χi=χi,1,χi,2,…,χi,niT,li,j=−eχi,j+12eχi,jλ˙i,j,2−λ˙i,j,1eχi,jλi,j,2+λi,j,1eχi,j+1,g¯i,j=−eχi,j+12eχi,jλi,j,2+λi,j,1gi,j,hi,j=eχi,j+12eχi,jλi,j,2+λi,j,1,Y¯=Y1,Y2,…YNT.Remark 2.
Although BLF is regarded as a strong tool for constrained issues, some limitations exist. The virtual controllers are required to meet the feasibility conditions, which means the optimal parameters have to be selected offline and additional computation is inevitable. By utilizing the above system transformation, the constrained closed-loop system is transformed into a novel closed-loop system without any constraint. The control design will become easier, and the control design for the novel closed-loop system can achieve the same control objective.Assumption 1.
The unknown nonlinear functionΔi,jY¯, i=1,2,…,N and j=1,2,…,ni, satisfies the following inequality, which is a common assumption for large-scale systems:(14)Δi,jY¯≤∑l=1Nρi,j,l2Yl,where ρi,j,l is an unknown smooth function satisfies ρi,j,l0=0.
Since system (1) contains unknown nonlinear dynamics, the FLSs are employed to approximate these unknown nonlinear dynamics.
For any continuous functionρυ defined over a compact set Λand any given positive constant ψ, there always exists an FLS ρ^υ|μ∗=μ∗Tϕυ such that(15)supυ∈Λρυ−μ∗Tϕυ≤ψ,where ϕiυ are always chosen as the Gaussian functions. ϕυ=ϕ1υ,ϕ2υ,…,ϕNυT/∑1Nϕiυ are the fuzzy basis function vectors and satisfy 0<ϕTυϕυ≤1. The ideal weight is defined as ρ=ρ1,ρ2,…,ρNT, and M is the fuzzy rule numbers.


## 3. Robust Constrained Control Design

In this section, we will carry out the control design for new system (12) with the new state vector χi and new nonlinear functions hi,j, fi,j, li,j, and g¯i,j. Combining backstepping design with the dead zone inverse technique, an adaptive fuzzy decentralized controller is established, and based on the Lyapunov stability theory, the stability of the closed-loop system is proved. The ni steps backstepping control design process is based on the following coordinate transformation:(16)zi,1=Yi−Yi,rzi,j=χi,j−αi,j−1,j=2,3,…,ni,where Yi,r=logλi,1,1+yi,r/λi,1,2−yi,r is the new reference signal with yi,r being the original reference signal, zi,j denotes the tracking error, and αi,j represents the virtual control signal which will be designed in each step.Step 1: from (12) and (16), the derivative of zi,1 is expressed as(17)dzi,1=hi,1χi,2+fi,1χi+Δi,1Y¯+li,1−Y˙i,rdt+g¯i,1χidw.Consider the following Lyapunov function candidate:(18)V1=∑i=1NVi,1=∑i=1N14zi,14+12γi,1Θ˜i,12+12ςiW˜i2,where γi,1>0 is a design parameter, Θ˜i,j=Θi,j−Θ^i,j, Θ^i,j is the estimation of Θi,j, Θi,j=θi,j2, W˜i=Wi−W^i, W^i is the estimation of Wi, and Wi=wi4. The definitions of wi and θi,j will be given later.Combining (17) and (18), it can be obtained that(19)ℓV1≤∑i=1Nzi,13hi,1zi,2+αi,1+fi,1χi+Δi,1Y¯+li,1−Y˙i,r+32zi,12g¯i,1Tg¯i,1−1γi,1Θ˜i,1Θ^˙i,1−1ςiW˜iW^˙i.Based on Young’s inequality, we have(20)zi,13hi,1zi,2≤14zi,24+34zi,14hi,14/3,(21)32zi,12g¯i,1Tg¯i,1≤34zi,14g¯i,14+34,(22)∑i=1Nzi,13Δi,1Y¯≤∑i=1N12zi,16+∑i=1NYi2∑l=1N12ρ¯l,1,i2Yi.Substituting (20)–(22) into (19) yields(23)ℓV1≤∑i=1Nzi,13hi,1αi,1+f¯i,1+YiFi−1ςiW˜iW^˙+14zi,24+34−1γi,1Θ˜i,1Θ^˙i,1−Yi2∑l=1N∑j=1nl∑k=1j12ρ¯l,k,i2Yi,where f¯i,1=hi,1fi,1χi+3/4zi,1hi,14/3+1/2zi,13+3/4zi,13g¯i,14+li,1−Y˙i,r and Fi=Yi∑l=1N∑j=1nl∑k=1j1/2ρ¯l,k,i2Yi.Then, the following FLSs are used to approximatef¯i,1 and Fi as(24)f¯i,1Xi,1=θi,1Tφi,1Xi,1+εi,1,(25)FiYi=wiTϕiYi+δi,where the unknown constants εi,1 and δi denote the approximated errors and satisfy εi,1≤εi,1∗, δi≤δi∗. Xi,1=χi,Yi,r.Substituting (24) and (25) into (23) yields(26)ℓV1≤∑i=1Nzi,13hi,1αi,1+θi,1Tφi,1Xi,1+εi,1+14zi,24−Yi2∑l=1N∑j=1nl∑k=1j12ρ¯l,k,i2Yi+YiwiTϕiYi+δi−1γi,1Θ˜i,1Θ^˙i,1−1ςiW˜iW^˙+34.By completing the squares, one has(27)zi,13θi,1Tφi,1Xi,1+εi,1≤zi,16Θi,14τiφi,1TX¯i,1φi,1X¯i,1+τi1+εi,1∗2+zi,164τi,(28)YiwiTϕiYi+δi≤zi,14Wi4τi+zi,144τi+3τi41+2δi∗4/3+Wi4τi+3τi4Yi,r2+Yi,r44τi,where X¯i,1=χi,1,Yi,r.Substituting (27) and (28) into (26) yields(29)ℓV1≤∑i=1Nzi,13hi,1αi,1+zi,13Θ^i,14τiφi,1TX¯i,1φi,1X¯i,1+zi,1W^i4τi+zi,13+zi,14τi−Yi2∑l=1N∑j=1nl∑k=1j12ρ¯l,k,i2Yi+1γi,1Θ˜i,1γi,1zi,164τiφi,1TX¯i,1φi,1X¯i,1−Θ^˙i,1+1ςiW˜iςizi,144τi−W^˙+14zi,24+Di,1,where(30)Di,1=τi1+εi,1∗2+3τi41+2δi∗4/3+Wi4τi+3τi4Yi,rmax2+Yi,rmax44τi+34.Design the virtual control functionαi,1 and adaptive laws of Θ^i,1 and W^i as below:(31)αi,1=hi,1−1−ci,1zi,1−zi,13Θ^i,14τiφi,1TX¯i,1φi,1X¯i,1−zi,1W^i4τi−zi,13+zi,14τi,(32)Θ^˙i,1=γi,1zi,164τiφi,1TX¯i,1φi,1X¯i,1−σi,1Θ^i,1,(33)W^˙i=ςizi,144τi−σ¯iW^i.Substituting (31)–(33) into (29) yields(34)ℓV1≤∑i=1N−ci,1zi,14−Yi2∑l=1N∑j=1nl∑k=1j12ρ¯l,k,i2Yi+14zi,24+σi,1γi,1Θ˜i,1Θ^i,1+σ¯iςiW˜iW^+Di,1.Stepiandj1≤i≤Nand2≤j≤ni: based on (12) and (16), the derivative of zi,j is expressed as(35)dzi,j=hi,jχi,j+1+fi,jχi+Δi,jY¯+li,j−Lαi,j−∑k=1j−1αi,j−1χi,khi,kΔi,kY¯dt+g¯i,jχidw,where(36)Lαi,j=∑k=1j−1∂αi,j−1∂χi,khi,kχi,k+2+fi,kχi+li,k+∑k=1j−1∂αi,j−1∂Yi,rkYi,rk+1+∑k=1i−1∂αi,j−1Θ^i,kΘ^˙i,k+∂αi,j−1∂W^iW^˙i+12∑p1,p2=1j−1∂2αi,j−1∂χp1∂χp2g¯i,p1g¯i,p2T.Consider the following Lyapunov function candidate:(37)Vj=∑i=1NVi,j=∑i=1NVi,j−1+zi,j4+12γi,jΘ˜i,j2,where γi,j>0 is a design parameter.Combining (35) and (37), it is shown that(38)ℓVj≤∑i=1N−∑k=1j−1ci,kzi,k4−Yi2∑l=1N∑q=ni−j+1nl∑k=1q12ρ¯l,k,i2Yi+14zi,j4+∑k=1j−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^+Di,j−1+zi,j3hi,jzi,j+1+αi,j+fi,jχi+Δi,jY¯+li,j−Lαi,j−∑k=1j−1αi,j−1χi,khi,kΔi,kY¯+3zi,j32g¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,kg¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,kT−1yi,jΘ˜i,jΘ^˙i,j,where Di,j−1=Di,1+3j−2/4+∑k=2j−11+τiεi,k∗2.Based on Young’s inequality, we obtain(39)zi,j3hi,jzi,j+1≤14zi,j+14+34zi,j4hi,j4/3,3zi,j32g¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,kg¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,kT≤3zi,j64g¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,k4+34,∑i=1Nzi,j3hi,jΔi,jY¯≤∑i=1N12zi,j6hi,j2+∑i=1NYi2∑l=1N12ρ¯l,j,i2Yi,−∑i=1Nzi,j3∑k=1j−1αi,j−1χi,khi,kΔi,kY¯≤∑i=1N∑k=1j−112αi,j−1χi,k2zi,j6hi,k2+∑i=1NYi2∑k=1j−1∑l=1N12ρ¯l,k,i2Yi.Then, we have(40)ℓVj≤∑i=1N−∑k=1j−1ci,kzi,k4−Yi2∑l=1N∑q=ni−j+1nl∑k=1q12ρ¯l,k,i2Yi+14zi,j4+∑k=1j−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^+14zi,j+14+34zi,j4hi,j4/3+zi,j3hi,jαi,j+fi,jχi+Δi,jY¯+li,j−Lαi,j−∑k=1j−1αi,j−1χi,khi,kΔi,kY¯+1γi,jΘ˜i,jΘ^˙i,j+3zi,j64g¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,k4+34+Di,j−1,ℓVj≤∑i=1N−∑k=1j−1ci,kzi,k4−Yi2∑l=1N∑q=ni−j+1nl∑k=1q12ρ¯l,k,i2Yi+14zi,j4+∑k=1j−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^+14zi,j+14+34zi,j4hi,j4/3+12zi,j6hi,j2+Yi2∑l=1N12ρ¯l,j,i2Yi+34+zi,j3hi,jαi,j+fi,jχi+li,j−Lαi,j−∑k=1j−1αi,j−1χi,khi,kΔi,kY¯+1γi,jΘ˜i,jΘ^˙i,j+3zi,j64g¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,k4+Di,j−1.We finally yield(41)ℓVj≤∑i=1N−∑k=1j−1ci,kzi,k4−Yi2∑l=1N∑q=ni−jnl∑k=1q12ρ¯l,k,i2Yi+14zi,j+14+∑k=1j−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^+Di,j−1+34+zi,j3hi,jαi,j+f¯i,j−1γi,jΘ˜i,jΘ^˙i,j,where(42)f¯i,j=hi,jfi,jχi+34zi,jhi,j4/3+14zi,j+12zi,j3hi,j2+∑k=1j−112αi,j−1χi,k2zi,j3hi,k2+li,j−Lαi,j+3zi,j34g¯i,j−∑k=1j−1∂αi,j−1∂χi,kg¯i,k4.Then,f¯i,j is approximated by the FLS:(43)f¯i,jXi,j=θi,jTφi,jXi,j+εi,j,where Xi,j=χi,W^i,Θ^i,1,…,Θ^j−1,Yi,r,Yi,r1,…,Yi,rj and εi,j denotes the approximate error and satisfies εi,j≤εi,j∗ with εi,j∗ being an unknown constant.Substituting (43) into (41) yields(44)ℓVj≤∑i=1N−∑k=1j−1ci,kzi,k4−Yi2∑l=1N∑q=ni−jnl∑k=1q12ρ¯l,k,i2Yi+14zi,j+14+∑k=1j−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^+Di,j−1+34+zi,j3hi,jαi,j+θi,jTφi,jXi,j+εi,j−1γi,jΘ˜i,jΘ^˙i,j.Note thatφi,jTX¯i,jφi,jX¯i,j=1; based on Young’s inequality, we have(45)zi,j3θi,jTφi,jXi,j+εi,j=zi,j3θi,jTφi,jXi,j+zi,j3εi,j≤14τiφi,jTX¯i,jφi,jX¯i,jzi,j6θi,j2+τiφi,jTX¯i,jφi,jX¯i,j+τiεi,j∗2+zi,j64τi.Then, we obtain(46)zi,j3θi,jTφi,jXi,j+εi,j≤zi,j6Θi,j4τiφi,jTX¯i,jφi,jX¯i,j+τi1+εi,j∗2+zi,j64τi,where X¯i,j=χ¯i,j,W^i,Θ^i,1,…,Θ^i,j−1,Yi,r,Yi.r1,…,Yi.rj.Combining (44) and (46), we obtain(47)ℓVj≤∑i=1N−∑k=1j−1ci,kzi,k4−Yi2∑l=1N∑q=ni−jnl∑k=1q12ρ¯l,k,i2Yi+14zi,j+14+∑k=1j−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^i+zi,j3hi,jαi,j+zi,j3Θ^i,j4τiφi,jTX¯i,jφi,jX¯i,j+zi,j34τi+1γi,jΘ˜i,jγi,jzi,j64τiφi,jTX¯i,jφi,jX¯i,j−Θ^˙i,j+Di,j,where Di,j=Di,j−1+τi1+εi,j∗2+3/4.Design the virtual controllerαi,j and adaptive law of Θ^i,j as(48)αi,j=hi,j−1−ci,jzi,j−zi,j3Θ^i,j4τiφi,jTX¯i,jφi,jX¯i,j−zi,j34τi,(49)Θ^˙i,j=γi,jzi,j64τiφi,jTX¯i,jφi,jX¯i,j−σi,jΘ^i,j.Substituting (48) and (49) into (41) yields(50)ℓVj≤∑i=1N−∑k=1jci,kzi,k4−Yi2∑l=1N∑q=ni−jnl∑k=1q12ρ¯l,k,i2Yi+14zi,j+14+∑k=1jσi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^i+Di,j.Stepiandni: in the final step, the actual controller is constructed. Similar to the above steps, the derivative of zi,ni is expressed as(51)dzi,ni=hi,niDiui+fi,niχi+Δi,niY¯+li,ni−Lαi,ni−∑k=1ni−1αi,ni−1χi,khi,kΔi,kY¯dt+g¯i,niχidw,where(52)Lαi,ni=∑k=1ni−1∂αi,ni−1∂χi,khi,kχi,k+2+fi,kχi+li,k+∑k=1ni−1∂αi,ni−1∂Yi,rkYi,rk+1+∑k=1ni−1∂αi,ni−1Θ^i,kΘ^˙i,k+∂αi,ni−1∂W^iW^˙i+12∑p1,p2=1ni−1∂2αi,ni−1∂χp1∂χp2g¯i,p1g¯i,p2T.Consider the following Lyapunov function candidate:(53)V=∑i=1NVi,ni=∑i=1NVi,ni−1+14zi,ni4+12γi,niΘ˜i,ni2,V=∑i=1NVi,ni−1+14zi,ni4+12γi,niΘ˜i,ni2+12ξi,1m˜i,r2+12ξi,2m˜i,l2+12ξi,3d˜i,r,m2+12ξi,4d˜i,l,m2,where γi,ni>0, ξi,1>0, ξi,2>0, ξi,3>0, and ξi,4>0 are designed parameters.Combining (51) and (53), it can be shown that(54)ℓV≤∑i=1N−∑k=1ni−1ci,kzi,k4+∑k=1ni−1σi,kγi,kΘ˜i,kΘ^i,k+σ¯iςiW˜iW^−1ξi,1m˜i,rm^˙i,r−1ξi,2m˜i,lm^˙i,l−1ξi,3d˜i,r,md^˙i,r,m−1ξi,4d˜i,l,md^˙i,l,m+zi,ni3hi,niδid˜i,r,m−ui∗+d^i,r,mm^i,rm˜i,r+d˜i,l,m−ui∗+d^i,l,mm^i,lm˜i,l1−δi+zi,ni3hi,niui∗+zi,ni3Θ^i,ni4τiφi,niTX¯i,niφi,niX¯i,ni+zi,ni34τi+12zi,ni3hi,ni2+1γi,niΘ˜i,niγi,nizi,ni64τiφi,niTX¯i,niφi,niX¯i,ni−Θ^˙i,ni+Di,ni,where Di,ni=Di,ni−1+τi1+εi,j∗2+3/4+1/2εiui∗2.Design the actual controllerui∗ and adaptive laws of Θ^i,ni, m^i,r, m^i,l, d^i,r,m, and d^i,r,l as(55)ui∗=hi,ni−1−ci,nizi,ni−zi,ni3Θ^i,ni4τiφi,niTX¯i,niφi,niX¯i,ni−zi,ni34τi−12zi,ni3hi,ni2,(56)Θ^˙i,ni=γi,nizi,ni64τiφi,niTX¯i,niφi,niX¯i,ni−σi,niΘ^i,ni,(57)m^˙i,r=ui∗+d^i,r,mm^i,rξi,1zi,ni3hi,niδi+ξ¯i,1m^i,r,(58)m^˙i,l=ui∗+d^i,l,mm^i,lξi,2zi,ni3hi,ni1−δi+ξ¯i,2m^i,l,(59)d^˙i,r,m=−ξi,3zi,ni3hi,niδi+ξ¯i,3d^i,r,m,(60)d^˙i,l,m=−ξi,4zi,ni3hi,ni1−δi+ξ¯i,4d^i,l,m.Substituting (55)–(60) into (54) yields(61)ℓV≤∑i=1N−∑k=1nici,kzi,k4−∑k=1niσi,k2γi,kΘ˜i,k2−σ¯i2ςiW˜i2+Di−ξ¯i,12ξi,1m˜i,r2−ξ¯i,22ξi,2m˜i,l2−ξ¯i,32ξi,3d˜i,r,m2−ξ¯i,42ξi,4d˜i,l,m2,where(62)Di=Di,j+∑k=1niσi,k2γi,kΘi,k2+σ¯i2ςiWi2+ξ¯i,12ξi,1mi,r2+ξ¯i,22ξi,2mi,l2+ξ¯i,32ξi,3di,r,m2+ξ¯i,42ξi,4di,l,m2.DefineCi=minci,k,σi,k,σ¯i,ξ¯i,1,ξ¯i,2,ξ¯i,3,ξ¯i,4; then, (61) is written as(63)ℓV≤∑i=1N−CiVi,ni+Di.LetC=min1≤i≤NCi and D=max1≤i≤NDi; then, (62) becomes(64)ℓV≤−CV+D.We summarize the above robust constrained control design as the following theorem.Theorem 1.
Consider nonlinear system (12); under Assumption 1, the virtual controllers are designed as (31) and (48), the actual controller is designed as (55), and adaptive laws are designed as (32), (33), (49), and (56)–(60); then, the developed control scheme can guarantee that all the signals in the closed-loop system are bounded, and the tracking errors converge to a small neighborhood of the origin.Proof.
Integrating (64) over 0,t results in(65)0≤EVt≤e−CtEV0+DC.
From (57), we can obtain that all the closed-loop signals χi,j, zi,j, Θ^i,j, W^i, m^i,r, m^i,l, d^i,r,m, and d^i,l,m are bounded, 1≤i≤N and 1≤j≤ni. The tracking error satisfies(66)zi,1≤2e−CtV0+DC.
Furthermore, since the novel closed-loop signalχi,j is bounded, according to the definition of the nonlinear state-dependent function in (8), the signal xi,j remains in the constrained region −λi,j,1,λi,j,2. Then, the proof of Theorem 1 has been completed.Remark 3.
The control design in this paper does not depend on the feasibility condition, and it is no need to utilize the BLFs, which can achieve the same control objective. Although the results in [21–33] can also solve the state constrained problem, the control designs appear much complicatedly.


## 4. Simulation Example

Consider an example of two inverted pendulums connected by a moving spring mounted on two cars. The input to each pendulum is the torqueuii=1,2 applied at the pivot point. We define the state vectors as(67)x1,1,x1,2T=θ1,θ˙1T,x2,1,x2,2T=θ2,θ˙2T.Defineat=sin5t, K1=sin2t, K2=sin3t+L, β1=msinx1,1/M, β2=msinx2,1/M, and c=m/m+M. The dynamics equations of the inverted two pendulums on cars are introduced in [3]:(68)x˙1,1=x1,2,x˙1,2=gcl−katat−clcml2x1,1+u1cml2+katat−clcml2x2,1+β1x1,22+katat−clcml2K1−K2,y1=x1,1,x˙2,1=x2,2,x˙2,2=gcl−katat−clcml2x2,1+u2cml2+katat−clcml2x1,1+β2x2,22+katat−clcml2K2−K1,y2=x2,1,where mkg, Mkg, lm, Lm, kN/m, and gm/s2 are the mass of the pendulum and the car, the length of the pole, the natural length of the connecting spring, the spring constant, and the gravity constant, respectively.Here, both stochastic disturbance and actuator dead zones are considered on the basis of the above systems:(69)x˙1,1=x1,2,x˙1,2=gcl−katat−clcml2x1,1+D1u1cml2+katat−clcml2x2,1+β1x1,22+katat−clcml2K1−K2dt+g1,2x1,1,x1,2dw,y1=x1,1,x˙2,1=x2,2,x˙2,2=gcl−katat−clcml2x2,1+D2u2cml2+katat−clcml2x1,1+β2x2,22+katat−clcml2K2−K1dt+g2,2x2,1,x2,2dw,y2=x2,1,where(70)g1,2x1,1,x1,2=cosx1,2−sinx1,1,g2,2x2,1,x2,2=sinx2,2−cosx2,1.Du1 and Du2 are the outputs of the unknown actuator dead zones with the dead-zone slopes being given as(71)m1,r=m2,r=1.1,m1,l=m2,l=1.7,d1,r=d2,r=0.3,d1,l=d2,l=0.4.The tracking signals are defined asy1,r=y2,r=0.45sint, and the states x1,1, x1,2, x2,1, and x2,2 are constrained in the regions:(72)−λ1,1,1t,λ1,1,2t,−λ1,2,1t,λ1,2,2t,−λ2,1,1t,λ2,1,2t,−λ2,2,1t,λ2,2,2t,with(73)λ1,1,1t=λ2,1,1t=0.5−0.4sint,λ1,1,2t=λ2,1,2t=0.6−0.1cost,λ1,2,1t=λ2,2,1t=0.4−0.3sint,λ1,2,2t=λ2,2,2t=0.5+0.1cost.From (10)–(12), the novel closed-loop system dynamics are described as(74)dχ1,1=h1,1χ1,2+l1,1dt,dχ1,2=h1,2gcl−katat−clcml2χ1,1+Du1cml2+katat−clcml2χ2,1+β¯1χ1,22+katat−clcml2K1−K2+l1,2dt+g¯1,2χ1,1,χ1,2dw,Y1=χ1,1,(75)dχ2,1=h2,1χ2,2+l2,1dt,dχ2,2=h2,2gcl−katat−clcml2χ2,1+Du2cml2+katat−clcml2χ1,1+β¯2χ2,22+katat−clcml2K2−K1+l2,2dt+g¯2,2χ2,1,χ2,2dw,Y2=χ2,1,where χi,j=logλi,j,1+xi,j/λi,j,2−xi,j, g¯i,j=−eχi,j+12/eχi,jλi,j,2+λi,j,1gi,j, li,j=−eχi,j+12eχi,jλ˙i,j,2−λ˙i,j,1/eχi,jλi,j,2+λi,j,1eχi,j+1, and hi,j=eχi,j+12/eχi,jλi,j,2+λi,j,1 with i=1and2 and j=1and2. The reference signals of the novel closed-loop system are given as(76)Y1,r=logλ1,1,1+y1,rλ1,1,2−y1,r,Y2,r=logλ2,1,1+y2,rλ2,1,2−y2,r.Choose the fuzzy membership functions as follows:(77)μFi,jk=e−0.5χi,j+6−2k2,i=1,2,j=1,2,…,5,andk=1,2,…,5.Then, the fuzzy basic functions can be constructed as(78)φi,j,kχ¯i,j=∏j=12exp−0.5χ¯i,j+6−2k2∑k=15∏j=12exp−0.5χ¯i,j+6−2k2.The virtual control functionsαi,1i=1and2, the actual control inputs ui∗i=1and2, and the adaptive laws of the parameters W^i,1i=1and2, Θ^i,ji=1and2andj=1and2, m^i,ri=1and2, m^i,li=1and2, d^i,r,mi=1and2, and d^i,l,mi=1and2 are designed as below:(79)αi,1=hi,1−1−ci,1zi,1−zi,13Θ^i,14τiφi,1TX¯i,1φi,1X¯i,1−zi,1W^i4τi−zi,13+zi,14τi,(80)ui∗=hi,2−1−ci,2zi,2−zi,23Θ^i,24τiφi,2TX¯i,2φi,2X¯i,2−zi,234τi−12zi,23hi,22,(81)Θ^˙i,1=γi,1zi,164τiφi,1TX¯i,1φi,1X¯i,1−σi,1Θ^i,1,(82)Θ^˙i,2=γi,2zi,264τiφi,2TX¯i,2φi,2X¯i,2−σi,2Θ^i,2,(83)W^˙i=ςizi,144τi−σ¯iW^i,(84)m^˙i,r=ui∗+d^i,r,mm^i,rξi,1zi,23hi,2δi+ξ¯i,1m^i,r,(85)m^˙i,l=ui∗+d^i,l,mm^i,lξi,2zi,23hi,21−δi+ξ¯i,2m^i,l,(86)d^˙i,r,m=−ξi,3zi,23hi,2δi+ξ¯i,3d^i,r,m,(87)d^˙i,l,m=−ξi,4zi,23hi,21−δi+ξ¯i,4d^i,l,m.In the simulation, the initial values are chosen as(88)χ1,10,χ1,20,χ2,10,χ2,20T=0.01,0.02,0.03,0.01T,Θ^1,10,Θ^1,20,Θ^2,10,Θ^2,20T=0.1,0.2,0.3,0.4T,W^10,W^20T=0.3,0.2T,m^1,r0,m^1,l0,d^1,r,m0,d^1,l,m0,m^2,r0,m^2,l0,d^2,r,m0,d^2,l,m0T=1,1,1,1,1,1,1,1T.Remark 4.
Note that the two inverted pendulums’ systems introduced above are free of dead zones and did not consider the stochastic disturbance. However, the stochastic disturbance and dead zones are evitable in practical engineering due to the restrictions of external physical conditions.
The design parameters are selected as(89)c1,1=c2,1=30,c1,2=c2,2=60,γ1,1=γ1,2=γ2,1=γ2,2=0.01,σ1,1=σ1,2=σ2,1=σ2,2=0.01,ς1=ς2=0.01,σ¯1=σ¯2=0.1,ξ1,1=ξ1,2=ξ1,3=ξ1,4=0.02,ξ¯1,1=ξ¯1,2=ξ¯1,3=ξ¯1,4=0.001,ξ2,1=ξ2,2=ξ2,3=ξ2,4=0.02,ξ¯2,1=ξ¯2,2=ξ¯2,3=ξ¯2,4=0.001.
Insert the control schemes (79)–(87) into systems (74) and (75), and the simulation results are reflected by Figures 1–10. Figure 1 shows the curves of outputs Yii=1and2 and their reference signals Yi,ri=1and2, and the trajectory of outputs Yi track the given signals during the whole period; Figure 2 shows the curves of state variables χi,2i=1and2, and it is obvious that χi,2 are bounded; both the tracking errors showed in Figures 1 and 2 can be directly reflected by Figures 9 and 10. Figure 3 shows the curves of control inputs uii=1and2 and dead zones Diuii=1and2; Figure 4 shows the curves of adaptive parameters Θ^i,ji=1and2andj=1and2 and W^ii=1and2; Figure 5 shows the curves of output y1 and its reference signals y1,r in the time-varying constrained closed-loop system; Figure 6 shows the curve of state variable x1,2 in the closed-loop system, and it is constrained by time-varying bounds; Figure 7 shows the curves of output y2 and its reference signal y2,r; Figure 8 shows that the curve of state variable x2,2 is constrained by time-varying bounds.Figure 1
Trajectories of outputsYi and reference signals Yi,r.Figure 2
Trajectories of state variablesχi,2.Figure 3
Trajectories of control inputsui∗ and dead zones Diui.Figure 4
Trajectories of adaptive parametersΘ^i,j and W^i.Figure 5
Trajectories of outputy1 and reference signals y1,r.Figure 6
Trajectories of state variablex1,2.Figure 7
Trajectories of outputy2 and reference signals y2,r.Figure 8
Trajectories of state variablex2,2.Figure 9
Trajectories of tracking errors betweenY1 and Y1,r.Figure 10
Trajectories of tracking errors betweenY2 and Y2,r.


## 5. Conclusion

In this article, the robust adaptive fuzzy control design for stochastic state-constrained nonlinear large-scale systems has been studied. The constrained closed-loop system is transformed into a novel system without any constraint by utilizing the variable transformation method, which can achieve the same control objective. Then, FLSs are used to approximate the unknown nonlinear functions, and the dead zone inverse technique is used to compensate the dead zone effect. Combining backstepping design with the decentralized control method, a robust adaptive fuzzy controller is constructed ultimately. In addition, the states of the controlled system remain in their constrained bounds all the time.



---

*Source: 1491995-2021-10-16.xml*