# Adaptive Neural Tracking Control for Discrete-Time Switched Nonlinear Systems with Dead Zone Inputs

Authors: Jidong Wang; Lengxue Zhu; Xiaoping Si
Journal: Complexity
 (2017)

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## Abstract

In this paper, the adaptive neural controllers of subsystems are proposed for a class of discrete-time switched nonlinear systems with dead zone inputs under arbitrary switching signals. Due to the complicated framework of the discrete-time switched nonlinear systems and the existence of the dead zone, it brings about difficulties for controlling such a class of systems. In addition, the radial basis function neural networks are employed to approximate the unknown terms of each subsystem. Switched update laws are designed while the parameter estimation is invariable until its corresponding subsystem is active. Then, the closed-loop system is stable and all the signals are bounded. Finally, to illustrate the effectiveness of the proposed method, an example is employed.


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## Body

## 1. Introduction

In the past few decades, intelligent control of uncertain nonlinear systems has attracted much more attention. Based on the universal approximation properties of neural networks (NNs) and fuzzy logic systems (FLSs), they are always used to approximate the unknown system functions or the control inputs. So far, there are a lot of results in intelligent control [1–6]. For example, the NN-based adaptive control methods were proposed for uncertain nonlinear systems in [7, 8] by using backstepping technique. The above results are all about nonlinear systems in continuous-time form. On the contrary, many researchers devoted much effort to study the adaptive control problem of discrete-time systems by using intelligent methods on the basis of these works [9–11].However, the previous results do not consider the switching phenomenon of the system. Actually, the switching phenomenon often exists in practical systems. We call this class of systems switched systems. Switched systems, which are used to model a wide variety of physical systems, consist of a family of continuous-time or discrete-time subsystems and a switching rule to govern the switching between the subsystems. Generally, the stability and stabilisation problems are the main concerns in the study of switched systems [12–16]. Subsequently, adaptive control and intelligent control of switched systems have been studied more and more [17–22]. But to the best of our knowledge, there are no results on discrete-time switched systems, and let alone results on discrete-time switched nonlinear systems.In addition, the dead zone input, as one of the most important input nonlinearity, widely exists in lots of practical systems. The existence of the dead zone input may damage the system performance or even destroy the system stability. Thus, the related robust control attracted high attention [8, 23–25]. Since the dead zone parameters are poorly known in most practical systems, adaptive control techniques are naturally used to deal with this problem. An adaptive tracking control strategy was proposed for a nonlinear system with nonsymmetric dead zone input having unknown but bounded parameters in [26].Motivated by the above discussion, we study the adaptive neural control problem of a class of discrete-time switched nonlinear systems with dead zone inputs under arbitrary switching signals. We use the radial basis function neural networks to approximate the unknown terms of each subsystem. Dead zone inputs of subsystems and switched update laws are designed such that the closed-loop system is stable and all the signals are bounded. Finally, an example is employed to illustrate the effectiveness of the proposed method.The main contributions of this paper compared with the existing results on switched and nonswitched nonlinear systems are in two aspects:(1)
Compared with the existing results [8, 23–25], in which the considered systems are all nonswitched systems or continuous-time switched systems, in this paper, we study a discrete-time switched uncertain nonlinear system.(2)
The dead zone input is considered in the discrete-time switched uncertain nonlinear system. As far as we know, there are no results on discrete-time switched systems with dead zone inputs. This is mainly due to the complicated framework of the discrete-time switched nonlinear systems, the existence of the dead zone, and the interaction between the system structure and switching.


## 2. System Description

In this paper, we consider the following discrete-time switched uncertain nonlinear system:(1)ξ1k+1=ξ2k,⋮ξn-1k+1=ξnk,ξnk+1=fσξk+gσξkuσk,yk=ξ1k,where ξ(k)=[ξ1(k),ξ2(k),…,ξn(k)]T∈Rn and y∈R are the system state and output, respectively. Moreover, the state of switched system (1) is assumed not to jump at the switching instants, which is a standard assumption in the switched system [27]. σ is the switching signal taking values in P={1,2,…,p} with p being the subsystems number of switched system (1). fi(x(k)) and gi(x(k)), i∈P are unknown smooth functions. ui(k) is the dead zone output of the ith subsystem, which is described by(2)uik=Dvik=hirvik-wir,ifvik≥wir,0,if-wil<vik<wir,hilvik+wil,ifvik≤-wil.In (2), vi(k) is the dead zone input of the ith subsystem. hir and hil stand for the right and left slopes of the dead zone characteristic, respectively. wir and wil represent the breakpoints of the input nonlinearity.According to [28], the dead zone can be further expressed as(3)uik=Dvik=hikvik+wik,i∈P,where (4)hik=hir,ifvik>0,hil,ifvik≤0,wik=-hirwir,ifvik≥wir,-hikvik,if-wil<vik<wir,hilwil,ifvik≤-wil.Assumption 1.
The parameters of dead zone are all positive, and they are unknown but bounded, that is,h_i=min⁡hir,hil, h¯i=max⁡hir,hil, and wik≤w¯i.From Assumption1, we can easily know that (5)h_i≤hik≤h¯i,wik≤w¯i=max⁡hirwir,hilwil.Assumption 2 (see [29–31]).
g
i
(
ξ
(
k
)
), i∈P, are strictly positive or negative and the signs of gi(ξ(k)) are known. Without loss of generality, we suppose that gi(ξ(k)) are positive with g_i and g¯i being the lower bound and upper bound, respectively. It implies that g¯i≥g_i>0.The control objective of this paper is to design dead zone inputsvi(k) of subsystems, such that(1)
the system stateξ(k) can track a given signal ξd(k)=[ξ1d(k),ξ2d(k),…,ξnd(k)]T, or equivalently speaking, the output can track a desired trajectory yd(k),(2)
all the signals in the closed-loop system are bounded under arbitrary switching signals.Assumption 3 (see [32]).
The desired trajectory of the system stateξd(k)=[ξ1d(k),ξ2d(k),…,ξnd(k)]T∈Rn satisfies the condition ξjd(k+1)=ξj+1d(k), j=1,…,n-1. Moreover yd(k) is a known smooth bounded function.According to Assumption2 and system (1), we can obtain that ξjd(k)=yd(k+j-1), j=1,…,n.


## 3. Neural Networks

Similar to FLSs, NNs also have the approximation properties. In this paper, we consider the case that there exist unknown functions in the systems. Here, we use RBFNNs to approximate the unknown functions of subsystems.For any unknown continuous functionF(X) defined on Ω⊂Rn, there exists NN θTφ(X), such that(6)FX=θTφX+εX,where X∈Ω⊂Rn is the input variable of the NN, θ=[θ1,…,θl]T is the weight vector with l being the node number of NNs, ε(X) is the approximation error, and φ(X)=[ϕ1(X),…,ϕl(X)]T is the basis function. In RBFNNs, ϕi(X), i=1,…,l, are chosen as Gaussian functions(7)ϕiX=exp⁡-X-ciTX-ciμi2,where ci=[ci1,…,cin]T and μi, i=1,…,l, are the centers and widths of the Gaussian functions, respectively.According to (7), we have 0<ϕi(X)≤1. Thus, we can further obtain that ϕi2(X)≤1, i=1,…,l. This leads to the following inequality:(8)φTXφX≤l.In this paper, since there are unknown functions in (11), we cannot obtain vi0, i=1,…,n, directly. Thus, the neural networks are used to approximate the unknown function vi0, i=1,…,n. In fact, the neural networks in this paper are used for identification.


## 4. Dead Zone Input Design

Define the tracking error at instantk as(9)ejk=ξjk-ξjdk+1=ξjk-ydk+j-1.Then, we can get the nth state error en(k)=ξn(k)-yd(k+n-1). Furthermore, we have(10)enk+1=fiξk+giξkuik-ydk+n=fik+gikhikvik+gikwik-ydk+n,where fi(k) and gi(k) are the abbreviations of fi(ξ(k)) and gi(ξ(k)), respectively.Define functions(11)vi0k=-1gikmikfik-ydk+n,i∈P.Because of the existence of the unknown functions in (11), vi0(k) cannot be directly obtained. Here, we use RBFNNs to approximate them(12)vi0k=θiTφSk+εiSk,i∈P,where θi are the weight matrices which are assumed to be bounded, that is, θi≤θ¯i, and φ(S(k)) and εi(S(k)) are the basis function vector and the approximation errors, respectively. Furthermore, φ(S(k))2≤l and εi(S(k))≤ε¯i. S(k)=[ξT(k),ydT(k),ydT(k+n)]T is the input vector of RBFNNs.By adding and subtractinggi(k)hi(k)vi0(k) on the right side of equation (10), we can conclude that(13)enk+1=gikhikvik-θiTφk-εik+wikhik,where φ(k) and εi(k) are the shorthand for φ(S(k)) and εi∗(S(k)), respectively. Denote that wi(k)/hi(k)≤w¯i/h_i≜ηi and ηi, i∈P, are unknown constants. Let η^i be the estimation of ηi and lwt η~i(k)=η^i(k)-ηi be the estimation error.According to (13), we design the controllers as(14)vik=θ^iTkφk+η^ik,where θ^i(k) are the estimation of θi(k). Let the estimation error satisfy θ~i(k)=θ^i(k)-θi.The switched update laws are designed as(15)θ^ik+1=θ^ik-Γiδiθ^ik+φkenk+1,σ=i,0,σ≠i,η^ik+1=η^ik-μiτiη^ik+enk+1,σ=i,0,σ≠i,where Γi and μi are positive matrices and positive design parameters, respectively.Based on (14), (13) becomes(16)enk+1=gikhikθ~iTφk-εik+η^ik+wikhik=gikhikθ~iTφk+η~ik-εik+ηi+wikhik.


## 5. Stability Analysis

In this section, the stability of switched system (1) and the boundedness of all the signals in the closed-loop system are proved under arbitrary switching signals.Theorem 4.
Consider the discrete-time switched system (1), the dead zone (3), the switched update laws given by (15), and the input of the ith dead zone selected as in (14) under Assumptions 1–3. If the design parameters satisfy the conditions(17)γσ<1-μσg¯h¯3+lg¯h¯+l+μσ,δσ<1γσ+γσg¯h¯,μσ<γσγσ+g¯h¯τσ,then the boundedness of all the signals in the closed-loop switched system can be ensured and the state ξ(k) can track the reference signal ξd(k) and the system output can follow the desired trajectory yd(k).Proof.
Choose the following common Lyapunov function for system (1) as(18)Vk=1g¯h¯en2k+∑i=1ptr⁡θ~iTkΓi-1θ~ik+∑i=1p1μiη~i2k,i∈P,where g¯=maxi∈Pg¯i and h¯=maxi∈Ph¯i.
The first difference ofV(k) is(19)ΔVk=1g¯h¯en2k+1-en2k+∑i=1ptr⁡θ~iTk+1Γi-1θ~ik+1-∑i=1ptr⁡θ~iTkΓi-1θ~ik+∑i=1p1μiη~i2k+1-∑i=1p1μiη~i2k.
Substituting (15) into (19), one has(20)ΔVk=1g¯h¯en2k+1-en2k-2tr⁡θ~σTkδσθ^σk+φkenk+1+tr⁡δσθ^σk+φkenk+1TΓσδσθ^σk+φkenk+1-2η~σkτση^σk+enk+1+μστση^σk+enk+12=1g¯h¯en2k+1-en2k-2δσθ~σTkθ^σk-2θ~σTkφkenk+1+tr⁡δσ2θ^σTkΓσθ^σk+tr⁡enk+1φTkΓσφkenk+1+2tr⁡δσθ^σTkΓσφkenk+1-2τση~σkη^σk+μστση^σ2k-2η~σkenk+1+μσen2k+1+2μστση^σkenk+1,with γσ being the maximum eigenvalue of Γσ.
According to (16), we can conclude that(21)θ~iTkφk+η~ik=enk+1gikhik+εik-ηi-wikhik.
Then,ΔV(k) becomes(22)ΔVk≤1g¯h¯en2k+1-en2k-2g¯σkh¯σken2k+1-2εσkenk+1+2ησenk+1+wσkhσkenk+1-2δσθ~σTkθ^σk+μστση^σ2k+γσδσ2θ^σk2+γσlen2k+1+2γσδσθ^σTkφkenk+1-2τση~σkη^σk+μσen2k+1+2μστση^σkenk+1.
Based on Assumption2, it holds that (23)-2gikhiken2k+1≤-2g¯h¯en2k+1.
By using Young’s inequality, we have the following inequalities:(24)-2εσkenk+1≤g¯h¯γσε¯σ2+γσg¯h¯en2k+1,2ησenk+1≤g¯h¯γσησ2+γσg¯h¯en2k+1,2wikhikenk+1≤g¯h¯γσηi2+γσg¯h¯en2k+1,2γσδσθ^σTkφkenk+1≤g¯h¯γσδσ2θ^σk2+γσlg¯h¯en2k+1,2μστση^σkenk+1≤g¯h¯μστσ2γση^σ2k+γσμσg¯h¯en2k+1,and we consider the following equations: (25)2θ~σTkθ^σk=θ~σk2+θ^σk2-θσ2,2η~σkη^σk=η~σ2k+η^σ2k-ησ2.
Then, we have(26)ΔVk≤1g¯h¯en2k+1-en2k-2g¯h¯en2k+1+g¯h¯γσε¯σ2+γσg¯h¯en2k+1+2g¯h¯γσησ2+2γσg¯h¯en2k+1-δσθ~σk2-δσθ^σk2+δσθσ2+μστση^σ2k+γσδσ2θ^σk2+γσlen2k+1+g¯h¯γσδσ2θ^σk2+γσlg¯h¯en2k+1-τση~σ2k-τση^σ2k+τσησ2+μσen2k+1g¯h¯μστσ2γση^σ2k+γσμσg¯h¯en2k+1≤-1g¯h¯M1en2k+1-δσM2θ^σk2-τσM3η^σ2k-1g¯h¯en2k-δσθ~σk2-τση~σ2k+β,where (27)M1=1-3γσ-γσlg¯h¯-γσl-μσg¯h¯-γσμσ,M2=1-γσδσ-γσδσg¯h¯,M3=1-μσ-g¯h¯μστσγσ,β=g¯h¯γσε¯σ2+2g¯h¯γσησ2+δσθσ2+τσησ2.
If the design parameters satisfy condition (17), then we can deduce that(28)ΔVik≤-1g¯h¯en2k-δσθ~σk2-τση~σ2k+β.
If|en(k)|>g¯h¯β or θ~σ(k)>β/δσ or |η~σ(k)|>β/τσ holds, we have ΔVi(k)<0. According to the standard Lyapunov extension theorem, the tracking error en(k) and the estimation errors θ~σ(k) and η~σ(k) are bounded. Since ej(k)=ξj(k)-ξjd(k+1)=en(k-n+j), j=1,…,n, we can easily know that ej(k), j=1,…,n-1, are also bounded. Furthermore, because θσ and ησ are bounded, the boundedness of the estimated parameters θ^σ(k) and η^σ(k) is guaranteed by θ~σ(k)=θ^σ(k)-θσ and η~σ(k)=η^σ(k)-ησ. Thus, from (14), we know that the dead zone inputs vσ are bounded. Based on the definition of dead zone, one can conclude that the control inputs uσ are also bounded. Therefore, all the signals in the closed-loop system are bounded under arbitrary switching signals.Remark 5.
As we all know that a switched system might be unstable even if all the subsystems are stable, arbitrary switching method can guarantee the stability of the switched system in this case. A sufficient condition for analysing the stability of switched systems under arbitrary switching signals is that there exists a common Lyapunov function. In many cases, it is easy to find a Lyapunov function for each subsystem. However, a common Lyapunov is hard to find for switched systems. In this paper, we have successfully found a common Lyapunov function shown as in (18).Remark 6.
There are many parameters in the design procedure.hir, hil, wir, and wil, i=1,2, are parameters of dead zone. They are all positive based on Assumption 1. In addition, Γi, δi, μi, and τi, i=1,2, are introduced to improve the flexibility of the update laws. In this paper, they are chosen to be positive. But how to choose a set of optimal ones to ensure the satisfactory performance is still an open problem; they need to be flexibly selected. In the following, a constructive suggestion on how to choose these parameters is given. From the above stability analysis, we can see that the bounds of en(k), θ~σ(k), and η~σk depend on the values of g¯h¯, β, δσ(k), and τσ(k). Actually, we can select smaller g¯h¯ and larger δσ(k) and τσ(k) to obtain smaller bounds of en(k), θ~σ(k), and η~σ(k) to further get good performance. But, it is important to note that these parameters must satisfy conditions (17). In the simulation, a trial-and-error method is used.


## 6. Simulation

In this section, we apply the proposed approach to the following discrete-time switched nonlinear system:(29)ξ1k+1=ξ2k,ξ2k+1=fσ1ξk+gσ1ξkuσk,ξ3k+1=ξ4k,ξ4k+1=fσ2ξk+gσ2ξkuσk,y1k=ξ1k,y2k=ξ3k,where f11(ξ¯(k))=ξ1(k)+ξ22(k), f12(ξ¯(k))=1/2(ξ1(k)+ξ2)(k), g11(ξ¯(k))=ξ1(k)ξ2(k), g12(ξ¯(k))=ξ1(k)ξ2(k)+ξ1(k)ξ22(k), f21(ξ¯(k))=ξ3(k)+ξ12(k), f22(ξ¯(k))=ξ1(k)+ξ3(k), g21(ξ¯(k))=ξ1(k)ξ32(k)+ξ3(k)ξ42(k), and g22(ξ¯(k))=ξ1(k)ξ3(k)+ξ3(k)ξ4(k). The switching signal takes values in P={1,2}.The dead zone is described as(30)uik=Dvik=hirvik-wir,ifvik≥wir,0,if-wil<vik<wir,hilvik+wil,ifvik≤-wil.The parameters of the dead zone inputs are selected ash1r=0.3, h2r=0.4, h1l=0.2, h2l=0.3, w1r=0.5, w2r=0.7, w1l=0.6, and w2l=0.8.Define the tracking errorse1(k)=y1(k)-yd1(k)=ξ1(k)-yd1(k) and e2(k)=y2(k)-yd2(k)=ξ3(k)-yd2(k). The desired signals yd1(k) and yd2(k) are selected as yd1(k)=0.28[sin⁡kπ/10+cos⁡(kπ/10)] and yd2(k)=0.22[sin⁡kπ/10+cos⁡(kπ/10)], respectively. The initial values for the system states are ξ1(0)=0.01, ξ2(0)=0.01, ξ1(0)=0.01, and ξ2(0)=0.01.The dead zone input is designed for each subsystem(31)vik=θ^iTkφk+η^ik,where φ(k) is the Gaussian function.The switched update laws are designed fori=1,2, as(32)θ^ik+1=θ^ik-Γiδiθ^ik+φkenk+1,σ=i,0,σ≠i,η^ik+1=η^ik-μiτiη^ik+enk+1,σ=i,0,σ≠i,where Γ1=0.5I, Γ2=0.2I, δ1=2, δ2=2.5, μ1=0.5, μ2=0.7, τ1=2.3, and τ2=1.8.Furthermore, we give the simulation results in Figures1–7. Figure 1 gives the curves of y1 and yd1. Figure 2 shows the trajectories of y2 and yd2. Figures 3 and 4 display the tracking errors e1 and e2, respectively. From Figures 1–4, we can observe that a good tracking performance is obtained. Moreover, the dead zone inputs are shown in Figures 5 and 6, which explain the boundedness of the dead zone inputs. Figure 7 is the switching signal.Figure 1
The trajectories ofy1 and yd1.Figure 2
The trajectories ofy2 and yd2.Figure 3
The trajectory of tracking errore1.Figure 4
The trajectory of tracking errore2.Figure 5
Control inputu1.Figure 6
Control inputu2.Figure 7
Switching signalσ.


## 7. Conclusion

We design a controller of each subsystem for a class of discrete-time switched nonlinear systems with dead zone inputs under arbitrary switching signals in this paper. The unknown functions are approximated by RBFNNs. Switched update laws are designed while the parameter estimation is invariable until its corresponding subsystem is active. Then, the closed-loop system is stable and all the signals are bounded. Due to the complicated framework of the discrete-time switched nonlinear systems and the existence of the dead zone, it brings about great difficulties for controlling such a class of systems.



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