# Adaptive Neural Control with Prespecified Tracking Accuracy for a Class of Switched Systems Subject to Input Delay

Authors: Xu Zhang; Jian Wu; Wu Ai; Jing Li
Journal: Complexity
 (2019)

---

## Abstract

This paper is concerned with the adaptive tracking control design for a class of uncertain switched systems subject to input delay. Unlike the existing results on uncertain switched systems, the new proposed control scheme ensures that the tracking error converges to the accuracy given a priori according to the requirement. To achieve this aim, some nonnegative switching functions are introduced to replace the conventional Lyapunov function. In addition, neural networks are used to approximate the unknown simultaneous domination functions. By combining the backstepping technique and some common nonnegative switching functions, a stable adaptive neural controller is established. It can be shown that the closed-loop system is semiglobally uniformly ultimately bounded (SGUUB) and the tracking error satisfies the predefined accuracy. The effectiveness of the proposed control scheme is verified by a simulation example.


---

## Body

## 1. Introduction

As we all know, switched systems are a special class of hybrid systems, which have been paid much attention due to their practical applications. Many practical engineering systems are well described by switched systems such as networked system, robotic system, and traffic surveillance and control system [1, 2]. It provides a strong motivation for investigating the switched systems. Over the past two decades, a lot of scholars focused on dealing with the control issues of the switched systems, and a number of interesting and meaningful results have been presented in [3–10]. In [3, 4], based on the multiple linear copositive Lyapunov function method, the stability problem for switched positive linear systems with the average dwell time switching is studied. By using the technique of adding a power integrator, Fu et al. [5] discuss the finite-time control problem, and an adaptive controller is constructed to achieve the globally finite-time stabilization of the switched nonlinear systems with the powers of positive odd rational numbers. It is worth pointing out that the mentioned references focus on dealing with the control problem for the switched systems without unknown system functions.It is noted that in many practical industrial processes, not all the system functions can be accurately modeled. Thus, an important problem is to overcome the foregoing uncertainties. In recent years, some approaches have been proposed to overcome the restriction and a large amount of encouraging results have been reported [11–24]. When the considered systems are unknown, neural networks (NNs)/fuzzy logic systems are often used to approximate the uncertain function to construct a controller. Most works on adaptive NN control are based on backstepping technique in [15–24]. For example, a NN-based adaptive controller has been designed with backstepping method, and NN-based design methods warrant the semiglobal stability for the nonlinear systems in a strict-feedback form in [15]. In [16], a backstepping controller containing a recurrent-neural-network-based uncertainty observer and a robust controller has been developed to address the position control problem for induction servomotors. By utilizing backstepping technique and NN appropriate method in [17, 18], the stability problem of a class of stochastic nonlinear strict-feedback systems has been explored. It is worth noting that the aforementioned results on adaptive NN control design only consider the uncertain nonswitched nonlinear systems. Recently, several works have reported on uncertain switched nonlinear systems [25–27]. For uncertain nonlinear multiagent systems, distributed finite-time control scheme and cooperative adaptive neural partial tracking errors constrained control have been addressed in [28] and [29], respectively.In addition, input delay plays an increasing important role in the real engineering systems. It has been attracted much attention, and numerous results have been presented in [30–38]. For example, in [30], the augmented complete Lyapunov-Krasovskii functional method has been adopted to solve the stability issue for linear systems with input delay. Based on the stability conditions proposed in [35], a state transformation technique has been presented to tackle the stability problem for linear systems with input delays. In [31], an improved reciprocally convex approach has been proposed to achieve the asymptotic stabilization of the continuous-time systems with time-varying input delays. With respect to the switched systems with input delay, a few results have been reported in [37, 38].Inspired by the above discussions, this paper addresses the adaptive neural tracking control problem for a class of uncertain switched systems subject to input delay. The main work of this paper can be summarized as follows. (i) By introducing some nonnegative switching functions, the desired adaptive neural controller is developed, and it is guaranteed that the tracking error can achieve the accuracy which can be adjusted based on the actual demands. (ii) The Pade approximation approach is borrowed to deal with the problem of input delay in the considered systems. (iii) In each backstepping design, NN is employed to approximate the simultaneous domination function rather than the switched system function. Thus, the number of adaptive learning parameters can be reduced.


## 2. Preliminaries

This section introduces some preliminaries on the system stability and NN approximation theory.Definition 1 ((SGUUB) [39]).
For a general nonlinear system(1)x˙=fx,t,xt0=x0, where x(t)∈Rn is the system state, f:Rn×[t0,∞]→Rn is a continuous vector-valued function, and t0 and x0∈Rn denote the initial time and the initial state vector, respectively. If there exists a compact set U∈Rn such that for all x0∈U, there exists a δ>0 and a number T(δ,x0) such that x(t)<δ for all t≥t0+T(δ,x0); we say that the solution of this system is semiglobally uniformly ultimately bounded (SGUUB).Lemma 2 ((Young’s inequality) [39]).
For∀(x,y)∈R2, the following inequality holds: (2)xy≤ιppxp+1qιqyq,where ι>0, p>1, q>1, and (p-1)(q-1)=1.
The following switching functions are introduced and they will be used to design the desired adaptive neural controller.(3)Nmϵe=e-ϵmm!,e>ϵ0,e≤ϵand(4)Smϵe=sgn⁡e,e≥ϵ1-2cosm⁡π2sinmπ4ϵe+ϵ,e<ϵ,where ϵ>0 is the given accuracy and m is a positive integer.The main characteristics of functionsNmϵ(e) and Smϵ(e) are given by the following lemma.Lemma 3 (see [40]).
FunctionsNmϵ(e) and Smϵ(e) have the following properties:
(‎1)Nmϵ(e)∈Cm and Smϵ(e)∈Cm, where Cm is the set of m-th order continuously differentiable functions.
(‎2) Forl=1,2,…,m, the l-th order derivatives of Nm+1ϵ(e) are (5)dlNm+1ϵedel=Nm-l+1ϵeSm-l+1ϵel.
(‎3)Nm+1ϵ(e) is a nonnegative function, and when and only when e≤ϵ, Nm+1ϵ(e)=0.
(‎4) Forl=1,2,…,m, Nlϵ(e)[Slϵ(e)]2=Nlϵ(e).
In addition, in this paper, some unknown continuous functionsFi(Xi), i=1,2,…,m, which will be defined later, are adopted to design the desired controller, and some RBF NNs are used to approximate these functions on a compact set Ωi, i.e.,(6)FiXi=WiTSiXi+εiXi,where the input vectors Xi∈Ωi⊂Rm, weight vectors Wi∈Rl, the NN node number l>1, εi(Xi) are the NN inherent approximation errors which are bounded over the compact sets, i.e., εiXi≤εi∗ with unknown constants εi∗, and S(Xi)=[s1(Xi),…,sl(Xi)]T:Ωi→Rl are known smooth vector functions with sq(Xi) being chosen as the commonly used Gaussian functions, which have the form (7)sqXi=exp⁡-Xi-μqTXi-μqη2,q=1,…,l,where μq=[μq1,…,μqm]T are the centers of the receptive field and ηq>0 are the spreads of the Gaussian functions. The optimal weight vector Wi is defined as (8)Wi=arg⁡minW^i∈Rl⁡supXi∈Ωi⁡FXi-W^iTSiXi,where Wi^ is the estimate of Wi.


## 3. Problem Formulation, Control Design, and Stability Analysis

### 3.1. Problem Description

Consider a class of switched nonlinear systems described by(9)χ˙i=χi+1+fiσtχ¯i,i=1,…,m-1χ˙m=ut-τ+fmσtχ¯my=χ1where χ=[χ1,χ2,…,χm]T∈Rm is the state vector. y∈R and u∈R denote the system output and the control input, respectively. σ(t):[0,+∞)→M={1,2,…,n} represents a switching signal, and fik:Ri→R are unknown continuous functions for i=1,2,…,m, k=1,2,…,n. τ represents the delayed time.To deal with the problem of input delay in system (55), the Pade approximation approach used in [35] is introduced. Subsequently, we can have (10)Lut-τ=exp⁡-τvLut=exp⁡-τv/2exp⁡τv/2Lut≈1-τv/21+τv/2Lut,where L[u(t)] is the Laplace transform of u(t) and v is Laplace variable. For further analysis, a new variable χm+1 is proposed and it conforms to the following relation (11)Lut1-τv/21+τv/2=Lχm+1t-Lut.Then the following equation is obtained (12)u-τ2u˙=χm+1+τ2χ˙m+1-u-τ2u˙. That is, we have the following equation(13)χ˙m+1=-κχm+1+2κu,where κ=2/τ.Remark 4.
In this paper, Pade approximation method is introduced to deal with small delay. Since Pade approximation has some limitations in handling delay, the proposed scheme cannot work in large-delay case. Relaxed control design for systems with long delay and actuators saturation deserves further investigation.
Based on the above transformation, system (6) can be rewritten as follows(14)χ˙i=χi+1+fiσtχ¯i,i=1,…,m-1χ˙m=χm+1-u+fmσtχ¯mχ˙m+1=-κχm+1+2κuy=χ1.
The control objective of this paper is addressed as follows: for a given reference signalyr(t), to develop an adaptive neural controller u such that (i)
all the closed-loop signals remain SGUUB;(ii)
the tracking errore1=y-yr≤ϵ as t→∞.Remark 5.
In fact, some control schemes have been proposed for uncertain systems to achieve the objective of ensuring that the tracking error converges to the accuracy given a priori according to the requirement; e.g., see [6, 40]. However, this paper studies the control problem for a class of uncertain switched systems with input delay. For such more general system, the existing control methods cannot be directly employed to solve this control issue. As far as the authors know, this is the first work to address the practical control for uncertain switched systems with input delay.To design the desired adaptive neural controller, the following assumption is given.Assumption 6.
The reference signalyr(t) and its derivative yr(i)(t) are continuous and bounded for i=1,2,…,m.


### 3.2. Adaptive Neural Controller Design and Stability Analysis

To develop the desired control scheme for system (14), the following error variables are defined(15)e1=χ1-yr,ei=χi-αi-1,i=2,…,m-1em=χm-αm-1+1κχm+1,where αi,i=1,…,m-1, denote the virtual control variables and will be designed later.Following the adaptive backstepping control method, we present the design process of the desired controller and the adaptation laws.Step‎ 1. Consider the error variable e1 in (15) and the first subsystem in (14). Then, we have(16)e˙1=χ˙1-y˙r=χ2+f1σtχ¯1-y˙r.Select the following nonnegative function(17)V1=m!Nm+1ϵe1,and then along the trajectory of (16), by using Lemma 3, the time derivative of V1 is given as(18)dV1dt=dNm+1ϵe1de1e˙1m!=Nmϵe1Smϵe1m!χ2+f1σtχ¯1-y˙r≤Nmϵe1Smϵe1m!e2+α1+Smϵe1G1χ¯1-y˙r,where G1(χ¯1) is a simultaneous domination function which is continuous and unknown. For example, here it can be selected as G1(χ¯1)=∑1nf1kχ¯12+1. Since f1k(χ¯1) is unknown, it is obvious that G1(χ¯1) is unknown and it cannot be used to design the controller directly. As shown in (6), an RBF NN is employed to approximate F1(X1) online, and then one can have(19)dV1dt≤Nmϵe1Smϵe1m!e2+α1+Smϵe1G1χ¯1-y˙r=Nmϵe1Smϵe1m!e2+α1+F1X1=Nmϵe1Smϵe1m!e2+α1+W1TS1X1+ε1X1,where we define F1(X1)=Smϵ(e1)G1(χ¯1)-y˙r with X1=[χ1,yr,y˙r]T∈Ω1⊆R3.The Young’s inequality is used for the following analysis:(20)Nmϵe1Smϵe1m!W1TS1X1≤Nmϵe1Smϵe12m!W1S1X1≤Nmϵe1Smϵe12m!12a12ψ1S1X12+12a12,where ψ1≔W12 and a1>0 is a design parameter, and(21)Nmϵe1Smϵe1m!ε1X1≤Nmϵe1Smϵe12m!ε1∗.Substituting (20) and (21) into (19) yields(22)dV1dt≤Nmϵe1Smϵe1m!e2+α1+12a12ψ1S1X12Smϵe1+12a12Smϵe1+Smϵe1ε1∗=Nmϵe1Smϵe1m!e2+α1+12a12ψ1S1X12Smϵe1+Smϵe1ρ1,where ρ1≔1/2a12+ε1∗ is an unknown constant.The first virtual controller is designed from the above information.(23)α1=-k1+14Nmϵe1Smϵe1m!-12a12ψ1^S1X12Smϵe1-ρ1^Smϵe1-ϵ+1Smϵe1,where k1 is a positive constant and ψ1^ and ρ1^ denote the estimates of ψ1 and ρ1.Obviously, (22) can be changed into(24)dV1dt≤Nmϵe1Smϵe1m!e2-k1+14Nmϵe1Smϵe1m!-12a12ψ1^S1X12Smϵe1-ρ1^Smϵe1-ϵ+1Smϵe1+12a12ψ1S1X12Smϵe1+ρ1Smϵe1≤-k1+14Nmϵe1m!2+12a12ψ~1Nmϵe1Smϵe12m!S1X12+ρ~1Nmϵe1Smϵe12m!+Nmϵe1m!e2-ϵ+1with estimate errors ψ~1≔ψ1-ψ^1 and ρ~1≔ρ1-ρ^1.For further study, we establish the nonnegative function as below(25)V¯1=V1+12λ1ψ~12+12γ1ρ~12,where λ1>0 and γ1>0 are design parameters. We choose the adaptation laws as follows:(26)ψ^˙1=λ12a12Nmϵe1S1X12Smϵe12m!,ρ^˙1=γ1Nmϵe1Smϵe12m!.Then we can obtain(27)dV¯1dt=dV1dt-1λ1ψ~1ψ^˙1-1γ1ρ~1ρ^˙1≤-k1+14Nmϵe1m!2-1λ1ψ~1ψ^˙1-λ12a12S1X12Nmϵe1Smϵe12m!-1γ1ρ~1ρ^˙1-γ1Nmϵe1Smϵe12m!+Nmϵe1m!e2-ϵ+1=-k1+14Nmϵe1m!2+Nmϵe1m!e2-ϵ+1.Step i   ( i = 2,3 , … , m - 1 ) .The method of recursion is used in each step when the value of i increases from 2 to m-1. Similar to Step‎ 1, the time derivative of ei is(28)e˙i=χ˙i-α˙i-1=χi+1+fiσtχ¯i-α˙i-1,where α˙i-1=∑j=1i-1∂αi-1/∂χjχj+1+fjσtχ¯j+Λi-1 with Λi-1=∑j=0i-1∂αi-1/∂yd(j)yd(j+1) + ∑j=1i-1∂αi-1/∂ψ^jψ^˙j + ∑j=1i-1∂αi-1/∂ρ^jρ^˙j.Consider the following nonnegative functions(29)Vi=V¯i-1+Nm-i+2ϵeim-i+1!,V¯i=Vi+12λiψ~i2+12γiρ~i2,where λi>0 and γi>0 are design parameters, estimate errors ψ~i≔ψi-ψ^i and ρ~i≔ρi-ρ^i, and ψi^ and ρi^ denote the estimates of ψi and ρi which will be defined later.Then we can obtain(30)dVidt=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!e˙i=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+fiσtχ¯i-Λi-1-∑j=1i-1∂αi-1∂χjχj+1+fjσtχ¯j.Meanwhile the following inequality holds(31)-Nm-i+1ϵeiSm-i+1ϵeim-i+1!∑j=1i-1∂αi-1∂χjfjσtχ¯j≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!∑j=1i-1∂αi-1∂χjfjσtχ¯j2+1.Substituting (31) into (30), we gain(32)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+Sm-i+1ϵeiGiχ¯i+∑j=1i-1Γjχj¯-Λi-1-∑j=1i-1∂αi-1∂χjχj+1where Gi(χ¯i) and Γj(χj¯) are simultaneous domination functions which are continuous and unknown. For example, here they can be chosen as Gi(χ¯i)=∑k=1n[fik(χi¯)]2+1, Γj(χ¯j)=∑k=1n[∂αi-1/∂χjfjk(χ¯j)]2+1. Define Fi(Xi)=Sm-i+1ϵeiGiχ¯i+∑j=1i-1Γjχj¯-Λi-1-∑j=1i-1∂αi-1/∂χjχj+1 with Xi=[χ¯iT,αi-1,∂αi-1/∂χ1,…,∂αi-1/∂χi-1,Λi-1]T∈Ωi⊆R2i+1, and an RBF NN (6) is used to approximate Fi(Xi) online. Based on this information, (32) can be changed as below(33)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+FiXi=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+WiTSiXi+εiXi.Similar to (20) and (21), by using the Young’s inequality, inequalities (34) and (35) can be gained easily:​(34)Nm-i+1ϵeiSm-i+1ϵeim-i+1!WiTSiXi≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!WiTSiXi≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!12ai2ψiSiXi2+12ai2 where ψi=Wi2, ai>0 is a design parameter, and(35)Nm-i+1ϵeiSm-i+1ϵeim-i+1!εiXi≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!εi∗.Substituting (34) and (35) into (33) yields(36)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+12ai2ψiSiXi2+12ai2Sm-i+1ϵei+Sm-i+1ϵeiεi∗=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+12ai2ψiSiXi2Sm-i+1ϵei+Sm-i+1ϵeiρi,where ρi=1/2ai2+εi∗ is an unknown constant.We construct thei-th virtual controller as follows:(37)αi=-ki+54Nm-i+1ϵeiSm-i+1ϵeim-i+1!-12ai2ψ^iSiXi2Sm-i+1ϵei-Sm-i+1ϵeiρ^i-ϵ+1Sm-i+1ϵei.Obviously, based on (37), the undermentioned inequality is true:(38)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+12ai2ψ~iSiXi2Sm-i+1ϵei+Sm-i+1ϵeiρ~i-ϵ+1Sm-i+1ϵei-ki+54Nm-i+1ϵeiSm-i+1ϵeim-i+1!≤-∑j=1ikjNm-j+1ϵejm-j+1!2+Nm-i+1ϵeiSm-i+1ϵei2m-i+1!×12ai2ψ~iSiXi2+ρ~i-14Nm-i+2ϵei-1m-i+22+Nm-i+2ϵeim-i+2!ei-ϵ+1-54Nm-i+1ϵeim-i+12+Nm-i+1ϵeim-i+1!ei+1-ϵ+1.Let(39)Ui=-14Nm-i+2ϵei-1m-i+22+Nm-i+2ϵei-1m-i+2!ei-ϵ+1-Nm-i+1ϵeim-i+12.It is obvious thatUi≤0 when ei≤(ϵ+1). When ei>(ϵ+1), using the Young’s inequality, we obtain(40)Ui≤ei-ϵ+12-Nm-i+1ϵeim-i+12=ei-ϵ+12-ei-ϵ2m-i+1≤ei-ϵ2-ei-ϵ2m-i+1=ei-ϵ21-ei-ϵ2m-i+1≤0.Hence, Ui≤0 holds all the time.Based on the above information, we choose(41)ψ^˙i=λi2ai2Nm-i+1ϵeiSm-i+1ϵei2m-i+1!SiXi2ρ^˙i=γiNm-i+1ϵeiSm-i+1ϵei2m-i+1!.Thus, we can have(42)dV¯idt=dVidt-1λiψ~iψ^˙i-1γiρ~iρ^˙i≤-∑j=1ikjNm-j+1ϵejm-j+12-1λiψ~ψ^˙i-λi2ai2Nm-i+1ϵeiSm-i+1ϵei2m-i+1!SiXi2-1γiρ~iρ^˙i-γiNm-i+1ϵeiSm-i+1ϵei2m-i+1!-14Nm-i+1ϵeim-i+1!2+Nm-i+1ϵeim-i+1!ei+1-ϵ+1≤-∑j=1ikjNm-j+1ϵejm-j+12-14Nm-i+1ϵeim-i+1!2+Nm-i+1ϵeim-i+1!ei+1-ϵ+1.Step m . Combining (14) and (15), we have(43)e˙m=χ˙m-α˙m-1+1κχ˙m+1=χm+1-u+fmσtχ¯m-α˙m-1+1κχ˙m+1,where α˙m-1=∑j=1m-1∂αm-1/∂χj[χj+1+fjσ(t)(χ¯j)]+Λm-1 with Λm-1=∑j=0m-1∂αm-1/∂yd(j)yd(j+1)+∑j=1m-1∂αm-1/∂ψ^jψ^˙j+∑j=1m-1∂αm-1/∂ρ^jρ^˙j.ChooseV¯m=Vm+1/2λmψ~m2+1/2γmρ~m2, Vm=V¯m-1+N2ϵ(em), where λm>0 and γm>0 are design parameters, and design the actual controller as below(44)u=-km+1N1ϵem-12am2ψ^m2SmXm2S1ϵem-S1ϵemρ^m-ϵ+1S1ϵem,where am>0 and km>0 are design parameters, estimate errors ψ~m≔ψm-ψ^m and ρ~m≔ρm-ρ^m, and ψ^m and ρ^m denote the estimates of ψm and ρm which will be defined later.Similar to the design process of Stepi, let i=m. Then we can have ψm=Wm2 and ρm=1/2am2+εm∗ where εm∗ is the upper bound of the RBF NN approximation error. The following adaptive laws are chosen(45)ψ^˙m=λm2am2N1ϵemS1ϵem2SmXm2ρ^˙m=γmN1ϵemS1ϵem2,where we define Xm=[χ¯mT,αm-1, ∂αm-1/∂χ1,…,∂αm-1/∂χm-1, Λm-1]T ∈ Ωm⊆R2m+1.According to the above information, we have(46)dV¯mdt≤-∑j=1mkjNm-j+1ϵejm-j+1!2-14N2ϵem-12!2-N1ϵem2+N2ϵem-12!em-ϵ+1≤-∑j=1mkjNm-j+1ϵejm-j+1!2.Based on inequality (46), the main result of this paper is summarized by the following theorem.Theorem 7.
Under Assumption6, consider the closed-loop system that comprises the plant (9), virtual control variables (23) and (37), and actual control law (44) with adaptive laws (26), (41), and (45). For the bounded initial condition on a compact set Ω0, assume there exist sufficiently large compact sets Ωi⊆R2i+1, i=1,2,…,m, such that Xi∈Ωi for all t≥0; the following statements hold. (i)
The tracking errore1=y-yr≤ϵ as t→∞.(ii)
All the closed-loop signals remain semiglobal bounded.Proof.
(i) From (46), by employing Barbalat’s Lemma [41], we can easily conclude that the tracking error satisfies e1=y-yr≤ϵ as t→∞.
(ii) Considering inequality (46), we can see that V¯m is nonincreasing. Therefore, it can be concluded that, for all 1≤i≤m, error signals including ei(t),ρ~i(t) and ψ~i(t) are bounded. To prove the boundedness of all the closed-loop signals, it can be shown that ρ^i(t)=ρi-ρ~i(t) and ψ^i(t)=ψi-ψi(t) are bounded according to the boundedness of ρ~i(t) and ψ~i(t). Next, it follows from the boundedness of e1(t) and yr(t) that χ1(t)=yr(t)+e1(t) is also bounded. Then, the boundedness of α1 given in (23) can be easily obtained. The boundedness of e2 and α1 further implies that χ2=e2+α1 is bounded. Continuing in the same fashion, we conclude that all the closed-loop signals are bounded. In addition, similar to the existing works on adaptive neural network control, the proposed control scheme just guarantees that the closed-loop system is semiglobally stable since the initial condition of the controlled system must remain on a compact set Ω0.Remark 8.
To show the semiglobal stability of the controlled system, the following analysis is presented. Firstly, fromχ1=e1+yr, we have χ1≤e1+yr≤ϵ+Y, where Y is the upper bound of reference signal yr. Thus, the boundedness of χ1 is obtained as t→∞. Furthermore, from the above analysis, it can be seen that ψ^1 and ρ^1 are bounded. Assume that ψ^1≤Ψ1 and ρ^1≤Φ1 with positive constants Ψ1 and Φ1. Since χ2=e2+α1, then we can obtain the following inequality for t→∞,(47)χ2≤e2+α1≤2ϵ+k1+14Nmϵe1m!+12a12Ψ1S¯1+Φ1+1=2ϵ+12a12Ψ1S¯1+Φ1+1,where -1≤Smϵ(e1)≤1, Nmϵ(e1)=0 when e1≤ϵ, and basis function S1(X1) satisfies S1(X1)2≤S¯1 with a positive constant S¯1. From the above inequality, it can be seen that we may increase the size of the attraction region with increasing the gain 1/a12.
By the same way, we can show that the signalsχ3,…,χm are semiglobally bounded.Remark 9.
For system (9) without input delay, some adaptive NN/fuzzy control schemes have been reported; e.g., see [25–27, 42–44]. For the control problem considered in this paper, two differences are summarized as follows. First, for the controlled switching system, the input delay phenomenon is considered. In addition, all the existing control methods for the uncertain switched system just can guarantee the tracking error converging to a small compact set, and the accurate size cannot be determined. This paper addresses the adaptive tracking control issue with the known tracking accuracy.


## 3.1. Problem Description

Consider a class of switched nonlinear systems described by(9)χ˙i=χi+1+fiσtχ¯i,i=1,…,m-1χ˙m=ut-τ+fmσtχ¯my=χ1where χ=[χ1,χ2,…,χm]T∈Rm is the state vector. y∈R and u∈R denote the system output and the control input, respectively. σ(t):[0,+∞)→M={1,2,…,n} represents a switching signal, and fik:Ri→R are unknown continuous functions for i=1,2,…,m, k=1,2,…,n. τ represents the delayed time.To deal with the problem of input delay in system (55), the Pade approximation approach used in [35] is introduced. Subsequently, we can have (10)Lut-τ=exp⁡-τvLut=exp⁡-τv/2exp⁡τv/2Lut≈1-τv/21+τv/2Lut,where L[u(t)] is the Laplace transform of u(t) and v is Laplace variable. For further analysis, a new variable χm+1 is proposed and it conforms to the following relation (11)Lut1-τv/21+τv/2=Lχm+1t-Lut.Then the following equation is obtained (12)u-τ2u˙=χm+1+τ2χ˙m+1-u-τ2u˙. That is, we have the following equation(13)χ˙m+1=-κχm+1+2κu,where κ=2/τ.Remark 4.
In this paper, Pade approximation method is introduced to deal with small delay. Since Pade approximation has some limitations in handling delay, the proposed scheme cannot work in large-delay case. Relaxed control design for systems with long delay and actuators saturation deserves further investigation.
Based on the above transformation, system (6) can be rewritten as follows(14)χ˙i=χi+1+fiσtχ¯i,i=1,…,m-1χ˙m=χm+1-u+fmσtχ¯mχ˙m+1=-κχm+1+2κuy=χ1.
The control objective of this paper is addressed as follows: for a given reference signalyr(t), to develop an adaptive neural controller u such that (i)
all the closed-loop signals remain SGUUB;(ii)
the tracking errore1=y-yr≤ϵ as t→∞.Remark 5.
In fact, some control schemes have been proposed for uncertain systems to achieve the objective of ensuring that the tracking error converges to the accuracy given a priori according to the requirement; e.g., see [6, 40]. However, this paper studies the control problem for a class of uncertain switched systems with input delay. For such more general system, the existing control methods cannot be directly employed to solve this control issue. As far as the authors know, this is the first work to address the practical control for uncertain switched systems with input delay.To design the desired adaptive neural controller, the following assumption is given.Assumption 6.
The reference signalyr(t) and its derivative yr(i)(t) are continuous and bounded for i=1,2,…,m.


## 3.2. Adaptive Neural Controller Design and Stability Analysis

To develop the desired control scheme for system (14), the following error variables are defined(15)e1=χ1-yr,ei=χi-αi-1,i=2,…,m-1em=χm-αm-1+1κχm+1,where αi,i=1,…,m-1, denote the virtual control variables and will be designed later.Following the adaptive backstepping control method, we present the design process of the desired controller and the adaptation laws.Step‎ 1. Consider the error variable e1 in (15) and the first subsystem in (14). Then, we have(16)e˙1=χ˙1-y˙r=χ2+f1σtχ¯1-y˙r.Select the following nonnegative function(17)V1=m!Nm+1ϵe1,and then along the trajectory of (16), by using Lemma 3, the time derivative of V1 is given as(18)dV1dt=dNm+1ϵe1de1e˙1m!=Nmϵe1Smϵe1m!χ2+f1σtχ¯1-y˙r≤Nmϵe1Smϵe1m!e2+α1+Smϵe1G1χ¯1-y˙r,where G1(χ¯1) is a simultaneous domination function which is continuous and unknown. For example, here it can be selected as G1(χ¯1)=∑1nf1kχ¯12+1. Since f1k(χ¯1) is unknown, it is obvious that G1(χ¯1) is unknown and it cannot be used to design the controller directly. As shown in (6), an RBF NN is employed to approximate F1(X1) online, and then one can have(19)dV1dt≤Nmϵe1Smϵe1m!e2+α1+Smϵe1G1χ¯1-y˙r=Nmϵe1Smϵe1m!e2+α1+F1X1=Nmϵe1Smϵe1m!e2+α1+W1TS1X1+ε1X1,where we define F1(X1)=Smϵ(e1)G1(χ¯1)-y˙r with X1=[χ1,yr,y˙r]T∈Ω1⊆R3.The Young’s inequality is used for the following analysis:(20)Nmϵe1Smϵe1m!W1TS1X1≤Nmϵe1Smϵe12m!W1S1X1≤Nmϵe1Smϵe12m!12a12ψ1S1X12+12a12,where ψ1≔W12 and a1>0 is a design parameter, and(21)Nmϵe1Smϵe1m!ε1X1≤Nmϵe1Smϵe12m!ε1∗.Substituting (20) and (21) into (19) yields(22)dV1dt≤Nmϵe1Smϵe1m!e2+α1+12a12ψ1S1X12Smϵe1+12a12Smϵe1+Smϵe1ε1∗=Nmϵe1Smϵe1m!e2+α1+12a12ψ1S1X12Smϵe1+Smϵe1ρ1,where ρ1≔1/2a12+ε1∗ is an unknown constant.The first virtual controller is designed from the above information.(23)α1=-k1+14Nmϵe1Smϵe1m!-12a12ψ1^S1X12Smϵe1-ρ1^Smϵe1-ϵ+1Smϵe1,where k1 is a positive constant and ψ1^ and ρ1^ denote the estimates of ψ1 and ρ1.Obviously, (22) can be changed into(24)dV1dt≤Nmϵe1Smϵe1m!e2-k1+14Nmϵe1Smϵe1m!-12a12ψ1^S1X12Smϵe1-ρ1^Smϵe1-ϵ+1Smϵe1+12a12ψ1S1X12Smϵe1+ρ1Smϵe1≤-k1+14Nmϵe1m!2+12a12ψ~1Nmϵe1Smϵe12m!S1X12+ρ~1Nmϵe1Smϵe12m!+Nmϵe1m!e2-ϵ+1with estimate errors ψ~1≔ψ1-ψ^1 and ρ~1≔ρ1-ρ^1.For further study, we establish the nonnegative function as below(25)V¯1=V1+12λ1ψ~12+12γ1ρ~12,where λ1>0 and γ1>0 are design parameters. We choose the adaptation laws as follows:(26)ψ^˙1=λ12a12Nmϵe1S1X12Smϵe12m!,ρ^˙1=γ1Nmϵe1Smϵe12m!.Then we can obtain(27)dV¯1dt=dV1dt-1λ1ψ~1ψ^˙1-1γ1ρ~1ρ^˙1≤-k1+14Nmϵe1m!2-1λ1ψ~1ψ^˙1-λ12a12S1X12Nmϵe1Smϵe12m!-1γ1ρ~1ρ^˙1-γ1Nmϵe1Smϵe12m!+Nmϵe1m!e2-ϵ+1=-k1+14Nmϵe1m!2+Nmϵe1m!e2-ϵ+1.Step i   ( i = 2,3 , … , m - 1 ) .The method of recursion is used in each step when the value of i increases from 2 to m-1. Similar to Step‎ 1, the time derivative of ei is(28)e˙i=χ˙i-α˙i-1=χi+1+fiσtχ¯i-α˙i-1,where α˙i-1=∑j=1i-1∂αi-1/∂χjχj+1+fjσtχ¯j+Λi-1 with Λi-1=∑j=0i-1∂αi-1/∂yd(j)yd(j+1) + ∑j=1i-1∂αi-1/∂ψ^jψ^˙j + ∑j=1i-1∂αi-1/∂ρ^jρ^˙j.Consider the following nonnegative functions(29)Vi=V¯i-1+Nm-i+2ϵeim-i+1!,V¯i=Vi+12λiψ~i2+12γiρ~i2,where λi>0 and γi>0 are design parameters, estimate errors ψ~i≔ψi-ψ^i and ρ~i≔ρi-ρ^i, and ψi^ and ρi^ denote the estimates of ψi and ρi which will be defined later.Then we can obtain(30)dVidt=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!e˙i=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+fiσtχ¯i-Λi-1-∑j=1i-1∂αi-1∂χjχj+1+fjσtχ¯j.Meanwhile the following inequality holds(31)-Nm-i+1ϵeiSm-i+1ϵeim-i+1!∑j=1i-1∂αi-1∂χjfjσtχ¯j≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!∑j=1i-1∂αi-1∂χjfjσtχ¯j2+1.Substituting (31) into (30), we gain(32)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+Sm-i+1ϵeiGiχ¯i+∑j=1i-1Γjχj¯-Λi-1-∑j=1i-1∂αi-1∂χjχj+1where Gi(χ¯i) and Γj(χj¯) are simultaneous domination functions which are continuous and unknown. For example, here they can be chosen as Gi(χ¯i)=∑k=1n[fik(χi¯)]2+1, Γj(χ¯j)=∑k=1n[∂αi-1/∂χjfjk(χ¯j)]2+1. Define Fi(Xi)=Sm-i+1ϵeiGiχ¯i+∑j=1i-1Γjχj¯-Λi-1-∑j=1i-1∂αi-1/∂χjχj+1 with Xi=[χ¯iT,αi-1,∂αi-1/∂χ1,…,∂αi-1/∂χi-1,Λi-1]T∈Ωi⊆R2i+1, and an RBF NN (6) is used to approximate Fi(Xi) online. Based on this information, (32) can be changed as below(33)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+FiXi=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+WiTSiXi+εiXi.Similar to (20) and (21), by using the Young’s inequality, inequalities (34) and (35) can be gained easily:​(34)Nm-i+1ϵeiSm-i+1ϵeim-i+1!WiTSiXi≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!WiTSiXi≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!12ai2ψiSiXi2+12ai2 where ψi=Wi2, ai>0 is a design parameter, and(35)Nm-i+1ϵeiSm-i+1ϵeim-i+1!εiXi≤Nm-i+1ϵeiSm-i+1ϵei2m-i+1!εi∗.Substituting (34) and (35) into (33) yields(36)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+12ai2ψiSiXi2+12ai2Sm-i+1ϵei+Sm-i+1ϵeiεi∗=dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+αi+12ai2ψiSiXi2Sm-i+1ϵei+Sm-i+1ϵeiρi,where ρi=1/2ai2+εi∗ is an unknown constant.We construct thei-th virtual controller as follows:(37)αi=-ki+54Nm-i+1ϵeiSm-i+1ϵeim-i+1!-12ai2ψ^iSiXi2Sm-i+1ϵei-Sm-i+1ϵeiρ^i-ϵ+1Sm-i+1ϵei.Obviously, based on (37), the undermentioned inequality is true:(38)dVidt≤dV¯i-1dt+Nm-i+1ϵeiSm-i+1ϵeim-i+1!ei+1+12ai2ψ~iSiXi2Sm-i+1ϵei+Sm-i+1ϵeiρ~i-ϵ+1Sm-i+1ϵei-ki+54Nm-i+1ϵeiSm-i+1ϵeim-i+1!≤-∑j=1ikjNm-j+1ϵejm-j+1!2+Nm-i+1ϵeiSm-i+1ϵei2m-i+1!×12ai2ψ~iSiXi2+ρ~i-14Nm-i+2ϵei-1m-i+22+Nm-i+2ϵeim-i+2!ei-ϵ+1-54Nm-i+1ϵeim-i+12+Nm-i+1ϵeim-i+1!ei+1-ϵ+1.Let(39)Ui=-14Nm-i+2ϵei-1m-i+22+Nm-i+2ϵei-1m-i+2!ei-ϵ+1-Nm-i+1ϵeim-i+12.It is obvious thatUi≤0 when ei≤(ϵ+1). When ei>(ϵ+1), using the Young’s inequality, we obtain(40)Ui≤ei-ϵ+12-Nm-i+1ϵeim-i+12=ei-ϵ+12-ei-ϵ2m-i+1≤ei-ϵ2-ei-ϵ2m-i+1=ei-ϵ21-ei-ϵ2m-i+1≤0.Hence, Ui≤0 holds all the time.Based on the above information, we choose(41)ψ^˙i=λi2ai2Nm-i+1ϵeiSm-i+1ϵei2m-i+1!SiXi2ρ^˙i=γiNm-i+1ϵeiSm-i+1ϵei2m-i+1!.Thus, we can have(42)dV¯idt=dVidt-1λiψ~iψ^˙i-1γiρ~iρ^˙i≤-∑j=1ikjNm-j+1ϵejm-j+12-1λiψ~ψ^˙i-λi2ai2Nm-i+1ϵeiSm-i+1ϵei2m-i+1!SiXi2-1γiρ~iρ^˙i-γiNm-i+1ϵeiSm-i+1ϵei2m-i+1!-14Nm-i+1ϵeim-i+1!2+Nm-i+1ϵeim-i+1!ei+1-ϵ+1≤-∑j=1ikjNm-j+1ϵejm-j+12-14Nm-i+1ϵeim-i+1!2+Nm-i+1ϵeim-i+1!ei+1-ϵ+1.Step m . Combining (14) and (15), we have(43)e˙m=χ˙m-α˙m-1+1κχ˙m+1=χm+1-u+fmσtχ¯m-α˙m-1+1κχ˙m+1,where α˙m-1=∑j=1m-1∂αm-1/∂χj[χj+1+fjσ(t)(χ¯j)]+Λm-1 with Λm-1=∑j=0m-1∂αm-1/∂yd(j)yd(j+1)+∑j=1m-1∂αm-1/∂ψ^jψ^˙j+∑j=1m-1∂αm-1/∂ρ^jρ^˙j.ChooseV¯m=Vm+1/2λmψ~m2+1/2γmρ~m2, Vm=V¯m-1+N2ϵ(em), where λm>0 and γm>0 are design parameters, and design the actual controller as below(44)u=-km+1N1ϵem-12am2ψ^m2SmXm2S1ϵem-S1ϵemρ^m-ϵ+1S1ϵem,where am>0 and km>0 are design parameters, estimate errors ψ~m≔ψm-ψ^m and ρ~m≔ρm-ρ^m, and ψ^m and ρ^m denote the estimates of ψm and ρm which will be defined later.Similar to the design process of Stepi, let i=m. Then we can have ψm=Wm2 and ρm=1/2am2+εm∗ where εm∗ is the upper bound of the RBF NN approximation error. The following adaptive laws are chosen(45)ψ^˙m=λm2am2N1ϵemS1ϵem2SmXm2ρ^˙m=γmN1ϵemS1ϵem2,where we define Xm=[χ¯mT,αm-1, ∂αm-1/∂χ1,…,∂αm-1/∂χm-1, Λm-1]T ∈ Ωm⊆R2m+1.According to the above information, we have(46)dV¯mdt≤-∑j=1mkjNm-j+1ϵejm-j+1!2-14N2ϵem-12!2-N1ϵem2+N2ϵem-12!em-ϵ+1≤-∑j=1mkjNm-j+1ϵejm-j+1!2.Based on inequality (46), the main result of this paper is summarized by the following theorem.Theorem 7.
Under Assumption6, consider the closed-loop system that comprises the plant (9), virtual control variables (23) and (37), and actual control law (44) with adaptive laws (26), (41), and (45). For the bounded initial condition on a compact set Ω0, assume there exist sufficiently large compact sets Ωi⊆R2i+1, i=1,2,…,m, such that Xi∈Ωi for all t≥0; the following statements hold. (i)
The tracking errore1=y-yr≤ϵ as t→∞.(ii)
All the closed-loop signals remain semiglobal bounded.Proof.
(i) From (46), by employing Barbalat’s Lemma [41], we can easily conclude that the tracking error satisfies e1=y-yr≤ϵ as t→∞.
(ii) Considering inequality (46), we can see that V¯m is nonincreasing. Therefore, it can be concluded that, for all 1≤i≤m, error signals including ei(t),ρ~i(t) and ψ~i(t) are bounded. To prove the boundedness of all the closed-loop signals, it can be shown that ρ^i(t)=ρi-ρ~i(t) and ψ^i(t)=ψi-ψi(t) are bounded according to the boundedness of ρ~i(t) and ψ~i(t). Next, it follows from the boundedness of e1(t) and yr(t) that χ1(t)=yr(t)+e1(t) is also bounded. Then, the boundedness of α1 given in (23) can be easily obtained. The boundedness of e2 and α1 further implies that χ2=e2+α1 is bounded. Continuing in the same fashion, we conclude that all the closed-loop signals are bounded. In addition, similar to the existing works on adaptive neural network control, the proposed control scheme just guarantees that the closed-loop system is semiglobally stable since the initial condition of the controlled system must remain on a compact set Ω0.Remark 8.
To show the semiglobal stability of the controlled system, the following analysis is presented. Firstly, fromχ1=e1+yr, we have χ1≤e1+yr≤ϵ+Y, where Y is the upper bound of reference signal yr. Thus, the boundedness of χ1 is obtained as t→∞. Furthermore, from the above analysis, it can be seen that ψ^1 and ρ^1 are bounded. Assume that ψ^1≤Ψ1 and ρ^1≤Φ1 with positive constants Ψ1 and Φ1. Since χ2=e2+α1, then we can obtain the following inequality for t→∞,(47)χ2≤e2+α1≤2ϵ+k1+14Nmϵe1m!+12a12Ψ1S¯1+Φ1+1=2ϵ+12a12Ψ1S¯1+Φ1+1,where -1≤Smϵ(e1)≤1, Nmϵ(e1)=0 when e1≤ϵ, and basis function S1(X1) satisfies S1(X1)2≤S¯1 with a positive constant S¯1. From the above inequality, it can be seen that we may increase the size of the attraction region with increasing the gain 1/a12.
By the same way, we can show that the signalsχ3,…,χm are semiglobally bounded.Remark 9.
For system (9) without input delay, some adaptive NN/fuzzy control schemes have been reported; e.g., see [25–27, 42–44]. For the control problem considered in this paper, two differences are summarized as follows. First, for the controlled switching system, the input delay phenomenon is considered. In addition, all the existing control methods for the uncertain switched system just can guarantee the tracking error converging to a small compact set, and the accurate size cannot be determined. This paper addresses the adaptive tracking control issue with the known tracking accuracy.


## 4. Simulation Examples

Example 1.
Consider the following uncertain switched system(48)χ˙1=χ2+f1σtχ¯1,χ˙2=ut-τ+f2σtχ¯2y=χ1,where σ(t)=1,2, f11(χ1)=sin(χ1)cos(χ1), f21(χ1)=χ12, f12(χ1,χ2)=sin(χ1)cos(χ2)+χ22, f22(χ1,χ2)=χ1χ2+sin(χ1). The reference signal is given as yr(t)=2+sin(2t)+0.5cos⁡(t). The tracking accuracy is assigned as ϵ=0.05, i.e., yt-yrt≤0.05 as t→∞.According to the design process of Section3, the virtual controller and the actual controller are presented as(49)α1=-k1+14N2ϵe1S2ϵe12!-12a12ψ1^S1X12S2ϵe1-ρ1^S2ϵe1-ϵ+1S2ϵe1,and(50)u=-k2+1N1ϵe2-12a22ψ^22SiXi2S1ϵe2-S1ϵe2ρ^2-ϵ+1S1ϵe2.The adaptive laws are designed as(51)ψ^˙i=λi2ai2N3-iϵeiS3-iϵei23-i!SiXi2ρ^˙i=γiN3-iϵeiS3-iϵei23-i!,i=1,2.The design parameters of the adaptive neural controllers and learning laws are chosen ask1=k2=3, a1=a2=1, γ1=7, γ2=10, and λ1=λ2=15. The input delay is τ=0.0012. In the simulation, two RBF NNs are employed to approximate uncertain functions in the control scheme. The first RBF vector S1(X1) contains 125 nodes with the centers μq(q=1,2,…,125) evenly placed on [-1.7,1.7]×[-3,3]×[-3,3] and the width η=0.62. The second RBF vector S2(X2) contains 1875 nodes with the centers μq(q=1,2,…,1875) evenly placed on [-1.7,1.7]×[-3,3]×[-3,3]×[-7,7]×[-7,7] and the width η=0.38.The initial conditions are given asχ1(0)=0.1, χ2(0)=0.5, and ρ^i=ψ^i=0, i=1,2. The simulation results are shown in Figures 1–4, and Figure 2 shows that the tracking error achieves the given accuracy after 20 seconds, which implies that the control objective is achieved by using the proposed control scheme.Figure 1
Trajectories of system output signaly(t) (solid line) and reference signal yr(t) (dash-dotted line).Figure 2
Trajectory of tracking errore1(t).Figure 3
Trajectory of control input signalu(t).Figure 4
Trajectory of switching signalσ(t).To further show the superiority of the proposed control method, a comparison simulation is presented. Considering the control scheme developed in [42], we have the following controllers and parameters learning laws(52)u=θ^22ζ2,min2+λ2z2,(53)α1=θ^12ζ1,min2+λ1z1, and(54)θ^˙i=ri2ζi,min2-βiθ^i,i=1,2,where the definitions and values of zi,λi,ri,βi,ζi,min, and θ^i can be found in [42]. For the simulation, the parameters are chosen as ri=ζi,min=1 and βi=0. The rest of the conditions remain the same as before. The simulation results are shown in Figures 5 and 6, from which it can be seen that by using the method proposed in [42], the boundedness of the tracking error can be ensured, but the accurate size cannot be determined. This drawback can be overcome by employing the control scheme established in this paper.Figure 5
Trajectories of output signaly(t) (dash-dotted line) and reference signal yr(t) (solid line) using method [42].Figure 6
Trajectory of tracking errore1(t) using method [42].Example 2.
Consider a switched RCL circuit system [45] as shown in Figure 7, which can be described as(55)χ˙1=χ2+f1σtχ¯1,χ˙2=ut-τ+f2σtχ¯2where switching signal σ(t)∈{1,2} is described as Figure 4. f11=f21=(1/L)χ2-χ1, f12=-(1/C1)χ1-(R/L)χ2, f22=-(1/C2)χ1-(R/L)χ2,χ1=qc stands for the charge in capacitor, and χ2=ϕL denotes the flux in the inductance in the circuit. Ci is the ith capacitor, L is the inductance, R is the resistance, and u means the voltage. The related parameters are selected as L=1.2,R=1,C1=50, and C2=100. The input delay is chosen as τ=0.0001. The reference signal is given as yr(t)=1.5+0.5∗sin(t) and the tracking accuracy is assigned as ϵ=0.02.
According to the design process of Section3, the design parameters of the adaptive neural controllers and learning laws are chosen as k1=1.5, k2=2, a1=1.2, a2=1.7, γ1=5, γ2=7, and λ1=λ2=10. Two RBF NNs shown in Example 1 are employed to approximate uncertain functions in this example.
The initial conditions are given asχ1(0)=0.1, χ2(0)=-0.1, and ρ^i=ψ^i=0, i=1,2. The simulation results are shown in Figures 8–10, and from Figure 9, it can be seen that the tracking error achieves the given accuracy after 20 seconds.Figure 7
Switched RCL circuit [45].Figure 8
Trajectories of system output signaly(t) (dash-dotted line) and reference signal yr(t) (solid line).Figure 9
Trajectory of tracking errore1(t).Figure 10
Trajectory of control input signalu(t).


## 5. Conclusions

This paper has addressed the adaptive tracking control problem for a class of uncertain switched systems with input delay. The main contribution of this work is that the proposed adaptive neural controller ensures the tracking error converging to the accuracy predefined a priori according to the requirement for the considered switched nonlinear system. To achieve this control performance, some nonnegative switching functions are introduced to design the desired controller. Of course, the proposed control scheme has some improvements and this topic can be further considered in the future, for example, how to develop the asymptotic tracking control scheme for the considered system (9), how to extend the proposed control method to system (9) with stochastic disturbances and nonlinearly input, etc. In addition, for the uncertain switched systems with the actuators saturation, it is a practical and challenging control problem to design an adaptive NN controller with the tracking accuracy known a priori.



---

*Source: 6920372-2019-07-09.xml*