# Adaptive Asymptotic Tracking Control for a Class of Uncertain Switched Systems via Dynamic Surface Technique

Authors: Xuemiao Chen; Qianjin Zhao; Chunsheng Zhang; Jian Wu
Journal: Complexity
 (2019)

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## Abstract

A novel adaptive tracking control scheme is proposed for a class of uncertain nonlinear switched systems with perturbations in this paper. The common Lyapunov function method is introduced to handle the switched system in the design process of the desired adaptive controller. In addition, a dynamic surface control method is proposed by employing a nonlinear filter such that the “explosion of complexity” problem existing in the conventional backstepping design can be overcome. Under the presented adaptive controller, all the closed-loop signals are semiglobally bounded, and especially the output signal of the controlled system can follow the given reference signal asymptotically. To show the availability of the presented control scheme, a simulation is given in this paper.


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## Body

## 1. Introduction

Switched systems, as a typical class of hybrid systems, have attracted many researchers’ sights in the past decades (e.g., see [1–14]) since many physical systems can be mathematically modeled by the switched systems. Owing to the large-scale applications, the research on switched systems never stops and great achievements have been made (e.g., see [1–6]). The stabilization problem for a class of slowly switched systems based on unstable subsystems is studied in [6]. The optimal control scheme of the switching positive system is presented in [7]. The stabilization problem of switched linear systems with mode-dependent average dwell time is addressed in [8]. Most significant issues on switched systems have been acquired for multitudinous switched systems under arbitrary switching or constraint switching (e.g., see [8–11]), and references therein.Adaptive control of uncertain nonlinear systems has achieved significant research results (e.g., see [15–25]) including adaptive feedback linearization [15], adaptive backstepping [16], immersion and invariance adaptive control [17], and adaptive neural network/fuzzy-logic control [21, 25]. Very recently, some novel adaptive control schemes have been established for uncertain systems with the unknown control directions or the parameter estimator triggering [26–28]. However, the “explosion of complexity” problem exists in the repeated differentiations of virtual control variables [18, 19] in the backstepping approach. Thus, with the system order increasing, the computation of the backstepping controller will be more complicate.To overcome the “explosion of complexity” problem in the backstepping design process, for uncertain strict-feedback/nonstrict-feedback nonlinear systems, the dynamic surface control (DSC) technique is established (e.g., see [29–38]). In literature studies [29, 30], a virtual control law is designed in each design step by using a low-pass filter, thus it can avoid the derivative of the virtual controllers. Another advantage of the DSC approach is that it can reduce the requirement on the smoothness of plant functions and desired signal obviously. Therefore, the DSC technique has a large-scale application in the process of designing simplified adaptive controllers for uncertain nonlinear systems. The adaptive robust DSC with composite adaptation laws is designed for the uncertain system with the semistrict feedback form in [31]. Based on multiple models, the result is applied to enhance the transient response of an adaptive DSC system in [32]. An adaptive DSC scheme is presented by using composite learning to ensure parameter convergence without the persistent excitation condition in [33]. Various adaptive backstepping schemes using neural networks (NNs) or fuzzy logic systems as approximators are proposed to deal with the time delays, dynamic uncertainties, and output dead zone for stochastic large-scale nonlinear systems in [34–37]. Very recently, a novel DSC method with the nonlinear filter is proposed for a class of uncertain systems, and the asymptotic tracking control performance is achieved in [38].In the above discussions, some adaptive DSC schemes have been developed for uncertain switched systems. Nevertheless, these control schemes can just guarantee the semiglobal boundedness for the controlled systems, and the asymptotic stability cannot be achieved since the stability analysis cannot be completed by the conventional DSC technique with a linear low-pass filter. In this paper, we try to address this issue by employing the DSC method with a nonlinear filter. The main contributions of this work are summarized as following.(i)
As far as we know now, this is the first work to address the asymptotic tracking control problem for uncertain switched systems by using the DSC method to solve the “explosion of complexity” problem existing in the conventional backstepping design.(ii)
By introducing the DSC method with a nonlinear filter proposed in [38], the desired controller is developed based on the common Lyapunov function, and then the stability analysis of the closed-loop switching control system is completed according to Barbalat’s lemma.This paper is organized as follows. The problem statement and some preliminaries are introduced and adaptive DSC scheme with a detailed stability analysis are presented in Section2. Then, a simulation example is given in Section 3. A conclusion is drawn in Section 4.


## 2. Problem Statement and Main Result

### 2.1. Problem Statement and Some Preliminaries

Taking the following class of uncertain strict-feedback nonlinear switched systems into account,(1)x˙i=xi+1+θifi,ϱtxi¯+dit,i=1,…,n−1,x˙n=u+θnfn,ϱtxn¯+dnt,y=x1,where x¯i=x1,…,xiT∈Ri,i=1,…,n are the states of the system, u∈R is the control input, and y∈R is the system output. For i=1,…,n, θi are the unknown constants, dit are the perturbations, and fi⋅:Rn⟶R are the known continuously differentiable functions. ϱt:0,∞ is the switching signal, and p denotes the number of subsystems in the switched system.The control objective of this paper for system (1) is to design an adaptive DSC law u such that the output yt asymptotically tracks a desired trajectory yrt, and the boundedness of all the signals in the closed-loop system is guaranteed.Assumption 1.
The desired trajectoryyrt and its derivatives y˙rt and y¨rt are bounded and available.Assumption 2.
The perturbationsdit,i=1,…,n are bounded, i.e., dit≤Wi with the constants Wi>0.Lemma 1 (see [39]).
For anyε>0 and z∈R, the following inequality can be obtained:(2)0≤z−z2z2+ε2<ε.


### 2.2. Adaptive Controller Design

The presented adaptive DSC scheme is similar to the backstepping technique, and it containsn steps as follows. Define the estimate error ∗˜=∗−∗⌃, where ∗⌃ is the estimate of ∗⌃. Let εi=θi,i=1,…,n.Step 1.
The first surface error is defined asz1=x1−yr, and the time derivative of z1 is(3)z˙1=x˙1−y˙r=x2+θ1f1,ϱtx1¯+d1t−y˙r.
Then, the Lyapunov function candidateV1 is defined as(4)V1=12z12+12γ1ε˜12+12β1W˜12,where γ1 and β1 are positive design parameters.
Considering equations (1)–(4), the following equation can be obtained:(5)V˙1=z1x2+θ1f1,ϱtx1¯+d1t−y˙r−1γ1ε˜1ε^˙−1β1W˜1W⌃˙1=z1x2+θ1f1,ϱtx1¯+d1t−y˙r+α1−α1−1γ1ε˜1ε⌃˙−1β1W˜1W⌃˙1.
On the basis offi,ϱt, we can obtain the following inequality:(6)fi,ϱt≤∑j=1pfi,j2.
The above inequality is introduced to design the desired controller by employing the common Lyapunov function method. In order to develop an adaptive controller without the switching signalϱt, the subsystem function fi,ϱt is bounded by an upper bound function which is without the switching signals.
LetDi=∑j=1pfi,j2. The following inequality can be obtained:(7)z1θ1f1,ϱt≤z1θ1fi,ϱt≤z1θ1D1≤ε1z12D12z12D12+σt2+ε1σt,and the following inequality can be obtained by using Assumption 2(8)z1d1t≤z1d1t≤z1W1=z1W⌃1+W⌃1≤z12W⌃12z12W⌃12+σt2+σt+z1W˜1.
On designing the virtual control lawα1 as(9)α1=−k1z1+y˙r−ε⌃1z1D12z12D12+σt2−z1W⌃12z12W⌃12+σt2,the update laws of ε⌃1 and W⌃1 can be designed as(10)ε⌃˙1=γ1z12D12z12D12+σt2,(11)W⌃˙1=β1z1.
In view of (5)–(11), one has(12)V˙1=z1x2+θ1f1,ϱtx1¯+d1t−y˙r+−k1z1+y˙r−ε⌃1z1D12z12D12+σt2−z1W^12z12W^12+σt2−α1−1γ1ε˜1γ1z12D12z12D12+σt2−1β1W⌃1β1z1≤−k1z12+x2−α1z1+1+ε1σt.
Design the second error signal in the backstepping asx2−α1 such that it can avoid the “explosion of complexity” problem. Let α1 pass through the following novel nonlinear filter to obtain a filtered virtual controller s1:(13)τ1s˙1=−e1−τ1M^12e1M^12e12+σ2t−τ1z1,s10=α10,where e1=s1−α1 is the first boundary layer error, τ1>0 is a design parameter, M^1 is the estimate of M1 and it will be clarified later, and σt is any positive uniform continuous and bounded function, which satisfies(14)limt⟶∞∫​0tστdτ≤σ1<+∞,σ˙t≤σ2<+∞,where σ1 and σ2 are any positive constants.Remark 1.
In this paper, the nonlinear filter (13) is introduced to construct the desired controller. Compared with the reported results on adaptive DSC with a linear low-pass filter (e.g., see [30–37]), the advantage of our deign is that the asymptotic tracking control performance can be guaranteed, and the stability analysis can be completed successfully under the proposed controller.Stepi i=2,…,n−1.
Theith surface error is designed as zi=xi−si−1; then, we get the following equation:(15)z˙i=xi+1+θifi,ϱt+dit+M^i−12ei−1M^i−12ei−12+σ2t+zi−1+ei−1τi−1.
Design the virtual control lawαi and the update laws ε^i and W^i as follows:(16)αi=−kizi−2zi−1−ε⌃iziDi2zi2Di2+σ2t−ziW⌃i2zi2W⌃i2+σ2t−M⌃i−12ei−1M⌃i−12ei−12+σ2t−ei−1τi−1,(17)ε⌃˙1=γizi2Di2zi2Di2+σ2t,(18)W⌃˙1=βizi,where ki, γi, and βi are the positive design parameters.
Design the Lyapunov function candidateVi as(19)Vi=Vi−1+12zi2+12γiε˜i2+12βiW˜i2,where γi and βi are the positive design parameters.
In view of equations (15)–(19), consider the time derivative of Vi as(20)V˙i=V˙i−1+ziz˙i−1γiε˜iε^˙i−1βiW˜iW⌃˙i≤−∑j=1ikjzj2+∑j=1i−1zjej+xi+1−αizi+i+∑j=1iεjσt.
Letαi pass through the following nonlinear filter to obtain a filtered virtual controller si(21)τis˙i=−ei−τiM⌃i2eiM⌃i2ei2+σ2t−τizi,si0=αi0,and define(22)ei=si−αi,where ei means the ith boundary layer error and τi is a filter time constant.Stepn.
Considering thenth surface error as zn=xn−sn−1, one has(23)z˙n=u+θnfn,ϱt+dnt+M^n−12en−1M^n−12en−12+σ2t+zn−1+en−1τn−1.
Design the actual control lawu as(24)u=−knzn−2zn−1−ε⌃nznDn2zn2Dn2+σ2t−znW⌃n2zn2W⌃n2+σ2t−M⌃n−12en−1M⌃n−12en−12+σ2t−en−1τn−1,and consider the update laws for ε^i and W^n as follows:(25)ε⌃˙n=γnzn2Dn2zn2Dn2+σ2t,(26)W⌃˙n=βnzn,where kn, βn, and γn are the positive design parameters.
Design the Lyapunov function candidateVn as follows:(27)Vn=Vn−1+12zn2+12γnε˜n2+12βnW˜n2,where γn and βn are the positive design parameters.
Then, considering equations (23)–(27), we have(28)V˙n=V˙n−1+znz˙n−1γnε˜nε^˙n−1βnW˜nW⌃˙n≤−∑j=1nkjzj2+∑j=1n−1zjej+n+∑j=1nεjσt.


### 2.3. Stability Analysis

Based on inequality (28), the main result of this paper is presented by the following theorem.Theorem 1.
Consider the closed-loop system consisting of the plant (1), the nonlinear filters (13), (22), the actual controller (24), and the adaptive laws (10), (11), (17), (18), (25), and(26). Suppose that Assumptions 1-2 hold, for any initial conditions satisfying V0≤q, where q is a given constant, there exit design parameters ki, βi, γi, i=1,…,n, τj, and ηj, j=1,…,n−1, such that the following statements hold:(i)
All the resulting closed-loop signals are semiglobally bounded(ii)
The tracking errorz1=y−yr converges to zero asymptoticallyProof.
The compact sets are defined as(29)Ω1=yr,y˙r,y¨rT:yr2+y˙r2+y¨r2≤B0,Ω2=Vt≤q,where B0 is a known positive constant. Note that set Ω1×Ω2 is also a compact in R4n+1. σt and σ˙t are the bounded functions.
Differentiating the boundary layer errorsei=si−αi yields(30)e˙i=−eiτi−M⌃i2eiM⌃i2ei2+σ2t−zi+Biz1,…,zi+1,e1,…,ei,ε⌃1,…,ε⌃i,M⌃1,…,M⌃i,W⌃1,…,W⌃i,yr,y¨r,y¨r,σt,σ˙t,i=1,…,n−1,where(31)B1⋅=−α˙1=−∂α1∂x1x˙1−∂α1∂ε^1ε⌃˙1−∂α1∂yry˙r−∂α1∂yr˙y¨r−∂α1∂W^1W⌃˙1,Bi⋅=−α˙i=−∑j=1i∂αi∂xjx˙j−∂αi∂εj^ε⌃˙j−∂αi∂ei−1e˙i−1−∂αi∂M^i−1M⌃˙i−1−∂αi∂W^iW⌃˙i−∂αi∂yry˙r−∂αi∂y˙ry¨r−∂αi∂σtσ˙t,are the continuous functions on Ω1×Ω2. As a consequence, a positive constant Mi can be obtained such that Bi⋅≤Mi, where Mi>0 is an unknown constant.
We consider the Lyapunov function candidate as follows:(32)V=Vn+∑i=1n−112ei2+∑i=1n−112ηiM˜i2,where ηi, i=1,…,n−1 are the positive design parameters.
The time derivative ofV is(33)V˙=V˙n+∑i=1n−1eie˙i−∑i=1n−11ηiM˜iM⌃˙i≤−∑i=1nkizi2−∑i=1n−1ei2τi−∑i=1n−1M⌃i2eiM⌃i2ei2+σ2t+∑i=1n−1Miei−∑i=1n−11ηiM˜iM⌃˙i+n+∑j=1nεjσt.
We can obtain the following inequality in view of Lemma 1:(34)Miei=M⌃iei+M⌃iei≤M⌃i2ei2M⌃i2ei2+σ2t+σt+M˜iei,and then, we have(35)V˙≤−∑i=1nkizi2−∑i=1n−1ei2τi−∑i=1n−11ηiM˜iM⌃˙i−βiei+ϖσt,where ϖ≔2n+∑j=1nεj−1 is a constant.
Consider the update laws forM^i as(36)M⌃˙i=ηiei,i=1,…,n−1.
From equation (35), the following inequality can be obtained:(37)V˙≤−∑i=1nkizi2−∑i=1n−1ei2τi+ϖσt.
Integrating inequality (37) over 0,t yields(38)Vt≤V0−∫0t∑i=1nkizi2κ+∑i=1nei2κτidκ+ϖ∫0tσκdκ≤V0+ϖσ1,which means that zi, zn, ε˜i, ε^n, W^i, W^n, ei, and M^i ,i=1,…,n−1 are bounded. Therefore, we also conclude that xi, xn, si, αi, and u, i=1,…,n−1 are bounded. From inequality (38), we can obtain the following inequality:(39)∫0t∑i=1nkizi2κdκ≤V0+ϖσ1.
By applying Barbalat’s lemma [40, 41] to inequality (39), it is concluded that(40)limt⟶∞z1=0,that is to say, the asymptotic stability is accomplished.Remark 2.
For uncertain switched systems (e.g., see [21–23]), lots of adaptive control schemes have been developed by the conventional backstepping method. This paper introduces the adaptive DSC method to solve the “explosion of complexity” problem existing in the aforementioned literature studies, and the asymptotic tracking performance has been achieved under the developed adaptive controller.Remark 3.
Recently, some novel adaptive control schemes have been proposed for uncertain systems with time-varying performance bounds and actuator failures in [42, 43], where the “explosion of complexity” problem cannot be well solved and the asymptotic tracking performance is not achieved. Of course, this paper does not consider the actuator failure phenomenon of the controlled system, which is the further study program in the future.


## 2.1. Problem Statement and Some Preliminaries

Taking the following class of uncertain strict-feedback nonlinear switched systems into account,(1)x˙i=xi+1+θifi,ϱtxi¯+dit,i=1,…,n−1,x˙n=u+θnfn,ϱtxn¯+dnt,y=x1,where x¯i=x1,…,xiT∈Ri,i=1,…,n are the states of the system, u∈R is the control input, and y∈R is the system output. For i=1,…,n, θi are the unknown constants, dit are the perturbations, and fi⋅:Rn⟶R are the known continuously differentiable functions. ϱt:0,∞ is the switching signal, and p denotes the number of subsystems in the switched system.The control objective of this paper for system (1) is to design an adaptive DSC law u such that the output yt asymptotically tracks a desired trajectory yrt, and the boundedness of all the signals in the closed-loop system is guaranteed.Assumption 1.
The desired trajectoryyrt and its derivatives y˙rt and y¨rt are bounded and available.Assumption 2.
The perturbationsdit,i=1,…,n are bounded, i.e., dit≤Wi with the constants Wi>0.Lemma 1 (see [39]).
For anyε>0 and z∈R, the following inequality can be obtained:(2)0≤z−z2z2+ε2<ε.


## 2.2. Adaptive Controller Design

The presented adaptive DSC scheme is similar to the backstepping technique, and it containsn steps as follows. Define the estimate error ∗˜=∗−∗⌃, where ∗⌃ is the estimate of ∗⌃. Let εi=θi,i=1,…,n.Step 1.
The first surface error is defined asz1=x1−yr, and the time derivative of z1 is(3)z˙1=x˙1−y˙r=x2+θ1f1,ϱtx1¯+d1t−y˙r.
Then, the Lyapunov function candidateV1 is defined as(4)V1=12z12+12γ1ε˜12+12β1W˜12,where γ1 and β1 are positive design parameters.
Considering equations (1)–(4), the following equation can be obtained:(5)V˙1=z1x2+θ1f1,ϱtx1¯+d1t−y˙r−1γ1ε˜1ε^˙−1β1W˜1W⌃˙1=z1x2+θ1f1,ϱtx1¯+d1t−y˙r+α1−α1−1γ1ε˜1ε⌃˙−1β1W˜1W⌃˙1.
On the basis offi,ϱt, we can obtain the following inequality:(6)fi,ϱt≤∑j=1pfi,j2.
The above inequality is introduced to design the desired controller by employing the common Lyapunov function method. In order to develop an adaptive controller without the switching signalϱt, the subsystem function fi,ϱt is bounded by an upper bound function which is without the switching signals.
LetDi=∑j=1pfi,j2. The following inequality can be obtained:(7)z1θ1f1,ϱt≤z1θ1fi,ϱt≤z1θ1D1≤ε1z12D12z12D12+σt2+ε1σt,and the following inequality can be obtained by using Assumption 2(8)z1d1t≤z1d1t≤z1W1=z1W⌃1+W⌃1≤z12W⌃12z12W⌃12+σt2+σt+z1W˜1.
On designing the virtual control lawα1 as(9)α1=−k1z1+y˙r−ε⌃1z1D12z12D12+σt2−z1W⌃12z12W⌃12+σt2,the update laws of ε⌃1 and W⌃1 can be designed as(10)ε⌃˙1=γ1z12D12z12D12+σt2,(11)W⌃˙1=β1z1.
In view of (5)–(11), one has(12)V˙1=z1x2+θ1f1,ϱtx1¯+d1t−y˙r+−k1z1+y˙r−ε⌃1z1D12z12D12+σt2−z1W^12z12W^12+σt2−α1−1γ1ε˜1γ1z12D12z12D12+σt2−1β1W⌃1β1z1≤−k1z12+x2−α1z1+1+ε1σt.
Design the second error signal in the backstepping asx2−α1 such that it can avoid the “explosion of complexity” problem. Let α1 pass through the following novel nonlinear filter to obtain a filtered virtual controller s1:(13)τ1s˙1=−e1−τ1M^12e1M^12e12+σ2t−τ1z1,s10=α10,where e1=s1−α1 is the first boundary layer error, τ1>0 is a design parameter, M^1 is the estimate of M1 and it will be clarified later, and σt is any positive uniform continuous and bounded function, which satisfies(14)limt⟶∞∫​0tστdτ≤σ1<+∞,σ˙t≤σ2<+∞,where σ1 and σ2 are any positive constants.Remark 1.
In this paper, the nonlinear filter (13) is introduced to construct the desired controller. Compared with the reported results on adaptive DSC with a linear low-pass filter (e.g., see [30–37]), the advantage of our deign is that the asymptotic tracking control performance can be guaranteed, and the stability analysis can be completed successfully under the proposed controller.Stepi i=2,…,n−1.
Theith surface error is designed as zi=xi−si−1; then, we get the following equation:(15)z˙i=xi+1+θifi,ϱt+dit+M^i−12ei−1M^i−12ei−12+σ2t+zi−1+ei−1τi−1.
Design the virtual control lawαi and the update laws ε^i and W^i as follows:(16)αi=−kizi−2zi−1−ε⌃iziDi2zi2Di2+σ2t−ziW⌃i2zi2W⌃i2+σ2t−M⌃i−12ei−1M⌃i−12ei−12+σ2t−ei−1τi−1,(17)ε⌃˙1=γizi2Di2zi2Di2+σ2t,(18)W⌃˙1=βizi,where ki, γi, and βi are the positive design parameters.
Design the Lyapunov function candidateVi as(19)Vi=Vi−1+12zi2+12γiε˜i2+12βiW˜i2,where γi and βi are the positive design parameters.
In view of equations (15)–(19), consider the time derivative of Vi as(20)V˙i=V˙i−1+ziz˙i−1γiε˜iε^˙i−1βiW˜iW⌃˙i≤−∑j=1ikjzj2+∑j=1i−1zjej+xi+1−αizi+i+∑j=1iεjσt.
Letαi pass through the following nonlinear filter to obtain a filtered virtual controller si(21)τis˙i=−ei−τiM⌃i2eiM⌃i2ei2+σ2t−τizi,si0=αi0,and define(22)ei=si−αi,where ei means the ith boundary layer error and τi is a filter time constant.Stepn.
Considering thenth surface error as zn=xn−sn−1, one has(23)z˙n=u+θnfn,ϱt+dnt+M^n−12en−1M^n−12en−12+σ2t+zn−1+en−1τn−1.
Design the actual control lawu as(24)u=−knzn−2zn−1−ε⌃nznDn2zn2Dn2+σ2t−znW⌃n2zn2W⌃n2+σ2t−M⌃n−12en−1M⌃n−12en−12+σ2t−en−1τn−1,and consider the update laws for ε^i and W^n as follows:(25)ε⌃˙n=γnzn2Dn2zn2Dn2+σ2t,(26)W⌃˙n=βnzn,where kn, βn, and γn are the positive design parameters.
Design the Lyapunov function candidateVn as follows:(27)Vn=Vn−1+12zn2+12γnε˜n2+12βnW˜n2,where γn and βn are the positive design parameters.
Then, considering equations (23)–(27), we have(28)V˙n=V˙n−1+znz˙n−1γnε˜nε^˙n−1βnW˜nW⌃˙n≤−∑j=1nkjzj2+∑j=1n−1zjej+n+∑j=1nεjσt.


## 2.3. Stability Analysis

Based on inequality (28), the main result of this paper is presented by the following theorem.Theorem 1.
Consider the closed-loop system consisting of the plant (1), the nonlinear filters (13), (22), the actual controller (24), and the adaptive laws (10), (11), (17), (18), (25), and(26). Suppose that Assumptions 1-2 hold, for any initial conditions satisfying V0≤q, where q is a given constant, there exit design parameters ki, βi, γi, i=1,…,n, τj, and ηj, j=1,…,n−1, such that the following statements hold:(i)
All the resulting closed-loop signals are semiglobally bounded(ii)
The tracking errorz1=y−yr converges to zero asymptoticallyProof.
The compact sets are defined as(29)Ω1=yr,y˙r,y¨rT:yr2+y˙r2+y¨r2≤B0,Ω2=Vt≤q,where B0 is a known positive constant. Note that set Ω1×Ω2 is also a compact in R4n+1. σt and σ˙t are the bounded functions.
Differentiating the boundary layer errorsei=si−αi yields(30)e˙i=−eiτi−M⌃i2eiM⌃i2ei2+σ2t−zi+Biz1,…,zi+1,e1,…,ei,ε⌃1,…,ε⌃i,M⌃1,…,M⌃i,W⌃1,…,W⌃i,yr,y¨r,y¨r,σt,σ˙t,i=1,…,n−1,where(31)B1⋅=−α˙1=−∂α1∂x1x˙1−∂α1∂ε^1ε⌃˙1−∂α1∂yry˙r−∂α1∂yr˙y¨r−∂α1∂W^1W⌃˙1,Bi⋅=−α˙i=−∑j=1i∂αi∂xjx˙j−∂αi∂εj^ε⌃˙j−∂αi∂ei−1e˙i−1−∂αi∂M^i−1M⌃˙i−1−∂αi∂W^iW⌃˙i−∂αi∂yry˙r−∂αi∂y˙ry¨r−∂αi∂σtσ˙t,are the continuous functions on Ω1×Ω2. As a consequence, a positive constant Mi can be obtained such that Bi⋅≤Mi, where Mi>0 is an unknown constant.
We consider the Lyapunov function candidate as follows:(32)V=Vn+∑i=1n−112ei2+∑i=1n−112ηiM˜i2,where ηi, i=1,…,n−1 are the positive design parameters.
The time derivative ofV is(33)V˙=V˙n+∑i=1n−1eie˙i−∑i=1n−11ηiM˜iM⌃˙i≤−∑i=1nkizi2−∑i=1n−1ei2τi−∑i=1n−1M⌃i2eiM⌃i2ei2+σ2t+∑i=1n−1Miei−∑i=1n−11ηiM˜iM⌃˙i+n+∑j=1nεjσt.
We can obtain the following inequality in view of Lemma 1:(34)Miei=M⌃iei+M⌃iei≤M⌃i2ei2M⌃i2ei2+σ2t+σt+M˜iei,and then, we have(35)V˙≤−∑i=1nkizi2−∑i=1n−1ei2τi−∑i=1n−11ηiM˜iM⌃˙i−βiei+ϖσt,where ϖ≔2n+∑j=1nεj−1 is a constant.
Consider the update laws forM^i as(36)M⌃˙i=ηiei,i=1,…,n−1.
From equation (35), the following inequality can be obtained:(37)V˙≤−∑i=1nkizi2−∑i=1n−1ei2τi+ϖσt.
Integrating inequality (37) over 0,t yields(38)Vt≤V0−∫0t∑i=1nkizi2κ+∑i=1nei2κτidκ+ϖ∫0tσκdκ≤V0+ϖσ1,which means that zi, zn, ε˜i, ε^n, W^i, W^n, ei, and M^i ,i=1,…,n−1 are bounded. Therefore, we also conclude that xi, xn, si, αi, and u, i=1,…,n−1 are bounded. From inequality (38), we can obtain the following inequality:(39)∫0t∑i=1nkizi2κdκ≤V0+ϖσ1.
By applying Barbalat’s lemma [40, 41] to inequality (39), it is concluded that(40)limt⟶∞z1=0,that is to say, the asymptotic stability is accomplished.Remark 2.
For uncertain switched systems (e.g., see [21–23]), lots of adaptive control schemes have been developed by the conventional backstepping method. This paper introduces the adaptive DSC method to solve the “explosion of complexity” problem existing in the aforementioned literature studies, and the asymptotic tracking performance has been achieved under the developed adaptive controller.Remark 3.
Recently, some novel adaptive control schemes have been proposed for uncertain systems with time-varying performance bounds and actuator failures in [42, 43], where the “explosion of complexity” problem cannot be well solved and the asymptotic tracking performance is not achieved. Of course, this paper does not consider the actuator failure phenomenon of the controlled system, which is the further study program in the future.


## 3. Simulation Example

Consider the following nonlinear switched system:(41)x˙1=x2+θ1f1,ϱtx1¯+d1t,x˙2=u+θ2f2,ϱtx2¯+d2t,y=x1,where θ1 and θ2 are the unknown parameters; d1t=cost and d2t=sint; ϱt∈1,2; and the switched functions are designed as follows: f1,1=x1e−0.5x1, f1,2=sinx1, f2,1=sinx1x2, and f2,2=x1x22. When the output y satisfies y>0, fi,σt=fi,1,i=1,2. Otherwise, fi,σt=fi,2,i=1,2. For simulation purpose, it is assumed that θ1=1 and θ2=2.5. The objective of control is that the output yt asymptotically tracks the desired trajectory yrt=sint via the proposed adaptive controller u.The design parameters are selected ask1=1.2, k2=2, γ1=1.12, γ2=1.02, β1=β2=3, η1=0.25, τ1=1, and σt=3e−0.0001t. The initial conditions of this switched system are chosen as x10,x20T=0,0T, ε⌃10=0.1, ε^20=1, W⌃10=W⌃20=0, and M⌃10=0. The simulation results are displayed in Figures 1–5. Figure 1 shows the output tracking performance under the presented DSC scheme, from which it can be analyzed that the asymptotic output tracking has been achieved. Figure 2 shows the control signal u. Figures 3 and 4 show the adaptive parameters. The state x2 is shown in Figure 5. The switching signal is presented in Figure 6. All these simulation results show that the closed-loop signals are bounded.Figure 1
The reference signalyr and the output signal y.Figure 2
The control signalu.Figure 3
Adaptive parametersθ^1, θ^2, and M^1.Figure 4
Adaptive parametersW^1 and W^2.Figure 5
Statex2.Figure 6
Switching signalϱt.


## 4. Conclusions

This paper studies the adaptive tracking control problem for a class of uncertain nonlinear systems with perturbations and switching signal. To deal with the switched signals of the considered system, the common Lyapunov function method is employed to deign the desired controller. Furthermore, according to a nonlinear filter, a DSC method is presented to overcome the “explosion of complexity” problem. Under the proposed controller, it has been expressed that all the closed-loop signals remain semiglobally bounded, and the tracking error converges to zero asymptotically. In the end of this paper, to checkout the validity of the control scheme, a simulation example is presented.



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