# Finite-Time Observer-Based Adaptive Control of Switched System with Unknown Backlash-Like Hysteresis

Authors: Guofa Sun; Yaming Xu
Journal: Complexity
 (2019)

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## Abstract

This work investigates a finite-time observer problem for a class of uncertain switched nonlinear systems in strict-feedback form, preceded by unknown hysteresis. By using a finite-time performance function, a finite-time switched state observer (FTSO) is derived using radial basis function neural networks (RBFNNs) to estimate the unmeasured states. An adaptive feedback neural network tracking control is derived based on the backstepping technique, which guarantees that all the signals of the closed-loop system are bounded, the output tracking error converges to zero, and the observer error converges to a prescribed arbitrarily small region within a finite-time interval. In addition, two simulation studies and an experiment test are provided to verify the feasibility and effectiveness of the theoretical finding in this study.


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## Body

## 1. Introduction

In the past decades, great attentions and developments have been gained in finite-time adaptive control design, which developed a great number of typical design approaches in the literature (see, e.g., [1–15]). A variety of different types of complex nonlinear systems have been explored by using neural networks or the fuzzy logic system-based adaptive backstepping technique. For instance, in [1–3], by utilizing the finite-time stability theory, barrier Lyapunov functions, and the adaptive backstepping method, various finite-time adaptive control strategies were proposed with unknown dead-zone, unknown hysteresis, and unknown time-delayed in the nonlinear system, respectively. Considering the full-state constraints in the nonlinear systems, an adaptive fuzzy controller was constructed to address the finite-time tracking control problem for a class of strict-feedback nonlinear systems in [4]. Besides, Wang et al. [5] developed a novel adaptive neural finite-time control strategy which considered quantized problems for single-input and single-output quantized nonlinear systems. Meanwhile, growth condition assumption was removed under the presented control approach. That assumption was also eliminated in [6]. Li et al. [6] presented a finite-time fuzzy adaptive output feedback control for the first time for nonstrict-feedback nonlinear systems, while it overcame the problem of explosion of complexity by using the dynamic surface control technique. Moreover, Yu et al. [7] investigated an adaptive finite-time quantized tracking problem with input and output quantization and asymmetric actuator saturation, and they designed an observer-based adaptive finite-time quantized tracking controller at the same time. In [8], Liu et al. studied a finite-time adaptive fuzzy tracking control problem and proposed a finite-time adaptive state feedback fuzzy tracking controller with a new concept named finite-time performance function. Although a novel assessment standard of the finite-time semiglobal practical stability (SGPS) was found in [1–8], the main innovations of those literatures are designed controllers for nonswitched systems. Sui et al. [9] solved a finite-time switching control issue for nonstrict-feedback nonlinear switched systems, and they also constructed a novel finite-time SGPS for switched systems. Meanwhile, by utilizing a comparison theorem and a mean value theorem of integrals, a significant finite-time stability criterion for stochastic nonlinear systems was first set up in [10]. Chen et al. [11] studied an adaptive finite-time synchronization controller given for multiple robotic manipulators. It was worth noting that there existed some controllers to solve finite-time control problems in [12–15].As a class of nonlinear inputs, hysteresis widely exists in the industrial system. Besides, the nonlinear characteristics of backlash-like hysteresis could seriously affect tracking performance, and it may cause a severe effect on the whole system. To capture hysteresis dynamics and achieve precise control, many scholars have made efforts to eliminate its effect. The hysteresis models can be roughly divided into two classes in [16, 17]: the operator-based hysteresis model [17–19] and the differential equation-based hysteresis model, such as Bouc–Wen model [20, 21] and backlash-like model [22–26]. For instance, in [18], Mayergoyz described a new approach to the scalar Preisach model of hysteresis which emphasized its phenomenological nature and mathematical generality. Zhang et al. [19] designed a hysteresis compensator to compensate the hysteresis nonlinearity described by a generalized PI hysteresis model, and the Krasnoselskii–Pokrovskii model was applied in [17]. The modeling accuracy can be guaranteed by increasing the number of superposed elementary operators for the operator-based hysteresis model, but also cause the computational burden during implementation. We continue to investigate the backlash-like hysteresis model in this paper since it has fewer parameters and has analytical solution. In [22], Ma et al. dealt with adaptive control for a class of switched nonlinear systems preceded by unknown backlash-like hysteresis, where the hysteresis was modeled by a differential equation. Similarly, Li et al. [23] investigated an adaptive neural output feedback control for a class of nonlinear systems with unknown backlash-like hysteresis of the actuator. Zhang and Lin [24] proposed a robust adaptive DSC control for a class of uncertain perturbed strict-feedback nonlinear systems preceded by unknown backlash-like hysteresis. On one hand, it is an active issue to study the systems with hysteresis nonlinearities which exist widely in the practical systems, such as manipulator system [22, 27], four-motor servomechanism [28, 29], and electrical circuit [30]. On the other hand, to date, there are no systematic methods to achieve a satisfactory result on finite-time switched state estimate of switched nonlinear systems with backlash-like hysteresis, which motivates our research interest. Hence, considering the inevitability of the backlash-like hysteresis constraint in many practical applications, it is worth further studying how to design a finite-time switched state observer (FTSO) for switched strict-feedback nonlinear systems with unknown backlash-like hysteresis.During the development of control and applications, there exists a design algorithm which links observer design and hysteresis compensation. In [31], a novel hysteresis compensation method based on extended high-gain observer was presented without any specific hysteresis models. Besides, there exist several seminal observers designed with a system of hysteresis results [30, 32–34]. Huo et al. proposed an MIMO switched fuzzy observer to estimate the system states with hysteresis nonlinearities, which was less conservative than using a common observer for all subsystems. As stated in [33], they addressed the consensus tracking problem of a class of nonlinear multiagent system with hysteresis. And then, they developed a distributed adaptive neural output feedback control scheme proposed by constructing a state observer and using the backstepping technique. In order to improve the functional approximation capability and disturbance compensation ability for a system with unknown backlash-like hysteresis, Wang et al. [34] considered the coupler design between the radial basis function neural network and observer. Wei et al. [30] presented an extremum seeking algorithm to accurately estimate the state-of-power by an electrical circuit incorporating hysteresis effect. However, those contributions in [30–34] do not include finite-time convergence of observers, which further improves the tracking effect of the control strategy.This study considers the finite-time adaptive switched observer problem for a kind of nonlinear systems with unknown hysteresis from a new perspective. Based on RBFNN approximation and finite-time performance function, the FTSO is proposed, which guarantees that observer error converge to a small neighborhood of the origin point in finite-time. Meanwhile, an adaptive tracking neural network controller is designed by the backstepping technique, which guarantees that all the signals of the switched closed-loop system are bounded and the observation error is converged to a small neighborhood of the origin point in a finite time. Compared with the existing results, the prominent contributions of this work are as follows:(1)
This paper proposes design process of a novel finite-time switched state observer which is designed to improve control performances for the switched system with unknown backlash-like hysteresis. In contrast with [1–15] which studies finite-time adaptive tracking controller, the smaller tracking error will be achieved with an FTSO for the same controller. In addition, with the presented control approach, all known state assumptions are removed by using the FTSO.(2)
To the best of our knowledge, it is the first time that the finite-time convergence problem of observer error is taken into consideration in a class of uncertain switched nonlinear systems with unknown backlash-like hysteresis, by using a finite-time performance function to obtain a better tracking performance. For switched systems with unknown backlash-like hysteresis, a control strategy with the FTSO can be applied more effectively and exactly.This paper is organized as follows. Sections2 presents the problem and the preliminaries. In Section 3, an adaptive finite-time switched state observer is designed. Section 4 draws control design and stability analysis. Simulation studies are conducted in Section 5 to illustrate the effectiveness of the proposed scheme. Finally, the paper is ended by concluding remarks in Section 6.


## 2. Problem Formulation and Preliminaries

This section begins by presenting the control problem for a class of uncertain switched nonlinear systems with unknown backlash-like hysteresis. Then, it reviews some preliminaries about RBFNNs to facilitate the proposed observer and controller design procedure. Finally, a finite-time performance function is given to ensure observer error finite-time convergence.


### 2.1. Proposed System Descriptions

Consider a class of switched strict-feedback nonlinear systems with unknown hysteresis in the form of(1)x˙i=gi,σtxi+1+fi,σtx¯i+di,σtt,x˙n=gn,σtuv+fn,σtx¯n+dn,σtt,y=x1,where x¯i=x1,x2,⋯,xiT∈Ri, i=1,2,⋯,n−1, denotes the vectors of system states, y∈R is the output of the switched nonlinear system, fi⋅ are smooth unknown nonlinear functions, and gi are known constants and gi≠0. σt:0,∞⟶M=1,…,m expressing the switching signal, which is a piecewise right continuous function. In addition, σt=j denotes the jth subsystem. dit stands for the bounded external disturbance. uv∈R is the output signal of backlash-like hysteresis described as(2)u˙=αv˙cv−u+bv˙,where v indicates the input of the unknown backlash-like hysteresis. α,c, and b are unknown constants with c>b.It follows from the analysis in [22, 23, 35] that the solution of system (2) is expressed as(3)ut=cvt+lv,where lv is bounded and satisfies(4)lv=u0−cv0e−αv−v0sgnv˙+e−αvsgnv˙∫v0vb−ceαξsgnv˙dξ,where u0 and v0 are the initial values of u and v, respectively.Figure1 shows that a typical backlash-like hysteresis is generated. By (3), rewrite (1) as(5)x˙i=gi,jxi+1+fi,jx¯i+di,jt,x˙n=b0,jvt+Fn,jx¯n+dn,jt,y=x1,with Fnx¯n,j=fn,jx¯n+gn,jlv and b0,j=gn,jc.Figure 1
Hysteresis curves given by (2) with α=1,c=3.1635, and b=0.345 for vt=ksin2.3t with k=2.6 (red solid line) and k=3.6 (dashed line) to approximate backlash (blue solid line).This paper aims to derive an FTSO-based adaptive neural control signalv for nonlinear system (1) with unknown backlash-like hysteresis (2), so that the following objectives can be achieved:(1)
Observer errors converge to an arbitrarily small specified neighborhood at a finite-time(2)
All the variables of the closed-loop system are bounded, and the tracking errors could converge to an arbitrarily small specified neighborhoodLemma 1 (see [36, 37]).
For a nonlinear switching system oft∈0,Tf, Nσt,Tf denotes the number of times and σt switches within a time interval 0,Tf. If(6)Nσt,Tf≤N0+Tf−tτa,where τa and N0 are positive constants, then τa is average dwell time. In order to facilitate calculation, the buffeting bound N0 is generally taken.Lemma 2 (see [36, 37]).
A Lyapunov function candidate is defined as(7)VP=xTPσtx,∀x∈Rn,where Pσt is switched in accordance with the piecewise constant switching signal σt. Then, the following properties are obtained:(1)
EachVPσt=xTPσtx in (7) is continuous. There exist constant scalars χ2≥χ2≥0 such that(8)χ1x≤VPjx≤χ2x,∀x∈Rn,∀j∈M.(2)
There exist constant scalarsζ≥1 such that(9)VPix≤ζVPjx,∀x∈Rn,∀i,j∈M.


### 2.2. Radial Basis Function Neural Networks

The RBFNN has the ability to approximate the unknown smooth functionFX:Rq⟶R, and the form of FX is as follows:(10)FX=θTϕX+ηX,where X=X1,⋯,XqT∈Rq and θ∈Rq are the weight vectors of the neural network, ϕX=ϕ1X,⋯,ϕqXT∈Rq are the basis function vector of the neural network, and ηX is approximation error of neural network function, and ηX satisfies that ηX<η∗, where η∗ is the upper bound of the estimated error.In the following,ϕ is used to represent ϕX and η is used to represent ηX, where the estimated error of the observer and the controller are expressed as ηo and ηC, respectively.The RBFNN is an approximator for nonlinear smooth functionfix¯i which is approximated as(11)f^ix¯^iθ^i=θiTϕix¯^i,x¯^i=y,x2,⋯,xnT,where x¯^i is the estimation of x¯i.The ideal parameter vectorsθi∗ are defined as(12)θi∗=argminθi∈Ωisupx¯^i∈Uif^ix¯^iθ^i−fix¯i,where Ui and Ωi are compact regions for x^i and θi, respectively. Define the minimum approximation error ηi as(13)ηi=f^ix¯^iθi∗−fix¯i.Lemma 3 (see [38]).
The following inequality holds(14)ϕTXϕX≤1ϕTx¯iϕx¯i,for any inter i≤n, where ϕx¯i=π1x¯i,π2x¯i,⋯,πNx¯iT/∑k=1Nπkx¯i and πkx¯i=exp−x¯i−liTx¯i−li/2γi2.


### 2.3. Finite-Time Performance Function

Definition 1.
Different from a classical prescribed performance function in [39], a smooth function ξt is called finite-time performance function (FTPF) for all ξ0=ξ0. There exists a settling time 0<T0<∞, such that ξt<ξT0=ξT0, which is an arbitrarily small positive number. ∀t≥T0=ξt, in this work, is defined by(15)ξt=ξ0−τλt1/τ+ξT0,0≤t≤T0,ξT0,t>T0,where λ, τ, ξ0, and ξT0 are chosen as positive constants, and let τ=q/p∈0,1, where p and q are chosen positive constants. Moreover, it has T0=ξ0/τλ.


## 2.1. Proposed System Descriptions

Consider a class of switched strict-feedback nonlinear systems with unknown hysteresis in the form of(1)x˙i=gi,σtxi+1+fi,σtx¯i+di,σtt,x˙n=gn,σtuv+fn,σtx¯n+dn,σtt,y=x1,where x¯i=x1,x2,⋯,xiT∈Ri, i=1,2,⋯,n−1, denotes the vectors of system states, y∈R is the output of the switched nonlinear system, fi⋅ are smooth unknown nonlinear functions, and gi are known constants and gi≠0. σt:0,∞⟶M=1,…,m expressing the switching signal, which is a piecewise right continuous function. In addition, σt=j denotes the jth subsystem. dit stands for the bounded external disturbance. uv∈R is the output signal of backlash-like hysteresis described as(2)u˙=αv˙cv−u+bv˙,where v indicates the input of the unknown backlash-like hysteresis. α,c, and b are unknown constants with c>b.It follows from the analysis in [22, 23, 35] that the solution of system (2) is expressed as(3)ut=cvt+lv,where lv is bounded and satisfies(4)lv=u0−cv0e−αv−v0sgnv˙+e−αvsgnv˙∫v0vb−ceαξsgnv˙dξ,where u0 and v0 are the initial values of u and v, respectively.Figure1 shows that a typical backlash-like hysteresis is generated. By (3), rewrite (1) as(5)x˙i=gi,jxi+1+fi,jx¯i+di,jt,x˙n=b0,jvt+Fn,jx¯n+dn,jt,y=x1,with Fnx¯n,j=fn,jx¯n+gn,jlv and b0,j=gn,jc.Figure 1
Hysteresis curves given by (2) with α=1,c=3.1635, and b=0.345 for vt=ksin2.3t with k=2.6 (red solid line) and k=3.6 (dashed line) to approximate backlash (blue solid line).This paper aims to derive an FTSO-based adaptive neural control signalv for nonlinear system (1) with unknown backlash-like hysteresis (2), so that the following objectives can be achieved:(1)
Observer errors converge to an arbitrarily small specified neighborhood at a finite-time(2)
All the variables of the closed-loop system are bounded, and the tracking errors could converge to an arbitrarily small specified neighborhoodLemma 1 (see [36, 37]).
For a nonlinear switching system oft∈0,Tf, Nσt,Tf denotes the number of times and σt switches within a time interval 0,Tf. If(6)Nσt,Tf≤N0+Tf−tτa,where τa and N0 are positive constants, then τa is average dwell time. In order to facilitate calculation, the buffeting bound N0 is generally taken.Lemma 2 (see [36, 37]).
A Lyapunov function candidate is defined as(7)VP=xTPσtx,∀x∈Rn,where Pσt is switched in accordance with the piecewise constant switching signal σt. Then, the following properties are obtained:(1)
EachVPσt=xTPσtx in (7) is continuous. There exist constant scalars χ2≥χ2≥0 such that(8)χ1x≤VPjx≤χ2x,∀x∈Rn,∀j∈M.(2)
There exist constant scalarsζ≥1 such that(9)VPix≤ζVPjx,∀x∈Rn,∀i,j∈M.


## 2.2. Radial Basis Function Neural Networks

The RBFNN has the ability to approximate the unknown smooth functionFX:Rq⟶R, and the form of FX is as follows:(10)FX=θTϕX+ηX,where X=X1,⋯,XqT∈Rq and θ∈Rq are the weight vectors of the neural network, ϕX=ϕ1X,⋯,ϕqXT∈Rq are the basis function vector of the neural network, and ηX is approximation error of neural network function, and ηX satisfies that ηX<η∗, where η∗ is the upper bound of the estimated error.In the following,ϕ is used to represent ϕX and η is used to represent ηX, where the estimated error of the observer and the controller are expressed as ηo and ηC, respectively.The RBFNN is an approximator for nonlinear smooth functionfix¯i which is approximated as(11)f^ix¯^iθ^i=θiTϕix¯^i,x¯^i=y,x2,⋯,xnT,where x¯^i is the estimation of x¯i.The ideal parameter vectorsθi∗ are defined as(12)θi∗=argminθi∈Ωisupx¯^i∈Uif^ix¯^iθ^i−fix¯i,where Ui and Ωi are compact regions for x^i and θi, respectively. Define the minimum approximation error ηi as(13)ηi=f^ix¯^iθi∗−fix¯i.Lemma 3 (see [38]).
The following inequality holds(14)ϕTXϕX≤1ϕTx¯iϕx¯i,for any inter i≤n, where ϕx¯i=π1x¯i,π2x¯i,⋯,πNx¯iT/∑k=1Nπkx¯i and πkx¯i=exp−x¯i−liTx¯i−li/2γi2.


## 2.3. Finite-Time Performance Function

Definition 1.
Different from a classical prescribed performance function in [39], a smooth function ξt is called finite-time performance function (FTPF) for all ξ0=ξ0. There exists a settling time 0<T0<∞, such that ξt<ξT0=ξT0, which is an arbitrarily small positive number. ∀t≥T0=ξt, in this work, is defined by(15)ξt=ξ0−τλt1/τ+ξT0,0≤t≤T0,ξT0,t>T0,where λ, τ, ξ0, and ξT0 are chosen as positive constants, and let τ=q/p∈0,1, where p and q are chosen positive constants. Moreover, it has T0=ξ0/τλ.


## 3. Finite-Time Switched State Observer Design

In this section, an FTSO is derived, involving the use of FTPF. Stability of the developed observer, estimating the states of switched nonlinear system (1) with unknown hysteresis (2), is proved by Lyapunov function.For the switched nonlinear system (5), an FTSO is designed as follows:(16)ε1=x^1−x1,x^˙1=α1+f^1x¯^1θ^1,α1=g1,jx^2−k1,jε1,⋯εi=x^i−αi−1gi−1,j,x^˙i=αi+f^ix¯^iθ^i,αi=gi,jx^i+1−ki,jεi,i=2,⋯,n−1,⋯εn=x^n−αn−1gn−1,j,x^˙n=αn+F^nx¯^iθ^n+b0,jv,αn=−kn,jεn,where x^=x^1,…,x^nT is the estimation of real system states. gi,j=g1,j,…,gn,jT and ki,j=k1,j,…,kn,jT,i=1,…,n, are parameters to be designed. Furthermore, combining (16), the transfer error of the observer εi are further expressed as(17)ε1=x^1−x1,εi=αi−1ki−1,jgi−1,jεi−1,εn=αn−1kn−1,jgn−1,jεn−1.Further, define the observer errore¯o as(18)eo1=x^1−x1,eoi=x^i−xi,eon=x^n−xn.From (16) and (18), the differential with respect to t of e¯o becomes(19)e˙o1=g1,jeo2+θ˜1Tϕ1x¯^1+η1−k1,jε1,e˙oi=gi,jeoi+1+θ˜iTϕix¯^i+ηi−ki,jεi,e˙on=θ˜nTϕnx¯^n+ηn−kn,jεn,where θ˜i=θi−θi∗i=1,…,n is the adaptive parameter error.Define the error transform functionhzi as(20)eoi=ξhzi,where hzi=2/πarctanzi. hzi is a smooth and strictly increasing function and hzi∈−1,1. It also satisfies limzi⟶+∞hzi=1 and limzi⟶−∞hzi=−1. Define a new function as(21)Lzi=12dhzidzizi+hzi,which satisfies Lza>Lzb, when za>zb. Combining (20) and (21),(22)12zi+hzi>Lzi>12hzi.Differentiating (20) with respect to t yields(23)e˙oi=ξ˙hzi+ξ∂hzi∂ziz˙i.Then, the dynamic system errors are expressed as(24)zi˙=Mξ,zigi,jξhzi+1−∏i=1n−1ki,jξhz1−ξ˙hzi+Mξ,ziθ˜iTϕix¯^i+ηi,zn˙=Mξ,zi−∏i=1nki,jξhz1−ξ˙hzn+Mξ,znθ˜nTϕnx¯^n+ηn,where Mξ,zi=1/ξ∂hzi/∂zi.Step 1.
Construct a Lyapunov function candidate as(25)Vi=1Mzihzi+1Mzi2,i=1,…,n−1.
Differentiating (25),(26)V˙i=−1M2M˙zihzi+1Mz˙ihzi+ziz˙ih˙zi−M˙M2zi2+1Mziz˙i=−M˙M2zihzi−M˙M2zi2+gi,jξhzi+1−∏i=1ikiξhz1−ξ˙hzihzi+zih˙zi+θ˜iTϕix¯^i+ηihzi+zih˙zi+zigi,jξhzi+1−∏i=1iki,jξhz1−ξ˙hzi+ziθ˜iTϕix¯^i+ηi.
Computinggi,jξhzi+1−∏i=1iki,jξhz1−ξ˙hzihzi+zih˙zi gives(27)gi,jξhzi+1−∏i=1iki,jξhz1−ξ˙hzihzi+zih˙zi=2gi,jξhzi+1−∏i=1iξhz1−ξ˙hziLzi≤gi,jξhzi+1zi+hzi−∏i=1iki,jξhz1+ξ˙hzihzi.
Then,θ˜1Tϕ1x¯^1+η1hz1+z1h˙z1 is rewritten as(28)θ˜iTϕix¯^i+ηihzi+zih˙zi≤θ˜iTϕix¯^i+ηihzi+zi.
Computingzigi,jξhzi+1−∏i=1iki,jξhz1−ξ˙hzi gives(29)zigi,jξhzi+1−∏i=1iki,jξhz1−ξ˙hzi≤−∏i=1iki,jξzihz1−ξ˙zihzi+gi,jξzihzi+1.
Combining (27)–(29) into (26),(30)V˙i≤−M˙M2zihzi−M˙M2zi2+gi,jξhzi+1zi+hzi−∏i=1iki,jξhz1+ξ˙hzihzi−∏i=1iki,jξzihz1−ξ˙zihzi+gi,jξzihzi+1+θ˜iTϕix¯^i+ηihzi+2zi.
In the light of Young’s inequality and the characteristic ofh., we have(31)2hzi+1zi+hzi≤2h2zi+1+zi2+h2zi,2zihzi+1≤h2zi+1+zi2,h2zi≤zi2,θ˜iTϕix¯^i+ηihzi+2zi≤θi˜2+ηi2+3zi2.
By substituting (31) into (30), it follows(32)V˙i≤−M˙M2+ξ˙zihzi−M˙M2+ξ˙−3gi,jξ2−3zi2+3gi,jξ2zi+12−∏i=1iki,jξhz1hzi−∏i=1iki,jξzihz1+θi˜2+ηi2.Step 2.
Consider the Lyapunov function candidate as(33)Vn=1Mznhzn+1Mzn2.
Similar to Step 1,(34)V˙n≤−M˙M2znhzn−M˙M2zn2−∏i=1nki,jξhz1+ξ˙hznhzn−∏i=1nki,jξznhz1−ξ˙znhzn+θ˜nTϕnx¯^n+ηnhzn+2zn≤−M˙M2+ξ˙znhzn−M˙M2+ξ˙−3zn2−∏i=1nki,jξhz1hzn−∏i=1nki,jξznhz1+θn˜2+ηn2.
We augment the Lyapunov candidateVo as(35)Vo=∑i=1nVi.
Synthesizing (32), (34), and (35),(36)V˙o≤−∑i=1nM˙M2+ξ˙zihzi−M˙M2+ξ˙−3g1,jξ2−3z12−∑i=2n−1M˙M2+ξ˙−3gi,jξ2−3gi−1,jξ2−3zi2−M˙M2+ξ˙−3gn−1,jξ2−3zn2−∑i=1n∏i=1nki,jξhz1hzi+zihz1+∑i=1nθ˜n2+ηn2.
Stability analysis is described in the following two parts.Case 1.
z
1 and zi have the same sign (such as both z1 and zi are positive or negative).
Computing−hz1hzi+zihz1 gives(37)−hz1hzi+zihz1≤0.
Substituting (37) into (36), one obtains(38)V˙o≤−∑i=1nM˙M2+ξ˙zihzi−M˙M2+ξ˙−3g1,jξ2−3z12−∑i=2n−1M˙M2+ξ˙−3gi,jξ2−3gi−1,jξ2−3zi2−M˙M2+ξ˙−3gn−1,jξ2−3zn2−∑i=1nθ˜n2+ηn2.Case 2.
z
1 and zi have opposite signs.Computing−hz1hzi+zihz1 gives(39)−hz1hzi+zihz1≤12−hz12+h2zi+zi2+−hz12≤z12+zi2.Theorem 1.
Consider system (5) with unknown backlash-like hysteresis under Definition 1 and error transform function (20), applying the proposed FTSO mechanism in (16), observer errors (18) converge to a small neighborhood of the origin point in a settling time T0.Proof.
Substituting (39) into (36), one obtains(40)V˙o≤−∑i=1nM˙M2+ξ˙zihzi−M˙M2+ξ˙−3g1,jξ2−3−2∑i=1n∏i=1nki,jξz12−∑i=2n−1M˙M2+ξ˙−3gi,jξ2−3gi−1,jξ2−3−∑i=1n∏i=1nki,jξzi2−M˙M2+ξ˙−3gn−1,jξ2−3−∑i=1n∏i=1nki,jξzn2+∑i=1nθ˜n2+ηn2.
For simplification of the expression, reconfigure the parametersλ1=M˙/M2+ξ˙−3g1,jξ/2−3−2∑i=1n∏i=1nki,jξ>0, λi=∑i=2n−1M˙/M2+ξ˙−3gi,jξ/2−3gi−1,jξ/2−3−∑i=1n∏i=1nki,jξ>0,i=2,…,n, and λn=M˙/M2+ξ˙−3gn−1,jξ/2−3−∑i=1n∏i=1nki,jξ>0. Futhermore, denote λ=minM˙/M2+ξ˙,λ1,…,λn and Do=∑i=1nθ˜n2+ηn2. The Laypunov function Vo consequently takes the form(41)V˙o≤λVo+D.
Integrating (41) from 0 to t produces(42)0≤Vot≤Dλ+Vo0e−λt.
Therefore, the observer errors converge to a small neighborhood of the origin point in a finite-time interval0,T0, and all of the closed-loop systems signals are bounded.
This completes the proof of Theorem 1.


## 4. Adaptive Controller Design and Stability Analysis

In this section, an adaptive control scheme is developed for nonlinear system with unknown backlash-like hysteresis by using the above FTSO and the backstepping technique. To facilitate the design procedure, the following errors are defined as(43)ec1=x^1−xd,eci=x^i−γi−1,i=2,⋯,n,where γi−1 are virtual intermediate control signals which will be defined later.The feedback control laws (44), (45), and adaptive laws (46) are chosen as(44)γ1=−ec1θ^14ϕ1Tx¯^1ϕ1x¯^1−h1,jec1,(45)γi=−eciθ^i4ϕiTx¯^iϕix¯^i−hi,jeci+γ˙i−1gi,j,i=2,…,n,(46)θ^˙i=digi,jeci24ϕnTx¯^iϕix¯^i−piθ^i,i=1,…,n,where v=γn. h1,j>1, d1>0, p1>0, hi,j>1, di>0, and pi>0 are the design parameters.Step 1.
Consider a Lyapunov candidate function in the form of(47)Vc1=12ec12+12d1θ˜12,where θ˜1=θ^1−θ1∗, with θ^1 being the estimation of θ1∗=θ^12.(48)V˙c1=e1e˙1+1d1θ˜1Tθ^˙1.
From (16) and (43), one obtains(49)e˙c1=g1,jx^2−k1,jε1+f^1x¯^1θ^1−x˙d=g1,jx^2−k1,jε1+θ^1Tϕ1x¯^1+η1−x˙d=g1,jec2+γ1−k1,jε1+θ^1Tϕ1x¯^1+η1−x˙d.
From (48) and (49), it is obtained that V˙c1 is(50)V˙c1=ec1g1,jec2+γ1+θ^1Tϕ1x¯^1+ec1−k1,jε1+η1−x˙d+1d1θ˜1Tθ^˙1.
Based on Young’s inequality and Lemma 3, the following inequalities are obtained(51)ec1θ^1Tϕ1x¯^1≤g1,jec12θ12ϕ1Tx¯^1ϕ1x¯^14+1g1,j≤g1,jec12θ1∗4ϕ1Tx¯^1ϕ1x¯^1+1g1,j.
Substituting (44), (46), and (51) into (50) gives(52)V˙c1=ec1g1,jec2+γ1+g1,jec1θ^1−θ˜14ϕ1Tx¯^1ϕ1x¯^1+ec1−k1,jε1+η1−x˙d+1d1θ˜1Tθ^˙1+ec10g1,j+ec1g1,jec2−h1,jec1+ec1−k1,jε1+η1−x˙d+p1d1θ˜1θ^1+ec10g1,j.
Via Young’s inequality, we have(53)−p1d1θ˜1θ^1≤−p12d1θ˜12+p12d1θ1∗2,(54)ec1−k1,jε1+η1−x˙d≤12ec12+12ς2,where ς=−k1,jε1+η1−x˙d. ε1, η1, and x˙d are bounded.
Thus, (52) is modified as(55)V˙c1=−h1,jg1,j−12ec12−p12d1θ˜12+g1,jec1ec2+E1,where E1=p1/2d1θ1∗2+1/2ς2+ec10/g1,j.
Stepi (i=2,⋯,n−1). Similar to Step 1, consider a Lyapunov candidate function in the form of(56)Vci=12eci2+12diθ˜i2,where θ˜i=θ^i−θi∗.(57)V˙ci=eie˙i+1diθ˜iTθ^˙i.
From (43) and (45), one obtains(58)e˙ci=gi,jeci+1+γi−∏i=1iki,jε1+θ^iTϕnx¯^i+ηi−γ˙i−1.
From (57) and (58), V˙ci is given by(59)V˙ci=ecigi,jeci+1+γi+θ^iTϕix¯^i+eci−∏i=1iki,jε1+ηi−γ˙i−1+1diθ˜iTθ^˙i.
Based on Young’s inequality and Lemma 3, the following inequalities are obtained:(60)eciθ^iTϕix¯^i≤gi,jeci2θi2ϕiTx¯^iϕix¯^i4+1gi,j≤gi,jeci2θi∗4ϕiTx¯^iϕix¯^i+1gi,j.
Thus,(61)V˙ci=ecigi,jeci+1−hi,jeci+eci−∏i=1iki,jε1+ηi−pidiθ˜iθ^i+eci0gi,j.
From Young’s inequality,(62)−pidiθ˜iθ^i≤−pi2diθ˜i2+pi2diθi∗2,(63)eci−∏i=1n−1ki,jε1+ηi≤12eci2+12ςi2,where ςi=−∏i=1iki,jε1+ηi and ηi are bounded.
Substituting (62) and (63) into (61) gives(64)V˙ci=−hi,jgi,j−12eci2−pi2diθ˜i2+gi,jecieci+1+Ei,where Ei=pi/2diθi∗2+1/2ςi2+eci0/gi,j.Stepn.
Similar to Stepi, consider a Lyapunov candidate function in the form of (65)Vcn=12ecn2+12dnθ˜n2,where θ˜n=θ^n−θn∗.
Similar to Stepi, one obtains(66)e˙cn=−∏i=1nki,jε1+θ^nTϕnx¯^n+ηn−γ˙n−1+b0,jv.
Moreover, the time-derivative ofVcn is given by(67)V˙cn=ecnb0,jυ+θ^nTϕnx¯^n+ecn−∏i=1nki,jε1+ηn−γ˙n−1+1dnθ˜nTθ^˙n.
Based on Young’s inequality and Lemma 3, the following inequalities are obtained:(68)ecnθ^nTϕnx¯^n≤b0,jecn2θn2ϕnTx¯^nϕnx¯^n4+1b0,j≤b0,jecn2θn∗4ϕnTx¯^nϕnx¯^n+1b0,j.
Substituting (45), (67), and (68) gives(69)V˙cn=ecnb0,j−hn,jecn+ecn−∏i=1nkn,jε1+ηn−pndnθ˜nθ^n+ecn0b0,j.
By using Young’s inequality,(70)−pndnθ˜nθ^n≤−pn2dnθ˜n2+pn2dnθn∗2,(71)ecn−∏i=1nki,jε1+ηn≤12ecn2+12ςn2,where ςn=−∏i=1nki,jε1+ηn and ηn are bounded.
Thus, (69) is modified to be(72)V˙cn=−hn,jb0,j−12ecn2−pn2dnθ˜n2+En,where En=pn/2dnθn∗2+1/2ςn2+ecn0/b0,j.
A Lyapunov functionVc is chosen as(73)Vc=∑i=1nVci=∑i=1n12eci2+∑i=1n12diθ˜i2.
The differentiation of equation (73) is(74)V˙c=∑i=1nV˙ci≤−∑i=1n−1hi,jgi,j−12eci2−∑i=1npi2diθ˜i2+∑i=1n−1gi,jecieci+1+∑i=1nEi−hn,jb0,j−12ecn2.
Based on Young’s inequality, it is obtained that(75)gi,jecieci+1≤gi,jhi,j2eci2+12hieci+12≤gi,jhi,j2eci2+12eci+12,with hi,j>1.
Thus, equation (74) is modified as(76)V˙c≤−∑i=1n−1hi,jgi,j−12eci2−∑i=1npi2diθ˜i2+∑i=1n−1gi,jhi,j2eci2+12eci+12+∑i=1nEi−hn,jb0,j−12ecn2≤−12∑i=1n−1hn,jgi,j−2eci2−122hn,jb0,j−2ecn2−∑i=1npi2diθ˜i2+∑i=1nEi.
Furthermore, denoteμ=minh1,jg1,j−2,⋯,hn−1,jgn−1,j−2,2hn,jb0,j−2,p1,p2,⋯,pn and Ec=∑i=1nEi. The Laypunov function Vc consequently takes the form of(77)V˙c≤−μVc+Ec.Theorem 2.
For switched strict-feedback nonlinear systems with unknown hysteresis (1), FTSO (16), feedback control laws (44) and (45), and adaptive laws (46), if the average dwell time of the switching signal σt satisfies the condition τa>lnζ/μζ=maxλmaxPk/λminPl,k,l∈M, the control scheme based on FTSO ensures that all variables of the closed-loop system are bounded. By properly selecting the parameters, the controller and observer error converge to origin, such that limT⟶∞z12≤0 and limT⟶∞e12≤0.Proof.
There is a piecewise differentiable functionWt=eμtVjεt in any interval ti−1,ti. Then, its time differentiation on ti−1,ti is(78)W˙t=μeμtVjεt+eμtV˙jεt.
From (76) and (78),(79)W˙t≤Eceμt.
Then, noting that at the switching pointVti≤ζVti∗ from Lemma 2, where ti∗=limt⟶tit, it follows that(80)Wti=eμtiVσtiεti≤ζeμtiVσti−1+εti=ζWti∗≤ζWti−1+∫ti−1tiEceμtdt.
LetTf>t0=0 and i∈1,N0,Tf, then equation (80) is further expressed as(81)WTf∗≤WtN0,Tf+∫tN0,TfTfEceμtdt≤ζWtN0,Tf−1+∫tN0,Tf−1tN0,TfEceμtdt+∫tN0,TfTfEceμtdt≤ζ2WtN0,Tf−2+∫tN0,Tf−2tN0,Tf−1Eceμtdt+ζ∫tN0,Tf−1tN0,TfEceμtdt+∫tN0,TfTfEceμtdt≤⋯≤ζN0,TfW0+ζN0,Tf∑i=1N0,Tf+1ζ1−i∫ti−1tiEceμtdt.
Sinceτa>lnζ/μ and Lemma 1, for any Tf>0,(82)N0,Tf≤N0+μTflnζ.
Substituting (82) from (81),(83)WTf∗≤ζN0,TfW0+ζN0,Tf∫0TfEceμtdt≤ζN0e−μTfW0+ζN0e−μTfEcμeμTf−1.
By Lemma 2 and (83), one has(84)χ1εT≤VσT∗εT∗=ζN0e−μTW0+ζN0Ecμ1−e−μTeμt≤eN0lnζe−2μTχ2ε0+ζN0e−μTEcμ1−e−μT.
Therefore, from (84), it follows that(85)limT⟶∞z12≤eN0lnζe−2μTχ2ε0+ζN0e−μTκμ1−e−μT=0.
Similarly, one obtains(86)limT⟶∞e12≤0.That is, using (85) and (86), observation errors and tracking errors converge to zero.


## 5. Simulation Results

In this section, three examples are considered to test the proposed control method. Consider the following second-order nonlinear system:(87)x˙1=g1,σtx2+f1,σtx¯1+d1,σtt,x˙2=g2,σtu+f2,σtx¯2+d2,σtt,where g1,1=1, g1,2=1, f1,1x¯1=x1−x13/1+x14, f1,1x¯1=x1−x12/1+x15, g2,1=1, g2,2=1, f2,1x¯2=−exp−x12sin5x2, f2,1x¯2=−sin5x2, ut=vt+lv, lv=u0−v0exp−5v−v0sgnv˙+exp−5vsgnv˙∫v0v0.345−1exp5ξsgnv˙dξ, d1,1t=0.2sint, d1,2t=0.3sint, d2,1t=0.2cost, and d2,2t=0.3cost.The selection guidelines of the design parameters in the proposed control design method are summarized in the following:Step 1. During selection of design parameters of observer (16), the parameter ki,j>0 are selected before ki+1,j>0. Then, the values are usually chosen to ensure ki,j<ki+1,j.Step 2. Determine the parameters of finite-time performance function (15) in FTSO (16). In this step, one needs to choose finite-time T0=0.3, the initial values ξ0=0.99, and the final value ξT0=0.01. Then, τ=0.5 and λ=6.66 are chosen. Moreover, it is obtained that T0=ξ0/τλ.Step 3. Select controller parameters of hi,j>1, di>0, and pi>0 to ensure μ>0, which is defined in (76).Step 4. Initial values of the system states and observer are set according to the proof of Theorem 1. The appropriate initial conditions are given in order to make the contrast effect of control and observation more clear. In this simulation, x0=00T and x^0=00T.For better illustration, we consider such two cases: (1) the proposed adaptive controller scheme with FTSO and (2) the proposed adaptive controller scheme with the observer in [12]. The reference signals xd are selected for xd=2 and xd=1.5sin2t. The basis function vector ϕx is chosen as the form of the Gaussian function:(88)Φix¯=exp−x¯−μiTx¯−μi2γi2,with five nodes, the width γ1=0.5 and γ2=0.8, and the centers μii=1,2 are equally distributed between 2,2.The chosen control parameters ared1=1, d2=1, p1=2, p2=2, h1,1=30, h1,2=30, h2,1=3, and h2,2=9 and x0=00T, x^0=00T, and θ^0=00T. The FTSO parameters are selected as k1,1=5.1, k1,2=5.2, k2,1=9, and k2,2=9.5, and the other observer parameters are k1,1=10, k1,2=11, k2,1=21, and k2,2=20.Example 1.
x
d
=
2.
Switching signalσt is shown in Figure 2, and the observer errors eo1 with the FTSO and the observer in [12] are shown in Figure 3. The tracking performance of the switched system are shown in Figure 4, while the tracking errors ec1 and the control input v of two schemes are shown in Figures 5 and 6, respectively.Figure 2
Switching signalσt in simulation.Figure 3
The observer compared results ofeo1.Figure 4
The observer compared results ofx1.Figure 5
The observer compared results ofec1.Figure 6
Control inputv.Example 2.
x
d
=
1.5
sin
2
t.
The tracking responses and errors with the FTSO and the contrastive observer are depicted in Figure7. Furthermore, in order to further analyze the performance of the tracking effect with the same controller in Figure 7, the performance evaluation, including average tracking error and average fluctuation of tracking error, is summarized in Table 1.
Figures8 and 9 show the observer error eo1 compared and the control input v for two cases, respectively.Figure 7
The output tracking responses ofx1 and the tracking error ec1.Table 1
Position tracking performance evaluation.
Method
Average tracking error
Fluctuation of tracking error
With the FTSO
0.0362
0.5220
With the observer in [12]
0.0571
1.6707Figure 8
The observer compared results ofeo1.Figure 9
Control inputv.Example 3.
In order to provide the convictive proof of the proposed control approach, a test is proceeded on a servo test rig (as shown in Figure10), which is given as follows:(89)Jmx¨1=−bmx˙1−τs+τm,Jlx¨2=−blx˙2+τs−τd,where x1 and x2 denote the displacement of motor subsystem and the load subsystem, respectively. τs is the transmitted torque. τm is the driving torque of the system, and τd is the disturbance torque at the driven gear. Jm=0.01, Jl=0.05, bm=0.1, and bl=0.12.
The reference signalyd=0.8sintrad, where parameters are selected as h1=4, h2=6, k1=200, and k2=200. The initial state of the switched system and observer are x10=0, x20=0, x^10=0, and x^20=0.
The trajectory shown in Figure11 is the estimation of output x^1 and reference signal xd and the tracking error. Figure 12 shows the trajectories of system output x1 and x^1 and the error between system output and estimation of output.Figure 10
Motor driving test rig.Figure 11
Tracking effect and tracking error.Figure 12
Observer effect and error.


## 6. Conclusion

In this paper, the adaptive switched control with a finite-time switched state observer has been investigated for a class of uncertain nonlinear systems with unknown backlash-like hysteresis in a strict-feedback form. A finite-time switched state observer has been adopted to estimate the unmeasurable states and obtain a better tracking performance, with the help of finite-time performance function and neural network technologies. By transforming the tracking error into a virtual variable and based on the backstepping technique, an adaptive control method has been proposed. It has been proved that the observer error is bounded and finite time is convergent with the proposed control scheme. Moreover, the signals of the resulting system are bounded and the output tracking error is convergent to a small compact set of the origin.



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