，，，

1.Equipment Management and Unmanned Aerial Vehicle Engineering College，Air Force Engineering University，Xi’an 710051，P.R.China；2.Department of Military and Political Foundation，Air Force Engineering University，Xi’an 710051，P.R.China

Abstract: An adaptive backstepping multi-sliding mode approximation variable structure control scheme is proposed for a class of uncertain nonlinear systems. An actuator model with compound nonlinear characteristics is established based on the model decomposition method. The unmodeled dynamic term of the radial basis function neural network approximation system is presented. The Nussbaum gain design technique is utilized to overcome the problem that the control gain is unknown. The adaptive law estimation is used to estimate the upper boundary of neural network approximation and uncertain interference. The adaptive approximate variable structure control effectively weakens the control signal chattering while enhancing the robustness of the controller.Based on the Lyapunov stability theory，the stability of the entire control system is proved. The main advantage of the designed controller is that the compound nonlinear characteristics are considered and solved. Finally，simulation results are given to show the validity of the control scheme.

Key words：compound nonlinearities；saturation；hysteresis；adaptive backstepping control；radial basis function（RBF）neural network

0 Introduction

The inherent characteristics of physical devic?es，mechanical design and manufacturing deviations make nonlinear characteristics，like saturation and hysteresis，inevitably exist in the actual control sys?tems，including mechanical systems，servo sys?tems，and piezoelectric systems，affecting the over?all performance. It may even cause instability in the system，like divergence and shock. With the devel?opment and application of information technology，the new material technology and the continuous im?provement of system control performance require?ments，it is necessary to adopt certain methods to eliminate or reduce the influence of nonlinear charac?teristics during the design and analysis of the control system.

In recent years，the problems of uncertain sys?tem control with nonlinear characteristics of actua?tors have received considerable attentions and be?come an active research area［1-10］. But there are also some limitations，e.q.，requiring nonlinear model parameter information to be partially known. The method of model decomposition requires that the nonlinear types are clear，and most works study spe?cific nonlinear input-output constraints in control de?sign procedure［2］. In engineering practice，the non?linear characteristics of the actuator are often diffi?cult to accurately judge，and sometimes they are a mixture of multiple situations［11］.

The inputs of the actual systems are limited by uncertain factors，but there are few studies on the control problems of uncertain systems considering the inputs being affected by compound nonlineari?ties. In this paper，a strict feedback nonlinear sys?tem with compound nonlinear characteristics and un?known control gain is considered. Its robust control?ler is constructed，by utilizing the adaptive backstep?ping sliding mode control method，dynamic surface control technology and radial basis function（RBF）neural network approximation technology.

1 Problem Statement

1.1 Actuator model with saturated nonlinear characteristics

An actuator model with saturated nonlinear characteristics is described as

wherevmax＞0 andvmin＜0 represent saturation nonlinearities. Its input-output relationship is shown in Fig.1.

Fig.1 Input-output relationship of saturated nonlinear actuator

A piecewise smoothing function to approxi?mate the saturation function is described as［10］

The saturation function in Eq.（1）can be ex?pressed as

whered1(t)is a bounded function.

Fig.2 shows the input-output relationship of the approximate smooth saturation functiong(u(t)).

Fig.2 Input-output relationship of smooth saturation function

According to the median theorem，for the con?stantλ，we have

where

Whenu0=0

Considering Eqs.（3，5），we have

In control engineering，the control inputu(t)would be increased indefinitely. The following as?sumption exists.

Assumption 1 Coefficientguλis unknown but bounded

wheregmis positive.

It should be noted thatguλis handled as an un?known control gain.

1.2 Actuator model with hysteretic nonlinear characteristics

Currently hysteresis nonlinear models are main?ly divided into two categories. One is a rate-indepen?dent hysteresis model，including Dual model，Lu?Gre model，Backlash-like model，Prandtl-Ishlinskii model，Preisach model，etc. The other is the ratedependent hysteresis model，which mainly includes the semi-linear Duhem model，and the modified Prandtl-Ishlinskii model［12-17］. The hysteresis nonlin?earity of this paper is characterized by the Backlashlike model as follows

whereA，B，Care constants，C＞0 andC＞B.

According to the analysis in Ref.［18］，it leads to

where(u)is bounded.

The input-output relationship is shown in Fig.3.

Fig.3 Input-output relationship of hysteresis nonlin?ear actuator

1.3 Actuator model with compound nonlinear characteristics

According to the model decomposition，a uni?fied actuator model with nonlinear characteristics of saturation and hysteresis is established as

wherev1(u) is the hysteresis nonlinearity，v2(v1)the input saturation nonlinearity，φ(u，t)=φ2(u，t)φ1(u，t) the linear coefficient，andd(t)=φ2(u，t)d1(t)+d2(t)the nonlinear part of the mod?el.

1.4 A class of uncertain systems with com?pound nonlinear characteristics

Consider the following class of uncertain sys?tems with compound nonlinear characteristics.

Considering Eqs.（14，15），we can deduce that

Assumption 2 Time-varying perturbationd(t)is bounded，and there is unknown positive con?stantD0＞0

Assumption 3 Control gaing(xˉn) is un?known but bounded

Assumption 4 The reference command sig?nalyr(t) and its derivativesexist and are bounded.

According to Eqs.（17—19），we have

whereD＞0 is an unknown positive constant.

This paper uses the Nussbaum gain design technique to overcome the problem that the control gainis unknown. Define continuous functionsN(ζ)：R→R，if the following conditions are satisfied.

whereN(ζ)is the Nussbaum function

For the Nussbaum functionN(ζ)，the follow?ing lemma exists.

Lemma 1［19］If bothV(?) andζ(?) are smooth functions on[0，tf)，V(t)≥0，t∈[0，tf)，and the fol?lowing inequality exists.

ThenV(t)，ζ(t) andare bounded on the interval [0，tf). Herec0is constant，c1＞0，andh(x(τ)) is an arbitrary function whose value is bounded in the interval [l-，l+]，

2 Design of Adaptive Backstepping Control Scheme and Stability Analysis

2.1 Design of adaptive backstepping control scheme

Define tracking error

wheree1is the system tracking error，andβi-1the virtual control signal of thei-1 order subsystem.

Step 1Differentiatinge1yields

The adaptive RBF neural network is used to approximate the unknown nonlinear functionf1(x1).For the compact setΩ?R，there exists an optimal weight vectorW?1

whereε1is the approach error andDefine

whereis the estimate of the optimal weight of the neural network andthe estimation error.The adaptive law of neural network weight vector is taken as

whereσ10＞0 is the parameter to be designed，Γ1=is the gain matrix to be designed and elements of the matrix are all positive.

whereσ11＞0 is the parameter to be designed andγ1the adaptive gain coefficient to be designed. The es?timated error is

The first error surface is defined ass1=e1.Then the virtual control law is chosen

whereυ1＞0 andk1＞0 are parameters to be de?signed.

To avoid repeatedly differentiating virtual con?trollers，which will lead to the so-called“explosion of complexity”，we employ the dynamic surface control technique to eliminate this problem. We in?troduce a first-order filterβ1，and letα1pass through it with the time constantτ1

By defining the output error of this filter asω1=β1-α1，we have

Define the Lyapunov function

DifferentiatingV1yields

According to Eqs.（26，27，31），we obtain

Differentiatingω1yields

Substituting Eqs.（36，37）into Eq.（35）yields

According to Eqs.（29，30），and boundary in?equalities，we obtain

StepiThe derivative ofei（i=2，…，n-1）is

whereβi-1is the output of the first-order filter

whereωi-1=βi-1-αi-1，αi-1is the input of the first-order filter andτi-1the time constant. Substi?tuting Eq.（41）into Eq.（40）yields

The adaptive RBF neural network is used to approximate the unknown nonlinear functionFor the compact setΩxˉi?Ri，there exists an opti?mal weight vectorW?i

whereεiis the approach error，the esti?mate of the optimal weight of the neural network andThe adaptive law of neural net?work weight vector is chosen as

whereσi0＞0 is the parameter to be designed，Γi=is the gain matrix to be designed and elements of the matrix are all positive.

whereσi1＞0 is the parameter to be designed，andγithe adaptive gain coefficient to be designed. The estimated error is

The virtual control law of theith order subsys?tem is chosen

whereυi＞0 andki＞0 are the parameters to be de?signed.

Using the similar way，we introduce a first-or?der filterβi，and letαipass through it with the time constantτi

Define the Lyapunov function

Similarly，it can be obtained as

whereφi(?)is a continuous function.

StepnThenth order subsystem’s error of the system isen=xn-βn-1.The derivative ofenis

whereωn-1=βn-1-αn-1andτn-1is the time con?stant.

The adaptive RBF neural network is used to approximate the unknown nonlinear function termof the system. For the compact setthere exists an optimal weight vectorW?n

whereεnis the approach error，the es?timate of the optimal weight of the neural network andThe adaptive law of neural network weight vector is chosen as

whereσn0＞0 is the parameter to be designed andis the gain matrix to be designed and ele?ments of the matrix are all positive.

The boundary value is defined asand the adaptive law to estimateD'nis chosen.

whereσn1＞0 is the parameter to be designed，andγnthe adaptive gain coefficient to be designed. The estimated error is

Define the sliding surfaces=en. The control law is designed as

Design the control law using Nussbaum

whereυn＞0 andkn＞0 are parameters to be de?signed.

2.2 Stability analysis

Theorem 1Consider the uncertain nonlinear systems（16），the controller（55），and the corre?sponding adaptive law. If the proposed control sys?tem satisfies Assumptions 1—5，for the system with any bounded initial state，all signals of the closed-loop system are semiglobally uniformly bounded. And，by tuning the designed parameters，the system tracking errore1converges to a small neighborhood near the origin.

Define the Lyapunov function

The derivative ofVis

According to Eq.（50），we obtain

According to the control law（55），we have

Substituting Eq.（59）into Eq.（58）yields

Substituting Eqs.（52，53，60）into Eq.（57）yields

Invoking the boundary inequality yields

Therefore

Substituting Eq.（49）into Eq.（63）yields

By Assumption 4 and the initial state of the sys?tem bounded，we have

According to Young’s inequality

Substituting Eqs.（66—70）into Eq.（64）yields

When

wherer1＞0.

Multiply both sides of Eq.（79）by eμtand inte?grate

According to Assumption 3， we haveIt can be proved by Lemma 1 thatV(t)，ζ(t)，andare bounded on the interval[0，tf).When

Eq.（80）can be rewritten as

Thus，V(t) is bounded. According to Eq.（56），the closed-loop system signalsei，ωi，，andare bounded by a semi-global uniform termi?nation，andandare bounded. By Assumption 4，the state of the closed-loop systemxiis bounded.

According to Eqs.（56，82），we obtain

Therefore， the convergence radiusof the steady-state tracking errore1of the system can be reduced by choosing parame?ters.

3 Simulation

Consider the following uncertain nonlinear sys?tem

The center of the Gaussian radial basis function of the RBF neural networkThe width isη1=2，(0)=0. The center of the Gaussian radial basis function of the RBF neural networkisThe width isThe center of the Gaussian radial basis function of the RBF neural network{-1，0，1}×{-1，0，1}，The width isη3=2，and

The reference command signal isyd=0.5sint+ 0.5sin(0.5t) . The initial values are[x1(0)，x2(0)，x3(0)]T= [ 0.25，0.25，0.25]T，(0)=0 (i=1，2，3) andζ(0)=1. Time con?stants areτ1=τ2=0.04. The parameters to be de?signed are set toki=2，Γi=diag[0.5]，υi=10，σi0=σi1=0.2，andγi=0.5.

The simulation results are shown in Figs.4—6. Adaptive backstepping multi-sliding mode vari?able control without RBF neural network approxi?mation is conducted as a comparative simulation result. The simulation result in Fig.4 shows that the scheme of this paper has better tracking con?trol effect. It can be seen that the designed control?ler can stably track the reference command while the actuator has nonlinear compound characteris?tics of hysteresis and input saturation，and the tracking error remains within a certain range. Ac?cording to Figs.7—9，variables of the closed-loop system state are bounded.

Fig.4 Tracking reference command signal curves

Fig.5 Control signal curve of the system u

Fig.6 Actual control signal curve of the system v(u(t))

Fig.7 Curves of neural network weight norms

Fig.8 Curves of the adaptive parameters

Fig.9 Curve of the adaptive parameter ζ

4 Conclusions

A class of uncertain nonlinear systems with compound nonlinear characteristics has been stud?ied. Combined with RBF neural network approxima?tion and adaptive control theory，an adaptive back?stepping multi-sliding mode variable structure con?troller scheme is presented. By utilizing the model decomposition method，a nonlinear actuator model of compound nonlinear characteristics is estab?lished，so that the inverse solution of nonlinear fea?tures is not needed in the controller design process.It has been proved that all signals of the closedloop system are semi-globally uniformly bounded. A simulation example has been conducted to show the validity of the proposed scheme.