Short communication
Novel design of adaptive neural network controller for a class of non-affine nonlinear systems

https://doi.org/10.1016/j.cnsns.2011.08.005Get rights and content

Abstract

The tracking control problem is studied for a class of uncertain non-affine systems. Based on the principle of sliding mode control (SMC), using the neural networks (NNs) and the property of the basis function, a novel adaptive design scheme is proposed. A novel Lyapunov function, which depends on both system states and control input variable, is used for the development of the control law and the adaptive law. The approach overcomes the drawback in the literature. In addition, the lumped disturbances are taken in account. By theoretical analysis, it is proved that tracking errors asymptotically converge to zero. Finally, simulation results demonstrate the effectiveness of the proposed approach.

Highlights

► A novel adaptive design scheme is proposed for a class of uncertain non-affine systems. ► Using the property of the basis function of the Neural Networks, a novel adaptive sliding mode control scheme is proposed. ► A novel Lyaounov function is used for the development of the control law and the adaptive law. ► The approach overcomes the drawback in the literature. In addition, the lumped disturbances are taken in account.

Introduction

Over the past two decades, stability analysis and stabilization of nonlinear systems have been developed successfully, and many excellent results have been obtained [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Speaking in general, according to the systems with or without the actuator nonlinearity, the considered systems can be divided into two parts: the healthy systems and the faulty systems. In [1], [2], [3], variable fuzzy controllers were proposed for the nonlinear healthy systems. For a class of interconnected systems, the problem of stable fuzzy control was studied, and a design scheme of an adaptive fuzzy controller was proposed under the condition that the control input should be linear [4]. However, actuator nonlinearity, such as dead-zone, nonlinear input, and so on, is quite common in practice systems and devises. The existence of actuator nonlinearity renders the control problem much more complex and difficult, and might lead to the controlled systems instability. Designing a controller for the systems with actuator nonlinearity, the effects of the nonlinearity must be taken into account. To solve the stability problem for uncertain nonlinear systems with actuator nonlinearity, many researchers have devoted a lot of efforts, and proposed many control approaches. Tao and Kokotovic [5] compensated the dead-zone by constructing adaptive dead-zone inverse. In [6], fuzzy sliding mode controller was designed for a class of uncertain time-delayed systems with nonlinear input. Shen et al. [7] studied the chaos tracking control problem for a class of uncertain time-delay chaotic systems with dead-zone and saturating input, and proposed a novel adaptive sliding mode controller. For uncertain systems with various input nonlinearity, many control schemes were given in [8], [9], [10], [11], [12]. However, the above-mentioned studies had the condition that the systems or subsystems should be affine.

In practice, there are many nonlinear systems with non-affine structure, such as biochemical process [13], dynamic model in pendulum control [14], etc. In [15], [16], [17], using the approximation property of the neural networks, different adaptive design schemes were proposed. However, we know, the approximation property of the neural networks is effective just under the assumption that all input parameters of the neural networks are inside the given bounded set, thus, not giving the proof of the system states boundedness, it is unreasonable to think that the up boundedness of the approximation errors exists. In addition, [16], [17] did not consider the external disturbance. In [18], an adaptive fuzzy control for a class of non-affine nonlinear systems was discussed with the condition that the control input should be bounded. In [19], the authors proposed direct and indirect adaptive fuzzy controllers for a class of nonlinear systems. However, the systems must satisfy the assumption |b˙u/bu|β, which was restrictive. For a class of SISO non-affine nonlinear dynamic systems, [20] presented a direct adaptive fuzzy scheme. But, since the proposed adaptive law contained e(n), which was unknown, the control scheme did not work in practical applications. In [21], an adaptive fuzzy control approach was proposed for a class of multiple-input-multiple-output (MIMO) nonlinear systems with completely unknown non-affine functions.

In this paper, the tracking control problem is studied for a class of uncertain non-affine systems. Firstly, based on the principle of sliding mode control, using the neural networks and the property of the basis function, a novel adaptive design scheme is proposed. The approach overcomes the drawback in the literature. In addition, the lumped disturbances are taken in account. By theoretical analysis, it is proved that tracking errors asymptotically converge to zero.

This paper is organized as follows. In Section 2, the preliminaries of this paper, which includes the description of the problem and systems, are stated. Section 3 introduces the adaptive neural networks control design, and proposes the overall design methodology, and analyzes the stability of the derived closed-loop system in Lyapunov sense. As a simulation study, the designed controller is applied to the tracking control of numerical system in Section 4. Finally, in Section 5, we conclude this paper.

Section snippets

Preliminaries and systems description

Consider the following nonlinear dynamic systems described by the following formx˙i=xi+1,i=1,2,n-1x˙n=f(x,u)+d(x,t),where x(t)=[x1(t),x2(t),,xn(t)]T=[x(t),x˙(t),,x(n-1)(t)]TRn is the state vector, and u  R is the control input. The non-affine function f(x, u)  R is unknown and smooth, d(x, t) denotes the external disturbance.

The control problem is how to design an adaptive controller u such that the state can track the given target orbit xd(t), where xd(t)=[xd1(t),xd2(t),,xdn(t)]T=[xd(t),x˙d(t)

Design of adaptive neural network controller and stability analysis

From (1), (4), we havee˙s=f(x,u)+d(x,t)+γ,where γ=i=1n-1ciei+1-xd(n).

If d(x, t) = 0, then (10) can be rewritten as follows:e˙s=f(x,u)+γ.Because of f(x,u)u0, according to the implicit function theorem in [23], there exists an ideal control u(x), such that f(x, u(x)) = 0.

By the mean value theorem, we obtainf(x,u)=f(x,u)+01f(x,uλ)uλdλ(u-u),where λ  [0, 1], anduλ=λu+(1-λ)u.Hence, (10) can be rewritten as follows:e˙s=f(x,u)+01f(x,uλ)uλdλ(u-u)+γ+d(x,t)=b(x,u)(u-u)+γ+d(t)b(x,u),whereb(x,u)=0

Numerical simulation

In this example, a system is described as follows:x˙1=x2x˙2=1-e-x11+e-x1-(x22+2x1)sin(x2)+(1-0.5sin(x2))u+0.4sin(u))+d(x,t)y=x1,where d(x, t) = 0.5sin(x1 + 10t). The desired reference signal yd = sint + cos(0.5t). It is easy to find that b0 = 0.1, b2 = 1.9, and b3 = 0.4. For this work, the following parameters are given as follows: ηε = ηW = 2, ηi = 2, i = 1, 2, 3, x(0) = (0.6, 0.5)T, W  R10 are taken randomly in interval (0, 1], the sample time is 0.08 s. Simulation results are shown in Fig. 1, Fig. 2, Fig. 3. From Fig. 1,

Conclusion

In this paper, an adaptive tracking control for a class of non-affine nonlinear systems has been investigated. Based on the principle of SMC, a novel design scheme of adaptive NN controller has been presented. The approach not only overcomes the drawback in the literature, but also takes the lumped disturbances in account. It has been proven that the closed-loop control system is semi-globally uniformly ultimately bounded with tracking errors converging to zero.

Acknowledgements

The authors are grateful for the support of the National Natural Science Foundation of China (Grant Nos. 60904030, 60874045 and 61174046) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 10KJB510027).

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    This work was partially supported by the National Natural Science Foundation of China (Nos. 60904030, 60874045, 61174046) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 10KJB510027).

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