Elsevier

Neurocomputing

Volume 78, Issue 1, 15 February 2012, Pages 83-88
Neurocomputing

Adaptive neural control of nonlinear MIMO systems with unknown time delays

https://doi.org/10.1016/j.neucom.2011.04.043Get rights and content

Abstract

In this paper, a novel adaptive NN control scheme is proposed for a class of uncertain multi-input and multi-output (MIMO) nonlinear time-delay systems. RBF NNs are used to tackle unknown nonlinear functions, then the adaptive NN tracking controller is constructed by combining Lyapunov–Krasovskii functionals and the dynamic surface control (DSC) technique along with the minimal-learning-parameters (MLP) algorithm. The proposed controller guarantees uniform ultimate boundedness (UUB) of all the signals in the closed-loop system, while the tracking error converges to a small neighborhood of the origin. An advantage of the proposed control scheme lies in that the number of adaptive parameters for each subsystem is reduced to one, triple problems of “explosion of complexity”, “curse of dimension” and “controller singularity” are solved, respectively. Finally, a numerical simulation is presented to demonstrate the effectiveness and performance of the proposed scheme.

Introduction

In many practical applications on the control of uncertain non-linear systems, neural network (NN)-based control methods are shown to be more efficient compared with other modern control techniques, such as classical adaptive control and robust control, for these methods generally require much more knowledge of the system model that may not be available in practice [1]. So, in the past decades, adaptive neural control of uncertain nonlinear systems has been achieved significant progress with the help of NN approximation. Many positive results have shown that semi-global uniform ultimate boundedness (SGUUB) of the closed-loop adaptive control system can be achieved and the output of the system is proven to converge to a small neighborhood of the desired trajectory, refer to [1], [2] and the references therein for details. But these methods are limited only to the nonlinear systems with the matching conditions. With the development of a powerful recursive-design procedure, i.e., adaptive-backstepping technique [3], adaptive neural control approach based on backstepping design has gained a remarkable progress for classes of single-input single-output(SISO) uncertain nonlinear systems without the requirement of matching conditions(see [4], [5], [6], [7] and the references therein).

For multi-input multi-output(MIMO) nonlinear systems, the control design is, as well known in general, very difficult due to the couplings among various inputs and outputs. The problem becomes even worse when there exist uncertainties and/or unknown nonlinear functions in the system matrices [8]. Hence, compared with the vast amount of results on controller design for SISO nonlinear systems, there are relatively few results available for the broader class of MIMO nonlinear systems. Recently, much research effort has been devoted to the development of systematic design methods for adaptive control of MIMO nonlinear systems with unstructured uncertainty based on the backstepping technique and the universal approximations, and some remarkable results have been developed [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. However, as stated out in [19], these schemes suffered from two major problems, the first one is dimensionality curse, i.e., to achieve a better approximation result, the number of parameters to be tuned online in the Fuzzy system or NN approximator-based adaptive-control schemes should be very large, in particular for high-dimensional systems, and the learning time tends to become unacceptably long when implemented. The second one is explosion of complexity, which is inherent in the conventional backstepping technique. This problem is caused by the repeated differentiations of virtual controllers and inevitably leads to a complicated algorithm with heavy computation burden. Especially, the complexity of controller grows drastically as the order of the system increases [20]. These two problems limit severely the implementation and application of those control algorithms in practice. Fortunately, by fusion of the DSC approach [21] and the MLP approach [22], [23], two novel adaptive control schemes were developed for classes of strongly interconnected MIMO systems by virtue of the universal approximations of NNs in [24] and T-S fuzzy systems [25]. Both above limitations were elegantly circumvented simultaneously in [24], [25]. Nevertheless, the aforementioned methods would fail when time delay is considered.

In practice, time delay is frequently encountered in models of engineering systems, natural phenomena, and biological systems. Delay may occur in the feedback loop of a plant, either in the states, inputs or outputs [26]. The existence of time delays may degrade the control performance and make the control problem become more difficult and challenging [27], [28]. Some of the useful tools in robust stability analysis for time-delay systems are based on the Lyapunov's second method, the Lyapunov–Krasovskii theorem, and the Lyapunov–Razumikhin theorem [29]. More recently, some positive results were gained for classes of uncertain nonlinear MIMO systems with time delays by combination of backstepping technique and NN approximators [29], [30], [31], [32], [33]. However, these methods also suffered from either computational explosion or dimension curse or both of them.

In this paper, based on above observations, the combination of MLP and DSC approaches proposed in [24], [25] was extended to a class of MIMO nonlinear time delay systems with arbitrary uncertainties. A stable time-delay-free adaptive NN tracking control scheme is developed based on the Lyapunov–Krasovskii functions. In the proposed method, both the tracking performance and the SGUUB stability of the closed-loop system are guaranteed. And it also performs three distinguished properties, that is, (i) the number of the updated parameters is independent of the order of the system, only one on-line learning parameter needs to be updated for each subsystem, (ii) both problems of dimensionality curse and explosion of complexity are avoided simultaneously, which leads to an easy-to-implement controller with much less computational burden, and, (iii) the RBF NNs are only used to deal with those unknown system functions, those uncertain virtual control gain functions and the actual control gain function are not required to be approximated. Thus, the potential controller singularity problem can be overcome.

The rest of this paper is organized as follows. Section 2 contains the problem formulation and some necessary preliminary results. In Section 3, the control design and its stability analysis are given. In Section 4, a numerical simulation example is used to demonstrate the performance of the scheme. The final section contains conclusions.

Section snippets

Problem formulation

Consider a class of large-scale nonlinear systems composed of n subsystems, where the ith subsystem is given as:x˙i,j=fi,j(x¯i,j)+gi,j(x¯i,j)xi,j+1+Δi,j(Y)+hi,j(x¯i,j(tτi,j)),x˙i,ni=fi,ni(x¯i,ni)+gi,ni(x¯i,ni)ui+Δi,ni(Y)+hi,ni(x¯i,ni(tτi,ni)),yi=xi,1,i=1,,N,j=1,,ni1,where x¯i=[xi,1,,xi,ni]TRni denotes the vector of state variables, and uiR and yiR represent the input and output of the ith sub-system, respectively. x¯i,j=[xi,1,,xi,j]TRj; Y=[Y1,,YN]TRN. fi,j, gi,j and hi,j are

Controller design and stability analysis

In this section, we will develop a decentralized design procedure by fusion of DSC and MLP approaches [19] for (1).

At first, we introduce a useful lemma on RBF NN approximation in the control design as follows.

Lemma 1

Yang et al. [22] and Yang and Wang [23]

For any given real continuous function fi,j(x¯i,j) with fi,j(0)=0, when the continuous function separation technique in [35] and RBF NN approximation technique are used, then fi,j(x¯i,j) can be denoted as follows:fi,j(x¯i,j)=ξi,j(x¯i,j)Wi,jx¯i,j+ɛi,j,where ξi,j=[ξi,j1(x),ξi,j2(x),,ξi,jl(

Simulation example

In this section, we will present an example of a second-order time-delay MIMO nonlinear system in a general form to reveal the control performance of the proposed algorithm.x˙i,1=fi,1(xi,1)+gi,1(xi,1)xi,2+Δi,1(Y)+hi,1(xi,1(tτi,1)),x˙i,2=fi,2(x¯i,2)+gi,2(x¯i,2)ui+Δi,2(Y)+hi,2(x¯i,2(tτi,2)),yi=xi,1,i=1,2,where f1,1=x1,1e0.5x1,1,g1,1=1+sin(x1,12), Δ1,1=x2,12+x1,1x2,1; f1,2=x1,1x1,2+0.5sin(x1,2), g1,2=[2+cos(x1,1x1,2)], Δ1,2=x1,1x2,1; f2,1=2x2,12,g2,1=[2+sin(x1,1x2,1)], Δ2,1=x1,1+x2,1; f2,2=(x2,1

Conclusion

In this paper, the tracking control problem has been considered for a class of MIMO strict-feedback uncertain nonlinear systems with time delays. Combining DSC technique with MLP algorithm, an adaptive NN tracking control scheme is developed based on the Lyapunov–Krasovskii method. It is shown that the whole closed-loop system is UUB. The main features of the proposed algorithms are that the adaptive mechanism with minimal learning parameters is achieved, and both problems of “explosion of

Acknowledgments

This work is supported in part by the China Postdoctoral Science Foundation (Grant no. 200902241), National Natural Science Foundation of China (Grant nos. 51179019, 60874056 and 61074017), Natural Science Foundation of Liaoning Province (Grant no. 20102012).

Tieshan Li received the BS degree from Ocean University of China, Qingdao, China, in 1992, and the PhD degrees in Transportation Information Engineering & Control from Dalian Maritime University (DMU), China in 2005. Currently, he is an Associate Professor in DMU. From March 2007 to February 2010, he worked as a post-doctoral scholar at Shanghai Jiao Tong University. He also visited City University of Hong Kong as a Senior Research Associate (SRA) from November 2008 to February 2009. His

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    Tieshan Li received the BS degree from Ocean University of China, Qingdao, China, in 1992, and the PhD degrees in Transportation Information Engineering & Control from Dalian Maritime University (DMU), China in 2005. Currently, he is an Associate Professor in DMU. From March 2007 to February 2010, he worked as a post-doctoral scholar at Shanghai Jiao Tong University. He also visited City University of Hong Kong as a Senior Research Associate (SRA) from November 2008 to February 2009. His research interests include decentralized adaptive control, fuzzy control and neural-network control for nonlinear systems and their applications to marine control.

    Ronghui Li received the BS and MS degrees in Marine Engineering from Dalian Maritime University (DMU), China, in 2005. Currently, he is an Associate Professor in DMU, and also pursuing his PhD degree in Transportation Information Engineering & Control in DMU. His research interests include adaptive control, robust control, and neural-network control for nonlinear systems and their applications to marine control.

    Dan Wang received the BE degree in industrial automation engineering from Dalian University of Technology, Dalian, China, in 1982, the ME degree in marine automation engineering from Dalian Maritime University (DMU), Dalian, China, in 1987, and the PhD degree in mechanical and automation engineering from The Chinese University of Hong Kong in 2001. From 1987 to 1998, he was a Lecturer, and later an Associate Professor (1992) at DMU, Dalian, China. From 2001 to 2005, he was a Research Scientist at Temasek Laboratories, National University of Singapore. Since 2006, he has been with DMU where he is a Professor in the Department of Marine Electrical Engineering, Marine Engineering College. Wang's research interests include nonlinear control theory and applications, neural networks, adaptive control, robust control, fault detection and isolation, and system identification.

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