Adaptive recurrent neural control for nonlinear system tracking ...

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fication error. The proposed controller is based on sliding mode techniques. Our main result, stated as a theorem, concerns tracking error asymptotic stability.
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 30, NO. 6, DECEMBER 2000

Adaptive Recurrent Neural Control for Nonlinear System Tracking Edgar N. Sanchez and Miguel A. Bernal

Abstract—We present a new indirect adaptive control law based on recurrent neural networks, which are linear on the input. For the identifier, we adapt a recently published algorithm to fit the neural network type used for identification; this algorithm ensures exponential stability for the identification error. The proposed controller is based on sliding mode techniques. Our main result, stated as a theorem, concerns tracking error asymptotic stability. Applicability of the proposed scheme is tested via simulations. Index Terms—Control, identification, mechanical application, neural networks, nonmodel based.

I. INTRODUCTION Powerful approximation capabilities of neural networks have recently motivated intensive research for their application in nonlinear identification and control; the earlier publication [1] deserves much of the credit for that. Most of the known results are based on static neural network of different architectures such as multilayer perceptron or radial basis functions. For this kind of neural networks, the response depends only on its current inputs and the synaptic weight values; hence they lack dynamic memory. Stability properties for identification and control schemes using static neural networks is by now well known [2]. A recent approach introduces high-order static neural networks to robust nonlinear control [3]. Dynamic or recurrent neural networks, which include feedback, handle dynamic nonlinear mapping more efficiently that static ones; for the same dynamic mapping representation, the latter could be equivalent to a very large or possibly infinite static structure. Properties of dynamic neural networks for information storage have been extensively analyzed [4]; nevertheless they are not frequently applied in nonlinear system identification and control. One of the most promising approaches is to use the recurrent neural network for identifying on-line a model of the plant in order to implement an adaptive control law. In this direction, there already exist some results. In [5], an adaptive scheme, which is suited to nonlinear systems with a different number of inputs and states, is presented. This scheme is able to drive the system state from any initial condition to zero. That publication is a continuation of previous work by the same authors; in [6] they discuss an indirect adaptive controller to force the nonlinear system to follow a linear model, and in [7], they develop a direct adaptive regulation scheme for affine in the control nonlinear systems. In both papers, they illustrate the applicability of the respective approach by a D.C. motor speed control. Based on optimal control techniques, an indirect adaptive control scheme, for nonlinear system trajectory tracking, is developed in [8]. This scheme is robustified for the presence of an additive perturbation term, which takes into account system uncertainties, on the right-hand side (RHS) of the differential equation describing the nonlinear system [9]. The stability analyses developed for all of these adaptive schemes are based on Lyapunov methodology. A recent result, which ensures exponential convergence of the identi-

Manuscript received March 1, 1999; revised May 31, 2000. This work was supported by CONACYT, Mexico Grants 0652A9606 and 32059A. This paper was recommended by Associate Editor S. Lakshmivarahan. The authors are with CINVESTAV, Unidad Guadalajara, Guadalajara, Jalisco C.P. 45091, Mexico (e-mail: [email protected]). Publisher Item Identifier S 1083-4419(00)08802-6.

fication error, is presented in [10]; the respective analysis, instead of using a Lyapunov function, is based on differential equation solutions. Other nonadaptive control schemes, based on recurrent neural networks, have also been reported. In [11], an SISO affine control representation of the nonlinear system is used; a recurrent neural network approximates this model. This neural network is linearized by an inner loop designed based on differential geometry techniques; the outer control law is implemented using a PID controller. Continuing in the same direction, in [12], inverting the recurrent neural network which represents the nonlinear system, an internal model control (IMC) structure is proposed. Stability of the closed loop is analyzed by linearization of both neural networks. The so-called NLq theory is discussed in [13]; this theory enables to design robust controller based upon identified models by recurrent neural network structures. A publication, directly related to nonlinear system control by recurrent neural network, is [14], in which controllability conditions for this kind of neural networks are established. In this paper, we present a new indirect adaptive control scheme based on recurrent neural networks, which are linear on the input; an early version of the paper was presented in [15]. The structure of this scheme is similar to the one introduced in [8]. For identification, we were strongly inspired by [10]. In fact, we modify the most powerful algorithm reported there, in order to fit the neural network type we use. The control law to ensure trajectory tracking is based on sliding mode techniques. To the best of our knowledge, this is the first time these techniques are used to implement adaptive control based upon plant models identified on-line by recurrent neural networks. Our main result is the proof of tracking error asymptotic stability and it is stated as a theorem. The paper outline is as follows. First, we present the mathematical models for both the nonlinear system and the recurrent neural network; then we explain the identifier main facts. To complete the structure of the proposed indirect adaptive controller, trajectory tracking is developed, including the theorem establishing our main result. Once the control scheme is complete, its applicability is tested by simulation of an inverted pendulum trajectory tracking. Finally, we present relevant conclusions. II. MATHEMATICAL MODELS We consider general nonlinear systems represented as

_ = F (; u)

(1)

where  2