**1. Introduction**

114 Recurrent Neural Networks and Soft Computing

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The design goal of a control system is to influence the behavior of dynamic systems to achieve some pre-determinate objectives. A control system is usually designed on the premise that an accurate knowledge of a given object and environment cannot be obtained in advance. It usually requires suitable methods to address the problems related to uncertain and highly complicated dynamic system identification. As a matter of fact, system identification is an important branch of research in the automatic control domain. However, the majority of methods for system identification and parameters' adjustment are based on linear analysis: therefore it is difficult to extend them to complex non-linear systems. Normally, a large amount of approximations and simplifications have to be performed and, unavoidably, they have a negative impact on the desired accuracy. Fortunately the characteristics of the Artificial Neural Network (ANN) approach, namely non-linear transformation and support to highly parallel operation, provide effective techniques for system identification and control, especially for non-linear systems [1-9]. The ANN approach has a high potential for identification and control applications mainly because: (1) it can approximate the nonlinear input-output mapping of a dynamic system [10]; (2) it enables to model the complex systems' behavior and to achieve an accurate control through training, without a priori information about the structures or parameters of systems. Due to these characteristics, there has been a growing interest, in recent years, in the application of neural networks to dynamic system identification and control.

"Depth" and "resolution ratio" are the main characteristics to measure the dynamic memory performance of neural networks [11]. "Depth" denotes how far information can be memorized; "resolution ratio" denotes how much information in input sequences of neural networks can be retained. The memory of time-delay units is of lower depth and higher resolution ratio, while most recurrent neural networks, such as Elman neural networks, are higher depth and lower resolution ratio. The popular neural networks have much defect on dynamic memory performance. This chapter proposed a novel time-delay recurrent network model which has far more "depth" and "resolution ratio" in memory for

<sup>\*</sup> Corresponding author

Recurrent Neural Network-Based Adaptive Controller Design for Nonlinear Dynamical Systems 117

( ) ( 1) ( 1) ( 1,2, , ) *Cl Cl <sup>l</sup> y k y k yk l m*

where ( ) *Cl y k* and ( ) *<sup>l</sup> y k* are, respectively, the outputs of the *l*th context unit and the *l*th

nodes in the input layer, *n* nodes in the hidden layer, and *m* nodes in the output layer and context layers respectively, then the input *u* is an *r* dimensional vector, the output *x* of the hidden layer is *n* dimensional vector, the output *y* of the output layer and the output *Cy* of the context nodes are *m* dimensional vectors, and the weights *W* <sup>1</sup> , *W* <sup>2</sup> and *W* 3 are *n r*,

( ) ( 1) ( 1) *C C y k y k yk*

1 ( ) ( ) ( ) ( 1) *i zk uk uk i zk* 

<sup>1</sup> ( ) <sup>1</sup> *<sup>x</sup> f x e*

( ) ( ) ( ) (0) *k*

From Eq.(8) it can be seen that the memory neurons in the input layer include all the previous input information and the context nodes memorize previous activations of the output nodes, so the proposed TDRNN model has far higher memory depth than the popular neural networks. Furthermore, the neurons in the input layer can memory

*zk uk i uk i u*

 

*i k*

 

1, *z*(0) 0 , and

Taking expansion for *z k*( 1) , *z k*( 2) ,…, *z*(1) by using Eq.(5), then we have

0 1

*i i*

performance of popular recurrent neural networks. If the delay step

TDRNN possesses higher memory resolution ratio.

The mathematical model of the proposed TDRNN can be described as follows.

in the context nodes. Thus the output of the context nodes can be

) is the self-feedback coefficient. If we assume that there are *r*

. (1)

2 3 ( ) ( ( ) ( )) *<sup>C</sup> y k g W xk W y k* , (2)

, (3)

is the step number of time delay. *f* ( ) *x* is

. (6)

<sup>1</sup> *x k*( ) ( ( )) *f W zk* , (4)

. (5)

2 3 () () () *<sup>C</sup> y k W xk W y k* . (7)

. (8)

to time *k* , and this is quite different from the memory

is moderate large, the

with fixed coefficient

described by

output unit and

where 0 , , 1, 

 (0 1 

*m n* and *m m* dimensional matrices, respectively.

 

and *g x*( ) is often taken as a linear function, that is

often taken as the sigmoidal function

accurately the inputs from time *k*

identifying and controlling dynamic systems. The proposed identification and control schemes are examined by the numerical experiments for identifying and controlling some typical nonlinear systems.

The rest of this chapter is organized as follows. Section 2 proposes a novel time-delay recurrent neural network (TDRNN) by introducing the time-delay and recurrent mechanism; moreover, a dynamic recurrent back propagation algorithm is developed according to the gradient descent method. Section 3 derives the optimal adaptive learning rates to guarantee the global convergence in the sense of discrete-type Lyapunov stability. Thereafter, the proposed identification and control schemes based on TDRNN models are examined by numerical experiments in Section 4. Finally, some conclusions are made in Section 5.
