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Neural networks have become a well-established methodology as exempli?ed by their applications to identi?cation and control of general nonlinear and complex systems; the use of high order neural networks for modeling and learning has recently increased. Usingneuralnetworks,controlalgorithmscanbedevelopedtoberobustto uncertainties and modeling errors. The most used NN structures are Feedf- ward networks and Recurrent networks. The latter type o?ers a better suited tool to model and control of nonlinear systems. There exist di?erent training algorithms for neural networks, which, h- ever, normally encounter some technical problems such as local minima, slow learning, and high sensitivity to initial conditions, among others. As a viable alternative, new training algorithms, for example, those based on Kalman ?ltering, have been proposed. There already exists publications about trajectory tracking using neural networks; however, most of those works were developed for continuous-time systems. On the other hand, while extensive literature is available for linear discrete-timecontrolsystem,nonlineardiscrete-timecontroldesigntechniques have not been discussed to the same degree. Besides, discrete-time neural networks are better ?tted for real-time implementations.



1. Introduction

The ultimate goal of control engineering is to implement an automatic system that could operate with increasing independence from human actions in an unstructured and uncertain environment. Such a system may be named autonomous or intelligent. It would need only to be presented with a goal and would achieve its objective by learning through continuous interaction with its environment through feedback about its behavior [13].
One class of models that has the capability to implement this learning is the artificial neural networks. Indeed, the neural morphology of the nervous system is quite complex to analyze. Nevertheless, simplified analogies have been developed, which could be used for engineering applications. Based on these simplified understandings, artificial neural networks are built [6].
An artificial neural network is a massively parallel distributed processor, inspired from biological neural networks, which can store experimental knowledge and makes it available for use. An artificial neural network consists of a finite number of neurons (structural element), which are interconnected to each other. It has some similarities with the brain, such as knowledge is acquired through a learning process and interneuron connectivity named as synaptic weights are used to store this knowledge, among others [13].

2. Mathematical Preliminaries

In this chapter, important mathematical preliminaries, required in future chapters, are presented.

3. Discrete-Time Adaptive Neural Backstepping

This chapter deals with adaptive tracking for a class of MIMO discrete-time nonlinear systems in presence of bounded disturbances. In this chapter, a high order neural network structure is used to approximate a control law designed by the backstepping technique, applied to a block strict feedback form (BSFF). It also presents the respective stability analysis, on the basis of the Lyapunov approach, for the whole scheme including the extended Kalman filter (EKF)-based NN learning algorithm. Applicability of this scheme is illustrated via simulation for a discrete-time nonlinear model of an electric induction motor.
In recent adaptive and robust control literature, numerous approaches have been proposed for the design of nonlinear control systems. Among these, adaptive backstepping constitutes a major design methodology [6, 9]. The idea behind backstepping design is that some appropriate functions of state variables are selected recursively as virtual control inputs for lower dimension subsystems of the overall system [12]. Each backstepping stage results in a new virtual control designs from the preceding design stages. When the procedure ends, a feedback design for the true control input results, which achieves the original design objective. The backstepping technique provides a systematic framework for the design of tracking and regulation strategies, suitable for a large class of state feedback linearizable nonlinear systems [1, 9–11].

4. Discrete-Time Block Control

This chapter deals with the adaptive tracking problem for a class of MIMO discrete-time nonlinear systems in presence of bounded disturbances. In this chapter, a recurrent high order neural network is first used to identify the plant model, then based on this neural model, a discrete-time control law, which combines discrete-time block control and sliding modes techniques, is derived. The chapter also includes the respective stability analysis for the whole system. It is proposed too a strategy to avoid specific adaptive weights zero-crossing. Applicability of the proposed scheme is illustrated via simulation of a discretetime nonlinear controller for an induction motor.
Frequently, modern control systems require a very structured knowledge about the system to be controlled; such knowledge should be represented in terms of differential or difference equations. This mathematical description of the dynamic system is named as the model. Basically there are two ways to obtain a model; it can be derived in a deductive manner using physics laws, or it can be inferred from a set of data collected during a practical experiment. The first method can be simple, but in many cases it is excessively timeconsuming; some times, it would be unrealistic or impossible to obtain an accurate model in this way. The second method, which is commonly referred as system identification, could be a useful short cut for deriving mathematical models. Although system identification not always results in a equally accurate model, a satisfactory model can be often obtained with reasonable efforts. The main drawback is the requirement to conduct a practical experiment, which brings the system through its range of operation. Besides a certain knowledge about the plant is still required.

5. Discrete-Time Neural Observers

This chapter presents the design of an adaptive recurrent neural observer for nonlinear systems, whose mathematical model is assumed to be unknown. The observer is based on a recurrent high order neural network (RHONN), which estimates the state vector of the unknown plant dynamics and it has a Luenberger structure. The learning algorithm for the RHONN is implemented using an extended Kaiman filter (EKF). The respective stability analysis, on the basis of the Lyapunov approach, is included for the observer trained with an EKF and simulation results are included to illustrate the applicability of the proposed scheme.

6. Discrete-Time Output Trajectory Tracking

In this chapter, two schemes for trajectory tracking based on the backstepping and the block control techniques, respectively, are proposed, using an RHONO. This observer is based on a discrete-time recurrent high-order neural network (RHONN), which estimates the state of the unknown plant dynamics. The learning algorithm for the RHONN is based on an EKF. Once the neural network structure is determined, the backstepping and the block control techniques are used to develop the corresponding trajectory tracking controllers. The respective stability analyzes, using the Lyapunov approach, for the neural observer trained with the EKF and the controllers are included. Finally, the applicability of the proposed design is illustrated by an example: output trajectory tracking for an induction motor.

7. Real Time Implementation

In this chapter real time implementation is presented in order to validate the theoretical results discussed in previous chapters. The results presented in this chapter include the Neural Network Identification scheme presented in Chap. 4, the RHONO presented in Chap. 5, the Neural Backstepping Approach analyzed in Chap. 3, the Neural Bock Control Technique discussed in Chap. 4 and the modifications of the last two controllers treated in Chap. 6 to include the RHONO. All these applications was performed using a three phase induction motor.

8. Conclusions and Future Work

In this work, based on the neural network and feedback linearization techniques, a novel method to design robust control for a class of MIMO discretetime nonlinear uncertain systems is proposed. This method includes four different control schemes, which can be applied depending on the state vector measurement viability:
The first designed robust direct neural control scheme is based on the backstepping technique, approximated by a high order neural network. On the basis of the Lyapunov approach, the respective stability analysis, for the whole closed-loop system, including the extended Kalman filter (EKF)-based NN learning algorithm, is also performed.
The second robust indirect control is designed with a recurrent high order neural network, which enables to identify the plant model. A strategy to avoid specific adaptive weights zero-crossing and conserve the identifier controllability property is proposed. Based on this neural identifier and applying the discrete-time block control approach, a nonlinear sliding manifold with a desired asymptotically stable motions was formulated. Using a Lyapunov functions approach, a discrete-time sliding mode control that makes the designed sliding manifold to be attractive was introduced.


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