In this thesis, the problem of adaptively controlling a nonlinear discrete-time system using neural networks is considered. In the first part of the thesis, a new frame-work is proposed to establish the existence of solutions to stabilization, regulation and tracking problems, for the case when the state vector is accessible, as well as for the case when only the input and the output are accessible. In this framework, a nonlinear system is explicitly represented in terms of linear and higher order functions, so that the role played by the linearization of the nonlinear system around the equilibrium state is made transparent in establishing the existence of solutions. Refined results on the normal form of a nonlinear discrete-time system and its input-output representation (NARMA model) have not only furthered our understanding of the previously obtained results, but also led to new results. The first half of the thesis constitutes the algebraic part of the solution to an adaptive control problem. The second half of the thesis constitutes the analytic part, in which the parameters of identifiers and controllers are adjusted to achieve certain control objectives. e.g., tracking a desired signal. How to assure that all the signals in the adaptive system remain bounded is the problem considered. The solution proposed in the thesis is to combine the robustness of a linear adaptive controller with the effectiveness of a neural network based nonlinear adaptive controller, in the framework of multiple models, switching and tuning. At every instant of time, the performances of the linear identifier and the nonlinear identifier in predicting the output of the plant are computed, based on a carefully designed performance criterion, and the one which performs better (basically, with a smaller prediction error) is chosen to generate a certainty equivalence control input to the plant. It is proved that all the signals in the switching system will remain bounded. This result is independent of the structure, parameterization, and weight-adjusting mechanism of the neural network used as identifiers. Thus, the stability issue is decoupled from that of performance, so that different types of neural networks can be used to identify and compensate for the nonlinearity of the plant. Since the neural networks are universal approximators, when properly constructed and trained, they will eventually perform better than the linear robust adaptive controller, and hence both stability and performance can be achieved.
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