Issues On The Applications Of Wavelet Analysis In The Control Of Nonlinear Systems

This dissertation mainly discusses issues on the applications of wavelet analysis in the control of nonlinear systems. The main topics are:(1) The problem of modeling and identification of nonlinear systems in the high dimensional space using wavelet analysis is studied. In order to handle problem that the number of wavelet basis functions grows exponentially with the number of the dimension of input space, two wavelet models are presented. The former is a wavelet network constructed by single-scaling multidimensional wavelet frames. Its function approximation property is analyzed. In the view of generalization ability of wavelet network, a constructive method of designing wavelet network and a new learning algorithm are proposed to obtain improved network's performance. The latter is a multiresolution wavelet model based on multiresolution analysis theory. Following an iterative reduced-dimensional strategy, the problem of "dimension curse" can be solved effectively. In the last section of this chapter, a cascade wavelet model, which is combined with a static wavelet submodel and a dynamic linear submodel, is presented to identify the dynamic nonlinear systems.(2) A new method of designing a wavelet network-based adaptive observer for a class of MIMO nonlinear systems is proposed. The wavelet network weights are tuned on-line, with no off-line learning required. No exact knowledge of nonlinearities in the observed system is needed. Furthermore, no linearity with respect to unknown system parameters is assumed. The robustifying term in the observer can reduce the error brought by the wavelet network approximation greatly. The overall adaptive scheme is shown to be uniformly ultimately bounded.(3) For a class of fully linearizable affine nonlinear systems, a wavelet network-based direct adaptive output feedback controller is presented. The nonlinearities are not required to satisfy any global growth condition and are not restricted to be linear withsystem parameters. A wavelet network is ued to approximate the nonlinearities of the system. The state feedback controller is restricted to be bounded globally. A high-gain observer is implemented to recover the performance achieved under the state feedback one. The disturbance produced by the error of wavelet network approximation is considered and the method to reduce its effect is developed. The overall closed-loop nonlinear system is proved to be stable on any known compact convex set by Lyapunov theory.(4) For a class of fully linearizable general nonlinear systems, a wavelet network-based direct adaptive output feedback controller is proposed. The nonlinearities are not required to be affine and be linear with unknown system parameters and satisfy any global growth condition. A wavelet network is applied to construct an ideal implicit feedback linearization control to realize approximate linearizaton. The state feedback controller is restricted to be bounded globally. The high-gain observer is implemented to recover the performance achieved under the state feedback one. The overall closed-loop nonlinear system is shown to be uniformly ultimately bounded on any known compact convex set.Simulations are given to demonstrate the effectiveness and feasibility of all the proposed methods.