Issues On The Applications Of Wavelets Theory In System Modeling And Control

The perfect mathematical characteristics of wavelets and wavelets transform are winning more and more favor from control scientists and engineering personnel. In 1991, Astrom---a Sweden control scientist advanced that wavelets approximation would become the latest researching method in system modeling, identification and control. During the later period over one decade, following with the development of wavelet theory, wavelet has been broadly applied in system identification and control, and partially resolved some problems in system identification and control science. On the base of predecessors's achievements and my research work, some algorithms about the application of wavelets theory in system modeling and control are presented in this paper , the main contents as follows:1. The paper strictly proved the theorem that variance of white noise would decrease to 50% through orthogonal scale (wavelet) decomposition and reconstruction, therefore identification data would be decomposed and reconstructed through orthogonal scaling transform in order to filter noises of data. According to the theorem, non-orthogonal wavelets is taken as basis function of impulse response process and noises is removed through multi-scale decomposition, and good identification result is achieved.2. According to the theory of control-relevant identification, a band-wise identification method online to the linear time-varying system based on wavelets decomposition is presented. According to the method, weighting coefficients of sub-band decomposed by wavelets can be adjusted online if the time-varying system's work band is changed, so the time-varying control request can be implemented by weighting coefficients. The method is fit to some system especially such as servo system.3. The application of wavelets approximation to nonlinear dynamic system. The first is that an adaptive least square wavelets support vector machine whose scale parameter is adjusted and its application to nonlinear system identification. The second is that 3-order B-spline function formula of equidistant points is deduced, and 3-order B-spline wavelet is constructed via B-spline High-Pass filter, then B-spline wavelet is taken as basis function and applied to approximate to the nonlinear part of a Hammerstein model.4. Some research of control strategy based on wavelet approximation. The first is that discrete affine wavelet network is applied to approximation of nonlinear system with mismatched uncertainties, and adaptive control law is designed according to the theory of hyperstability, so the drawbacks of BP network such as large computation online, local minimum aiid uncertain convergency are overcomed. The second is that a predictive functional control using wavelets as basis function is presented. Because wavelets has the properties of compact support and multi-scale analysis, the number of wavelets functions and their location are set flexibly according to the system's controlling precision and the different requirements of approximation precision at different intervals of the horizon. So the target function of the whole horizon are optimized and some important points are emphasized, meanwhile the optimal parameters have been aggregated. 5. On the base of orthogonal wavelet (scaling function) decomposition and reconstruction algorithm, the paper presents the equivalent wavelet (scaling function) FIR filter, and provides two theorems about filter coefficients. The filters are applied to solve the next two problems: The first, aiming at that the relay is sensitive to noise and extracting characteristic parameters from output values polluted by noises is hard, applying equivalent wavelet filter to eliminate noise online from feedback signal, and denoising of output wave-forms offline using wavelet thresholding to extract characteristic parameters easily . The second, in order to eliminate measurement noises in system's outputs for dynamical matrix control (DMC), output measurements are filtered by wavelets denoising online, and the reduction coefficient of reference locus in DMC is automatically adjusted according to wavelets correlation values, so system's anti-disturb (such as load disturb) ability is enhanced.6. Based on operation matrix via orthogonal wavelets, two simplified control algorithm based on orthogonal wavelets transform are presented: The first, the integral operation matrix of Harr wavelets is applied to approximation of hierarchical control of linear large scale systems, the differential equations are converted into a set of linear algebraic equations, so complicated calculous operation has been transformed to simple matrix operation, and computation has been reduced effectively. The second, an iterative learning control algorithm based on Harr wavelets is presented to address the terminal controlproblem of a linear time-varying systems. By employing the orthogonality and boundary values of Harr wavelets, the differential equations are converted into a set of linear algebraic equations, and the control problem is simplified to finding Harr wavelets coefficients of control variables, so the method avoids computing the state transfer matrix, and the coefficients are determined iteratively by steepest descent learning algorithm.