The Stability Analysis And Adaptive Preview Control For A Class Of Nonlinear Systems

Purely linear system in the real world is almost non-existent. So, the problem of stability and control of nonlinear systems is always a research hotspot. In this dissertation, the problem of stability for a class of nonlinear systems is studied, and based on nonlinearities isolation method of the stability analysis for this nonlinear systems, the multi-model adaptive preview control for a class of nonlinear systems is investigated.First of all, the absolute stability of Lurie direct control systems with time-varying coefficients is considered, and the results are extended to Large-scale systems. In addition, multi-model adaptive preview control for piecewise linear systems with unknown parameters is studied. Based on it, by employing the idea of nonlinearities isolation method for the Lurie control systems, and taking the direct feedback linearization (DFL) technique, the problem of preview control for a class of nonlinear systems with large uncertainties is discussed. The main work of this dissertation includes the following aspects:(1) The absolute stability of Lurie direct control systems with time-varying coefficients is studied. Depending on the related methods, relative magnitude of the norm-unbounded coefficients was estimated. By knowledge of nonsingular M-matrix, the Lyapunov function was constructed, and some absolute stability criteria of this kind of systems were obtained. In addition, some simple and practical corollaries were derived from the theorem of Taussky. Furthermore, the results are extended to multiple nonlinearities. The criteria which we introduced allow for the situation that the norms of coefficient matrices are unbounded. The salient feature of the methodology is that even if the coefficients are norm-unbounded, by restricting their relative magnitude, the problem of negative definiteness for the derivative can also be changed into the problem of stability for a constant matrix.(2) For the large-scale systems, the absolute stability of large-scale Lurie direct control systems with time-varying coefficients is investigated. Based on the decomposition method for large-scale systems, we will not involve the concept of the isolated subsystem for brevity, but directly give the Lyapunov function by employing the isolated subsystem, and estimate the upper bound of its total derivative, some absolute stability criteria and practical corollaries are derived from the Gerschgorin disk theorem and Laplace theorem. Moreover, the results are extended to multiple nonlinearities. The criteria which we introduced also allow for the situation that the norms of coefficient matrices are unbounded.(3) The problem of multi-model adaptive preview control is proposed for linear systems with unknown piecewise constant coefficients. Following preview control theory, corresponding augmented error systems with preview information for the original systems are constructed. By making use of the Heine-Borel covering theorem, a finite set of fixed augmented systems covering the range of the unknown parameters is established, and associated fixed preview controllers are derived. Based on the core idea of multi-model adaptive control, switching law of the controllers is designed, and eventually, an adaptive preview controller is obtained. The exponential asymptotic stability of closed-loop systems with large uncertainties is discussed in detail. In addition, the range of each dividing set is discussed, that is, the robustness of preview controller is studied.(4) The multi-model adaptive preview control for a class of nonlinear systems with unknown parameters is considered. Based on the decomposition of nonlinear part with unknown parameters, the "weight matrix" and "basis vectors’" are separated. By employing the idea of nonlinearities isolation method for the Lurie control systems,"basis vectors" and the control input is isolated as a "virtual control input". Depending on the direct feedback linearization (DFL) technique, and adding nonlinear compensations with known parameters to the control input, the original nonlinear system has been transformed into a linear system with unknown piecewise constant coefficients. By taking the obtained multi-model adaptive preview control method, a finite set of fixed augmented systems covering the range of the "weight matrix" is constructed, and the multi-model adaptive preview controller for systems with unknown piecewise constant coefficients is designed, that is "virtual control input". According to the relationship between original control input and "virtual control input", the switching controller with preview feedforward compensation for the original nonlinear systems is derived. Finally, the problem of preview control for this uncertain nonlinear system is studied.The mathematical proofs of the theorems are provided, and simulation results are given to demonstrate the effectiveness of the theoretical results.