H_∞ Control For Polynomial Systems Based On Sum-of-Squares Optimization

Polynomial systems appear widely in practical applications. In particular, many control problems in motion control systems, mechartronic systems, process control, biological systems, and electric circuits and so on, can be modeled as, transformed into, or approximated by polynomial systems. Therefore, how to analyze and synthesize polynomial system is a promising work for nonlinear control theory development and engineering applications. In recent years, considerable attention has been devoted to the study of polynomial systems in numerical approach, especially using semidefinite programming and the sum of squares decomposition, which provide an efficient way for researchers to explore polynomial systems. The sum of squares decomposition has been used for nonlinear systems analysis successfully; however, control synthesis still remains a stubborn problem and needs further development.This thesis explores numerical solution for H∞control of polynomial systems. We introduce numerical computation of polynomial systems L2 gain, then consider state feedback H∞control of polynomial systems, finally, concentrate on polynomial systems with uncertainty. The main research contents are showed as follows:1. Numerical computation for polynomial systems L2 gain is introduced. Using sum of squares decomposition, we propose two algorithms for polynomial systems L2 gain computation, which based on Hamilton-Jacobi inequality and dissipative inequality.2. State feedback H∞, control of polynomial systems is studied. Basing on a proposed iterative synthesis strategy, we extend the method to general cases. Usually, a feedback controller is obtained for the system by the linearized way, so the proposed algorithm can work. However, for those systems where an initial controller could not be found by the linearized way, a new method to design the initial value for the iteration is suggested. If the system is local asymptotic stability and satisfies the L2 performance index, we then estimated the domain of attraction of the system.3. Robustness of performance index of a class of polynomial systems with uncertainty is addressed. Similar to polynomial systems, the reachable set with L2 disturbances, robustness of performance index of polynomial with uncertainty are investigated. Furthermore, robustness of state feedback control is studied.