On Stabilizing Control Of Polynomial Nonlinear Systems Via Semidefinite Programming

Polynomial nonlinear systems appear widely in practical applications. In particular, many control problems in motion control systems, mechatronic systems, process control, biological systems, electric circuits and so on, can be modeled as, transformed into, or approximated by polynomial nonlinear systems. Therefore, how to analyze and synthesize polynomial nonlinear systems 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 nonlinear systems in numerical approach, especially by using semidefinite programming and the sum of squares decomposition. Significant progress has been made in the stability analysis of polynomial nonlinear systems by those numerical approaches; however, control synthesis still remains a stubborn problem and needs further development.This thesis explores numerical solution for control synthesis of time-invariant polynomial nonlinear systems. The main research content and achievements are showed as follows:1. Numerical solution for polynomial positivity validation (PPV) is explored. Three algorithms for polynomial decomposition are proposed to check PPV problem of a given polynomial or to construct a polynomial with positive constraints. Furthermore, the software named PolyDecomp is developed based on polynomial decomposition algorithms and matrix inequalities solvers. PolyDecomp is applied for PPV and implemented in Matlab environment. Many polynomial optimization problems can be formulated into PPV problems and can be further solved by PolyDecomp.2. Stabilizing control scheme for time-invariant polynomial nonlinear systems based on Lyapunov theorem is studied. State-feedback stabilizing control of polynomial nonlinear systems is presented by constructing control Lyapunov functions (CLF) and transforming corresponding problems into the PPV type which can be solved by PolyDecomp. Controller constructed by full monomial base will be in numerous terms for relatively high-order systems and obstruct its application in practice. To overcome this problem, reduced-form controller is introduced. Control performance optimization is considered in terms of maximum convergence rate control and minimum variance control. Local stabilizing control is given in terms of designing controller to expand the domain of attraction with guaranteed performance. 3. Control scheme for polynomial nonlinear systems is expanded. A nonlinearity-cancelling control scheme is proposed, which is adaptable for more complex polynomial nonlinear systems. The key technique in this scheme is to construct an appropriate CLF to guarantee stability of the closed loop systems in some subspaces with zero control inputs. Stabilizing control scheme for 2-D polynomial nonlinear systems is considered based on negative eigenvalue placement, which has potential value for back-stepping control development. Chaos synchronization control is also developed and demonstrated in Chua's system and Lorenz system.4. Global attitude control for space rigid body is addressed. Several common description methods for rotational kinematics are discussed in control point, such as direction cosine matrix, Euler angles, quaternions and Rodrigues/Modified-Rodrigues parameters. Transformation relationships, advantages and disadvantages among those methods are discussed. Attitude control based on Rodrigues/Modified-Rodrigues parameters description is given and achieves global convergence. Attitude regulator is established by converting the corresponding problem to stabilizing control problem in Rodrigues/Modified-Rodrigues parameters. The proposed global attitude control scheme achieves effective performance as illustrated in simulation of the SAPPHIRE satellite.