Controller Design Of Polynomial Nonlinear Systems Using Sum Of Squares

Many practical systems have inherent nonlinearity that cannot be ignored. The anal-ysis and design of nonlinear systems are among the most challenging problems in systemsand control theory. In this thesis, we focus on nonlinear control synthesis for polynomialnonlinear systems using Sum of Squares (SOS) programming.SOS is a powerful technique which has been widely used in recent years. It providesan efcient way for researchers to explore polynomial nonlinear systems. Compared toquadratic Lyapunov function, it has the advantage of being able to find high-order Lya-punov function through sum of squares programming for polynomial nonlinear systems.SOS has been used for nonlinear systems analysis extensively. However, its potential forcontrol synthesis has not been fully explored. Therefore, this dissertation explores how touse SOS programming for controller design and its numerical solution for the followingthree systems: satellite attitude control, ship course control and ship roll control.A novel convergence criterion for the nonlinear systems has been recently derived byAnders Rantzer. The criterion uses a density function, instead of a Lyapunov function, toguarantee the systems trajectories asymptotically tending to the equilibrium. The criterionis used for transforming the problem of searching controller into a convex programming.However, it is not a convex programming problem to search control Lyapunov functionoriginally based on Lyapunov’s second theorem. The main work is as follows:Firstly, we prove that the existence of a density function with its time derivativegreater than zero ensures almost global stability and local asymptotical stability. Theresult has been used for estimating the domain of attraction for polynomial nonlinearsystems.Secondly, for satellite attitude control with three inputs, the controller design is ofa typical nature of polynomial nonlinear control systems. Nonlinear state feedback con-trollers are designed based on the aforementioned asymptotically stable result combinedwith the sum of squares technique. Lyapunov function of degree4is searched success-fully using sum of squares programming. Simulations demonstrate the validity of thesuggested controller design method.Thirdly, for systems with certain parameters, we present a nonlinear controller de-sign method and its numerical solution using the sum of squares technique and Rantzer’s convergence criterion. For ship course control, nonlinear static feedback controllers aredesigned based on the suggested method. Lyapunov function of the resulting closed-loopsystem is found using sum of squares programming, which verifies globally asymptoticalstability of the closed-loop system.Lastly, for systems with uncertain parameters, a robust nonlinear controller designmethod and its numerical solution is proposed based on the sum of squares technique andRantzer’s convergence criterion. For three cases of ship course control with uncertainparameters, uncertain ship course control with dynamic actuator, and ship roll controlwith uncertain parameters, the corresponding robust controllers are designed based on thesuggested design method. The simulation results show that the suggested design methodhas strong robustness for the systems with uncertain parameters.