Feedback Design for Saturated Polynomial Nonlinear Systems via Higher Order Lyapunov Functions

In this work, we develop a new control design approach to deal with saturated polynomial nonlinear systems by using higher order Lyapunov functions. By combining power transformation with Sum-of-Squares (SOS) techniques, we can augment the systems with more state variables representing higher order combinations of the original ones. Then, the search of higher order Lyapunov functions for original systems can be recast to the design of quadratic Lyapunov functions for augmented systems. By computing for higher order Lyapunov functions using norm-bounded differential inclusion (NDI) LMI conditions, the flexible representations of augmented systems can help us to achieve better performance than quadratic based method. Two examples illustrate the improvements to enlarge the region of attraction and to improve the ℋ_∞ performance for nonlinear systems subjected to saturation nonlinearity, respectively.