State feedback synthesis for polynomial systems with bounded disturbances

This paper deals with a state feedback synthesis for polynomial systems in the presence of disturbances with bounded peak or bounded energy. Positively invariant sets are composed of level sets of polynomial Lyapunov functions and are included in the region of the inputs and the state constraints under the disturbances. A two-step non-iterative design procedure is available by using matrix sum of squares (SOS) relaxations and semidefinite programming. At the first step, the matrix SOS technique is applied. Then to remove one of the causes of conservativeness of the first step, the polynomial annihilators are utilized in the second step. Numerical examples illustrate the presented design procedure.