Polynomial Feedback and Observer Design using Nonquadratic Lyapunov Functions

In polynomial state feedback and observer design, it is often assumed that the corresponding Lyapunov functions are quadratic. This assumption allows to guarantee global stability and to use semidefinite programming and the sum of squares decomposition. In the present paper, state feedback and observer design strategies based on semidefinite programming and the sum of squares decomposition are proposed which can deal with nonquadratic Lyapunov functions without jeopardizing global stability. In particular, homogeneous Lyapunov functions and generalized Krasovskii-type Lyapunov functions are studied for state feedback design and Lyapunov functions which are nonquadratic with respect to the control system output are studied for observer design.