A convex characterization of analysis and synthesis for nonlinear systems via extended quadratic Lyapunov function

Using an extended quadratic Lyapunov function of the form V(x)=x ^TP(x)x, we consider L₂-gain analysis and state feedback control synthesis for input-affine polynomial type nonlinear systems, and derive Riccati type matrix inequality conditions that depend on x. We show that the solution P(x) can be given by solving linear matrix inequalities as a polynomial type matrix. We also determine the domain of internal stability. We finally show that the proposed method is effective through a numerical example of bilinear systems