Design of state-feedback control for polynomial systems with quadratic performance criterion and control input constraints

This paper presents a novel convex optimization approach to design state-feedback control for polynomial systems. Design criteria are comprised of a quadratic cost function and bounded magnitudes of control inputs. Specifically, we formulate a control synthesis of closed-loop systems operated in a given bounded domain characterized by a semi-algebraic set. We consider an extended class of rational Lyapunov functions and derive an upper bound of the cost function, together with a state-feedback control law. By exploiting bounds on the control input magnitudes, the controller design condition can be cast as a parameter-dependent linear matrix inequality (PDLMI), which is convex optimization and can be efficiently solved by sum-of-squares (SOS) technique. In addition, we derive a sufficient condition to compute a lower bound of the cost function. When choosing polynomial structure of the solution candidate, the lower bound can also be written as PDLMI. Numerical examples are provided to illustrate the effectiveness of the proposed design.