Synthesis of nonlinear state feedback control via extended quadratic Lyapunov function – convex analysis approach using linear matrix inequalities

We show a synthesis method of nonlinear state feedback control for input-affine polynomial-type nonlinear systems, which is in a class that coefficients of linear time-invariant systems depend on state. The method consists of the following two steps. First, using an extended quadratic Lyapunov function, which is in a class that coefficient of a quadratic Lyapunov function depends on state, we give a condition for the control synthesis problem to be feasible by a pair of linear matrix inequality and equation which depend on state. Next, we obtain a solution satisfying the condition by solving a convex programming problem. We can construct the controller by enclosing a domain of state in a convex hull and solving linear matrix inequalities which are given at vertices of the convex hull. In the paper, finally, to demonstrate efficiency of the method, we also show a computer simulation for a bilinear model of continuous stirred tank reactor.