Control design and analysis for discrete time bilinear systems using Sum of Squares methods

In this paper, stabilization of discrete time bilinear systems is investigated by using Sum of Squares (SOS) programming methods and a quadratic Lyapunov function. Starting from the fact that global asymptotic stability cannot be proven with a quadratic Lyapunov function if the controller is polynomial in the states, the controller is instead proposed to be a ratio of two polynomials of the states. First, a simple one-step optimal controller is designed, and it is found that it is indeed defined as a ratio of two polynomials. However, this simple controller design does not result in any stability guarantees. For stability investigation, the Lyapunov difference inequality is converted to a SOS problem, and an optimization problem is proposed to design a controller which maximizes the region of convergence of the bilinear system. Input constraints can also be accounted for in the optimization problem.