Abstract
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.
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