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

Abstract

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.

Control design for discrete‐time bilinear systems using the scalarized Schur complement

“…The maximum region where YALMIP could find a controller to stabilize the system is given by γ =295. This should be compared to the value γ =150 obtained in . The difference is due to the use of the scalarized Schur complement in the present work.…”

“…To the best of the authors' knowledge, this is the first work specifically addressing the quadratic stabilization of discrete‐time bilinear systems using SOS programming, with the exception of our previous work which presents some preliminary results. The present work significantly extends the results of . Unlike the designs in , the design procedure developed here can handle both open loop unstable systems and systems with multiple inputs.…”

Lyapunov-based proportional-integral controller design with guaranteed region of convergence for dc-dc power converters

“…In this paper, the SOS programming based controller synthesis method for discrete-time bilinear systems developed in [4] is used for the P and PI controllers design for a class of power converters described with discrete-time bilinear averaged dynamic model. First, the general averaged discretetime model of the converter is developed and a coordinate transformation is performed to transform the equilibrium point of the model to the origin.…”

“…This section presents a modified version of the SOS-based controller design procedure proposed in [4], which is adjusted in order to calculate the coefficients of the P&PI controller that maximizes the region of convergence for a discrete-time bilinear system (15) with equilibrium point in the origin.…”

“…Reference [4] investigates the stabilization of the discrete-time bilinear systems by using SOS decomposition method. The input is defined as the ratio of polynomials and a quadratic Lyapunov function is used to prove the stability.…”