Stabilizing feedback design for time delayed polynomial systems using kinetic realizations

Abstract

A new stabilizing feedback design method is proposed in this paper for time delayed polynomial systems with a linear input structure. The task is to transform the open loop system to a time delayed complex balanced kinetic system by using a polynomial state feedback structure which guarantees stability with arbitrary time delays.It is shown that the required computations can be performed by simple linear programming, when the only goal is the semistability of the chosen equilibrium point. If one wants to achieve additionally the uniqueness of the closed loop equilibrium point to ensure local asymptotic stability, then the extended optimization problem requires the application of semidefinite programming. The existence of the solution and computability of the feedback do not depend on the magnitude of the delays.It is shown that involving additional monomials into the feedback beyond the ones contained in the open-loop model does not improve the solvability of the semistabilization problem, but it may ensure the uniqueness of the prescribed complex balanced equilibrium point. Thus, two variants of a systematic method are proposed to find appropriate extra monomials for the feedback. One of these requires to solve a linear programming optimization problem even in this extended case.