On the Feedback Synthesis for an Autonomous Linear System with Perturbations

The paper investigates the feedback synthesis problem for an autonomous linear system subject to unknown time-variant continuous bounded perturbations of the coefficients. Sufficient conditions for the existence of a bounded control input transferring the initial state of the system to the origin in a finite time are discussed in detail. Our approach is based on the controllability function method proposed by V.I. Korobov. The results show that the control designed by controllability function method can guarantee that the trajectory of the system steering to the origin in a finite time even under perturbations. This study shows the relations between the range of perturbations and the upper bound of the settling-time function. As an application of results given in the paper, the motion of a mass on a spring with an uncertainty in the stiffness is considered.