Pole Placement for Time-Delayed Systems Using Galerkin Approximations

Abstract

Many dynamic systems of practical interest have inherent time delays and thus are governed by delay differential equations (DDEs). Because DDEs are infinite dimensional, time-delayed systems may be difficult to stabilize using traditional controller design strategies. We apply the Galerkin approximation method using a new pseudo-inverse-based technique for embedding the boundary conditions, which results in a simpler mathematical derivation than has been presented previously. We then use the pole placement technique to design closed-loop feedback gains that stabilize time-delayed systems and verify our results through comparison to those reported in the literature. Finally, we perform experimental validation by applying our method to stabilize a rotary inverted pendulum system with inherent sensing delays as well as additional time delays that are introduced deliberately. The proposed approach is easily implemented and performs at least as well as existing methods.