Stabilization of linear undamped systems via position and delayed position feedbacks

This paper presents a systematic approach to stabilizing a kind of linear undamped systems of multiple degrees of freedom by using both position and delayed position feedbacks, namely, PDP feedbacks for short. For the fully actuated system, the approach enables one to complete the design of controller directly through the use of modal decoupling and a stability chart. For the under-actuated system, the approach includes two steps. The first step is to move all the eigenvalues of the system on the imaginary axis of the complex plane by using a position feedback, and the second step is to drag all the eigenvalues of the system to the left half open complex plane through the use of a delayed position feedback, which can be determined on the basis of sensitivity analysis of eigenvalues. Two examples, i.e., a fully actuated robotic manipulator and an under-actuated double inverted pendulum, are discussed in the paper to demonstrate the design of controllers for the two different types of systems and to support the efficacy of the proposed approach.