Pole assignment in vibrating systems with time delay: An approach embedding an a-priori stability condition based on Linear Matrix Inequality

The assignment of the poles of a second order vibrating system through state feedback is a problem that has been widely investigated and for which asymptotic stability has been precisely characterized. In the presence of time delayed input or output, however, the available methods often address the issue of stability only by a-posteriori verification, which is made difficult by the characteristic equation of the system being transcendent. A convenient way to characterize delay-independent stability of time delayed linear systems makes use of Linear Matrix Inequalities, which are advantageous also from the computational point of view, given the reliability that algorithms for semi-definite optimization have acquired over the recent years. In this paper, a method for the assignment of a subset of closed-loop poles, while the remaining ones are constrained in the left half of the complex plane, is presented. The method is numerically validated in four meaningful test-cases, comparable to the ones that are used in the current literature. The obtained results demonstrate that the proposed method can accurately assign the desired subset of poles and guarantee asymptotic stability in the presence of bounded time delay.