COMPUTING THE STABILITY REGION IN DELAY-SPACE OF A TDS USING POLYNOMIAL EIGENPROBLEMS

In this work we analyze stability properties of retarded linear time invariant multi-dimensional, multi-delay, time delay systems with respect to perturbations in the delay parameters. We analyze two methods which allow the computation of the critical delays, i.e., the points in delay-space which causes the system to have a purely imaginary eigenvalue. The critical delays are potential stability boundaries as the boundaries of the stability region is necessarily a subset of the critical delays.The two methods originates from a Lyapunov-type condition, which is completely self-contained in this work. The first method corresponds to the case of commensurate delays, for which the the Lyapunov-type condition reduces to a polynomial eigenvalue problem for which the first companion form is exactly the eigenvalue problem occurring in Chen et al. (1995). The second method is the result of a simple substitution which allows the computation of the critical delays of an incommensurate system by solving a quadratic eigenvalue problem. For the scalar multi-delay case we find a closed expression for the critical curves using this method. We confirm the methods by comparing it to previous work and published examples.