Nonrobustness of the Pole Placement Design under Small Perturbations in Time Delays

It is proved that, by using state feedback, it is not possible to place the poles of a linear time invariant system to the left of a certain critical negative real number if the system dynamics is subject to small uncertain perturbations which include small time delays. Given a nominal controllable system (A,B) and a real number h≫0 (no matter how small) there exists a positive real number r and a (small) perturbation of the nominal model (described by the introduction of time delays not exceeding h in the state and input variables) such that for any feedback matrix K placing the eigenvalues of A + BK to the left of Re(s) = -r there is an eigenvalue of the perturbed closed loop system with real part not less than -r. This shows that left half planes which are sufficiently remote from the imaginary axis cannot be robustly assignable under small perturbations in time delays.