Arbitrary precision computation of the set of stabilising gains for time-delay LTI systems

A new method is proposed to calculate the set of proportional controllers stabilising a given linear time-invariant system with a fixed time-delay. The method is based on the utilisation of Padé approximations for the time-delay, Lam's studies on the convergence of Padé approximations and a generalisation of the Nyquist theorem applicable to quasi-polynomials. The method allows determination of the closeness of the resulting set to the exact set of stabilising gains by tracing the uncertainty introduced by the Padé approximation. As a result, it is possible to approximate the resulting set arbitrarily close to the exact stabilising set.