Stability of linear systems with time-varying delays: A direct frequency domain approach

The stability of linear systems with uncertain bounded time-varying delays (without any constraints on the delay derivatives) is analyzed. It is assumed that the system is stable for some known constant values of the delays (but may be unstable for zero delay values). The existing (Lyapunov-based) stability methods are restricted to the case of a single non-zero constant delay value, and lead to complicated and restrictive results. In the present note for the first time a stability criterion is derived in the general multiple delay case without any constraints on the delay derivative. The simple sufficient stability condition is given in terms of the system matrices and the lengths of the delay segments. Different from the existing frequency domain methods which usually apply the small gain theorem, the suggested approach is based on the direct application of the Laplace transform to the transformed system and on the bounding technique in L₂. A numerical example illustrates the efficiency of the method.