Stability of linear systems with general sawtooth delay

It is well known that in many particular systems, the upper bound on a certain time-varying delay that preserves the stability may be higher than the corresponding bound for the constant delay. Moreover, sometimes oscillating delays improve the performance (Michiels, W., Van Assche, V. & Niculescu, S. (2005) Stabilization of time-delay systems with a controlled time-varying delays and applications. IEEE Trans. Automat. Control, 50, 493–504). Sawtooth delays τ with τ̇ = 1 (almost everywhere) can posses this property (Louisell, J. (1999) New examples of quenching in delay differential equations having time-varying delay. Proceedigns of the 5th ECC, Karlsruhe, Germany). In this paper, we show that general sawtooth delay, where τ̇ 6= 0 is constant (almost everywhere), also can posses this property. By the existing Lyapunov-based methods, the stability analysis of such systems can be performed in the framework of systems with bounded fast-varying delays. Our objective is to develop ‘qualitatively new methods’ that can guarantee the stability for sawtooth delay which may be not less than the analytical upper bound on the constant delay that preserves the stability. We suggest two methods. One method develops a novel input–output approach via a Wirtinger-type inequality. By this method, we recover the result by Mirkin (2007, Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Trans. Automat. Control, 52, 1109–1112) for τ̇ = 1 and we show that for any integer τ̇ , the same maximum bound that preserves the stability is achieved. Another method extends piecewise continuous (in time) Lyapunov functionals that have been recently suggested for the case of τ̇ = 1 in Fridman (2010, A refined input delay approach to sampled-data control. Automatica, 46, 421–427) to the general sawtooth delay. The time-dependent terms of the functionals improve the results for all values of τ̇ , though the most essential improvement corresponds to τ̇ = 1.