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**Stability of linear systems with general sawtooth delay.**
*(English)*
Zbl 1206.93080

Summary: 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 [W. Michiels, V. Van Assche and S. Niculescu, Stabilization of time-delay systems with a controlled time-varying delays and applications. IEEE Trans. Automat. Control 50, 493–504 (2005)]. Sawtooth delays \(\tau \) with \(\dot {\tau} = 1\) (almost everywhere) can possess this property [J. Louisell, New examples of quenching in delay differential equations having time-varying delay. Proceedigns of the 5th ECC, Karlsruhe, Germany (1999)]. In this paper, we show that general sawtooth delay, where \(\dot {\tau} \neq 0\) is constant (almost everywhere), also can possess 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 [Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Trans. Automat. Control 52, 1109–1112 (2007)] for \(\dot {\tau} =1\) and we show that for any integer \(\dot {\tau}\), 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 \(\dot {\tau} = 1\) in E. Fridman [Automatica 46, No. 2, 421–427 (2010; Zbl 1205.93099)] to the general sawtooth delay. The time-dependent terms of the functionals improve the results for all values of \(\dot {\tau}\), though the most essential improvement corresponds to \(\dot {\tau} = 1\).

### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93C05 | Linear systems in control theory |

93B15 | Realizations from input-output data |

34H05 | Control problems involving ordinary differential equations |

93C15 | Control/observation systems governed by ordinary differential equations |