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Section: New Results

Time-varying systems with delay and Switched Systems

Participants : Frederic Mazenc, Michael Malisoff [LSU] , Saeed Ahmed [Inria] , Hitay Ozbay [Blikent University, Turkey] .

The family of the switched systems is frequently encountered in practice. It can be used to approximate time-varying systems to ease their stability analysis or control.

In [29] we provided theoretical results for the stability and robustness analysis of nonlinear switched time-varying systems with uncertainties and time-varying delays. The delays are allowed to be discontinuous and arbitrarily long with known upper bounds. We established the results via an adaptation of Halanay's inequality and a trajectory based technique. Also, we used the results for designing switched controllers that stabilize linear time-varying systems with time-varying delays.

The contribution [10] proposed a new technique of construction of observers making possible to stabilize by output feedback a class of continuous-time switched linear systems with a time-varying delay in the output. The motivation of this paper is strong: frequently measurements are affected by pointwise time-varying delays. For stability analysis, we developped an extension of the trajectory based approach. A stability condition is given in terms of the upper bound on the time-varying delay to ensure global exponential stability of the switched feedback systems. It is worth observing that the main result applies in cases where some of the subsystems of the switched system are not stabilizable and not detectable.

The paper [27] is also devoted to classes of nonlinear time-varying continuous-time systems with outputs. For a first family of systems, we builded an observer in the case where a state dependent disturbance affects the linear approximation. A fundamental feature of our observer is the fact that it converges after a predetermined finite time. When the disturbances are the zero functions, it provides exact values of the state and it provides an approximate estimate when there are nonzero disturbances. We used this construction to design a globally exponentially stabilizing dynamic output feedback for a second family of nonlinear systems whose outputs are only available on some finite time intervals. Our technique consists in switching between control laws. We applied the control design to the controlled Mathieu equation, which arises in the study of vibrations of an elliptic membrane.

The paper [38] is devoted to a stability analysis for a class of nonlinear systems with a time-varying delay taking both large and small values in an alternating manner, precluding the application of most of the classical control design techniques. The type of assumption we imposed is the following: we imposed on the delay to be "small" on "long" time intervals and possibly "large" on "small" time-intervals. Bearing in mind this key property, we first introduced the concept of delay-hybrid-dependent stability, which grasps the features of the delays described above and represented the studied system as a system with a switched delay. Then by using switching techniques and Lyapunov-Krasovskii functionals (LKFs), we provided a new stability criterion.