Abstract
This paper investigates the stability properties of a class of dynamic linear systems possessing several linear time-invariant parameterizations (or configurations) which conform a linear time-varying polytopic dynamic system with a finite number of time-varying time-differentiable point delays. The parameterizations may be timevarying and with bounded discontinuities and they can be subject to mixed regular plus impulsive controls within a sequence of time instants of zero measure. The polytopic parameterization for the dynamics associated with each delay is specific, so that(q+1)polytopic parameterizations are considered for a system withqdelays being also subject to delay-free dynamics. The considered general dynamic system includes, as particular cases, a wide class of switched linear systems whose individual parameterizations are timeinvariant which are governed by a switching rule. However, the dynamic system under consideration is viewed as much more general since it is time-varying with timevarying delays and the bounded discontinuous changes of active parameterizations are generated by impulsive controls in the dynamics and, at the same time, there is not a prescribed set of candidate potential parameterizations.
Highlights
The stabilization of dynamic systems is a very important question since it is the first requirement for most of applications
Powerful techniques for studying the stability of dynamic systems are Lyapunov stability theory and fixed point theory which can be extended from the linear time-invariant case to the time-varying one as well as to functional differential equations, as those arising, for instance, from the presence of internal delays, and to certain classes of nonlinear systems, 1, 2
Dynamic systems which are of increasing interest are the so-called switched systems which consist of a set of individual parameterizations and a switching law which selects along time, which parameterization
Summary
The stabilization of dynamic systems is a very important question since it is the first requirement for most of applications. The dynamic system under investigation is a linear polytopic system subject to internal point delays and feedback state-dependent impulsive controls. Both parameters and delays are assumed to be timevarying in the most general case. An important key point is that if the system is stabilizable, it can be stabilized via impulsive controls without requiring the delay-free dynamics of the system as it is shown in some of the given examples. For a given interimpulse time interval, the impulsive amplitudes are larger as the instability degree becomes larger, and the signs of the control components should be appropriate, in order to compensate it by the stabilization procedure Such a property will hold for nonpolytopic parameterizations. Imp : {ti ∈ R0 : ti 1 > ti}, where an impulsive control u ti δ t − ti occurs with δ · being the Dirac delta of the Dirac distribution
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