Robust Stability Analysis for Feedback Interconnections of Time-Varying Linear Systems

Abstract

A generalization of Vinnicombe's ν-gap metric and corresponding robust feedback stability results are proposed for a class of linear time-varying (LTV) systems in "Robust stability analysis of time-varying linear systems,"

Transitivity in simultaneous stabilization for a family of time-varying systems

“…This approach is a generalization of H ∞ control in the sense that 2-norm is used to quantify the size of signals, the mathematical framework will be operator theoretic as described in [4]. This interpretation of dynamical systems has generated significant research interest over the last decade, as witnessed by recent contributions such as [5][6][7][8][9][10]. For the time-varying linear systems the stabilization problem was first (as far as we know) formulated in [11], the class of all causal time-varying linear systems which are stabilizable has been shown to be those which allow doubly coprime factorizations and there is a Youla type parametrization of all stabilizers which is conceptually similar to the classical result for linear time-invariant systems.…”

Time-domain ν-gap robustness analysis for shift-invariant systems

“…The Vinnicombe's ν-gap metric [1], [2] for finitedimensional linear time-invariant (LTI) systems and its generalisations to infinite-dimensional LTI systems [3], [4] and linear time-varying (LTV) systems [5], have traditionally been defined in terms of specific representations for the graph of a dynamical system as a subspace of the signal space L 2 R . The success in generalising the standard gap metric from a function-theoretic setting [6] to a time-varying setting via a geometric approach [7] suggests that the same may be doable for the ν-gap.…”

“…The success in generalising the standard gap metric from a function-theoretic setting [6] to a time-varying setting via a geometric approach [7] suggests that the same may be doable for the ν-gap. This paper makes the first attempt in this direction for the class of stabilisable LTI systems in the time domain, borrowing the underlying ideas from [5].…”

“…A generalisation of the ν-gap was recently introduced in [5] for causal LTV systems in a continuous-time setting. It contains certain extensions of known LTI robustness results [1], [3].…”

“…However, a corresponding robust stability result is lacking therein. The papers [8], [5] provide an approach to patching this up via the use of a compactness assumption on a Hankel operator defined with respect to strong system graph representations, known to hold for various classes of linear systems, including that considered in [3]. Supported This paper defines a ν-gap distance in terms of a Fredholm index condition without making use of strong graph representations for LTI systems, potentially broadening the class of systems to which the ν-gap theory applies.…”