PhD Thesis, March 1998.

**Abstract**

Towards the development of a sensible framework for SD, control-system design, a new notion of frequency response is established for LPTV systems. It is defined in terms of the average-power of the asymptotic response to sinusoidal signals of a single frequency and can be characterised in terms of the singular values of a frequency-dependent, finite-dimensional matrix. The performance indicating properties of this new notion of frequency response are intuitive and it is used to derive bounds that facilitate the design of parametric weights employed in a new H-infinity loopshaping based procedure proposed for SD, control-system development.

Quantitative and qualitative results that characterise the uncertainty types to which LPTV, closed-loop systems can be desensitised are established in terms of the gap metric, which is a measure of the distance between the graphs of two systems. To this end, a formula for the directed-gap between two LPTV systems is obtained in terms of an optimisation over H-infinity. This plays a crucial role in the development of the main quantitative robustness result, which identifies the largest gap-ball of LPTV plants, centred at a nominal plant, that a nominal, stabilising controller is guaranteed to stabilise. A similar result that deals with simultaneous gap-perturbations to the plant and controller is also given. Qualitatively, it is established that the topology induced by the gap metric on quite a general class of LPTV systems, is the weakest with respect to which closed-loop performance varies continuously and closed-loop stability is a robust property. All of the results accommodate infinite-dimensional input/output spaces and apply to the special case of LPTV, SD control-systems.

The existence of particular representations of a class of LPTV systems is central to the framework used to obtain robustness results. Specifically, it is shown that the graph of a stabilisable, LPTV system can be expressed as the range and kernel of stable, LPTV systems that are respectively, left and right invertible by stable, LPTV systems. These representations of the graph resemble the coprime-factor representations known to exist for linear, time-invariant systems and lead to a useful characterisation of closed-loop stability. This in turn, yields a Youla-style parameterisation of stabilising controllers.

A numerical procedure is developed for computing the gap, to any desired accuracy, between LPTV systems that admit stabilisable and detectable, state-space realisations. This involves determining the existence of a solution to a related linear, shift-invariant, discrete-time, full-information, l2-synthesis problem, for which computationally-tractable necessary and sufficient conditions are obtained.

To complete this work, a new, compact derivation of state-space formula for H-infinity, SD synthesis is presented. Related numerical issues are discussed and it is shown how to restructure the formulae derived for numerical robustness.