Robust stability for nonlinear control: state-space and input-output synthesis

Abstract

In this thesis we consider the development of a general nonlinear input-output theory which encompasses systems with initial conditions.

Appropriate signal spaces (i.e., interval spaces, extended spaces and ambient spaces) are introduced with some fundamental assumptions to constitute a framework for the study of input-output systems with abstract initial conditions. Both systems and closed-loop systems are defined in a set theoretic manner from input-output pairs on a doubly infinite time axis, and a general construction of the initial conditions (i.e., a state at time zero) is given in terms of an equivalence class of trajectories on the negative time axis. Fundamental properties (such as existence, uniqueness, well-posedness and causality) of both systems and closed-loop systems are defined and discussed from a very natural point of view. Input-output operators are then defined for given initial conditions, and a suitable notion of input-output stability on the positive time axis with initial conditions is given. This notion of stability is closely related to the ISS/IOS concepts of Sontag.

A fundamental robust stability theorem is derived which represents a generalisation of the input-output operator robust stability theorem of Georgiou and Smith to include the case of initial conditions; and can also be viewed as a generalisation of the ISS approach to enable a realistic treatment of robust stability in the context of perturbations which fundamentally change the structure of the state space. This includes a suitable generalisation of the nonlinear gap metric. Generalisations of this robust stability result are also extended to finite-time reachable systems and to systems with potential for finite-time escape by extending signals on extended spaces to a wider space (ambient space). Some linear and nonlinear applications are given to show the effects of the robust stability results.

We also present a generalised nonlinear ISS-type small-gain result in this input-output structure set up in this thesis, which is established without extra observability conditions and with complete disconnection between the stability property and the existence, uniqueness properties of systems.

Connections between Georgiou and Smith's robust stability type theorems and the nonlinear small-gain theorems are also discussed. An equivalence between a small-gain theorem and a slight variation on the fundamental robust stability result of Georgiou and Smith is shown.

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