Adaptive Control of Functionally Uncertain Systems

PhD Thesis, University of Southampton, UK. 1998.

By Mark French, Supervisor Eric Rogers.

Absract. In this thesis we develop performance bounds for adaptive control designs applicable to functionally uncertain systems in either the affine, strict feedback and output feedback normal forms. L^2, weighted L^2, L^\infty and weighted L^\infty models of uncertainty are introduced. These allow us to specify the uncertainty in the system independently of the model structure chosen for the adaptive controller. A modified LQ cost functional, incoporating both control effort and state terms, is introduced and its interpretation is discussed. The main results bound this cost functional in terms of the uncertainty models for a class of Lyapunov based adaptive controllers. One class of results shows that a finite dimensional model suffices for stabilization given a sufficiently large adaption gain. A second class of results considers global uncertainties. It is shown that unless the adaption rate is faster than the uncertainty growth, instability may occur. An unbounded adaptive rate structure is described which allows global results to be given, using a special class of physically realisable global models, under the assumption that the uncertainty growth is known, but not the uncertainty level.

An example is given of a standard spline network based controller whose performance degrades as the resolution of the model increases. We then give a sufficient condition on the model for the performance to be asymptotically bounded independent of the size of the model. Under further L^\infty assumptions on the uncertainty model, we construct a control schema which has the required performance scalability. It is argued that average case performance costs are more natural than worst case costs for the adaptive designs, and it is shown how the average case performance costs can be estimated from the worst case bounds. The open problem of a structurally adaptive scheme which is stable in the absence of smoothness information is also solved.

The techniques we have developed allow a performance based adaptive theory to be developed; and ultimately this should lead towards a synthesis of the two main strands of modern control theory: robust control and adaptive control.

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