Uncertainty , Performance , and Model Dependency Approximate Adaptive Nonlinear Control

We consider systems satisfying a matching condition which are functionally known up to a L2 measure of uncertainty. A modified L2 performance measure is given, and the performance of a class of model based adaptive controllers is studied. An upper perfor­ mance bound is derived in terms of the uncertainty measure and measures of the approximation error of the model. Asymptotic analyses of the bounds under increasing model size are undertaken, and sufficient conditions are given on the model that ensure the performance bounds are bounded independent of the model size. Despite the simplicity of the systems under consider­ ation in neuro-control, little attention has been paid to performance and uncertainty aspects of the var­ ious i'olutions-ego for which functional uncertain­ ties arE; the designs stable; can transient performance be estimated a-priori? In this paper we consider a limited class of systems (feedback linearisable and satisfying a matching condition). We provide up­ per bounds on L2 performance measures of both the state vector and the control. L2 measures of uncer­ tainty are considered: these allow us to completely specify the uncertainty under consideration indepen­ dantly of the model chosen for the adaptive design, and these measures will be shown to be natural for obtaining conditions on the uncertainty for conver­ gence of the state vector to residual sets; stability in the large; and also for bounding the state vector part of performance measures. L oo measures will be used to bound the control effort terms in the perfor­ mance measures. The first major result shows that if sufficiently high adaption gains are utilised then a sufficiently large model sufffices for semi-global sta­ bilisation. This design requires knowledge of the L2 uncertainty level; conditions for stability can be ob­ tained and the state performance measure can be explicitly bounded. With LOO information, the con­ trol effort performance measure can also be bounded. However if the uncertainty level is unknown the state vector transient cannot be bounded a-priori, and we must consider global models and corresponding global uncertainty measures incoporating the uncer­ tainty growth. An example is constructed that shows that if the uncertainty growth is not known then sta­ bility cannot be guaranteed. The second major re­ sult shows that if the uncertainty growth is known, then a class of physically realisable, but non-finite dimensional (structurally adaptive) models suffices. This is proved using weighted L2 descriptions of the uncertainty, and the …