Adaptive switching supervisory control of nonlinear systems with no prior knowledge of noise bounds

1 I n t r o d u c t i o n One of the well-estabilished approaches for controlling time-varying uncertain plants is the introduction of adaptation in the feedback loop [2]. However, converttional continuous adaptation is not always capable of performing satisfactorily mainly because of its inherent difficulty in taking advantage of prior knowledge of potential plant changes. This is particulartly true whenever the plant switches among different modes of operation or in the absence of sufficient excitation. In both circumstances undesirable transients may typically arise due to slow adaptation. In recent years, adaptive switching supervisory control (SSC) has emerged as an alternative approach for tackling the problem [3, 4, 6, 9, 14] with its appealing inherent feature of resembling an adaptive version of classic gain-scheduling control which has been successful in so many applications. As a matter of fact, SSC aims at extending gain-scheduling control to cases where the supervisor has no full information on the current dynamical behaviour of the plant to be controlled. A typical situation is the one where only records of past plant I /O data are available in order to let the supervisor decide whether the current controller is adequate, and, in the negative, select another candidate controller. If for all potential changes the plant belongs to a prespecified set of models, the key idea of SSC is to have a, usually finite, family of candidate controllers Ci, such that each plant model in the set performs "satisfactorily" when controlled by at least one of the Ci's. Then, a suitably designed supervisory unit takes care of orchestrating the switching among the candidate controllers so as to preserve closed loop stability and, possibly, performance in the face of plant changes Switching mechanisms are usually based on a supervisory logic whereby a controller is falsified whenever the inferred behaviour of another controller turns out to be better than the one actually achieved by the currently operating controller. Whenever this happens, the candidate controller with the best inferred behaviour is switched on in feedback to the plant, replacing the currently operating controller. To the best of authors' knowledge, available controller falsification approaches rely on the assumption that either disturbance bounds are a priori given [1, 14] or the disturbance colour is known [10]. If such prior information is imprecise, either the falsification rate becomes too high (underestimated bounds or wrong colour) or the acting controller is falsified only after the plant I /O variables have become too large with respect to nominal performance specifications (overestimated disturbance amplitude). The main contribution of this paper is to present falsification and inference criteria integrated in a new supervisory switching logic, whereby no prior information on disturbance bounds is required and which are applicable to a wide class of linear and nonlinear plants. 2 P r o b l e m f o r m u l a t i o n Consider a discrete time nonlinear system of the following form: x(t + 1) f(x(t) ,u(t) ,d(t) ,O) (1)