Sequential continuous time adaptive control: a behavioral approach

We want to design a compensator for a behavior through an appropriate behavioral interconnection. The problem is that the behavior that we want to control is not known. All that is given is a desired interconnected behavior and the prior information that this desired behavior can indeed be achieved by means of regular interconnection. This problem calls for an adaptive flavored strategy. The strategy that we propose is as follows. Measurements are taken during successive time intervals of unit length. Each time a measurement is taken the most powerful unfalsified model of that measurement and the desired behavior is determined. Since this model contains the desired behavior it is possible to find additional constraints such that the desired behavior is achieved. Moreover these additional constraints can be chosen such that the corresponding interconnection is regular relative to the true unknown behavior. This regularity property makes it possible to invoke the additional constraints in the next time interval by incorporating a transient period. The new measurement therefore satisfies these additional constraints. The procedure is repeated for the new measurement and so on. The main result is that within a finite, though unknown, number of measurements the new measurements are constrained to the desired behavior.