Adaptive Control of Linear Time Varying Plants: The Case of "Jump" Parameter Variations

In this note we consider the adaptive control problem of linear, time varying, single-input singl-output plants with parameter discontinuities ("jump" parameter variations). In our formulation, the plant parameters are assumed to be uniformly bounded and piecewise continuous functions of time. Inside the intervals of continuity, the plant is assumed to satisfy some general conditions on observability, controllability and zero dynamics and the parameters are assumed to be smooth (or at least Lipschitz) but their speed of variation is not necessarily small. Under these, quite general, conditions it is established that a control law of the model reference type [5] which, in the case of know parameters, guarantees exponential and therefore internal stability of the closed loop, provided that in any time interval the average number of parameter discontinuities is small enough. In the case of partially unknown plant parameters, this control law is combined with an estimator. The resulting adaptive controller guarantees boundedness of the closed loop for bounded reference signals provided that: 1. in any time interval, the average number of parameter discontinuities is small enough and 2. inside the intervals of continuity, the functional dependence on time (or structure) of the fast time varying parameters is known up to a slowly varying component. The currently available results on the adaptive control problem of time-varying plants have been derived under stronger assumptions, typically requiring that the plant parameters are smooth and slowly time-varying [1,2,3] or piecewise continuous and slowly varying in the mean [4].