Adaptively controlling non‐linear systems, using trajectory approximations

The structure underlying most adaptive algorithms is a linear parametrized set. This set either represents the universe of controllers the adaptive algorithm can choose from in order to achieve the control task or it may represent a class of models the adaptive algorithm may exploit to approximate the open‐loop plant behaviour. In the latter approach the controller used to close the loop is selected via some non‐linear map of the identified model parameters. The emphasis is on approximating the vector field that generates the trajectories of the system. Alternatively we propose to predict the trajectories over a short period of time directly, not indirectly involving a representation of the underlying vector fields. The feasibility of such an approach using a one‐step‐ahead‐type algorithm for both prediction and control is analysed. The scheme is hybrid in that the plant is continuous‐time, whilst the control action is implemented in discrete time. The control action is of the model reference type. The algorithm is applied to a class of (non)‐linear time‐varying systems of a given structure (known relative degree) and possessing a stable inverse. Given input/output measurements only, the algorithm can enforce a desired response within guaranteed error bounds. Robustness properties with respect to partial state measurement (e.g. neglecting parasitic dynamics) and violation of the stable inverse assumption are investigated.