Implicit model techniques and their application to LQ adaptive control

Abstract

This chapter presents a general theory that is viewed as a basis for the proper use of ARX implicit models in adaptive control schemes. Necessary and sufficient conditions for the existence of such models have been given. The possible use of such models has been discussed to design adaptive algorithms based on finite and infinite horizon quadratic criteria. A particular way of exploiting the information provided by an implicit model has been considered. The main feature of this approach is that of obtaining, at every iteration of the algorithm, the new regulator not on the basis of the identified prediction model only, but also making use of the equation of the previously acting regulator. This is accomplished by solving a variational control problem and allows developing adaptive control algorithms based on either finite or infinite horizon quadratic optimization. Some theoretical results have been reported showing that the developed algorithm admits the optimal control law as an equilibrium point, and that a proper structuring of such algorithm assures that it admits a unique whitening equilibrium point coinciding with the optimal regulator. Furthermore, a technical way to test whether a generic convergence point coincides with this unique whitening equilibrium point has been indicated. Based on such a result, a modification to the basic algorithm has been described that ensures the enjoyment of the self-tuning property. Some simulation results show the effectiveness of the proposed approach.