Stability, robustness, and performance issues in second level adaptation

A new approach, described as second level adaptation, was introduced in [1] for the control of unknown linear time-invariant plants using multiple identification models. If θ_p ∈ Rⁿ, the unknown parameter vector of an LTI system lies in the convex hull P(t₀) of (n+1) vectors θ_i(t₀) (initial values of adaptive vectors) in parameter space, it was shown that it lies also in the convex hull of θ_i(t)(i = 1, 2, ..., n+1), of the adaptive parameters of the identification models. If the representations of the plants and models are in companion form, and all the state variables are accessible, simulation results were presented to demonstrate that the new method would result in much better performance than conventional adaptive control. In this paper, all aspects of second level adaptation are critically reviewed. Following this, an analysis of the stability and robustness of the approach is undertaken, and detailed reasons are provided for the observed improvement in performance. Due to space limitations, most of the results described in the paper pertain to plants in companion form, with all state variables accessible. Towards the end of the paper, an effort is made to indicate how the same concepts can be extended to more general cases. These include plants whose matrices are not in companion form, and systems in which only the input and the output of the plant are accessible. The details of the latter problems will be included in a forthcoming paper [7]. The authors believe that the two papers, together, will make a convincing case for the use of multiple models in adaptive control.