Towards a general complexity-based theory of identification and adaptive control

Abstract

This paper is based on the recent solution of two problems of control theory, which will be combined here to generate new approaches to H ^∞/ℓ¹ adaptive control, as well as to produce the rudiments of a general theory of adaptation and complexity-based or information-based learning.

The problems, which have been solved by the author and his co-workers, involve,

• 1. optimally fast identification to reduce plant uncertainty (e.g., to a weighted ball of radius ∈ in H ^∞/ℓ¹);

• 2. exact computation of feedback performance under large plant uncertainty (including such a ball).

By combining these two solutions and using frozen-time approximations to compute optimal feedbacks under time-varying data, we obtain control laws which conform to our definition of what the term “adaptive” should mean, and in fact are in a certain sense nearly optimally adaptive.

These results are concrete and lead to algorithms. However, they also provide a paradigm of a more general theory of adaptive control, which will be outlined. We propose definitions of the notions of machine adaptation and machine learning which are independent of the internal structure of, say, the controller in the case of a feedback system; and are independent of properties such as the presence or absence of nonlinearity, time-variation, or even feedback. Instead, they are based on external performance. They enable us to address such questions as: What should the term “adaptive” and “learning“ mean in the context of control? Is it possible to tell whether or not a black box is adaptive? Is adaptation synonymous with the presence of nonlinear feedback? More to the point, in design is it possible to determine beforehand whether it is necessary for a controller to adapt and learn in order to meet performance specifications, or is adaptation a matter of choice? We will claim that the answers are mostly positive.