Feedback organizations, learning and complexity in H∞

Recent developments in multivariable feedback theory are not only changing the way in which controls are designed, but have given us the means to systematize the role of organization, adaptation, and learning in Control. Some of these hitherto empyrical or anecdotal concepts are on the verge of yielding to axiomatic theory. In this survey we will start with concrete advances in feedback design, show how these can be extended to the optimization of adaptive controllers for slowly time-varying systems, and then proceed towards more speculative aspects of an incipient Organization Theory. Feedback and adaptation make it possible to achieve precise control of systems for which accurate or clearly parameterized apriori models may not be available. One of the more elusive and paradoxical issues in control engineering is to design such systems, or merely to capture their properties, without resorting to such unavailable models. For example, we would like to define the concept of adaptation in terms of purely external inputoutput behavior, without reference to the fact that one or more parameters are being twiddled. Indeed, the absence of a mathematical formalism or language for describing nonparametric (´unstructured´) model uncertainty has been a key roadblock in the path of feedbackadaptive theories. The H^∞ theory, in conjunction with concepts of metric complexity, has for the first time given us the rudiments of a suitable formalism.