Adaptive control: Myths and realities

It was found that all currently existing globally stable adaptive algorithms have three basic properties in common: positive realness of the error equation, square-integrability of the parameter adjustment law and, need for sufficient excitation for asymptotic parameter convergence. Of the three, the first property is of primary importance since it satisfies a sufficient condition for stabillity of the overall system, which is a baseline design objective. The second property has been instrumental in the proof of asymptotic error convergence to zero, while the third addresses the issue of parameter convergence. Positive-real error dynamics can be generated only if the relative degree (excess of poles over zeroes) of the process to be controlled is known exactly; this, in turn, implies perfect modeling. This and other assumptions, such as absence of nonminimum phase plant zeros on which the mathematical arguments are based, do not necessarily reflect properties of real systems. As a result, it is natural to inquire what happens to the designs under less than ideal assumptions. The issues arising from violation of the exact modeling assumption which is extremely restrictive in practice and impacts the most important system property, stability, are discussed.