Overcoming the Obstacle of High Relative Degree

Promulgated by some with religious-like fervor, viewed with skepticism by others, adaptive control has for almost forty years been one of the most alluring, intriguing, and often misunderstood areas within the eld of automatic control. In truth, despite its present shortcomings fe.g., its failure to adequately address performance issuesg adaptive control has come quite a long way since rst conceived. Once amounting to little more than a collection of seemingly unrelated heuristic ideas, adaptive control now rests on a bona de foundational theory which serves to explain basic concepts and constructions in a principled manner. An early advance contributing to the theory's development was the formulation and resolution of the now classical siso \model reference control problem." The main obstacle to the problem's resolution was dealing with nominal process models of high \relative degree." The assault on the relative degree problem involved many people and took place over a period of several years. The problem's rst solution appeared in the late 1970s and used what is now called \integrator backstepping." A second solution emerged about two years later and relied on the idea of \error normalization." The latter approach led to an overall control algorithm which was far simpler in form than that provided by backstepping. As a result the backstepping approach was totally eclipsed by the error normalization approach and remained so for more than a decade. Ironically, integrator backstepping has quite recently enjoyed renewed and considerable attention because of its apparently unique ability to deal with certain types of nonlinearities in both adaptive and nonadaptive systems. The aims of this paper are to explain what backstepping is, to chronicle the events leading to the discovery of the backstepping solution to the classical model reference problem and to discuss the signiicance of this work in the broader context of contemporary adaptive control. 1 The Classical Model Reference Adaptive Control Problem One of the rst problems in adaptive control to be studied in detail was an adaptive version of what Wolovich called the \exact model matching problem" 1]. The classical model reference adaptive control problem is to develop a dynamical controller capable of causing the output y of an imprecisely modelled siso process P to approach and track the output y R of a prespeciied reference model R with input r.