Adaptive Stabilization Theory

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Existing adaptive control algorithms which can be shown to be capable of "globally stabilizing" a system without persistent excitation are either applicable to process models satisfying assumptions which recent results show are unnecessarily restrictive, or perform very inefficiently and consequently are of little practical value. The underlying reason for this is that to be applicable to "large" classes of process models, the "parameterized design models" upon which "certainty equivalence control" is based, almost always have points in their parameter spaces at which the design model is unstabilizable - and at such points certainty equivalence controller design via "design model stabilization" is impossible. We propose to deal with this problem by exploiting the special capabilities if identifier- based adaptive controllers employing switching. We propose to study switching algorithms of two different types. The first, called "hysteresis switching" has already been used to successfully resolve several long standing problems from classical model reference adaptive control. The second, called "cyclic switching," is in an earlier stage of development, but has the potential of yielding adaptive controllers with capabilities well beyond those existing algorithms. Ever since the introduction of tuning error normalization by Edgart in 1978, the concept has been vigorously exploited. One of the main benefits of normalization is that it enables one to obtain stabilizing adaptive controllers which are very much simpler in structure than are algorithms which do not use normalization. It is largely for this reason that tuning error normalization in one form or another has been so widely utilized in parameter adaptive control. On the other hand, it has recently been observed that, in spite of their relative complexity, algorithms not using error normalization offer several clear advantages over algorithms which do. We propose to further develop the principles of normalization- free adaptive control, to study the capabilities of normalization-free algorithms, and to compare their performance with the performance of algorithms of the normalization- dependent type.