Adaptive Control? But is so Simple!

In a recent paper, a few pioneers of adaptive control review the classical model reference adaptive control (MRAC) concept, where the designer is basically supposed to conceive a model of the same order as the (possibly very large) plant, and then build an adaptive controller such that the plant is stable and ultimately follows the behavior of the model. Basically, adaptive control methods based on model following assume full-state feedback or full-order observers or identifiers. These assumptions, along with supplementary prior knowledge, allowed the first rigorous proofs of stability with adaptive controllers, which at the time was a very important first result. However, in order to obtain this important mathematical result, the developers of classical MRAC took the useful scalar Optimal Control feedback signal and made it into an adaptive gain-vector of basically of the same order as the plant, which again had to multiply the plant state-vector in order to finally end with another scalar adaptive control feedback signal. It is quite known today, however, what happens when this requirement is not satisfied, and when “unmodeled dynamics” distorts the controller based on these ideal assumptions. Even though much effort has been invested to maintain stability in spite of so-called “unmodeled dynamics,” in some applications, such as large flexible structures and other real-world applications, even if one can assume that the order of the plant is known, one just cannot implement a controller of the same order as the plant (or even a “nominal” or a “dominant” part of the plant), before even mentioning the complexity of such an adaptive controller. Without entering the argument around their special reserve in relation to claimed efficiency of the particular L1-Adaptive Control methodology, this paper first shows that, after the first successful proof of stability and even under the same basic full-state availability assumption, the adaptive control itself can be reduced to just one adaptive gain (which multiplies one error signal) in single-input-single-output (SISO) systems and, as a straightforward extension, an m*m gain matrix in an m-input-m-output (MIMO) plant. Not only is stability not affected, but actually the simplified scheme also gets rid of most seemingly “inherent” problems of the adaptive control represented by classical MRAC. Moreover, proofs of stability have all been based on the so-called Barbalat’s lemma which seems to require very strict uniform continuity of signals. The apparent implications are that any discontinuity, such as square-wave input commands or just some occasionally discontinuous disturbance, may put stability of adaptive control in danger, without even mentioning such things as impulse response. Instead, based on old yet amazingly unknown extensions of LaSalle’s Invariance Principle to nonautonomous nonlinear systems, recent developments in stability analysis of nonlinear systems have mitigated or even eliminated most apparently necessary prior conditions, thus adding confidence in the robustness of adaptive scheme in real world situations.