Robustness of adaptive control algorithms in the presence of unmodeled dynamics

This paper reports the outcome of an exhaustive analytical and numerical investigation of stability and robustness properties of a wide class of adaptive control algorithms in the presence of unmodeled dynamics and output disturbances. The class of adaptive algorithms considered are those commonly referred to as model-reference adaptive control algorithms, self-tuning controllers, and dead-beat adaptive controllers; they have been developed for both continuous-time systems and discrete-time systems. The existing adaptive control algorithms have been proven to be globally assymptotically stable under certain assumptions, the key ones being (a) that the number of poles and zeroes of the unknown plant are known, and (b) that the primary performance criterion is related to good command following. These theoretical assumptions are too restrictive from an engineering point of view. Real plants always contain unmodeled high-frequency dynamics and small delays, and hence no upper bound on the number of the plant poles and zeroes exists. Also real plants are always subject to unmeasurable output additive disturbances, although these may be quite small. Hence, it is important to critically examine the stability robustness properties of the existing adaptive algorithms when some of the theoretical assumptions are removed; in particular, their stability and performance properties in the presence of unmodeled dynamics and output disturbances. A unified analytical approach has been developed and documented in the recently completed Ph.D. thesis by Rohrs [15] that can be used to examine the class of existing adaptive algorithms. It was discovered that all existing algorithms contain an infinite-gain operator in the dynamic system that defines command reference errors and parameter errors; it is argued that such an infinite gain operator appears to be generic to all adaptive algorithms, whether they exhibit explicit or implicit parameter identification. The practical engineering consequences of the existence of the infinite-gain operator are disastrous. Analytical and simulation results demonstrate that sinusoidal reference inputs at specific frequencies and/or sinusoidal output disturbances at any frequency (including d.c.) cause the loop gain of the adaptive control system to increase without bound, thereby exciting the (unmodeled) plant dynamics, and yielding an unstable control system. Hence, it is concluded that none of the adaptive algorithms considered can be used with confidence in a practical control system design, because instability will set in with a high probability.