Transient performance and robustness of direct adaptive control

It is shown that a discrete adaptive controller with a fixed gain gradient estimator can be applied to linear systems that are undermodeled provided that the mismatch is small and the estimated parameters are projected into a compact convex set which contains stabilizing control parameters and excludes the possibility of dividing with small numbers. The latter implies that the sign and a lower bound for the high-frequency gain of the modeled part of the system are known. The performance of the adaptive system is related to the size of the external perturbations, and the ideal result as these tend to zero is obtained. The main conclusion is that while convergence, stability, and robustness may be achieved, the transient performance may not be acceptable and the gradient estimator should probably not be used unless modifications are added. Design guidelines are established. The results extend readily to indirect adaptive laws like pole assignment, linear quadratic optimal, and predictive controllers