Exponential parameter and tracking error convergence guarantees for adaptive controllers without persistency of excitation

In Model Reference Adaptive Control (MRAC) the modeling uncertainty is often assumed to be parameterized with time-invariant unknown ideal parameters. The convergence of parameters of the adaptive element to these ideal parameters is beneficial, as it guarantees exponential stability, and makes an online learned model of the system available. Most MRAC methods however, require Persistent Excitation (PE) of the states to guarantee that the adaptive parameters converge to the ideal values. Enforcing PE may be resource intensive and often infeasible in practice. This paper presents theoretical analysis and illustrative examples of an adaptive control method that leverages the increasing ability to record and process data online by using specifically selected and online recorded data concurrently with instantaneous data for adaptation. It is shown that when the system uncertainty can be modeled as a combination of known non-linear bases, simultaneous exponential tracking and parameter error convergence can be guaranteed if the system states are exciting over finite intervals such that rich data can be recorded online; PE is not required. Furthermore, the rate of convergence is directly proportional to the minimum singular value of the matrix containing online recorded data. Consequently, an online algorithm to record and forget data is presented and its effects on the resulting switched closed loop dynamics are analyzed. It is also shown that when radial basis function Neural Networks (NNs) are used as adaptive elements, the method guarantees exponential convergence of the NN parameters to a compact neighborhood of their ideal values without requiring PE. Flight-test results on a fixed wing unmanned aerial vehicle demonstrate the effectiveness of the method.