Using Persistent Excitation with Fixed Energy to Stabilize Adaptive Controllers and Obtain Hard Bounds for the Parameter Estimation Error

We show that persistent excitation with fixed and finite energy can be used as a tool to stabilize adaptive control algorithms and obtain hard bounds for the parameter estimation errors. Two important instability problems in certainty equivalence adaptive control are solved by applying excitation. The first instability is parameter drift due to an unstable manifold which appears when the excitation level is not high enough. This problem has previously been studied using averaging and local analysis. Our results are global. The second instability is numerical, and due to a division with zero in the adaptive law. In the paper we show that the system parameters are estimated when the level of excitation is sufficiently high relative to the magnitudes of the external disturbances and the unmodelled dynamics. It follows that the singularity problem only can occur during the transient. The consequence of this is that we can implement a direct adaptive controller without requiring knowledge of the sign of the high frequency gain. The approach can be generalized to more complex adaptive laws and this, together with the fact that we obtain hard bounds for the parameter estimation error opens up for the possibility of designing robust controllers that are adaptive.