Section: New Results

Certifying good performance for adaptive systems

Participants : Frédéric Mazenc, Michael Malisoff [Louisiana State University] .

The usual adaptive control problem is to design a controller that forces all trajectories of the system to track a prescribed trajectory, while keeping the estimator of the unknown constant parameter vector bounded. We studied the important and more difficult adaptive tracking and estimation problem of simultaneously (1) forcing the trajectories of the system to track a given trajectory and (2) identifying the parameter vector. This problem was known to be solvable when the regressor satisfies a persistency of excitation condition, but the known results did not provide a strict Lyapunov function for the augmented error dynamics and so could not certify good performance such as ISS with respect to uncertainty added to the controller.

Our main result from [118] covers adaptively controlled first-order nonlinear systems that satisfy the persistency of excitation condition and are affine in the unknown parameter vector. Our contribution consists in particular in constructing global strict Lyapunov functions for the augmented error dynamics In [28] , we extended [118] to adaptive tracking for nonlinear systems in feedback form with multiple inputs and unknown high-frequency gains multiplying the controllers. The control gains must be identified as part of the control objective. High-frequency gains are important for electric motors, flight dynamics, and robot manipulators. We used a persistency of excitation condition that again ensured tracking and parameter identification and led to the explicit construction of a global strict Lyapunov function for the closed loop augmented error dynamics. The strict Lyapunov function was key to proving integral ISS with respect to time varying uncertainties added to the unknown parameters.