Asymptotic Linearity of Optimal Control Modification Adaptive Law with Analytical Stability Margins

Optimal control modification has been developed to improve robustness to model-reference adaptive control. For systems with linear matched uncertainty, optimal control modification adaptive law can be shown by a singular perturbation argument to possess an outer solution that exhibits a linear asymptotic property. Analytical expressions of phase and time delay margins for the outer solution can be obtained that allows a free design parameter of the adaptive law to be selected to satisfy stability margins. Various modifications were developed to increase robustness of MRAC by adding damping to the adaptive law. Two well-known modifications in adaptive control are the s -modification 15 and e- modification. 17 These modifications have been used extensively in adaptive control. This paper introduces a new adaptive law based on an optimal control formulation to minimize the L2 norm of the tracking error. The optimality condition results in a damping term proportional to the persistent excitation. The analysis shows that the optimal control modification can allow fast adaptation with a large adaptive gain without causing high-frequency oscillations and can provide improved stability robustness while preserving the tracking performance. One of the features of this adaptive law is its asymptotic linear property for systems with linear matched uncertainty as exhbited by a singular perturbation argument. Due to the asymptotic linear property, the optimal control modification exhibits a linear mapping between the input and output. As a result, it is possible to compute analytically stability margins for this adaptive control scheme. With the use of the gradient projection operator, a free design parameter of the optimal control modification can be designed to satisfy stability margins for its asymptotic solution. Simulation results demonstrate the effectiveness of the method.