Estimated optimality and robustness of nonlinear adaptive control systems under bounded disturbances

TL;DR

This paper develops a framework for achieving suboptimal yet robust performance of adaptive control systems under bounded disturbances using speed-gradient (SG) algorithms. It formalizes the notion of estimated optimality within nonlinear and nonlinearly parametrized plants, introducing robustification via negative parametric feedback and a class of problems $\Xi_1(\Delta_*)$ to derive explicit bounds. The main result, Theorem 1, proves ultimate boundedness and estimated $\varepsilon$-optimality under suitable convexity and growth conditions, with a deadzone and saturated feedback variant to improve practical performance. An illustrative example on adaptive passification for linear systems demonstrates how the methodology yields explicit bounds on the steady-state error, highlighting the practical relevance for robust adaptive control in the presence of disturbances.

Abstract

The problem of suboptimality under bounded disturbances for the adaptive systems based on speed-graadient approach is discussed. A formulation of the estimated optimality of nonlinear nonlinearly parametrized adaptive control systems is given and sufficient conditions for the estimated optimality in a specified uncertainty class are described for algorithms robustified by negative parametric feedback. A special case of passification based adaptive control for linear time invariant systems is studied to demonstrate application of the paper results..