Towards replacing lyapunov’s “direct” method in adaptive control of nonlinear systems

Abstract

In adaptive nonlinear control Lyapunov’s 2nd or “Direct” method became a fundamental tool in control design due to the typical practical difficulties viz. a) most of the control problems do not have closed analytical solutions; b) from numerical calculations “well behaving within a finite period” the stability cannot be taken for granted. AccordingtoLyapunov, guaranteeing negative time-derivative of the Lyapunov function by relatively simple estimations the stability of the solution can theoretically be guaranteed. However, finding an appropriate Lyapunov function to a given problem is rather an “art” that cannot algorithmically be automated. Adaptivity normally requires slow tuning of numerous model parameters. This processis sensitive to unknown external disturbances, and the tuning ruleis determined by numerous other, more or less arbitrary “adaptive control parameters”.Furthermore making the necessaryestimations is a laborious, tedious work that normally results in “very strange conditions” to be met for guaranteeing stability of the solution. In the present paper the application of“Robust Fixed Point Transformations” is proposed instead of the Lyapunovtechnique.Itcan find the proper solution without any parameter tuning and depends on the setting only ofthree“adaptive control parameters”. As application example direct control of a“SingleInput’Single Output (SISO)” system, and a novel version of the “Model Reference Adaptive Control (MRAC)” of a “Multiple Input’Multiple Output(MIMO)”system is presented. Since this method cannot automatically guaranteeglobal stability, as a novelty, a possible adaptive tuning of one of the adaptive control parameters is proposed for SISO systems to keep the control within the local basin of attraction of the proper convergence. Its operation ispresentedvia simulations at first time in this paper.