2005/06/21 by Tyukin Ivan, Ivan, Tyukin, Danil Prokhorov +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #93C10 #93C40 #93C41 #93D25 #Adaptive Control of Nonlinear Systems #Chaos control and synchronization #Control Systems and Identification #Dynamical Systems (math.DS) #FOS: Mathematics #Optimization and Control (math.OC) #math.DS #math.OC #msc:93C10 #msc:93C40 #msc:93C41 #msc:93D25
paper · pdf · doi:10.48550/arxiv.math/0506419
40 pages,4 figures, 1 table
arxiv created 2005/06/21 · openalex publication_date 2005/06/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a technique for the design and analysis of adaptation algorithms in dynamical systems. The technique applies both to systems with conventional Lyapunov-stable target dynamics and to ones of which the desired dynamics around the target set is nonequilibrium and in general unstable in the Lyapunov sense. Mathematical models of uncertainties are allowed to be nonlinearly parametrized, smooth, and monotonic functions of linear functionals of the parameters. We illustrate with applications how the proposed method leads to control algorithms. In particular we show that the mere existence of nonlinear operator gains for the desired dynamics guarantees that system solutions are bounded, reach a neighborhood of the target set, and mismatches between the modeled uncertainties and uncertainty compensator vanish with time. The proposed class of algorithms can also serve as parameter identification procedures. In particular, standard persistent excitation suffices to ensure exponential convergence of the estimated to the actual values of the parameters. When a weak, nonlinear version of the persistent excitation condition is satisfied, convergence is asymptotic. The approach extends to a broader class of parameterizations where the monotonicity restriction holds only locally. In this case excitation with oscillations of sufficiently high frequency ensure convergence.