Inferring Global Exponential Stability Properties using Lie-bracket Approximations

Abstract

In the present paper, a novel result for inferring uniform global, not semi-global, exponential stability in the sense of Lyapunov with respect to input-affine systems from global uniform exponential stability properties with respect to their associated Lie-bracket systems is shown. The result is applied to adapt dither frequencies to find a sufficiently high gain in adaptive control of linear unknown systems, and a simple numerical example is simulated to support the theoretical findings.

Figures (2)

• Figure 1: Solutions $x_\omega(t;t_0,x_0)$ (\ref{['plot:trajectory-approximation:ai']})of \ref{['eqn:es-system']},solution $\bar{x}(t;t_0,x_0)$ (\ref{['plot:trajectory-approximation:lbs']})oftheassociatedLie-bracketsystem \ref{['eqn:lbs']},errorboundinterval $D\, \left\lVert x_0 \right\rVert$ (\ref{['plot:trajectory-approximation:error-interval']})overapproximationhorizon $\interval[]{t_0}{{t_0+t_f}}$andinsinuatedrestartat${t_0+t_f}$andcontinuationforsolutions $x_\omega(t;{t_0+t_f},x_\omega(t_0+t_f;t_0,x_0)),\bar{x}(t;{t_0+t_f},x_\omega({t_0+t_f};t_0,x_0))$andtheirrespectiveerrorboundinterval $D\, \left\lVert x(t_f;t_0,x_0) \right\rVert$ (\ref{['plot:trajectory-approximation:error-intervalc']}).
• Figure 2: Simulationresultsshowthesolution $x_{\omega}(t;t_0,x_0)$ (\ref{['plot:exponential-convergence:ai']})from \ref{['eqn:example-system']},thesolution $\bar{x}(t;t_0,x_0)$ (\ref{['plot:exponential-convergence:lbs']})oftheassociatedLie-bracketsystem \ref{['eqn:example:lbs']}foracommoninitialcondition$x_0$,theconservativeanalyticalupperbound $\alpha e^{-\beta t}$ (\ref{['plot:exponential-convergence:upperbound-analytical']})aswellastheempiricalbounds $\alpha e^{-\bar{\beta}t}$ (\ref{['plot:exponential-convergence:upperbound-emprical']})forparametersfrom\ref{['sec:simulation-example']}.