Extremum seeking-based tracking for unknown systems with unknown control directions

Abstract

Abstract-We develop controllers which perform trajectory tracking for a class of unknown linear and nonlinear systems. In the linear case we work under the assumption that the timevarying input vector, which is otherwise unknown, satisfies a persistency of excitation condition over a sufficiently short window. In the multi-input nonlinear case the square of the unknown time-varying input matrix has a known lower bound. Our design does not guarantee perfect regulation to the desired trajectory, but ensures semiglobal uniform ultimate boundedness of the error system and prevents large overshoots even when the initial estimate of the control direction is wrong. The stability analysis is inspired by the approach in a recent work by Dürr, Stankovic, and Johansson, which combines a Lie bracket second-order averaging result of Gurvits and Li with an extension of the perturbation theory semiglobal practical stability result of Moreau and Aeyels.

Non-C2 Lie Bracket Averaging for Nonsmooth Extremum Seekers

“…If, for all d > 0; wðtÞ is D-CT relative to w e ðtÞ on D ¼ R n nBð0; dÞ ¼ fx 2 R n : jxj ! dg, then if the origin is a GUAS equilibrium point of system (12), it is also a e-SPUAS equilibrium point of system (13). We choose D 2 2 ð0; minfD; c 1 gÞ, because the trajectory wðtÞ is D 2 -CT relative to w e ðtÞ on D, forT > T there existsê 1 such that for all e 2 ð0;ê 1 Þ, for all t 2 t 0 ; t 0 þT Â Ã for which kw e ðtÞk > d; kwðtÞ À w e ðtÞk < D 2 .…”

“…Recent work by D€ urr et al [6], where an innovative combination of certain Lie bracket-based averaging results of Gurvits and Li [7][8][9] was combined with results of Moreau and Aeyels [10], provided a technique for Lyapunov function based ES analysis. By employing a modified ES scheme for minimization of a known Lyapunov function for an unknown system, the problem of model-independent semiglobal exponential practical stabilization as well as ultimately bounded trajectory tracking for any linear time-varying single-input system was solved [11][12][13].…”

“…We then develop non-C 2 ES-based controllers which achieve the same stability results for the same family of systems as in Refs. [11][12][13]. Our newly developed controllers allow a system to settle at its equilibrium point if it is ever reached, removing the always present perturbation of typical ES schemes.…”

“…once the system's trajectory re-enters D. This allows us to develop non-C 2 controllers which achieve the same stability results as in Refs. [11,12], but have the advantage that control effort settles to zero when the system reaches equilibrium.…”

A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies

“…Such problems arise, for example, when a robot has to follow a moving target (tracking problem) and only the distance (but not the relative position) to the moving target can be measured. Although gradient-free control laws for the tracking of a moving target have been previously considered (see, e.g., [3,4,13,18,19,23,24,25,31,32,33]), the main advantage of the proposed family of extremum seeking laws is, on the one hand, the high flexibility in designing the control functions such that they meet further specifications like input constraints, and, on the other hand, the family of control functions ensures rigorous stability and tracking properties.…”