Persistent excitation in adaptive systems

Stability analysis of PE systems via Steklov's averaging technique

“…Up to now, this fact was established via Lyapunov-like arguments similar to [11] exemplified by, for example, Lemma 5 in [4]; see also Lemma 1 in [12]. However to the best of our knowledge, no Lyapunov function or functional is available so far that proves exponential stability of this system, though a design of such a functional might be of self-interest when treating, for example, system identification, adaptive control, or robustness issue.…”

Uniform Global Asymptotic Stability of a Class of Adaptively Controlled Nonlinear Systems

“…Persistency of excitation (PE) has been linked to the asymptotic stability of adaptive systems [13]. PE establishes that a necessary (and sometimes sufficient) condition for parameter identification is that the reference trajectory be sufficiently rich so that the regressor satisfies a PE inequality [3] along the reference trajectory.…”

“…For example, this property ensures stability in the face of persistent disturbances [2] and provides rate of convergence information [12]. In general, PE is neither necessary nor sufficient for uniform asymptotic stability [13].) One notable example is the nonlinear dynamics of robot manipulators, where PE ensures asymptotic parameter error convergence under the Slotine-Li adaptive controller [15].…”

Combined Adaptive Controller for UAV Guidance

“…The so-called persistent excitation condition is well known condition of convergence of the parameter estimates to their true values [7,10]. Fulfillment of this requirement is an open question for the considered system, because the UAV controlling input (the rudder deflection) is produced by the autopilot as a function on current UAV position and the command (reference) signal.…”