A sampled-data extremum-seeking approach for accurate setpoint control of motion systems with friction

Proportional–Integral–Derivative-Based Learning Control for High-Accuracy Repetitive Positioning of Frictional Motion Systems

“…In the remainder of this article, we opt for a linear spline basis function parameterization of k i (t), as illustrated in Example 2, as it yields a continuous control signal. In contrast, the steplike basis function parameterization, as illustrated in Example 1 and used in [24], results in discontinuities in the control signal, risking excitation of high-frequency system dynamics, which is also a well-known problem in reset control and impulsive control strategies developed for the control of frictional systems.…”

“…In particular, k i (t) then satisfies k i ≤ k i (t) ≤ k i , with k i = −0.2 × 10 8 , and k i = 1.2×10 8 . The augmented objective function is then given by Q(u) := Q(u) + μB(u) (24) with Q(u) as in (14), μ = 1 × 10 −4 the barrier parameter, and the logarithmic barrier function B given by…”

“…The dependence of the achievable setpoint accuracy, captured by the performance metric Q in (24) to be minimized, on the vector u is depicted by means of an measured input-output mapping Q in Fig. 6.…”

“…2) Time-Domain Results Obtained by Extremum Seeking: Consider Fig. 7, which shows the measured augmented performance cost Q(u k ) as in (24) and the corresponding vector of parameters u k as determined by the extremum-seeking controller, as a function of the controller updates, starting with initial parameter vector u 0 = [0.85, 0.175] • 10 8 . Moreover, Fig.…”

“…The third contribution is an experimental case study on an industrial high-precision motion stage of an electron microscope. This article builds upon preliminary results presented in [24], which contains a particular discontinuous integrator gain design and merely simulation examples. In addition to [24], this article contains a more generic controller structure, a novel parameterization of the integrator gain that leads to a continuous control signal (which is highly desirable in practice), a formal analysis of the existence of a static input-output map of the closed-loop system (allowing for the use of ESC), and an extensive experimental case study.…”