Hybrid PID control for transient performance improvement of motion systems with friction

Abstract

We present a novel reset control approach to improve transient performance of a PID-controlled motion system subject to friction. In particular, a reset integrator is applied to circumvent the depletion and refilling process of a linear integrator when the system overshoots the setpoint, thereby significantly reducing settling times. Moreover, robustness for unknown static friction levels is obtained. A hybrid closed-loop system formulation is derived, and stability follows from a discontinuous Lyapunov-like function and a meagre-limsup invariance argument. The working principle of the controller is illustrated by means of a numerical example.

To stick or to slip: A reset PID control perspective on positioning systems with friction

“…The results presented in this paper provide a unified and comprehensive overview of the research accomplishments reported in [22,17,18,21,15] and the preliminary works [16,19]. As compared to those works we provide here a unified development, highlighting the importance of building hybrid models comprising logic variables to allow for the construction of smooth or Lipschitz Lyapunov functions, in addition to including a novel understanding of the exponential convergence properties of certain solutions in the Coulomb friction case.…”

“…A second reason for using coordinates x in (16) is that these provide a simplified representation of the sets where solutions are in the stick phase (the intervals where the top plots of Figures 4 and 5 are flat, namely the intervals where v ≡ 0) or in the slip phase (the time intervals associated to the speed bumps in the middle plots of Figures 4 and 5). In particular, we may define…”

“…More specifically, the generalized controller state φ represents all the nonzero components of the control action at zero velocity (that is, the proportional and integral terms), and according to (20), the size of φ compared to (16), shown in the three top plots. The 3D plots in the middle of Figures 7 and 8 show the corresponding phase portraits and provide an insightful interpretation of the evolution of the solutions with respect to the attractor A in (18), which is represented as a dashed red segment.…”

“…The jump set D is expressed in (30c) in terms of x. The states φ and σ are not measurable in the case of an unknown mass m, as one can see from (16) and (7). However, even for an unknown mass m, we can define from (16) and 7the measurable states…”

“…we impose two different types of jump laws on the (generalized) controller state φ, whose net effect is to ensure that the constraint σφ ≥ kp ki σ 2 is always satisfied. We note that this constraint is equivalent to (s − r)x c ≥ 0 in the original coordinates z (see (6) and (16)), namely that the integrator state x c always points in the same direction as the position error s − r. As a consequence, we also impose bvφ ≥ 0 so that even when σ = 0, bv and φ must have matching signs.…”