Relay-based hybrid control of minimal-order mechanical systems with applications

Abstract

Relay-based hybrid control of minimal-order mechanical systems with applications. Automatica, Elsevier, 2018, 97, pp. AbstractThis work explores the potential of relay-based control on a one-degree-of-freedom nonlinear mechanical system, in the contexts of both sustaining and damping oscillations. For both cases we state our main results building upon a simple reset formulation (relay feedback) and providing intuitive basic equations from classical mechanics. With a more rigorous description following a hybrid system formalism, we establish then the global asymptotic stability of the corresponding (compact-set) attractors through hybrid Lyapunov tools. The aspects of sustaining and damping oscillation are seen as complementary, because they reduce to a suitable mirroring of the reset surface. Finally, we discuss two applications of our results to the case of a hopping mass and an automotive suspension.

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

“…This interplay between injected and dissipated energy can be proven to always lead to the occurrence of a nontrivial attractive hybrid periodic orbit by following similar steps to those reported in [23] (see also [53]). While a rigorous proof of this energy-based explanation of the hunting phenomenon is beyond the scope of this paper, the key mechanism behind it is that the injection of energy is always constant and equal to (F s − F ∞ ) 2 (due to the second line of (37)), whereas the dissipated energy during the flow phase is proportional to the value ofσ at the stick-to-slip transition.…”

“…Besides its intuitive relevance, function V is only a first step towards the proof of Theorem 1 given in [22], which requires nontrivial tools from nonsmooth analysis. This was a main motivation for introducing the hybrid extended model (23). Indeed, model (23) simplifies the Lyapunov characterization of the desirable behavior of solutions by way of introducing, in [21], an equivalent smooth version of function V , corresponding tō…”

“…In particular, as visible in the red curve of Figure 3, or in the lower diagram of Figure 17, as the magnitude of the velocity increases from zero, the magnitude of the friction force "weakens". Simulating model (23) it is possible to understand the net effect of the velocity weakening: at a stick-to-slip transition the control force exactly compensates for the static friction F s . Immediately after this transition the friction force u f decreases (due to the velocity weakening characteristic of Stribeck friction) and the control action dramatically overcompensates the friction.…”

“…The limiting shape, corresponding to the darkest curve in Figure 17, resembles a Coulomb friction contribution with amplitude F ∞ away from zero, but a larger value of static friction F s > F ∞ that must be overcome by the control input to exit a stick phase, thus causing a discontinuous drop of the friction force in Figure 18 at any stick-to-slip transition. This limiting phenomenon can be effectively modeled by adapting the hybrid automaton (23) with the flow mapf of (23d) replaced by the following selection:f…”

“…Instead, the quantity characterizing the stick and slip sets (q = 0 or |q| = 1, respectively) in (23f)-(23j) remains unchanged and equal to F s because in Figure 17 we clearly see that φ must overcome F s exit a stick phase (namely to compensate for the static component of the friction when v = 0). Simulating model (23), (36) we may further understand the net effect of the velocity weakening by inspecting the evolution of the following adaptation of the smooth function in (27), Figure 19: Evolution of the Lyapunov functionV in (37) along the solutions to (23), (36) and corresponding values ofq,φ andσ with the steepest (darkest) Stribeck function in Figure 17.…”