Dynamical Systems With Active Singularities of Elastic Type: A Modeling and Controller Synthesis Framework

Abstract

A new class of systems characterized by admitting impulsive control action within their singular motion phases, such as dimension changes, state discontinuities, and other irregularities, is introduced. This class, termed dynamical systems with active, or controlled, singularities, encompasses various applications with sensing and/or actuation ultra-fast in comparison with the natural system time scale, including mechanisms with impact-induced motion, power and sensor networks under faults, fast positioning devices, smart skins, and switching electronic circuits. The present work focuses on systems with impact-type singularities induced through system interaction with controlled, or active, state constraints. The latter are assumed to be characterized by parameter-dependent elastic-type constraint violation that vanishes in the limit as the parameter value tends to infinity. A physically well justified representation of this class of systems is proposed, but found to be not well suited for controller synthesis in the singular phase. To address this problem, two equations are derived -one describing controlled rigid impacts in auxiliary stretched time and compatible with the regular control design techniques, termed controlled infinitesimal dynamics equation, and another incorporating controlled impact dynamics in the infinitesimal form in terms of the shift operator along the trajectories of the first equation, termed the limit model. The latter is demonstrated to generate a unique isolated discontinuous system motion, i.e., to provide a tight and well-behaved description of the collision with rigid constraint. It is then shown that the corresponding paths generated by the original physically-based and the limit representations can be made arbitrarily close to each other uniformly except, possibly, in the vicinities of the jump points, by the appropriate choice of the value of the constraint violation parameter. This is shown to permit enforcing, for sufficiently large values of the parameter, the desired limit system behavior onto the original system by simply taking the control signals found through the limit representation, timerescaling them, and inserting the resulting signals directly into the original system. These features show that the procedure developed, in fact, provides a well-posed controller synthesis framework for the class of systems considered. Using this framework, a ball/racket rotationally controlled impact representation is developed, and on its basis, a singular phase control signal in the form of racket rotation velocity is designed, demonstrating that the soft racket provides the specified bounce-off angle increment under a noticeably lower racket rotation velocity and a longer rotation phase than the hard racket.Index Terms-Control of mechanical systems, controlled discrete transitions, generalized solutions, hybrid systems, impacts, impulsive control, nonsmooth analysis and synthesis, penalty method, singularities, systems with unilateral constraints.

Hybrid dynamical systems with controlled discrete transitions

“…As indicated in [4], the latter systems can be also viewed as a new class of hybrid systems characterized by impulsively controlled discrete transitions. In [4][5][6][7], and [16], a rigorous modeling and controller synthesis framework is developed for these systems under the full state accessibility. This framework is succinctly presented next, demonstrating the explicit representation of the controlled discrete transitions.…”

“…In [7], the limit system is obtained from (9)-(11) using a specially constructed topological map parametrized by µ -the space-time transformation of the form Fig. 4.…”

“…The map (12) and the system (14) are jointly referred to in [7] as the multi-scale motion representation of the original system (11). Introduce the equatioṅ y p (s) = F r p (x p (τ ), y v (s), τ ), y v (s) =F v (y p (s), y v (s), s, w τ (s), x p (τ ), τ ),…”

“…The emergence of systems with controlled discrete transitions can be traced to the introduction of systems with active mechanical constraints of elastic type in [3][4][5][6][7] and [15] in the context of impact systems. However, the full generality of this class of systems has not been realized until the recent work [32] and [31] on bumpless transfer, where controlled transitions have arisen in a completely different content of a combined numerical/physical dynamics.…”

Graph completions for impulsive feedback controls

“…The references [3,7,13,15,16,17] can only provide a partial sample. It is worth mentioning that graph completions have also been used in [11] and in [12] to define "nonconservative products" in connection with hyperbolic systems of PDEs.…”