Dynamical systems with active singularities of elastic type: A modeling and controller synthesis framework

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, time-rescaling 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.