We provide conditions under which trajectory outcomes in mechanical systems subject to unilateral constraints depend piecewise-differentiably on initial conditions, even as the sequence of constraint activations and deactivations varies. This builds on prior work that provided conditions ensuring existence, uniqueness, and continuity of trajectory outcomes, and extends previous differentiability results that applied only to fixed constraint (de)activation sequences. We discuss extensions of our result and implications for assessing stability and controllability.

Light detection and ranging (LIDAR) systems can be used to provide wind inflow information to a wind turbine controller before the wind reaches the turbine. Both fatigue and extreme load reduction are possible as a result; in this research, we propose a LIDAR-based controller designed to prevent emergency shutdowns caused by rotor overspeed. This switching controller consists of a disturbance accommodating control (DAC)-based baseline controller and a different DAC linearized about a reduced generator speed for extreme events, also referred to as an extreme event controller. Switching between the controllers was performed using linear interpolation over various transition times, depending on how early the extreme event could be detected. If a gust of wind is detected using LIDAR measurements evaluated by a one-sided cumulative summation algorithm, a relatively long transition time can be used. Switching can also be based on a large output estimation error, y , in which case the transition time is shorter. Once the extreme event passed, control is switched from the extreme event controller back to the baseline DAC. This switching controller resulted in fewer overspeeds when compared with the modified baseline controller, which is a gain scheduled DAC. By preventing overspeeds, the switching controller increased the mean power the wind turbine generated over a simulated 10 min period.

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When limbs are decoupled, we find that trajectory outcomes in mechanical systems subject to unilateral constraints vary differentiably with respect to initial conditions, even as the contact mode sequence varies. OrganizationWe begin in Sec. 2 by specifying the class of dynamical systems under consideration, namely, mechanical systems subject to unilateral constraints. Sec. 3 imposes conditions on the system dynamics and trajectories that enable us in Sec. 4 to report that trajectories vary differentiably with respect to initial conditions, even as the contact mode sequence varies. d k=1 (D k M m + D m M k − D M km ) [JK13, Eqn. 30]. 3 We let 2 n = {J ⊂ {1, . . . , n}} denote the power set (i.e. the set containing all subsets) of {1, . . . , n}.

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