Prescribed-Time Stabilization and Robust Safety

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In engineering applications ranging from power grids to robotics, the paradigm of using constant gains in feedback systems hasn’t changed since the pioneering days of Nyquist and Bode before WWII. Constant gains produce easy-to-analyze exponential rates of setpoint regulation but underutilize the actuation capability: starting far from the setpoint, the control signal begins large but then gets needlessly small (“lazy”) close to the setpoint. In many contemporary applications, more than the infinite-time exponential convergence is desired. Convergence in finite, user-prescribed time is the goal, along with the goal of smooth settling at the setpoint. From semiconductor manufacturing to energy systems, “rapid and smooth” transitions are the way of the future for control technology. The PI recently introduced time-varying feedback laws, with gains that grow over time. Such growing gains prevent the control signal from decaying prematurely and achieve ``prescribed-time stabilization’’ (PTS), in time that is independent of the initial condition. This project advances this idea in a number of directions. In addition to transformational advances in asymptotic performance, it pioneers the methodology of “prescribed-time safety” (PTSf), which enables control systems, like those in driverless cars, to be less conservative in avoiding collisions because, as human drivers know, excessive conservativeness incentivizes a breach of safety by others. The project will establish robustness of PTS feedback laws to measurement noise, design PTS controllers for stochastic nonlinear systems, and develop PTSf controllers for operation under disturbances. In particular, the research will target the following challenges: (1) systems whose safety is characterized by control barrier functions (CBFs) of relative degree higher than one, such as position constraints under force inputs, (2) design of safety filters which combine backstepping and quadratic programming (QP) approaches, and (3) the relaxation of the concept of safety from the infinite-time notion of ``safe forever'' (too conservative) to the notion of “safe over a user-prescribed time interval,” i.e., PTSf. Under disturbances of unknown bound, more will be achieved than with the current robust safety methods, which let the disturbance violate the barrier, by a tolerable amount, for all time. With PTSf, a complete disturbance rejection will be achieved by the terminal time. A system under disturbance, possibly even starting in the unsafe set, will be ``rescued to safety’’ by a time prescribed by the user, independent of how deeply in the unsafe set the system starts and how large the disturbance is. Designs will be experimentally tested on the 7 degree of freedom Baxter robot.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.