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Abstract
In this paper, we introduce the notion of periodic safety, which requires that the system trajectories periodically visit a subset of a forward-invariant safe set, and utilize it in a multi-rate framework where a high-level planner generates a reference trajectory that is tracked by a low-level controller under input constraints. We introduce the notion of fixed-time barrier functions which is leveraged by the proposed low-level controller in a quadratic programming framework. Then, we design a model predictive control policy for high-level planning with a bound on the rate of change for the reference trajectory to guarantee that periodic safety is achieved. We demonstrate the effectiveness of the proposed strategy on a simulation example, where the proposed fixed-time stabilizing low-level controller shows successful satisfaction of control objectives, whereas an exponentially stabilizing low-level controller fails.
Highlights
Constraints requiring the system trajectories to evolve in some safe set at all times while visiting some goal set(s) are common in safety-critical applications
Constraints pertaining to the convergence of the trajectories to certain sets within a fixed time often appear in time-critical applications, e.g., when a task must be completed within a given time interval
Most popular approaches on the control synthesis under such specifications include quadratic programming techniques, where the safety requirements are encoded via control barrier functions (CBFs) and convergence requirements via control Lyapunov functions (CLFs), see e.g. [1], [2], or via one function that encodes both the safety and convergence requirements [3], [4]
Summary
Constraints requiring the system trajectories to evolve in some safe set at all times while visiting some goal set(s) are common in safety-critical applications. As argued in the recent article [7], myopic control synthesis approaches relying solely on QPs are susceptible to infeasibility To circumvent this issue, combining a high-level planner. We combine the concepts of FxTS Lyapunov functions [5] and CBFs [1] to define the notion of fixed-time barrier functions We use it in a provably feasible QP, guaranteeing fixed-time convergence to a neighborhood of the reference trajectory from a region of attraction under input constraints. Compared to [12], we limit the rate of change for the planned trajectory so that the low-level controller is able to track the resulting reference trajectory within a predefined error bound.
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