Multi-Rate Control Design Under Input Constraints via Fixed-Time Barrier Functions

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.

Finite- and Fixed-Time Nonovershooting Stabilizers and Safety Filters by Homogeneous Feedback

“…2) Let us show that the homogeneous norm • d is a Lyapunov function of the closed-loop system (1), (37). Indeed, using the formula (11) and the identities d(s)A = e s d(s)A, d(s)B = e s B, ∀s ∈ R we derive…”

“…a) Finite-and fixed-time safety: Since the notion of fixed-time barrier functions has already appeared in the literature [11], let us explain what fixed-time safety (FxTSf) means to us. While the precise meaning of this property and the weaker finite-time safety (FnTSf) property will become clear in the statements of Theorems 3 and 2, respectively, here we only give a descriptive, intuitive definition.…”

“…Such a control can be employed in a "safety filter" framework to override a potentially unsafe nominal controller [1]. Nonovershooting and safety controllers have been designed for both linear [26], [23], [8] and nonlinear plants [19], [22], [11]. Inspired by robust non-overshooting control under deterministic disturbances in [19], i.e., by stabilization to an equilibrium at the barrier along with input-to-state safety (ISSf), mean-square stabilization of stochastic nonlinear systems to an equilibrium at the barrier, along with a guarantee of non-violation of the barrier in the mean sense, is solved in [21].…”

Simultaneous Lane-Keeping and Obstacle Avoidance by Combining Model Predictive Control and Control Barrier Functions

“…Zhe et al [14] design a CBF-CLF hard constraint for the MPC with proven recursive feasibility, producing stability with safety but without computational/feasibility considerations. Using a multi-rate formulation, in [15] a MPC planner is designed in conjunction with a CBF-QP formulation under the assumption that an ICCBF exists. For OA, the authors in [16] propose a continuoustime CBF as an additional state constraint for a discrete-time nonlinear MPC.…”

Control barrier function based attack-recovery with provable guarantees

“…When the system undergoes a single period of attack (i.e., t 2 1 = ∞), Assumption 3 can be relaxed by requiring the left-hand side of (16) to be upper-bounded by a positive constant that is inversely proportional to the maximum length of the attack T . In case of multiple attacks, it is possible to use the same relaxation by using the notion of periodic safety of the set S c with respect to the set S with time T na (see [14]), where…”