Predictive Control Barrier Functions for Discrete-Time Linear Systems with Unmodeled Delays

TL;DR

The paper tackles enforcing state constraints in discrete-time systems with unknown relative degree caused by input delays or unmodeled dynamics. It introduces Predictive Control Barrier Functions (PCBFs) by extending the prediction horizon to produce a relative-degree-one CBF for a lifted system, yielding a single linear inequality that defines the safe set. The approach is developed for linear DT systems, with a clear procedure to compute $A_{\\ell_{ m h}}$ and $B_{\\ell_{ m h}}$, and validated through simulations on a delayed double integrator and a bicopter, showing constraint satisfaction when the horizon exceeds the delay and robustness to unmodeled inner dynamics. This work simplifies safety-filter design in discrete-time control and integrates naturally with MPC/LQR-based frameworks, offering practical benefits for real-time constraint enforcement.

Abstract

This paper introduces a predictive control barrier function (PCBF) framework for enforcing state constraints in discrete-time systems with unknown relative degree, which can be caused by input delays or unmodeled input dynamics. Existing discrete-time CBF formulations typically require the construction of auxiliary barrier functions when the relative degree is greater than one, which complicates implementation and may yield conservative safe sets. The proposed PCBF framework addresses this challenge by extending the prediction horizon to construct a CBF for an associated system with relative degree one. As a result, the superlevel set of the PCBF coincides with the safe set, simplifying constraint enforcement and eliminating the need for auxiliary functions. The effectiveness of the proposed method is demonstrated on a discrete-time double integrator with input delay and a bicopter system with position constraints.

Figures (7)

• Figure 1: Sampled data implementation of discrete-time controller $G_{\rm c}$ applied to a continuous-time system $G$ with input $u,$ state $x,$ and output $y.$ The function $f_{\rm cbf}$ implements the CBF by modifying the output of $G_{\rm c}$ for state constraint enforcement. All sample-and-hold operations are synchronous.
• Figure 2: Feasible command. Closed-loop response of the discrete-time double integrator in the case of a feasible command with a delay of $m = 1.$ The command is given by $r_k \equiv 5,$ and the state constraints are defined by $x_{1,k} \in [-8, \ 8]$ and $x_{2, k} \in [-0.5, \ 0.5]$ for all $k \ge 0.$ The responses in the cases with no CBF and with predictive CBF with $\gamma = 0.6, \ell_{\rm h} = 1$ and $\gamma = 0.6, \ell_{\rm h} = 3$ are shown. The red-shaded areas correspond to periods of time during which the constraints are violated.
• Figure 3: Infeasible command. Closed-loop response of the discrete-time double integrator in the case of an infeasible command with a delay of $m = 1.$ The command is given by $r_k \equiv 5,$ and the state constraints are defined by $x_{1,k} \in [-4, \ 4]$ and $x_{2, k} \in [-0.5, \ 0.5]$ for all $k \ge 0.$ The responses in the cases with no CBF and with predictive CBF with $\gamma = 0.6, \ell_{\rm h} = 1$ and $\gamma = 0.6, \ell_{\rm h} = 3$ are shown. The red-shaded areas correspond to periods of time during which the constraints are violated.
• Figure 4: Diagram of bicopter in vertical plane
• Figure 5: Inner-loop, outer-loop control architecture considered for the bicopter example.

Theorems & Definitions (4)

• Theorem III.1
• Remark III.1
• Proposition III.2
• Proposition IV.1