Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A multiobjective approach to robust predictive control barrier functions for discrete-time systems

Alexandre Didier, Melanie N. Zeilinger

TL;DR

The paper addresses robust convergence to a target set for discrete-time nonlinear systems while optimizing a primary objective under bounded disturbances. It develops a robust Predictive Control Barrier Function (PCBF) using an RPI-based constraint tightening to guarantee recursive feasibility and robust stability, then extends to a single, multiobjective MPC by enforcing a decrease constraint relative to a warmstart PCBF solution with a tunable hyperparameter $c_\alpha$. This single-step approach reduces conservativeness and computational demand compared to the original two-step PCBF, while maintaining stability guarantees. Numerical demonstrations on a linear space rendezvous and a nonlinear lane-changing scenario illustrate substantial fuel savings and faster computation with the multiobjective formulation, validating the practical impact for safety-critical control with learning-based or human-in-the-loop components.

Abstract

We present an optimisation-based approach to ensure robust asymptotic stability stability of a desired set in the state space of nonlinear dynamical systems, while optimising a general control objective. The approach relies on the decrease of a robust predictive control barrier function (PCBF), which is defined as the optimal value function of a slack minimisation problem with respect to the target set. We show continuity of the proposed robust PCBF, allowing the introduction of a decrease constraint in the control objective minimisation. The PCBF decrease is given with respect to a warmstart value based on a feasible solution at the prior time step. Thereby, the control objective can be optimised while ensuring robust asymptotic stability of the target set. We demonstrate the effectiveness of the proposed formulation on a linear space rendezvous and nonlinear lane changing problem.

A multiobjective approach to robust predictive control barrier functions for discrete-time systems

TL;DR

The paper addresses robust convergence to a target set for discrete-time nonlinear systems while optimizing a primary objective under bounded disturbances. It develops a robust Predictive Control Barrier Function (PCBF) using an RPI-based constraint tightening to guarantee recursive feasibility and robust stability, then extends to a single, multiobjective MPC by enforcing a decrease constraint relative to a warmstart PCBF solution with a tunable hyperparameter . This single-step approach reduces conservativeness and computational demand compared to the original two-step PCBF, while maintaining stability guarantees. Numerical demonstrations on a linear space rendezvous and a nonlinear lane-changing scenario illustrate substantial fuel savings and faster computation with the multiobjective formulation, validating the practical impact for safety-critical control with learning-based or human-in-the-loop components.

Abstract

We present an optimisation-based approach to ensure robust asymptotic stability stability of a desired set in the state space of nonlinear dynamical systems, while optimising a general control objective. The approach relies on the decrease of a robust predictive control barrier function (PCBF), which is defined as the optimal value function of a slack minimisation problem with respect to the target set. We show continuity of the proposed robust PCBF, allowing the introduction of a decrease constraint in the control objective minimisation. The PCBF decrease is given with respect to a warmstart value based on a feasible solution at the prior time step. Thereby, the control objective can be optimised while ensuring robust asymptotic stability of the target set. We demonstrate the effectiveness of the proposed formulation on a linear space rendezvous and nonlinear lane changing problem.

Paper Structure

This paper contains 9 sections, 2 theorems, 27 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Preliminaries on Predictive Control Barrier Functions
  4. Robust predictive control barrier function
  5. Multiobjective Predictive Control Barrier Function
  6. NUMERICAL EXAMPLE
  7. Linear Space Rendezvous
  8. Lane Changing Example
  9. CONCLUSION

Key Result

Theorem 1

Let Assumptions ass:prel, ass:RPI and ass:rTermCBF hold. The optimisation problems eq:rPCBF and eq:rPCBFSF are recursively feasible and it holds that the set $\mathcal{S}\coloneqq\{x\in\mathbb{R}^n\mid h(x)\leq 0\}$ is robust asymptotically stable within $\mathcal{D}\coloneqq\{x\in\mathbb{R}^n \mid

Figures (5)

  • Figure 1: Closed-loop state evolution for a space rendezvous example using the proposed robust PCBF formulation in Algorithm \ref{['alg:rpcbf']} (red) and the proposed multiobjective formulation in Algorithm \ref{['alg:online']} (blue). Using both approaches, the states are guaranteed to converge to the constraints (black). An advantage of the multiobjective approach is to relax the required rate of convergence to the constraint set using the hyperparameter $c_\alpha\in[0,1)$.
  • Figure 2: Value of the predictive control barrier function for the robust approach in Algorithm \ref{['alg:rpcbf']} (red) and its upper bound for the multiobjective approach in Algorithm \ref{['alg:online']} (blue) The hyperparameter $c_\alpha$ allows tuning the required rate of decrease of $h$.
  • Figure 3: Closed-loop simulation of a human-operated vehicle with the proposed PCBF formulations, where the human driver aims to actively steer away from the desired lane.
  • Figure :
  • Figure :

Theorems & Definitions (4)

  • Definition 1: Control Barrier Function wabersich2022
  • Theorem 1
  • Theorem 2
  • Remark 1