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Enhancing Feasibility and Safety of Nonlinear Model Predictive Control with Discrete-Time Control Barrier Functions

Jun Zeng, Zhongyu Li, Koushil Sreenath

arXiv:2105.10596v2eess.SYcs.ROcs.SYmath.OC

TL;DR

Existing discrete-time CBF optimal-control approaches face a feasibility–safety tradeoff. This paper proposes NMPC formulations that relax CBF decay rates, and reports theoretical and numerical improvements in feasibility and safety over existing approaches.

  • Problem

    Existing discrete-time CBF formulations cannot theoretically enhance optimization feasibility and system safety at the same time.

    Source: A. Motivation · p. 1, 2) Feasibility & Safety · p. 1

  • Method

    The proposed formulations relax discrete-time CBF decay rates within NMPC to enhance feasibility and safety for one-step and multi-step input optimization.

    Source: C. Contributions · p. 2

  • Results

    The proposed formulations are reported to outperform state-of-the-art approaches in feasibility and safety, with theoretical analysis and numerical validation.

    Source: §VI · p. 8, A. Numerical Results for Feasibility · p. 8

  • Takeaways & Limitations

    The decay-rate relaxation is presented for both one-step and multi-step input optimization, rather than requiring only one-step constraints.

    Source: C. Contributions · p. 2

Abstract

from arXiv · show

Safety is one of the fundamental problems in robotics. Recently, one-step or multi-step optimal control problems for discrete-time nonlinear dynamical system were formulated to offer tracking stability using control Lyapunov functions (CLFs) while subject to input constraints as well as safety-critical constraints using control barrier functions (CBFs). The limitations of these existing approaches are mainly about feasibility and safety. In the existing approaches, the feasibility of the optimization and the system safety cannot be enhanced at the same time theoretically. In this paper, we propose two formulations that unifies CLFs and CBFs under the framework of nonlinear model predictive control (NMPC). In the proposed formulations, safety criteria is commonly formulated as CBF constraints and stability performance is ensured with either a terminal cost function or CLF constraints. Slack variables with relaxing technique are introduced on the CBF constraints to resolve the tradeoff between feasibility and safety so that they can be enhanced at the same. The advantages about feasibility and safety of proposed formulations compared with existing methods are analyzed theoretically and validated with numerical results.

I. INTRODUCTION

Discrete-time CBF optimal control combines safety constraints with CLF-based performance, but existing formulations face a tradeoff between feasibility and safety. The paper proposes NMPC formulations that relax CBF decay rates to address both concerns.

A. Motivation

  • The paper situates its work in robotic safety-critical control, where performance objectives and safety criteria may conflict.CLFs and CBFs are used to address stability and safety, respectively.Source: A. Motivation · p. 1

B. Related Work

  • Earlier discrete-time work includes current-input optimization with DCLF-DCBF and predictive safety constraints with MPC-DCBF.NMPC-DCLF instead uses discrete-time CLF constraints under NMPC.Source: 1) Optimal Control with Discrete-time… · p. 1

2) Feasibility & Safety:

  • Existing discrete-time CBF methods face feasibility challenges when reachable states and the CBF-defined safe region do not intersect.One-step constraints can improve feasibility, but do not remove the safety–feasibility tradeoff; the intersection can still be empty.Source: 2) Feasibility & Safety · pp. 1–2
  • The paper relaxes discrete-time CBF decay rates and supports both one-step and multi-step constraints, rather than requiring only the one-step approach of MPC-GCBF.The authors state that this can enhance feasibility and safety together.Source: 2) Feasibility & Safety · p. 2, C. Contributions · p. 2

C. Contributions

  • The proposed frameworks combine discrete-time CBF safety constraints with CLF stability criteria, represented by either a terminal cost or CLF constraints.The paper reports theoretical and numerical comparisons with existing approaches.Source: C. Contributions · p. 2

D. Paper Structure

The paper first introduces discrete-time CLF and CBF background, then presents its proposed NMPC formulations, analysis, and numerical simulations. The background defines the safe set and describes how discrete-time barrier and Lyapunov functions constrain safety and stability.

D. Paper Structure

  • The paper structure proceeds from discrete-time CLF and CBF background to proposed NMPC formulations, theoretical analysis, numerical simulations, and concluding remarks.Source: D. Paper Structure · p. 2

A. CLFs and CBFs

  • The background treats the safe set C as the superlevel set of h and describes it as the set satisfying distance constraints.Source: A. CLFs and CBFs · p. 2
  • The discrete-time CBF condition bounds h(x_k+1) below by (1−γ_k)h(x_k), while the CLF condition makes the upper bound of V decrease at rate 1−α_k.Source: A. CLFs and CBFs · p. 2

B. Existing Approaches Revisited

The section revisits discrete-time optimal-control formulations that combine CLFs and CBFs or impose predictive CBF constraints. It highlights their safety and stability mechanisms as well as feasibility limitations of predictive constraints.

B. Existing Approaches Revisited

1) DCLF-DCBF [3]:

  • DCLF-DCBF unifies discrete-time CLF and CBF constraints in one optimization, with a slack variable and cost that permit Lyapunov growth when the constraints conflict.The safe set is invariant when h(x0) ≥ 0 and 0 < γ_k ≤ 1.Source: 1) DCLF-DCBF [3] · p. 2, DCLF-DCBF · p. 3

2) MPC-DCBF [4]:

  • MPC-DCBF imposes CBF constraints across a prediction horizon and applies the first input of the optimized sequence in a receding-horizon strategy.Its larger number of constraints can reduce its feasibility region relative to DCLF-DCBF.Source: MPC-DCBF · p. 3, 2) MPC-DCBF [4] · p. 3

3) MPC-GCBF [18]:

MPC-GCBF reduces predictive CBF constraints to one step to improve feasibility and computational time. A generalization for high-relative-degree safety constraints poses constraints on two nonadjacent steps.

3) MPC-GCBF [18]:

  • A one-step CBF constraint can reduce computational complexity and improve feasibility by imposing fewer constraints.The approach was first presented in MPC-GCBF.Source: 3) MPC-GCBF [18] · p. 3
  • For a high-relative-degree safety constraint, the generalized formulation places constraints on two nonadjacent steps, with m denoting the relative degree.Source: 3) MPC-GCBF [18] · p. 3

4) CLF-NMPC [15]:

This section contrasts prior CLF-NMPC, which does not include safety criteria, with two proposed NMPC formulations that combine discrete-time CLFs and CBFs. The formulations relax CBF decay rates to improve feasibility, while using either a terminal CLF cost or CLF constraints for stability.

4) CLF-NMPC [15]:

  • CLF-NMPC incorporates stability constraints but does not consider safety criteria.Its CLF constraint may be imposed over one or multiple steps.Source: 4) CLF-NMPC [15] · p. 3

III. CLFS AND CBFS UNIFIED WITH NMPC

  • The proposed NMPC-DCBF and NMPC-DCLF-DCBF formulations unify discrete-time CLFs and CBFs within NMPC.Safety is expressed through CBF constraints, while stability is handled by a terminal cost or CLF constraints.Source: §III · p. 3

A. Formulations

  • One formulation uses the terminal CLF V(x_t+N|t) as a cost alongside cumulative stage cost.The passage notes that stability can usually be achieved without a terminal-state constraint if β is sufficiently large.Source: A. Formulations · p. 4
  • NMPC-DCBF optimizes relaxed CBF decay rates and guarantees safety for the first M_CBF open-loop steps, not the full horizon N.A separate CBF-constraint horizon M_CBF ≤ N allows its length to be chosen to reduce computational complexity.Source: A. Formulations · p. 4
  • Unlike one-step MPC-GCBF, NMPC-DCBF retains multi-step constraints, providing stronger set invariance with fewer assumptions for high-relative-degree systems.The stated requirement is M_CBF ≥ m, where m is the relative degree.Source: A. Formulations · p. 4
  • NMPC-DCLF-DCBF instead imposes CLF constraints, with slack variables and penalties to avoid infeasibility between CLF and CBF constraints.The CLF and CBF constraint horizons may be shorter than prediction horizon N; NMPC-DCBF also makes fixed CBF decay rates optimization variables, increasing feasibility.Source: A. Formulations · p. 4
  • Complexity depends on system dynamics and CBF nonlinearity; with nonlinear CBFs, shorter constraint horizons can outweigh added ω_k variables.The added optimization variables increase complexity, while shorter constraint horizons can reduce it.Source: A. Formulations · pp. 4–5

IV. THEORETICAL ANALYSIS

The section presents a theoretical comparison of the proposed approaches with state-of-the-art methods, summarized in Table I.

IV. THEORETICAL ANALYSIS

  • The section states that theoretical advantages in feasibility and safety are compared with state-of-the-art methods and summarized in Table I.Source: §IV · p. 5

A. Theoretical Analysis of Feasibility

The feasibility analysis compares reachable and safe regions for MPC-DCBF and NMPC-DCBF. It concludes that decay-rate relaxation makes the proposed formulations’ feasible regions larger and independent of the CBF decay-rate hyperparameters.

A. Theoretical Analysis of Feasibility

B. Theoretical Analysis of Safety

Reducing γ_k can improve safety while worsening feasibility in existing formulations, but the proposed NMPC-DCBF keeps its safety-constrained region independent of γ_k. An appropriate cost for the decay-rate slack variable can further support safety without harming feasibility.

B. Theoretical Analysis of Safety

  • Reducing γ_k improves safety but can make existing formulations more likely to become infeasible; in NMPC-DCBF, the safety region is independent of γ_k, allowing safety to improve without harming feasibility.The authors summarize that reducing γ_k with an appropriate slack-variable cost can outperform existing formulations in safety while not harming feasibility.Source: B. Theoretical Analysis of Safety · pp. 5–6
  • A larger P_ω tends to keep ω_k closer to 1, while a value that is too small may overrelax the CBF constraint.Keeping ω_k near 1 limits deviation from the nominal decay rate 1 − γ_k; the authors therefore do not recommend setting P_ω too small.Source: B. Theoretical Analysis of Safety · pp. 5–6
  • With ω_k = 0, the relaxed CBF constraint becomes a distance constraint, and MPC-DC needs a longer horizon for expected obstacle avoidance.Source: B. Theoretical Analysis of Safety · p. 6

V. NUMERICAL EXAMPLES & RESULTS

The numerical examples use a discrete-time linear triple-integrator system with bounded jerk inputs. Simulations use a 0.1-second sampling time and MATLAB, with Yalmip and IPOPT for formulation and solution.

V. NUMERICAL EXAMPLES & RESULTS

  • The numerical examples consider a discrete-time linear triple-integrator system.Source: §V · p. 6
  • The state is position, velocity, and acceleration, and the input is jerk constrained between j_min and j_max.Source: §V · p. 6
  • Simulations use Δt = 0.1 s and jerk bounds of −1 and 1 m/s^3; MATLAB, Yalmip, and IPOPT are used.Source: §V · p. 6

A. Numerical Results for Feasibility

In sampled feasibility comparisons, NMPC-DCBF outperforms MPC-DCBF and MPC-GCBF, while NMPC-DCLF-DCBF outperforms DCLF-DCBF. The proposed formulations’ feasible regions are independent of γk in the reported comparisons.

A. Numerical Results for Feasibility

B. Safety

Reducing γk made NMPC-DCBF safer without adversely affecting feasibility in the reported comparison. The conclusion presents the formulations as outperforming the state of the art in feasibility and safety, while identifying deployment and theoretical analyses as future work.

B. Safety

  • With γ = 0.05, NMPC-DCBF alone remained feasible from the initial condition; reducing γk slowed CBF decrease and improved safety without adversely affecting feasibility.MPC-DCBF and MPC-GCBF were infeasible initially at γ = 0.05, and MPC-GCBF became infeasible after around 5 seconds.Source: B. Safety · p. 8

VI. CONCLUSION & FUTURE WORK

  • The proposed formulations unify CLFs and CBFs in NMPC and outperform the state of the art in feasibility and safety, according to theoretical and numerical results.They relax the decay rate of CBF constraints and use different horizon lengths for cost functions and constraints.Source: §VI · p. 8
  • Real-time mobile-robot deployment still requires handling modelling uncertainty, disturbance, and noise, while formal recursive-feasibility and stability analysis remains future work.The authors identify implementation as either a mid-level planner or real-time controller as a future direction.Source: §VI · p. 8

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