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Enforcing Mixed State-Input Constraints with Multiple Backup Control Barrier Functions: A Projection-based Approach

Laszlo Gacsi, Adam K. Kiss, Ersin Das, Tamas G. Molnar

Abstract

Ensuring the safety of control systems often requires the satisfaction of constraints on states (such as position or velocity), control inputs (such as force), and a mixture of states and inputs (such as power that depends on both velocity and force). This paper presents a safety-critical control framework for enforcing mixed state-input constraints through a generalization of backup control barrier functions (backup CBFs). First, we extend the backup CBF approach to maintain multiple decoupled state and input constraints using a single backup set-backup controller pair. Second, we address mixed state-input constraints by converting them into state constraints using a projection from the state-input space to the state space along the backup controller. In the special case of decoupled state and input constraints, the proposed method simplifies the synthesis of backup CBFs by eliminating the need for saturating backup control laws. Finally, we demonstrate the efficacy of the proposed method on an inverted pendulum example, where constraints on the angle (state), torque (input), and power (mixture of state and input) are satisfied simultaneously.

Enforcing Mixed State-Input Constraints with Multiple Backup Control Barrier Functions: A Projection-based Approach

Abstract

Ensuring the safety of control systems often requires the satisfaction of constraints on states (such as position or velocity), control inputs (such as force), and a mixture of states and inputs (such as power that depends on both velocity and force). This paper presents a safety-critical control framework for enforcing mixed state-input constraints through a generalization of backup control barrier functions (backup CBFs). First, we extend the backup CBF approach to maintain multiple decoupled state and input constraints using a single backup set-backup controller pair. Second, we address mixed state-input constraints by converting them into state constraints using a projection from the state-input space to the state space along the backup controller. In the special case of decoupled state and input constraints, the proposed method simplifies the synthesis of backup CBFs by eliminating the need for saturating backup control laws. Finally, we demonstrate the efficacy of the proposed method on an inverted pendulum example, where constraints on the angle (state), torque (input), and power (mixture of state and input) are satisfied simultaneously.
Paper Structure (10 sections, 35 equations, 3 figures, 1 table)

This paper contains 10 sections, 35 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Input Constrained Safety
  3. Classical Control Barrier Function Framework
  4. Backup Control Barrier Functions
  5. Multiple Backup CBFs
  6. Mixed State-Input Constraints
  7. Projection with Single Constraint
  8. Projection with Multiple Constraints
  9. State-Input-Power Constrained Inverted Pendulum
  10. Conclusion

Figures (3)

  • Figure 1: Overview of the proposed projection-based backup CBF approach.
  • Figure 2: Set analysis for the inverted pendulum \ref{['eq:pendulum']} with mixed state-input constraints. Panel (a) depicts the six constraint sets from \ref{['eq:constr_pos']}-\ref{['eq:constr_pwr']} that are projected using the backup controller \ref{['eq:pend_backup_controller']} (light blue: state, gray: input, yellow: power constraints). Their intersection, $\mathcal{C}_{\mathrm{p}}$ from \ref{['eq:projected_set_multi']} (red), contains the backup set $\mathcal{S}_\mathrm{b}$ from \ref{['eq:pend_backup_set']} (blue). Panel (b) compares four sets: the zero-superlevel set $\mathcal{C}_{\rm e}$ of the HOCBF in \ref{['eq:pend_hocbf']}, the invariant set $\mathcal{S}_{\mathrm{I}}^{\varphi}$ for state constraints \ref{['eq:constr_pos']}, the set $\mathcal{S}_{\mathrm{I}}^{\varphi u}$ for state and input constraints \ref{['eq:constr_pos']}-\ref{['eq:constr_u']}, and the set $\mathcal{S}_{\mathrm{I}}^{\mathrm{p}}$ for all six constraints \ref{['eq:constr_pos']}-\ref{['eq:constr_pwr']}. The backup flows \ref{['eq:pend_flow']} launched from different initial conditions are also highlighted (red lines).
  • Figure 3: Simulation of the inverted pendulum \ref{['eq:pendulum']} using the proposed backup CBF-QP \ref{['eq:QP3']} (solid) and, for comparison, the HOCBF \ref{['eq:pend_hocbf']} (dotted). Panel (a) illustrates two trajectories (blue and orange) initiated from the invariant set $\mathcal{S}_{\mathrm{I}}^{\mathrm{p}}$ (green) where they remain throughout the simulation. The boundaries of the projected constraint set $\mathcal{C}_{\mathrm{p}}$ are indicated by dashed lines (light blue: state, black: input, yellow: power constraint). Panels (b)-(d) display the evolution of angle, input, and power signals, with dashed black lines representing their limits. The proposed method maintains all constraints.

Theorems & Definitions (5)

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