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Constructive Safety Control

Si Wu, Tengfei Liu, Zhong-Ping Jiang

TL;DR

This work addresses safety control for nonlinear uncertain cascade systems with multiple constraints by developing a QCQP-based safety controller for relative-degree-one plants and a novel feasible-set reshaping technique that guarantees Lipschitz continuity of the safety law. The method recursively stabilizes the cascade via virtual controls and a nonlinear small-gain synthesis, yielding input-to-state practical safety (ISpSf) under disturbances. When safety constraints are removed, the approach reduces to standard constructive nonlinear control, highlighting its generality. The VTOL experiment demonstrates the method's practical viability for real-world systems operating in tight spaces.

Abstract

This paper proposes a constructive approach to safety control of nonlinear cascade systems subject to multiple state constraints. New design ingredients include a unified characterization of safety and stability for systematic designs of safety controllers, and a novel technique of reshaping the feasible sets of quadratically constrained quadratic programming induced from safety control. The proposed method guarantees Lipschitz continuity of virtual control laws, enabling a stepwise constructive design. A refined nonlinear small-gain synthesis is employed to address the nonlinear uncertain interconnections between the resulting subsystems corresponding to different virtual control laws, and to guarantee the achievement of the safety control objective. When the safety constraints are removed, the proposed approach coincides with the standard constructive nonlinear control. The proposed safety-control algorithm is experimentally validated in a testbed involving a vertical takeoff and landing (VTOL) vehicle taking off in narrow spaces.

Constructive Safety Control

TL;DR

This work addresses safety control for nonlinear uncertain cascade systems with multiple constraints by developing a QCQP-based safety controller for relative-degree-one plants and a novel feasible-set reshaping technique that guarantees Lipschitz continuity of the safety law. The method recursively stabilizes the cascade via virtual controls and a nonlinear small-gain synthesis, yielding input-to-state practical safety (ISpSf) under disturbances. When safety constraints are removed, the approach reduces to standard constructive nonlinear control, highlighting its generality. The VTOL experiment demonstrates the method's practical viability for real-world systems operating in tight spaces.

Abstract

This paper proposes a constructive approach to safety control of nonlinear cascade systems subject to multiple state constraints. New design ingredients include a unified characterization of safety and stability for systematic designs of safety controllers, and a novel technique of reshaping the feasible sets of quadratically constrained quadratic programming induced from safety control. The proposed method guarantees Lipschitz continuity of virtual control laws, enabling a stepwise constructive design. A refined nonlinear small-gain synthesis is employed to address the nonlinear uncertain interconnections between the resulting subsystems corresponding to different virtual control laws, and to guarantee the achievement of the safety control objective. When the safety constraints are removed, the proposed approach coincides with the standard constructive nonlinear control. The proposed safety-control algorithm is experimentally validated in a testbed involving a vertical takeoff and landing (VTOL) vehicle taking off in narrow spaces.
Paper Structure (33 sections, 12 theorems, 117 equations, 12 figures)

This paper contains 33 sections, 12 theorems, 117 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Safety Control of Nonlinear Cascade Systems
  4. A Sub-Problem: Systems with Relative-Degree-One
  5. Design Ingredient: Safety Control of Nonlinear Plants with Relative-Degree-One
  6. Safety Controller Design
  7. Proof of Theorem \ref{['theorem_safety_relative_degree_one']}
  8. Property 1)
  9. Property 2)
  10. Design Ingredient: Reshaping Feasible Set of QCQP
  11. The Non-Lipschitz Issue with Normal QCQP
  12. Feasible-Set Reshaping
  13. Proof of Theorem \ref{['theorem_reshaping']}
  14. Property 1)
  15. Property 2)
  16. ...and 18 more sections

Key Result

Lemma 1

Consider the certificate function $V_j$ and the corresponding positive constant $v_j$ for $j=1,\ldots,n_c$. The satisfaction of condition eq_assume_nozero_gradient implies the existence of $\underaccent{\bar{}}{\alpha}_j,\bar{\alpha}_j\in\mathcal{K}^e$ defined on $(-a,b)$ such that where for $j=1,\ldots,n_c$.

Figures (12)

  • Figure 1: Block diagram of the proposed safety control system.
  • Figure 2: Block diagram of the safety control system.
  • Figure 3: $|\rho_c(x)|$ with respect to different $x$: $\rho_c$ may be non-Lipschitz or admit a Lipschitz constant depending on the configuration of the obstacles.
  • Figure 4: $|\rho_L(x)|$ with respect to different $x$. In the scenario of Example \ref{['example_non_Lipschitz']}, the Lipschitz continuity of the solution to the QP problem \ref{['eq_safety_control_law_varrhoL']} can be guaranteed by reshaping the feasible set.
  • Figure 5: The closed-loop system as an interconnected system.
  • ...and 7 more figures

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Theorem 1
  • Remark 4
  • Remark 5
  • Example 1
  • Lemma 2
  • Lemma 3
  • ...and 11 more