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Input-to-State Safe Backstepping: Robust Safety-Critical Control with Unmatched Uncertainties

Max H. Cohen, Pio Ong, Aaron D. Ames

TL;DR

This work tackles safety guarantees for nonlinear control systems subject to unmatched disturbances by integrating input-to-state safety with backstepping. It introduces Optimal-Decay CBFs (OD-CBFs) and extends them to the ISSf setting, yielding OD-ISSf-CBFs that relax verification and rely on disturbance direction via $w(x) d$. A constructive backstepping procedure is then developed for two system classes, strict-feedback and dual relative degree (DRD), enabling robust safety through a hierarchy of CBFs even when safety-related dynamics are not directly actuated. The approach is demonstrated on an inverted pendulum and planar quadrotor, illustrating how smaller $\varepsilon$ enhances robustness to unmatched disturbances and how safety expansions $\mathcal{S}_{\delta}$ quantify degraded safety under disturbances. Overall, the paper provides a practical framework for safety-critical control in the presence of unmatched uncertainties with theoretical guarantees and actionable synthesis steps.

Abstract

Guaranteeing safety in the presence of unmatched disturbances -- uncertainties that cannot be directly canceled by the control input -- remains a key challenge in nonlinear control. This paper presents a constructive approach to safety-critical control of nonlinear systems with unmatched disturbances. We first present a generalization of the input-to-state safety (ISSf) framework for systems with these uncertainties using the recently developed notion of an Optimal Decay CBF, which provides more flexibility for satisfying the associated Lyapunov-like conditions for safety. From there, we outline a procedure for constructing ISSf-CBFs for two relevant classes of systems with unmatched uncertainties: i) strict-feedback systems; ii) dual-relative-degree systems, which are similar to differentially flat systems. Our theoretical results are illustrated via numerical simulations of an inverted pendulum and planar quadrotor.

Input-to-State Safe Backstepping: Robust Safety-Critical Control with Unmatched Uncertainties

TL;DR

This work tackles safety guarantees for nonlinear control systems subject to unmatched disturbances by integrating input-to-state safety with backstepping. It introduces Optimal-Decay CBFs (OD-CBFs) and extends them to the ISSf setting, yielding OD-ISSf-CBFs that relax verification and rely on disturbance direction via . A constructive backstepping procedure is then developed for two system classes, strict-feedback and dual relative degree (DRD), enabling robust safety through a hierarchy of CBFs even when safety-related dynamics are not directly actuated. The approach is demonstrated on an inverted pendulum and planar quadrotor, illustrating how smaller enhances robustness to unmatched disturbances and how safety expansions quantify degraded safety under disturbances. Overall, the paper provides a practical framework for safety-critical control in the presence of unmatched uncertainties with theoretical guarantees and actionable synthesis steps.

Abstract

Guaranteeing safety in the presence of unmatched disturbances -- uncertainties that cannot be directly canceled by the control input -- remains a key challenge in nonlinear control. This paper presents a constructive approach to safety-critical control of nonlinear systems with unmatched disturbances. We first present a generalization of the input-to-state safety (ISSf) framework for systems with these uncertainties using the recently developed notion of an Optimal Decay CBF, which provides more flexibility for satisfying the associated Lyapunov-like conditions for safety. From there, we outline a procedure for constructing ISSf-CBFs for two relevant classes of systems with unmatched uncertainties: i) strict-feedback systems; ii) dual-relative-degree systems, which are similar to differentially flat systems. Our theoretical results are illustrated via numerical simulations of an inverted pendulum and planar quadrotor.
Paper Structure (8 sections, 8 theorems, 64 equations, 3 figures)

This paper contains 8 sections, 8 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Optimal Decay Input-to-State Safe CBFs
  4. Input-to-State Safe Backstepping
  5. Strict-Feedback Systems
  6. Dual Relative Degree Systems
  7. Examples
  8. Conclusions

Key Result

Lemma 1

A continuously differentiable function $h:\mathbb{R}^n \rightarrow \mathbb{R}$ is a CBF for sys:ctrl_affine on $\mathcal{D}\supset\mathcal{S}$ iff there exists an extended class $\mathcal{K}$ function $\alpha\,:\,(-b,c)\rightarrow\mathbb{R}$ such that:

Figures (3)

  • Figure 1: Left: Evolution of the pendulum's position under an ISSf safety filter, where the dashed black lines dente the cosntraint boundary. Right: Expansions of the pendulum's safe set using a virtual controller with $\varepsilon=1$. Here, the thick yellow curve corresponds to the boundary of $\mathcal{S}$ whereas the thinner curves corresponds to the boundary of $\mathcal{S}_{\delta}$ for different values of $\delta$.
  • Figure 2: Left: Evolution of the drone's $x$ position under the DRD-CBF-based controller for different values of $\varepsilon$ and disturbance input $\mathbf{d}=(1,0)$, where the gray region denotes unsafe states. Right: Evolution of the drone's orientation under the DRD-CBF-based controller for different values of $\varepsilon$, where the transparent lines denote the values of $\bm{\eta}_{d}$.
  • Figure 3: Evolution of the drone's $(x,y)$ position under different disturbances, where the gray arrows indicate the direction of the disturbance.

Theorems & Definitions (20)

  • Definition 1: AmesTAC17
  • Lemma 1: jankovic2018robust
  • Definition 2: AmesLCSS19
  • Definition 3: AndrewTaylor
  • Lemma 2: AndrewTaylor
  • Definition 4
  • Lemma 3
  • proof
  • Proposition 1
  • Definition 5: LopezLCSS21Contraction
  • ...and 10 more