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Secondary Safety Control for Systems with Sector Bounded Nonlinearities [Extended Version]

Yankai Lin, Michelle S. Chong, Carlos Murguia

TL;DR

The paper addresses safety verification for nonlinear systems with sector-bounded nonlinearities under sensor and actuator attacks by combining forward-invariant ellipsoids and barrier-certificate techniques. It introduces a secondary dynamic output-feedback controller operating on attack-free sensors to enforce safety, and derives LMI-based conditions via the S-procedure and projection lemma to certify an invariant unsafe-free region. A nonlinear general framework is provided, with a linear-state-independent-attacks special case and optimization perspectives to balance safety with performance, all illustrated by a numerical Josephson-junction case study. The results enable offline, convex design of attack-resilient controllers that guarantee safety within a prescribed safe set, potentially guiding cyber-physical security in safety-critical applications.

Abstract

We consider the problem of safety verification and safety-aware controller synthesis for systems with sector bounded nonlinearities. We aim to keep the states of the system within a given safe set under potential actuator and sensor attacks. Specifically, we adopt the setup that a controller has already been designed to stabilize the plant. Using invariant sets and barrier certificate theory, we first give sufficient conditions to verify the safety of the closed-loop system under attacks. Furthermore, by using a subset of sensors that are assumed to be free of attacks, we provide a synthesis method for a secondary controller that enhances the safety of the system. The sufficient conditions to verify safety are derived using Lyapunov-based tools and the S-procedure. Using the projection lemma, the conditions are then formulated as linear matrix inequality (LMI) problems which can be solved efficiently. Lastly, our theoretical results are illustrated through numerical simulations.

Secondary Safety Control for Systems with Sector Bounded Nonlinearities [Extended Version]

TL;DR

The paper addresses safety verification for nonlinear systems with sector-bounded nonlinearities under sensor and actuator attacks by combining forward-invariant ellipsoids and barrier-certificate techniques. It introduces a secondary dynamic output-feedback controller operating on attack-free sensors to enforce safety, and derives LMI-based conditions via the S-procedure and projection lemma to certify an invariant unsafe-free region. A nonlinear general framework is provided, with a linear-state-independent-attacks special case and optimization perspectives to balance safety with performance, all illustrated by a numerical Josephson-junction case study. The results enable offline, convex design of attack-resilient controllers that guarantee safety within a prescribed safe set, potentially guiding cyber-physical security in safety-critical applications.

Abstract

We consider the problem of safety verification and safety-aware controller synthesis for systems with sector bounded nonlinearities. We aim to keep the states of the system within a given safe set under potential actuator and sensor attacks. Specifically, we adopt the setup that a controller has already been designed to stabilize the plant. Using invariant sets and barrier certificate theory, we first give sufficient conditions to verify the safety of the closed-loop system under attacks. Furthermore, by using a subset of sensors that are assumed to be free of attacks, we provide a synthesis method for a secondary controller that enhances the safety of the system. The sufficient conditions to verify safety are derived using Lyapunov-based tools and the S-procedure. Using the projection lemma, the conditions are then formulated as linear matrix inequality (LMI) problems which can be solved efficiently. Lastly, our theoretical results are illustrated through numerical simulations.

Paper Structure

This paper contains 11 sections, 5 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. System description
  4. Safe sets and attack signals
  5. Main results for safety verification and secondary controller synthesis
  6. Safety verification
  7. Secondary controller synthesis
  8. Linear case with state-independent attacks
  9. Optimization perspectives
  10. Numerical Case Study
  11. Conclusions

Key Result

Lemma 1

Let $F_0,\ldots, F_p$ be quadratic functions of the variable $\xi\in\mathbb{R}^n$: where $T_i=T_i^\top$. If there exist $\tau_1\geq0,\ldots,\tau_p\geq0$ such that for all $\xi$, then, we have $F_0(\xi)\geq0$ for all $\xi$ such that $F_i(\xi)\geq0$, $i=1,\ldots,p$.

Figures (3)

  • Figure 1: Ensuring safety with a secondary controller, where $u_p$ ($u_s$) and $y_p$ ($u_p$) denote the input signal generated by the primary (secondary) controller and output of the plant fed back to the primary (secondary) controller respectively. The actuator (sensor) signal is denoted by $a_u$ ($a_y$).
  • Figure 2: Plots of the safe set and the computed RPI set using the primary controller only. The red ellipsoid is the safe set and the blue ellipsoid is the RPI set achieved by the primary controller.
  • Figure 3: Plots of the safe set and the computed RPI with the addition of the secondary controller. The red ellipsoid is the safe set and the blue ellipsoid is the RPI set achieved by the primary controller and the secondary controller.

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Lemma 1: Section 2.6.3 in boyd1994linear
  • Proposition 1
  • Remark 7
  • ...and 8 more