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A Compositional Resilience Index for Computationally Efficient Safety Analysis of Interconnected Systems

Luyao Niu, Abdullah Al Maruf, Andrew Clark, J. Sukarno Mertoguno, Radha Poovendran

TL;DR

This work tackles safety verification for large-scale interconnected systems subject to faults and attacks by introducing a compositional resilience index for each subsystem. The index is a quadruple $(d_i,\tau_i,\phi_i,\eta_i)$ that defines a forward-invariant safety subregion $\mathcal{D}_i$ and enforces timing and derivative bounds to guarantee safety under faults, recoveries, and interconnections. Safety after interconnection is reduced to checking a pair of linear-inequality conditions, $R_1$ or $R_2$, driven by the coupling terms and a computed $\delta_j$, enabling scalable analysis without reconfiguring the network. A sum-of-squares optimization computes the resilience index for individual subsystems, and a chemical-plant case study demonstrates that the approach maintains safety under faults while exposing how interconnections influence resilience. The combination of SOS-based index computation and linear interconnection criteria provides a practical, topology-agnostic framework for verifying safety in complex, fault-prone networks.

Abstract

Interconnected systems such as power systems and chemical processes are often required to satisfy safety properties in the presence of faults and attacks. Verifying safety of these systems, however, is computationally challenging due to nonlinear dynamics, high dimensionality, and combinatorial number of possible faults and attacks that can be incurred by the subsystems interconnected within the network. In this paper, we develop a compositional resilience index to verify safety properties of interconnected systems under faults and attacks. The resilience index is a tuple serving the following two purposes. First, it quantifies how a safety property is impacted when a subsystem is compromised by faults and attacks. Second, the resilience index characterizes the needed behavior of a subsystem during normal operations to ensure safety violations will not occur when future adverse events occur. We develop a set of sufficient conditions on the dynamics of each subsystem to satisfy its safety constraint, and leverage these conditions to formulate an optimization program to compute the resilience index. When multiple subsystems are interconnected and their resilience indices are given, we show that the safety constraints of the interconnected system can be efficiently verified by solving a system of linear inequalities. We demonstrate our developed resilience index using a numerical case study on chemical reactors connected in series.

A Compositional Resilience Index for Computationally Efficient Safety Analysis of Interconnected Systems

TL;DR

This work tackles safety verification for large-scale interconnected systems subject to faults and attacks by introducing a compositional resilience index for each subsystem. The index is a quadruple that defines a forward-invariant safety subregion and enforces timing and derivative bounds to guarantee safety under faults, recoveries, and interconnections. Safety after interconnection is reduced to checking a pair of linear-inequality conditions, or , driven by the coupling terms and a computed , enabling scalable analysis without reconfiguring the network. A sum-of-squares optimization computes the resilience index for individual subsystems, and a chemical-plant case study demonstrates that the approach maintains safety under faults while exposing how interconnections influence resilience. The combination of SOS-based index computation and linear interconnection criteria provides a practical, topology-agnostic framework for verifying safety in complex, fault-prone networks.

Abstract

Interconnected systems such as power systems and chemical processes are often required to satisfy safety properties in the presence of faults and attacks. Verifying safety of these systems, however, is computationally challenging due to nonlinear dynamics, high dimensionality, and combinatorial number of possible faults and attacks that can be incurred by the subsystems interconnected within the network. In this paper, we develop a compositional resilience index to verify safety properties of interconnected systems under faults and attacks. The resilience index is a tuple serving the following two purposes. First, it quantifies how a safety property is impacted when a subsystem is compromised by faults and attacks. Second, the resilience index characterizes the needed behavior of a subsystem during normal operations to ensure safety violations will not occur when future adverse events occur. We develop a set of sufficient conditions on the dynamics of each subsystem to satisfy its safety constraint, and leverage these conditions to formulate an optimization program to compute the resilience index. When multiple subsystems are interconnected and their resilience indices are given, we show that the safety constraints of the interconnected system can be efficiently verified by solving a system of linear inequalities. We demonstrate our developed resilience index using a numerical case study on chemical reactors connected in series.
Paper Structure (11 sections, 6 theorems, 21 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 11 sections, 6 theorems, 21 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Proposed Resilience Index
  5. Definition of Resilience Index
  6. Computation of Resilience Index
  7. Resilience Index After Interconnection
  8. Computation of Resilience Index After Interconnection
  9. Feasibility of Resilience Index for Interconnected System
  10. Case Study
  11. Conclusion

Key Result

Proposition 1

Consider a subsystem in Eqn. eq:subsystem dynamics under attack and a safety set $\mathcal{C}_i$. Let $h_i^d(x_i) = h_i(x_i)-d_i$ and $\mathcal{D}_i=\{x_i:h_i^d(x_i)\geq 0\}\subseteq\mathcal{C}_i$. Suppose $x_i(0)\in\mathcal{D}_i$. If there exist constants $d_i,\eta_i\geq0$, $\tau_i,\phi_i>0$, and a then subsystem $\mathcal{S}_i$ is safe with respect to $\mathcal{C}_i$.

Figures (1)

  • Figure 1: This figure presents the temperature in CSTRs $\mathcal{S}_1$ and $\mathcal{S}_2$ after they are interconnected in series. The temperature of $\mathcal{S}_1$ and $\mathcal{S}_2$ are represented in solid and dashed lines. The portion plotted in red represents that the CSTR is faulty or compromised, whereas the portion plotted in green denotes that the desired rate of heat input is available.

Theorems & Definitions (13)

  • Definition 1: Resilience Index of a Subsystem
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 3 more