resilient-output-containment-cps

name	resilient-output-containment-cps
description	Resilient output containment control for heterogeneous multi-agent systems under actuator cyber-attacks. Two-layer adaptive control architecture combining virtual-actuator reconfiguration with network-level adaptive protocols. Handles undisclosed leader dynamics, directed network topologies, and multiple attack types (state-correlated, input-correlated, exogenous). Use when designing fault-tolerant multi-agent coordination, resilient CPS control systems, or distributed containment tracking under adversarial conditions.
metadata	{"arxiv_id":"2606.27257","published":"2026-06-25","authors":"Mohammadreza Nematollahi, Khashayar Khorasani, Nader Meskin","title":"Resilient Output Containment under Undisclosed Leader Dynamics and Actuator Attacks","tags":["cyber-physical systems","multi-agent systems","resilient control","containment tracking","actuator attacks","adaptive control","virtual actuator","directed graphs"]}

Resilient Output Containment for Multi-Agent Systems Under Cyber-Attacks

Core Methodology

Two-Layer Control Architecture

The methodology separates network-level coordination from local execution resilience:

Layer 1: Network Interface (Task-Space Commands)

• Generates containment commands using only neighbor-exchanged network-interface states
• State dimensions match plant output dimensions (not full system state)
• No global graph knowledge required for parameter tuning
• No a priori knowledge of leader velocity bounds or motion envelopes
• Continuous adaptive protocol (eliminates chattering from sliding mode approaches)

Layer 2: Virtual-Actuator Reconfiguration (Local Execution)

• Compensates for actuator attacks in local tracking-error dynamics
• Uses partial state measurements (not full state)
• Inserts recovery block between nominal controller and compromised actuator
• Prevents attack propagation into coordination layer
• Ensures attacks remain local, non-propagating, and compensable

Key Innovations

1. Undisclosed Leader Dynamics: Leaders generate bounded trajectories but their dynamics, velocity bounds, and motion envelopes are unknown to followers
2. Comprehensive Attack Model: Handles state-correlated + input-correlated + bounded exogenous actuator false-data simultaneously
3. Nonsmooth Lyapunov Analysis: Proves asymptotic command containment (not just ultimate boundedness)
4. Directed Topology Support: Works with leader-rooted united spanning trees (no symmetry requirements)
5. Minimum-Phase Requirement: Only requires stable zero dynamics (Assumption 2), avoiding vulnerabilities to zero-dynamic attacks

Mathematical Foundation

System Model (per follower i):

ẋᵢ = Aᵢxᵢ + Bᵢ(Lᵢηᵢ + ψᵢuᵢ)
η̇ᵢ = Γᵢηᵢ + Λᵢyᵢ
yᵢ = Cᵢxᵢ

• xᵢ ∈ ℝ^(∑dₖᵢ): external state (outputs and derivatives up to order dₖᵢ-1)
• ηᵢ ∈ ℝ^(nᵢ-∑dₖᵢ): zero-dynamics coordinates
• ψᵢ ∈ ℝᵐˣᵐ: known invertible input matrix

Attack Model (per follower i):

uᵢᵃ = uᵢ + Kₐₓᵢ(xᵢ,t)xᵢ + Kₐᵤᵢ(t)uᵢ + wᵤᶜₐᵢ(t)

• Kₐₓᵢ(xᵢ,t)xᵢ: state-correlated attack (admissible growth structure via known envelope)
• Kₐᵤᵢ(t)uᵢ: input-correlated attack (bounded, preserves positive input authority)
• wᵤᶜₐᵢ(t): exogenous false-data injection (bounded)

Containment Objective:

• Command-level: Asymptotic containment (followers' commands converge to leader convex hull)
• Physical-level: Ultimate boundedness (physical outputs converge to leader convex hull up to residual)

Implementation Pattern

When to Apply

Use this methodology when designing:

• Multi-agent coordination under adversarial actuator attacks
• Resilient CPS systems with undisclosed leader dynamics
• Fault-tolerant distributed control over directed networks
• Containment tracking with heterogeneous agents (different relative degrees)

Design Workflow

1. Verify Assumptions:

   • United spanning tree rooted at leaders (Assumption 1)
   • Minimum-phase zero dynamics: Γᵢ Hurwitz (Assumption 2)
   • Partial state measurements available: xᵢ accessible (Assumption 3)
   • Leader trajectories bounded and locally absolutely continuous (Assumption 4)

2. Characterize Attack Envelopes:

   • State-correlated: Define admissible growth via RKHS kernel kₓᵢ or polynomial envelope
   • Input-correlated: Verify Δᵤᵢ(t) = Iₘ + ψᵢKₐᵤᵢ(t)ψᵢ⁻¹ satisfies sᵀΔᵤᵢ(t)s ≥ χᵢ‖s‖²
   • Exogenous: Establish bounded envelope for wᵤᶜₐᵢ(t)

3. Design Network Interface:

   • Define task-space command variables matching output dimensions
   • Implement continuous adaptive protocol using neighbor information only
   • Avoid discontinuous terms (prevents chattering)
   • No global graph Laplacian eigenvalue information needed

4. Design Virtual-Actuator Layer:

   • Insert recovery block between controller and actuator
   • Use partial state feedback (xᵢ only, not ηᵢ)
   • Compensate for all three attack components simultaneously
   • Ensure local stability without propagating to network layer

5. Stability Analysis:

   • Apply nonsmooth Lyapunov analysis for command containment
   • Prove asymptotic convergence at command level
   • Establish ultimate boundedness at physical output level
   • Quantify residual based on command-tracking controller performance

Critical Pitfalls

1. Do NOT require full state measurements: The methodology specifically uses partial state (xᵢ) to avoid needing zero-dynamics state ηᵢ. Full-state designs are more vulnerable and less practical.

2. Do NOT use discontinuous protocols: Sliding mode approaches introduce chattering. Continuous adaptive protocols recover asymptotic convergence.

3. Do NOT assume leader model knowledge: The strength is handling undisclosed leader dynamics. Requiring exosystem models or velocity bounds defeats the purpose.

4. Do NOT neglect zero-dynamic attack vulnerabilities: Unstable zero dynamics (non-minimum-phase) create attack surfaces. Verify Γᵢ is Hurwitz.

5. Do NOT confuse command vs. physical containment: Command containment is asymptotic; physical containment has residual due to heterogeneous relative degrees and attack compensation limits.

6. Do NOT require symmetric graphs: The methodology works with directed graphs (united spanning tree condition), not just undirected or detailed-balanced topologies.

Validation & Testing

Simulation Benchmarks

• Quadrotor networks with damped suspended loads: Demonstrates attack recovery and containment tracking
• Heterogeneous relative degrees: Test with different dᵢ vectors across followers
• Combined attacks: Simultaneous state-correlated + input-correlated + exogenous attacks
• Directed topology: Verify with non-symmetric adjacency matrices

Metrics

• Command containment error: Should converge to zero asymptotically
• Physical output containment error: Should be ultimately bounded with small residual
• Attack compensation: Virtual-actuator should isolate attacks from network layer
• Chattering: Continuous protocols should eliminate high-frequency oscillations

Related Patterns

• Layered Defense in Depth: Network layer + local execution layer separation
• Fault-Hiding Virtual Actuators: Recovery block insertion without controller redesign
• Adaptive Robust Control: Continuous adaptation without discontinuous switching
• Distributed Observer-Free Design: Avoid leader model reconstruction entirely

Applications

• Resilient drone swarms: Containment under GPS spoofing or actuator hijacking
• Microgrid coordination: Distributed energy resources under cyber-attacks
• Autonomous vehicle platooning: Leader-follower containment with compromised actuators
• Industrial multi-robot systems: Task-space coordination under local faults/attacks
• Satellite formation flying: Undisclosed leader maneuvers with actuator degradation

References

• Paper: arXiv:2606.27257
• Key concepts: Virtual actuators, fault-hiding, adaptive containment, directed graphs, cyber-physical security
• Mathematical tools: Nonsmooth Lyapunov analysis, normal-form coordinates, M-matrix theory