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Funnel-Based Online Recovery Control for Nonlinear Systems With Unknown Dynamics

Zihao Song, Shirantha Welikala, Panos J. Antsaklis, Hai Lin

TL;DR

The paper tackles recovering nonlinear systems from attacks or faults with unknown dynamics by integrating online learning and formal guarantees. It employs Recurrent Equilibrium Networks to learn time-varying disturbances with incremental IQC guarantees, enabling a virtual nominal model around which a funnel-based recovery controller is designed. The core theoretical result shows internal and $L_2$ stability via a discrete-time matrix inequality, and invariant ellipsoidal funnels are synthesized to bound state deviations along a nominal trajectory. The approach is validated through a DC microgrid simulation, demonstrating online applicability with REN training and funnel computation within the sampling interval, highlighting potential for real-time resilience in cyber-physical systems.

Abstract

In this paper, we focus on recovery control of nonlinear systems from attacks or failures. The main challenges of this problem lie in (1) learning the unknown dynamics caused by attacks or failures with formal guarantees, and (2) finding the invariant set of states to formally ensure the state deviations allowed from the nominal trajectory. To solve this problem, we propose to apply the Recurrent Equilibrium Networks (RENs) to learn the unknown dynamics using the data from the real-time system states. The input-output property of this REN model is guaranteed by incremental integral quadratic constraints (IQCs). Then, we propose a funnel-based control method to achieve system recovery from the deviated states. In particular, a sufficient condition for nominal trajectory stabilization is derived together with the invariant funnels along the nominal trajectory. Eventually, the effectiveness of our proposed control method is illustrated by a simulation example of a DC microgrid control application.

Funnel-Based Online Recovery Control for Nonlinear Systems With Unknown Dynamics

TL;DR

The paper tackles recovering nonlinear systems from attacks or faults with unknown dynamics by integrating online learning and formal guarantees. It employs Recurrent Equilibrium Networks to learn time-varying disturbances with incremental IQC guarantees, enabling a virtual nominal model around which a funnel-based recovery controller is designed. The core theoretical result shows internal and stability via a discrete-time matrix inequality, and invariant ellipsoidal funnels are synthesized to bound state deviations along a nominal trajectory. The approach is validated through a DC microgrid simulation, demonstrating online applicability with REN training and funnel computation within the sampling interval, highlighting potential for real-time resilience in cyber-physical systems.

Abstract

In this paper, we focus on recovery control of nonlinear systems from attacks or failures. The main challenges of this problem lie in (1) learning the unknown dynamics caused by attacks or failures with formal guarantees, and (2) finding the invariant set of states to formally ensure the state deviations allowed from the nominal trajectory. To solve this problem, we propose to apply the Recurrent Equilibrium Networks (RENs) to learn the unknown dynamics using the data from the real-time system states. The input-output property of this REN model is guaranteed by incremental integral quadratic constraints (IQCs). Then, we propose a funnel-based control method to achieve system recovery from the deviated states. In particular, a sufficient condition for nominal trajectory stabilization is derived together with the invariant funnels along the nominal trajectory. Eventually, the effectiveness of our proposed control method is illustrated by a simulation example of a DC microgrid control application.

Paper Structure

This paper contains 14 sections, 5 theorems, 57 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Related work
  4. Our approach and contributions
  5. Background
  6. Notations
  7. Problem Formulation
  8. REN-based Unknown Dynamics Identification
  9. Funnel-based Online Recovery Control Design
  10. Closed-Loop Dynamics
  11. Stability Analysis of the Closed-Loop System
  12. Invariant Funnel Synthesis
  13. Simulation Example
  14. Conclusion

Key Result

Theorem 1

revay2023recurrent The REN model Eq:state_transition_compact satisfies the incremental IQC from input deviations $\hat{x}^a-\hat{x}^b$ to output deviations $\hat{\Delta}^a-\hat{\Delta}^b$ characterized by ($Q,S,R$) and is well-posed, if for a given $\bar{\alpha}\in(0,1]$, there exists $P_{1}>0$ and holds, where $F:=2\Lambda_{w}-\Lambda_{w}D_{11}-D_{11}^\top\Lambda_{w}>0$ is the well-posedness con

Figures (14)

  • Figure 1: Our proposed recovery control framework, where $x_{n}(t)=A\hat{x}(t)+B\hat{u}_r(t)+Ep(t)$ is the states of the nominal system dynamics. The REN model is trained online using the real-time state data $\hat{x}_{t+1}=x_{nt}+\hat{\Delta}_t$. The actual and the virtual system share the same form of controller, but use different feedback states.
  • Figure 2: The architecture of the invariant funnels, where the funnels are constructed at each way point and the virtual system states $\hat{x}(t)$ are close to the nominal state trajectory $\bar{x}(t)$, for all $t\in\mathcal{I}_{N}^0$. The last way point at $t=N$ is the desired equilibrium point.
  • Figure 3: The architecture of our DC microgrid najafirad2025dissipativitynakano2025dissipativity.
  • Figure 4: Nominal voltages.
  • Figure 5: Nominal currents.
  • ...and 9 more figures

Theorems & Definitions (20)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 2
  • ...and 10 more