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Encrypted Observer-based Control for Linear Continuous-Time Systems

Hung Nguyen, Binh Nguyen, Hyung-Gohn Lee, Hyo-Sung Ahn

TL;DR

This work introduces an encrypted observer-based control framework for linear continuous-time systems using LWE-based encryption to protect signals and controller parameters. By discretizing the observer-based controller and introducing a continuous-time virtual controller, the authors cast the closed-loop dynamics as a linear sampled-data system with quantization- and encryption-induced uncertainties. Stability is established via a discontinuous Lyapunov functional and IQC, leading to LMIs that relate quantization gains and sampling intervals to global asymptotic stability. Numerical results on a DC motor demonstrate feasibility and provide guidance on selecting quantization and sampling parameters for stability. The approach enhances security without sacrificing stability in encrypted control applications with practical relevance to cloud-enabled control scenarios.

Abstract

This paper is concerned with the stability analysis of encrypted observer-based control for linear continuous-time systems. Since conventional encryption has limited ability to deploy in continuous-time integral computation, our work presents systematically a new design of encryption for a continuous-time observer-based control scheme. To be specific, in this paper, both control parameters and signals are encrypted by the learning-with-errors (LWE) encryption to avoid data eavesdropping. Furthermore, we propose encrypted computations for the observer-based controller based on its discrete-time model, and present a continuous-time virtual dynamics of the controller for further stability analysis. Accordingly, we present novel stability criteria by introducing linear matrix inequalities (LMIs)-based conditions associated with quantization gains and sampling intervals. The established stability criteria with theoretical proofs based on a discontinuous Lyapunov functional possibly provide a way to select quantization gains and sampling intervals to guarantee the stability of the closed-loop system. Numerical results on DC motor control corresponding to several quantization gains and sampling intervals demonstrate the validity of our method.

Encrypted Observer-based Control for Linear Continuous-Time Systems

TL;DR

This work introduces an encrypted observer-based control framework for linear continuous-time systems using LWE-based encryption to protect signals and controller parameters. By discretizing the observer-based controller and introducing a continuous-time virtual controller, the authors cast the closed-loop dynamics as a linear sampled-data system with quantization- and encryption-induced uncertainties. Stability is established via a discontinuous Lyapunov functional and IQC, leading to LMIs that relate quantization gains and sampling intervals to global asymptotic stability. Numerical results on a DC motor demonstrate feasibility and provide guidance on selecting quantization and sampling parameters for stability. The approach enhances security without sacrificing stability in encrypted control applications with practical relevance to cloud-enabled control scenarios.

Abstract

This paper is concerned with the stability analysis of encrypted observer-based control for linear continuous-time systems. Since conventional encryption has limited ability to deploy in continuous-time integral computation, our work presents systematically a new design of encryption for a continuous-time observer-based control scheme. To be specific, in this paper, both control parameters and signals are encrypted by the learning-with-errors (LWE) encryption to avoid data eavesdropping. Furthermore, we propose encrypted computations for the observer-based controller based on its discrete-time model, and present a continuous-time virtual dynamics of the controller for further stability analysis. Accordingly, we present novel stability criteria by introducing linear matrix inequalities (LMIs)-based conditions associated with quantization gains and sampling intervals. The established stability criteria with theoretical proofs based on a discontinuous Lyapunov functional possibly provide a way to select quantization gains and sampling intervals to guarantee the stability of the closed-loop system. Numerical results on DC motor control corresponding to several quantization gains and sampling intervals demonstrate the validity of our method.
Paper Structure (17 sections, 6 theorems, 57 equations, 5 figures, 3 tables)

This paper contains 17 sections, 6 theorems, 57 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Encryption
  4. Additive property
  5. Multiplication
  6. Quantization
  7. Sampled-data observer-based controller
  8. Proposed Encrypted Observer-based Control
  9. Observer-based secure control scheme
  10. Linear sampled-data system formulation
  11. Stability Analysis
  12. Modified discontinuous Lyapunov functional for sampled-data systems
  13. Stability analysis of the closed-loop system
  14. Evaluation
  15. Conclusions
  16. ...and 2 more sections

Key Result

Lemma 1

Let $\Gamma = \Gamma^\top, \Pi_1, \Pi_2, \Omega$ and $\Delta$ be the matrices with appropriate dimensions. Then, the inequality $\Gamma + \Pi_1 \Delta^\top \Pi_2 + \Pi_2^\top \Delta \Pi_1^\top + \Delta \Omega \Delta^\top \preceq 0$ holds if there exist $\epsilon, \kappa >0$ such that

Figures (5)

  • Figure 1: (a) A cloud-based control scheme without encryption, (b) a cloud-based control scheme with encryption-based secure communications and an unencrypted controller, and (c) a cloud-based control scheme with both encryption-based secure communications and controller. The red indicates the parts of the system vulnerable to the attack, and the blue represents the parts protected against the attack by encryption.
  • Figure 2: Encrypted control system diagram
  • Figure 3: The angular position error and control input of the system in three cases $h = 0.03, 0.05, 0.07$ in accordance with $\Lambda$ shown in Table \ref{['Table1']}, and $\Lambda_k = k^2$.
  • Figure 4: The angular position error $\theta_e [rad]$ with different time-varying $\Lambda_k$, $h = 0.05$, and $\Lambda = 9.88 \times 10^3$.
  • Figure 5: The angular position error $\theta_e [rad]$ with fixed values of $\Lambda_k$, $h = 0.05$, and $\Lambda = 9.88 \times 10^3$.

Theorems & Definitions (9)

  • Definition 1: Integral Quadratic Constraint fetzer2016general
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2