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Differential Privacy in Nonlinear Dynamical Systems with Tracking Performance Guarantees

Dhrubajit Chowdhury, Raman Goyal, Shantanu Rane

TL;DR

This work tackles protecting sensitive information in feedback-controlled nonlinear dynamical systems by achieving differential privacy for the tracking error $e(t)$. It couples funnel control, which confines $e(t)$ within a user-specified funnel $\\mathcal{F}_{\\varphi}$, with an Ornstein–Uhlenbeck (OU)–filtered noise process that perturbs the funnel boundary and thus the control input. The authors prove that the funnel boundary and the tracking error inherit $(\\epsilon,\\delta)$-differential privacy under a defined adjacency, and they validate the approach for both state- and output-feedback implementations using high-gain observers. The results provide a principled privacy-preserving framework for CPS tracking tasks, with explicit privacy loss bounds and demonstrated privacy-performance trade-offs in nonlinear settings.

Abstract

We introduce a novel approach to make the tracking error of a class of nonlinear systems differentially private in addition to guaranteeing the tracking error performance. We use funnel control to make the tracking error evolve within a performance funnel that is pre-specified by the user. We make the performance funnel differentially private by adding a bounded continuous noise generated from an Ornstein-Uhlenbeck-type process. Since the funnel controller is a function of the performance funnel, the noise adds randomized perturbation to the control input. We show that, as a consequence of the differential privacy of the performance funnel, the tracking error is also differentially private. As a result, the tracking error is bounded by the noisy funnel boundary while maintaining privacy. We show a simulation result to demonstrate the framework.

Differential Privacy in Nonlinear Dynamical Systems with Tracking Performance Guarantees

TL;DR

This work tackles protecting sensitive information in feedback-controlled nonlinear dynamical systems by achieving differential privacy for the tracking error . It couples funnel control, which confines within a user-specified funnel , with an Ornstein–Uhlenbeck (OU)–filtered noise process that perturbs the funnel boundary and thus the control input. The authors prove that the funnel boundary and the tracking error inherit -differential privacy under a defined adjacency, and they validate the approach for both state- and output-feedback implementations using high-gain observers. The results provide a principled privacy-preserving framework for CPS tracking tasks, with explicit privacy loss bounds and demonstrated privacy-performance trade-offs in nonlinear settings.

Abstract

We introduce a novel approach to make the tracking error of a class of nonlinear systems differentially private in addition to guaranteeing the tracking error performance. We use funnel control to make the tracking error evolve within a performance funnel that is pre-specified by the user. We make the performance funnel differentially private by adding a bounded continuous noise generated from an Ornstein-Uhlenbeck-type process. Since the funnel controller is a function of the performance funnel, the noise adds randomized perturbation to the control input. We show that, as a consequence of the differential privacy of the performance funnel, the tracking error is also differentially private. As a result, the tracking error is bounded by the noisy funnel boundary while maintaining privacy. We show a simulation result to demonstrate the framework.
Paper Structure (22 sections, 8 theorems, 66 equations, 7 figures, 1 table)

This paper contains 22 sections, 8 theorems, 66 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on Differential Privacy
  3. Main Contribution
  4. Tracking Using Funnel Control
  5. Funnel Control
  6. Tracking Problem Definition
  7. Funnel Control by Synthesizing Virtual Output
  8. State Feedback Funnel Controller Design
  9. Output Feedback Funnel Controller Design
  10. Differential Privacy Problem Setup
  11. Adjacency Relation
  12. Univariate Bounded Gaussian Noise
  13. Ornstein-Uhlenbeck Type Process
  14. Boundedness & Solutions of the Process
  15. Probability Density Function of the OU type Process
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the closed-loop system (eq:singularly_perturbed) obtained using the state feedback controller (eq:statefeedback1). Let $k_2$ to $k_{\rho}$ be chosen such that the matrix $F$ is Hurwitz. Suppose Assumptions 1-3 are satisfied. Let $\varphi(t) \in \bar{\Phi}$ and suppose the initial states sat yields a closed-loop system such that the solution is bounded for all $t\geq0$ and there exists $\v

Figures (7)

  • Figure 1: Design architecture for making the tracking error differentially private by adding privacy noise to the performance funnel.
  • Figure 2: Performance Funnel $\mathcal{F}_{\varphi}$
  • Figure 3: Probability density function of OU type process
  • Figure 4: Comparison of the histogram of $y$ and $w$
  • Figure 5: Tracking error evolution inside performance funnel
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • ...and 3 more