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Adversarial Observability and Performance Trade-offs in Optimal Control

Filippos Fotiadis, Ufuk Topcu

TL;DR

It is shown that the performance-constrained optimization of the Gramian's trace can be formulated as a one-shot semidefinite program, while the optimization of its inverse through sequential semidefinite programming.

Abstract

We develop a feedback controller that minimizes the observability of a set of adversarial sensors of a linear system, while adhering to strict closed-loop performance constraints. We quantify the effectiveness of adversarial sensors using the trace of their observability Gramian and its inverse, capturing both average observability and the least observable state directions of the system. We derive theoretical lower bounds on these metrics under performance constraints, characterizing the fundamental limits of observability reduction as a function of the performance trade-off. Finally, we show that the performance-constrained optimization of the Gramian's trace can be formulated as a one-shot semidefinite program, while we address the optimization of its inverse through sequential semidefinite programming. Simulations on an aircraft show how the proposed scheme yields controllers that deteriorate adversarial observability while having near-optimal performance.

Adversarial Observability and Performance Trade-offs in Optimal Control

TL;DR

It is shown that the performance-constrained optimization of the Gramian's trace can be formulated as a one-shot semidefinite program, while the optimization of its inverse through sequential semidefinite programming.

Abstract

We develop a feedback controller that minimizes the observability of a set of adversarial sensors of a linear system, while adhering to strict closed-loop performance constraints. We quantify the effectiveness of adversarial sensors using the trace of their observability Gramian and its inverse, capturing both average observability and the least observable state directions of the system. We derive theoretical lower bounds on these metrics under performance constraints, characterizing the fundamental limits of observability reduction as a function of the performance trade-off. Finally, we show that the performance-constrained optimization of the Gramian's trace can be formulated as a one-shot semidefinite program, while we address the optimization of its inverse through sequential semidefinite programming. Simulations on an aircraft show how the proposed scheme yields controllers that deteriorate adversarial observability while having near-optimal performance.

Paper Structure

This paper contains 14 sections, 7 theorems, 23 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Operator and Adversary Models
  4. Operator's Control Design Problem
  5. Adversarial Observability and Performance Trade-offs
  6. Quantitative Lower Bounds
  7. Robustness to Adversary's Sensing Uncertainty
  8. Algorithmic Solutions to Problems \ref{['pr:tr']} and \ref{['pr:trinv']}
  9. Solution to Problem \ref{['pr:tr']} using an SDP
  10. Approximate Solution to Problem \ref{['pr:trinv']} using Sequential SDP
  11. Numerical Examples
  12. Intuitive Example
  13. ADMIRE Aircraft
  14. Conclusion

Key Result

Lemma 1

It holds that $\|\tilde{K}\|\le f(\lambda)$, where

Figures (4)

  • Figure 1: Illustration of the function $f(\lambda)$.
  • Figure 2: Minimum values of Problems \ref{['pr:tr']} and \ref{['pr:trinv']} as computed by Algorithm \ref{['al:SDP']} and \ref{['al:SSPinv']}, as well as the corresponding lower bounds calculated in Section \ref{['sec:tradeoff']}, for various values of the trade-off parameter $\lambda$.
  • Figure 3: Evolution of the eigenvalues $\lambda_i(W)$, $i=1,\ldots,5$, of the observability Gramian $W$ and of the trace of its inverse, $\mathrm{tr}(W^{-1})$, for $(A+BK_j,C)$ during Algorithm \ref{['al:SSPinv']}.
  • Figure 4: Trajectories of the adversary's observation errors, and the closed-loop performance cost.

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Lemma 2
  • ...and 3 more