Minimum-norm Sparse Perturbations for Opacity in Linear Systems
Abstract: Opacity is a notion that describes an eavesdropper's inability to estimate a system's 'secret' states by observing the system's outputs. In this paper, we propose algorithms to compute the minimum sparse perturbation to be added to a system to make its initial states opaque. For these perturbations, we consider two sparsity constraints - structured and affine. We develop an algorithm to compute the global minimum-norm perturbation for the structured case. For the affine case, we use the global minimum solution of the structured case as initial point to compute a local minimum. Empirically, this local minimum is very close to the global minimum. We demonstrate our results via a running example.
- B. Ramasubramanian, R. Cleaveland and S. I. Marcus, “Notions of Centralized and Decentralized Opacity in Linear Systems,” IEEE Transactions on Automatic Control, 65(4):1442-1455, 2020.
- E. Nekouei, T. Tanaka, M. Skoglund, K. H. Johansson, “Information-theoretic approaches to privacy in estimation and control,” Annual Reviews in Control, 47:412-422, 2019.
- E. Nozari, P. Tallapragada and J. Cortés, “Differentially Private Distributed Convex Optimization Via Objective Perturbation,” IEEE American Control Conference, pp. 2061-2066, 2016.
- V. M. John and V. Katewa, “Opacity and its Trade-offs with Security in Linear Systems,” IEEE Conference on Decision and Control, pp. 5443-5449, 2022.
- S. Lam and E. J. Davison, “A Fast Algorithm to Compute the Controllability, Decentralized Fixed-Mode, and Minimum-phase Radius of LTI Systems,” IEEE Conference on Decision and Control, Cancun, Mexico, pp. 508-513, 2008.
- S. Lam and E. J. Davison, “The Transmission Zero at s Radius and the Minimum Phase Radius of LTI Systems,” IFAC World Congress, Seoul, Korea, 41(2):6371-6376, 2008.
- G. A. Watson, “The Smallest Perturbation of a Submatrix that Lowers the Rank of the Matrix,” IMA Journal of Numerical Analysis, Cancun, Mexico, 8(3):295-303, 1988.
- V. M. John and V. Katewa, “Minimum Sparse Perturbation for Opacity in Linear Systems,” arXiv Preprint, arXiv:XX.YY [eess.SY], 2023.
- S. C. Johnson, M. Wicks, M. Žefran and R. A. DeCarlo, “The Structured Distance to the Nearest System Without Property 𝒫𝒫\mathcal{P}caligraphic_P,” IEEE Transactions on Automatic Control, 63(9):2960-2975, 2018.
- Y. Zhang, Y. Xia and Y. Zhan, “On Real Structured Controllability/Stabilizability/Stability Radius: Complexity and Unified Rank-Relaxation Based Methods,” arXiv Preprint, arXiv:2201.01112 [eess.SY], 2022.
- B. Molinari, “Extended Controllability and Observability for Linear Systems,” IEEE Transactions on Automatic Control, 21(1):136-137, 1976.
- H. L. Trentelman, A. A. Stoorvogel and M. Hautus, “Control Theory for Linear Systems,” Springer, ch. 7, 2001.
- M. Karow, “Geometry of Spectral Value Sets,” PhD Thesis, University of Bremen, Germany, 2003.
- S. Lam, “Real Robustness Radii and Performance Limitations of LTI Control Systems,” PhD Thesis, University of Toronto, Canada, 2011.
- S. Lam and E. J. Davison, “The Real DFM Radius and Minimum Norm Plant Perturbation for General Control Information Flow Constraints,” IFAC World Congress, Seoul, Korea, 41(2):6365-6370, 2008.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.