Input and state estimation exploiting input sparsity

Motivated by cyber-physical security applications, we face the problem of estimating the state and the input of a linear system, where the input may represent the presence of adversarial attacks. We consider the case where classical filters cannot be used, because the number of measurements is too low, for example it is lower than the size of the input vector. If the input, although of large size, is known to be sparse, the problem can be tackled using techniques from compressed sensing theory. In this paper, we propose a recursive estimator, based on compressed sensing and Kalman-like filtering, which is able to reconstruct both the state and the input from noisy, compressed measurements. The proposed algorithm is proved to be feasible and numerically efficient, and simulations show a good recovery accuracy with respect to an oracle estimator.

[1]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[2]  Joel A. Tropp,et al.  Just relax: convex programming methods for identifying sparse signals in noise , 2006, IEEE Transactions on Information Theory.

[3]  Shreyas Sundaram,et al.  On Delayed Observers for Linear Systems with Unknown Inputs , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[4]  Georgios B. Giannakis,et al.  Tracking target signal strengths on a grid using sparsity , 2011, EURASIP J. Adv. Signal Process..

[5]  Florian Dörfler,et al.  A divide-and-conquer approach to distributed attack identification , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[6]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[7]  Namrata Vaswani,et al.  Kalman filtered Compressed Sensing , 2008, 2008 15th IEEE International Conference on Image Processing.

[8]  Dave Zachariah,et al.  Dynamic Iterative Pursuit , 2012, IEEE Transactions on Signal Processing.

[9]  Bart De Moor,et al.  Unbiased minimum-variance input and state estimation for linear discrete-time systems , 2007, Autom..

[10]  Emilio Frazzoli,et al.  A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems , 2013, Autom..

[11]  Federica Garin,et al.  Structural and Strongly Structural Input and State Observability of Linear Network Systems , 2018, IEEE Transactions on Control of Network Systems.

[12]  Francesco Bullo,et al.  Control-Theoretic Methods for Cyberphysical Security: Geometric Principles for Optimal Cross-Layer Resilient Control Systems , 2015, IEEE Control Systems.

[13]  Karl Henrik Johansson,et al.  A secure control framework for resource-limited adversaries , 2012, Autom..

[14]  Bruno Sinopoli,et al.  Robust structural analysis and design of distributed control systems to prevent zero dynamics attacks , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[15]  Sophie Fosson,et al.  Online Optimization in Dynamic Environments: A Regret Analysis for Sparse Problems , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[16]  Martin J. Wainwright,et al.  Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $\ell _{1}$ -Constrained Quadratic Programming (Lasso) , 2009, IEEE Transactions on Information Theory.

[17]  Pini Gurfil,et al.  Methods for Sparse Signal Recovery Using Kalman Filtering With Embedded Pseudo-Measurement Norms and Quasi-Norms , 2010, IEEE Transactions on Signal Processing.

[18]  Philip Schniter,et al.  Dynamic Compressive Sensing of Time-Varying Signals Via Approximate Message Passing , 2012, IEEE Transactions on Signal Processing.

[19]  Christopher J. Rozell,et al.  Dynamic Filtering of Time-Varying Sparse Signals via $\ell _1$ Minimization , 2015, IEEE Transactions on Signal Processing.

[20]  Guang-Hong Yang,et al.  Secure State Estimation Against Sparse Sensor Attacks With Adaptive Switching Mechanism , 2018, IEEE Transactions on Automatic Control.

[21]  Paulo Tabuada,et al.  Secure Estimation and Control for Cyber-Physical Systems Under Adversarial Attacks , 2012, IEEE Transactions on Automatic Control.

[22]  Florian Dörfler,et al.  Attack Detection and Identification in Cyber-Physical Systems -- Part II: Centralized and Distributed Monitor Design , 2012, ArXiv.

[23]  Bruno Sinopoli,et al.  A Graph-Theoretic Characterization of Perfect Attackability for Secure Design of Distributed Control Systems , 2017, IEEE Transactions on Control of Network Systems.

[24]  R. Tibshirani.  The Lasso Problem and Uniqueness , 2012, 1206.0313.

[25]  René Vidal,et al.  Linear systems with sparse inputs: Observability and input recovery , 2015, 2015 American Control Conference (ACC).

[26]  Jean-Jacques Fuchs,et al.  Recovery of exact sparse representations in the presence of bounded noise , 2005, IEEE Transactions on Information Theory.

[27]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[28]  Enrico Magli,et al.  GPU-accelerated algorithms for compressed signals recovery with application to astronomical imagery deblurring , 2017, ArXiv.

[29]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[30]  Chengnian Long,et al.  Dynamic State Recovery for Cyber-Physical Systems Under Switching Location Attacks , 2017, IEEE Transactions on Control of Network Systems.

[31]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[32]  Paulo Tabuada,et al.  Event-Triggered State Observers for Sparse Sensor Noise/Attacks , 2013, IEEE Transactions on Automatic Control.

[33]  Justin K. Romberg,et al.  Discrete and Continuous-Time Soft-Thresholding for Dynamic Signal Recovery , 2014, IEEE Transactions on Signal Processing.

[34]  Enrico Magli,et al.  Distributed ADMM for in-network reconstruction of sparse signals with innovations , 2014, 2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP).