The vast majority of today's critical infrastructure is supported by numerous feedback control loops and an attack on these control loops can have disastrous consequences. This is a major concern since modern control systems are becoming large and decentralized and thus more vulnerable to attacks. This paper is concerned with the estimation and control of linear systems when some of the sensors or actuators are corrupted by an attacker. We give a new simple characterization of the maximum number of attacks that can be detected and corrected as a function of the pair $(A,C)$ of the system and we show in particular that it is impossible to accurately reconstruct the state of a system if more than half the sensors are attacked. In addition, we show how the design of a secure local control loop can improve the resilience of the system. When the number of attacks is smaller than a threshold, we propose an efficient algorithm inspired from techniques in compressed sensing to estimate the state of the plant despite attacks. We give a theoretical characterization of the performance of this algorithm and we show on numerical simulations that the method is promising and allows to reconstruct the state accurately despite attacks. Finally, we consider the problem of designing output-feedback controllers that stabilize the system despite sensor attacks. We show that a principle of separation between estimation and control holds and that the design of resilient output feedback controllers can be reduced to the design of resilient state estimators.

Secure Estimation and Control for Cyber-Physical Systems Under Adversarial Attacks

Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

Critical infrastructures must continuously operate safely and reliably, despite a variety of potential system disturbances. Given their strict operating requirements, such systems are automated and controlled in real time by several digital controllers receiving measurements from sensors and transmitting control signals to actuators. Since these physical systems are often spatially distributed, there is a need for information technology (IT) infrastructures enabling the timely data flow between the system components. These networked control systems are ubiquitous in modern societies [1]. Examples include the electric power network, intelligent transport systems, and industrial processes.

Secure Control Systems: A Quantitative Risk Management Approach

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P ≠ NP.

Certifying the Restricted Isometry Property is Hard

The incoherence between measurement and sparsifying transform matrices and the restricted isometry property (RIP) of measurement matrix are two of the key factors in determining the performance of compressive sensing (CS). In CS-MRI, the randomly under-sampled Fourier matrix is used as the measurement matrix and the wavelet transform is usually used as sparsifying transform matrix. However, the incoherence between the randomly under-sampled Fourier matrix and the wavelet matrix is not optimal, which can deteriorate the performance of CS-MRI. Using the mathematical result that noiselets are maximally incoherent with wavelets, this paper introduces the noiselet unitary bases as the measurement matrix to improve the incoherence and RIP in CS-MRI. Based on an empirical RIP analysis that compares the multichannel noiselet and multichannel Fourier measurement matrices in CS-MRI, we propose a multichannel compressive sensing (MCS) framework to take the advantage of multichannel data acquisition used in MRI scanners. Simulations are presented in the MCS framework to compare the performance of noiselet encoding reconstructions and Fourier encoding reconstructions at different acceleration factors. The comparisons indicate that multichannel noiselet measurement matrix has better RIP than that of its Fourier counterpart, and that noiselet encoded MCS-MRI outperforms Fourier encoded MCS-MRI in preserving image resolution and can achieve higher acceleration factors. To demonstrate the feasibility of the proposed noiselet encoding scheme, a pulse sequences with tailored spatially selective RF excitation pulses was designed and implemented on a 3T scanner to acquire the data in the noiselet domain from a phantom and a human brain. The results indicate that noislet encoding preserves image resolution better than Fouirer encoding.

/pdf/multichannel-compressive-sensing-mri-using-noiselet-encoding-3x4rbcp2wg.pdf

Multichannel compressive sensing MRI using noiselet encoding.

This paper deals with the computational complexity of conditions which guarantee that the NP-hard problem of finding the sparsest solution to an underdetermined linear system can be solved by efficient algorithms. In the literature, several such conditions have been introduced. The most well-known ones are the mutual coherence, the restricted isometry property (RIP), and the nullspace property (NSP). While evaluating the mutual coherence of a given matrix is easy, it has been suspected for some time that evaluating RIP and NSP is computationally intractable in general. We confirm these conjectures by showing that for a given matrix A and positive integer k, computing the best constants for which the RIP or NSP hold is, in general, NP-hard. These results are based on the fact that determining the spark of a matrix is NP-hard, which is also established in this paper. Furthermore, we also give several complexity statements about problems related to the above concepts.

The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent phase retrieval algorithms exploit underlying sparsity of the unknown signal in order to improve recovery performance. In this work, we consider such a sparse signal prior in the context of phase retrieval, when the sparsifying dictionary is not known in advance. Our algorithm jointly reconstructs the unknown signal—possibly corrupted by noise—and learns a dictionary such that each patch of the estimated image can be sparsely represented. Numerical experiments demonstrate that our approach can obtain significantly better reconstructions for phase retrieval problems with noise than methods that cannot exploit such “hidden” sparsity. Moreover, on the theoretical side, we provide a convergence result for our method.

/pdf/dolphin-dictionary-learning-for-phase-retrieval-40vgerkoal.pdf

DOLPHIn—Dictionary Learning for Phase Retrieval

The efficient sparse coding and reconstruction of signal vectors via linear observations has received a tremendous amount of attention over the last decade. In this context, the automated learning of a suitable basis or overcomplete dictionary from training data sets of certain signal classes for use in sparse representations has turned out to be of particular importance regarding practical signal processing applications. Most popular dictionary learning algorithms involve NP-hard sparse recovery problems in each iteration, which may give some indication about the complexity of dictionary learning but does not constitute an actual proof of computational intractability. In this technical note, we show that learning a dictionary with which a given set of training signals can be represented as sparsely as possible is indeed \ssb NP-hard. Moreover, we also establish hardness of approximating the solution to within large factors of the optimal sparsity level. Furthermore, we give \ssb NP-hardness and non-approximability results for a recent dictionary learning variation called the sensor permutation problem. Along the way, we also obtain a new non-approximability result for the classical sparse recovery problem from compressed sensing.

On the Computational Intractability of Exact and Approximate Dictionary Learning

The efficient sparse coding and reconstruction of signal vectors via linear observations has received a tremendous amount of attention over the last decade. In this context, the automated learning of a suitable basis or overcomplete dictionary from training data sets of certain signal classes for use in sparse representations has turned out to be of particular importance regarding practical signal processing applications. Most popular dictionary learning algorithms involve NP-hard sparse recovery problems in each iteration, which may give some indication about the complexity of dictionary learning but does not constitute an actual proof of computational intractability. In this technical note, we show that learning a dictionary with which a given set of training signals can be represented as sparsely as possible is indeed NP-hard. Moreover, we also establish hardness of approximating the solution to within large factors of the optimal sparsity level. Furthermore, we give NP-hardness and non-approximability results for a recent dictionary learning variation called the sensor permutation problem. Along the way, we also obtain a new non-approximability result for the classical sparse recovery problem from compressed sensing.

Andreas M. Tillmann

Papers

The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

DOLPHIn—Dictionary Learning for Phase Retrieval

On the Computational Intractability of Exact and Approximate Dictionary Learning

On the Computational Intractability of Exact and Approximate Dictionary Learning