On the observability of linear systems from random, compressive measurements

Wakin, Michael B.; Sanandaji, Borhan M.; Vincent, Tyrone L.

doi:10.1109/cdc.2010.5718068

Cited by 32 publications

(39 citation statements)

References 13 publications

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“…2) RIP for Observability Matrices: As evidence of one other potential application, a recent paper in the control systems literature [36] reveals that our results for DBD and RBD matrices also have immediate extensions to the observability matrices that arise in the analysis of linear dynamical systems. Suppose that x j denotes the state of a system at time j, a matrix A describes the evolution of the system such that x j = Ax j−1 , and a matrix C describes the observation of the system such that y j = Cx j .…”

Section: ) Rip For Toeplitz Matricesmentioning

confidence: 99%

“…A recent paper [36], however, details how O in fact has a straightforward relationship to a block diagonal matrix; how in certain settings (such as when A is unitary and C is random) O can satisfy the RIP; and how this can enable the recovery of sparse initial states x 1 from far fewer measurements than conventional rank-based observability theory suggests.…”

Section: ) Rip For Toeplitz Matricesmentioning

confidence: 99%

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Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing

Park

Yap

Rozell

et al. 2011

IEEE Trans. Signal Process.

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Theoretical analysis of randomized, compressive operators often depends on a concentration of measure inequality for the operator in question. Typically, such inequalities quantify the likelihood that a random matrix will preserve the norm of a signal after multiplication. When this likelihood is very high for any signal, the random matrices have a variety of known uses in dimensionality reduction and Compressive Sensing. Concentration of measure results are well-established for unstructured compressive matrices, populated with independent and identically distributed (i.i.d.) random entries. Many real-world acquisition systems, however, are subject to architectural constraints that make such matrices impractical. In this paper we derive concentration of measure bounds for two types of block diagonal compressive matrices, one in which the blocks along the main diagonal are random and independent, and one in which the blocks are random but equal. For both types of matrices, we show that the likelihood of norm preservation depends on certain properties of the signal being measured, but that for the best case signals, both types of block diagonal matrices can offer concentration performance on par with their unstructured, i.i.d. counterparts. We support our theoretical results with illustrative simulations as well as (analytical and empirical) investigations of several signal classes that are highly amenable to measurement using block diagonal matrices. Finally, we discuss applications of these results in establishing performance guarantees for solving signal processing tasks in the compressed domain (e.g., signal detection), and in establishing the Restricted Isometry Property for the Toeplitz matrices that arise in compressive channel sensing.

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Section: ) Rip For Toeplitz Matricesmentioning

confidence: 99%

Section: ) Rip For Toeplitz Matricesmentioning

confidence: 99%

Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing

Park

Yap

Rozell

et al. 2011

IEEE Trans. Signal Process.

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“…In terms of a difference equation, for example, a sparse system may be a high-dimensional system with only a few non-zero coefficients or it may be a system with an impulse response that is long but contains only a few non-zero terms. Multipath propagation [1]- [3], sparse channel estimation [4]- [6], largescale interconnected systems with sparse graph flow [7]- [10], time varying systems with few piecewise-constant parameter changes [11], [12], and sparse initial state estimation [13], [14] are examples involving dynamical systems that are highdimensional in terms of their ambient dimension but have a sparse (low-order) representation.…”

Section: A High-dimensional But Sparse Dynamical Systemsmentioning

confidence: 99%

A tutorial on recovery conditions for compressive system identification of sparse channels

Sanandaji

Vincent

Poolla

et al. 2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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Abstract-In this tutorial, we review some of the recent results concerning Compressive System Identification (CSI) (identification from few measurements) of sparse channels (and in general, Finite Impulse Response (FIR) systems) when it is known a priori that the impulse response of the system under study is sparse (high-dimensional but with few nonzero entries) in an appropriate basis. For the systems under study in this tutorial, the system identification problem boils down to an inverse problem of the form Ax = b, where the vector x ∈ R N is high-dimensional but with K N nonzero entries and the matrix A ∈ R M ×N is underdetermined (i.e., M < N ). Over the past few years, several algorithms with corresponding recovery conditions have been proposed to perform such a recovery. These conditions provide the number of measurements sufficient for correct recovery. In this note, we review alternate approaches to derive such recovery conditions concerning CSI of FIR systems whose impulse response is known to be sparse.

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“…The observability matrix was analyzed in the framework of compressive sensing and the observability and identification of linear system were discussed in [12] and [13]. With a different motivation, the compressive sensing was used to sparsify the system states, and then the recovered states were used for feedback [14].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of non-vector space control via compressive feedbacks

Zhao

Sun

et al. 2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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A new non-vector space control method with compressive feedback is presented in this paper. Two problems are discussed. The first is the development of the non-vector space control method. It formulates the control problem in the space of sets instead of the vector space, and it provides a convenient way to model physical phenomenon such as shapes and images. For the non-vector space dynamical systems, the issues related to stability analysis and design of controllers are discussed in the paper. The second problem is to design controllers for non-vector space dynamic systems using compressive feedback, which is a new concept proposed in this paper. Compressive feedback, obtained from the original feedback, reduces the amount of data for the original feedback but retains the essential information to ensure the stability and other key performance measures for a control system. The design methods for controllers based on compressive feedback are developed in this paper. Furthermore, the non-vector space control system with compressive feedback has been successfully applied to visual servoing, and simulations results have clearly demonstrated the superb performance and advantages.

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On the observability of linear systems from random, compressive measurements

Cited by 32 publications

References 13 publications

Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing

Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing

A tutorial on recovery conditions for compressive system identification of sparse channels

Stability analysis of non-vector space control via compressive feedbacks

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