IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 149 Distributed Sparse Signal Estimation in Sensor Networks Using H ∞ H ∞ H ∞ -Consensus Filtering Haiyang Yu Yisha Liu Wei Wang Abstract—This paper is concerned with the sparse signal recovery problem in sensor networks, and the main purpose is to design a filter for each sensor node to estimate a sparse signal sequence using the measurements distributed over the whole network. A so-called ℓ1-regularized H∞ filter is established at first by introducing a pseudo-measurement equation, and the necessary and sufficient condition for existence of this filter is derived by means of Krein space Kalman filtering. By embedding a high-pass consensus filter into ℓ1-regularized H∞ filter in information form, a distributed filtering algorithm is developed, which ensures that all node filters can reach a consensus on the estimates of sparse signals asymptotically and satisfy the prescribed H∞ performance constraint. Finally, a numerical example is provided to demonstrate effectiveness and applicability of the proposed method. Index Terms—Sensor network, sparse signal, distributed H∞ filter, consensus filter. I. I NTRODUCTION I N recent years, the problems of sparse signal recovery have been widely investigated because of the emergence of new signal sampling theory which is known as compressed sampling or compressed sensing (CS) [1−3] . Using less mea- surements than those required in Shannon sample principle, a sparse signal can be recovered with overwhelming probability by solving a 1-norm minimization problem. This convex optimization problem can be solved by various methods, such as basis pursuit de-noising (BPDN), least absolute shrinkage and selection operator (LASSO), Dantzig selector (DS), etc. Many researchers have attempted to discuss this problem in the classic framework of signal estimation, such as Kalman filtering (KF). In [4], the Dantzig selector was used to estimate the support set of a sparse signal, and then a reduced-order Kalman filter was employed to recover the signal. In [5], the authors presented an algorithm based on a hierarchical probabilistic model that used re-weighted ℓ 1 minimization as its core computation and propagated second order statistics through time similar to classic Kalman filtering. The dual problem of 1-norm minimization was solved in [6] by intro- ducing a pseudo-measurement equation into Kalman filtering, Manuscript received June 19, 2013; accepted October 16, 2013. This work was supported by National Natural Science Foundation of China (61305128). Recommended by Associate Editor Jie Chen Citation: Haiyang Yu, Yisha Liu, Wei Wang. Distributed sparse signal estimation in sensor networks using H∞-consensus filtering. IEEE/CAA Journal of Automatica Sinica, 2014, 1(2): 149-154 Haiyang Yu is with the Research Center of Information and Con- trol, Dalian University of Technology, Dalian 116024, China (e-mail: yuhaiyang08@gmail.com). Yisha Liu is with the School of Information Science and Technology, Dalian Maritime University, Dalian 116026, China (e-mail: liuyisha@dlut.edu.cn). Wei Wang is with the Research Center of Information and Control, Dalian University of Technology, Dalian 116024, China (e-mail: wang- wei@dlut.edu.cn). and the Bayesian interpretation of this method was provided in [7]. Distributed sensor network is an important way of data acquisition in engineering, and the distributed estimation or fil- tering problems have been paid much attention recently [8−10] . In [11], three types of distributed Kalman filtering algorithms were proposed. A distributed high-pass consensus filter was used to fuse local sensor measurements, such that all nodes could track the average measurement of the whole network. These algorithms were established based on Kalman filtering in information form, and analysis of stability and performance of Kalman-consensus filter was provided in [12]. It is well known that robustness of Kalman filtering is not satisfac- tory. In [13], a design method of distributed H ∞ filtering for polynomial nonlinear stochastic systems was presented, and the filter parameters were derived in terms of a set of parameter-dependent linear matrix inequalities (PDLMIs) such that a desired H ∞ performance was achieved. A H ∞ - type performance index was established in [14] to measure the disagreement between adjacent nodes, and the distributed robust filtering problem was solved with a vector dissipativity method. Nevertheless, the distributed sparse signal estimation problem has not been adequately researched yet in the frame- work of H ∞ filtering. In this paper, we aim to combine the pseudo-measurement method with H ∞ filtering, and develop a distributed H ∞ filtering algorithm to estimate a sparse signal sequence using the measurements distributed over a sensor network. A ℓ 1 - regularized H ∞ filter is established at first and the pseudo- measurement equation can be interpreted as a 1-norm regular- ization term added to the classic H ∞ performance index. To develop the distributed algorithm, a high-pass consensus filter is embedded into ℓ 1 -regularized H ∞ filter in information form, such that all node filters can reach a consensus on the estimates of sparse signals asymptotically and satisfy the prescribed H ∞ performance constraint. The remainder of this paper is organized as follows. Section II gives a brief overview of basic problems in compressed sampling, and introduces the sparse signal recovery method us- ing Kalman filtering with embedded pseudo-measurement. In Section III, the centralized ℓ 1 -regularized H ∞ filtering method is established by means of Krein space Kalman filtering, and the corresponding information filter is derived. A high-pass consensus filter is employed to develop the distributed filtering algorithm in Section IV. Simulation results are given in Section V to demonstrate effectiveness of the proposed method, and Section VI provides some concluding remarks. Notation. The notation used here is fairly standard except where otherwise stated. The support set of x ∈ R n is defined as Supp{x} = {i|x(i) = 0}. ‖x‖ 0 is the cardinality of Supp{x}. The 1-norm and 2-norm of x are defined as