IN recent years, the problems of sparse signal recovery

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1 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL Distributed Sparse Signal Estimation in Sensor Networs Using H -Consensus Filtering Haiyang Yu Yisha Liu Wei Wang Abstract This paper is concerned with the sparse signal recovery problem in sensor networs, 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 networ. A so-called l 1-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 l 1-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 networ, sparse signal, distributed H filter, consensus filter. I. INTRODUCTION IN recent years, the problems of sparse signal recovery have been widely investigated because of the emergence of new signal sampling theory which is nown as compressed sampling or compressed sensing CS [1 3]. Using less measurements than those required in Shannon sample principle, a sparse signal can be recovered with overwhelg probability by solving a 1-norm imization problem. This convex optimization problem can be solved by various methods, such as basis pursuit de-noising BPDN, least absolute shrinage and selection operator LASSO, Dantzig selector DS, etc. Many researchers have attempted to discuss this problem in the classic framewor 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 imization as its core computation and propagated second order statistics through time similar to classic Kalman filtering. The dual problem of 1-norm imization was solved in [6] by introducing a pseudo-measurement equation into Kalman filtering, Manuscript received June 19, 2013; accepted October 16, This wor was supported by National Natural Science Foundation of China Recommended by Associate Editor Jie Chen Citation: Haiyang Yu, Yisha Liu, Wei Wang. Distributed sparse signal estimation in sensor networs using H -consensus filtering. IEEE/CAA Journal of Automatica Sinica, 2014, 12: Haiyang Yu is with the Research Center of Information and Control, Dalian University of Technology, Dalian , China yuhaiyang08@gmail.com. Yisha Liu is with the School of Information Science and Technology, Dalian Maritime University, Dalian , China liuyisha@dlut.edu.cn. Wei Wang is with the Research Center of Information and Control, Dalian University of Technology, Dalian , China wangwei@dlut.edu.cn. and the Bayesian interpretation of this method was provided in [7]. Distributed sensor networ is an important way of data acquisition in engineering, and the distributed estimation or filtering 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 trac the average measurement of the whole networ. 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 nown that robustness of Kalman filtering is not satisfactory. 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 framewor 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 networ. A l 1 - regularized H filter is established at first and the pseudomeasurement equation can be interpreted as a 1-norm regularization 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 using 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 remars. 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 xi 0}. x 0 is the cardinality of Supp{x}. The 1-norm and 2-norm of x are defined as

2 150 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 x 1 n i1 xi and x 2 x T x, respectively. x 2 P means the product x T P x. sgn is the sign function. E[ξ] defines the mathematical expectation of random variable ξ. Nµ, σ 2 stands for normal distribution with mean µ and variance σ 2. U int [a, b] defines the integer uniform distribution in the interval [a, b]. vec{x 1,, x n } : [ ] T. x T 1 x T n diag{a 1,, A n } denotes a bloc-diagonal matrix whose diagonal blocs are given by A 1,, A n. S 0 respectively, S 0 means a real symmetric matrix S is positive semidefinite respectively, positive definite. II. SPARSE SIGNAL ESTIMATION USING KALMAN FILTERING This section briefly overviews some basic issues in compressed sampling theory, and introduces the pseudomeasurement embedded Kalman filtering method proposed in [6]. More details can be found in [1 3, 6, 7]. A. Sparse Signal Recovery A signal x R n is sparse if x 0 n, and x is called s- sparse if x 0 s. Assume {x } N 0 is an unnown discretetime sparse signal sequence in R n and x evolves according to the following dynamic model x +1 A x + w, 1 where A R n n is the state transition matrix, {w } N 1 0 is a zero-mean white Gaussian sequence with covariance Q 0, and x 0 Nx 0 1, P 0 1. The m-dimensional linear measurement of x is y C x + v, 2 where C R m n is the measurement matrix, and {v } N 0 is a zero-mean white Gaussian sequence with covariance R 0. Extracting signal x from measurement y is an ill-posed problem in general when m < n. As shown in [1 3], x can be accurately recovered by solving the following optimization problem n ˆx 0, s.t. y C ˆx 2 2 ε. 3 ˆx R But the optimization problem 3 is NP-hard and cannot be solved effectively. Fortunately, the authors of [1] have shown that if the measurement matrix C obeys the so-called restricted isometry property RIP, the solution of 3 can be obtained with overwhelg probability by solving the following convex optimization problem n ˆx 1, s.t. y C ˆx 2 2 ε. 4 ˆx R This is a fundamental result in compressive sampling, and one of the deep results is that for a s-sparse signal in R n, only on the order of m s log n samples are needed to reconstruct it. B. Pseudo-measurement Embedded Kalman Filtering For the system given in 1 and 2, Kalman filtering can provide an estimate of x which is a solution of the following unconstrained l 2 imization problem [ ˆx R ne x y 1,,y x ˆx 2 2 ]. 5 In [6], the authors discussed the stochastic case of 4 [ ] n ˆx 1, s.t. E x y ˆx R 1,,y x ˆx 2 2 ε 6 and its dual problem [ ] ˆx R ne x y 1,,y x ˆx 2 2, s.t. ˆx 1 ε. 7 By constructing a pseudo-measurement equation 0 H x ε, 8 where H sgnx T, and ε N0, σ2 serves as the fictitious measurement noise, the constrained optimization problem 7 can be solved in the framewor of Kalman filtering and the specific method has been summarized as CSembedded KF with 1-norm constraint CSKF-1 algorithm in [6]. In the pseudo-measurement equation 8, the measurement matrix H is state-dependent, and it can be approximated by Ĥ sgnˆx T. The divergence of this approximation was discussed in [7]. Furthermore, σ is a tuneable parameter which deteres the tightness of constraint on 1-norm of the state estimate ˆx. III. l 1 -REGULARIZED H FILTERING In this section, the pseudo-measurement equation is combined with H filtering, and a l 1 -regularized H filtering is developed for estimating a sparse signal sequence. Define the augmented measurement equation using 2 and 8 as ȳy C x + v, 9 [ ] [ ] [ ] y C v where ȳy, C, v 0 H ε, and [ ] denote R R 0 0 σ 2. Consider the system described by 1 and 9 with Gramian matrix x 0 w,, v x 0 w j v j P Q δ j R δ j 10 where δ j is Kronecer delta function. We aim to design a full-order filter in the form of ˆx A 1ˆx 1 + K ȳy C A 1ˆx 1, 11 such that for all non-zero w and v, the filtering error x x ˆx satisfies the following l 1 -regularized H performance constraint N 1 x 2 2 < γ x 2 0 x w 2 + Q v 2 R σ 2 x 2 1, 12 where γ > 0 is a prescribed scalar. There exists a filter in the form of 11 achieving the performance 12, if and only if the filtering error covariance matrix P satisfies 1 + CT R 1 C + σ 2 HT H γ 2 I 0, 13

3 YU et al.: DISTRIBUTED SPARSE SIGNAL ESTIMATION IN SENSOR NETWORKS USING H -CONSENSUS FILTERING 151 for 0 < N, where the initial value is P 0 1, and the predicted error covariance matrix P 1 satisfies the Riccati recursion P 1 A 1 1 A T 1 + Q The filtered estimates ˆx are recursively computed as where ˆx ˆx 1 + K ȳy C ˆx 1, 15 K P 1 CT C P 1 CT + R 1, 16 and the predicted estimates ˆx 1 A 1ˆx Proof. Define the quadratic performance function N 1 J x 0 x w 2 + v Q R 1 0 σ 2 x 2 1 γ 2 x 0 x y Cx 2 R 1, 0 where y y 0 ˆx 0 x N 1 0 w 2 + Q 1, C C H I 0 and R R σ 2 0. Then the proposed filter is not hard 0 0 γ 2 I to obtain according to Krein space Kalman filtering in [15]. Details of the proof are omitted. In [7], the pseudo-measurement equation 8 was interpreted in Bayesian filtering framewor, and a semi-gaussian prior distribution was discussed. Here, according to Theorem 1, the pseudo-measurement equation in H filtering can be interpreted as a 1-norm regularization term added to the classic H performance index. To establish a distributed filtering algorithm, the result in Theorem 1 will be rebuilt in information form, so by denoting z C T 1 R ȳy C T R 1 y, 18 S [1] S [2] CT R 1 C, 19 σ 2 HT H, 20 we can obtain the l 1 -regularized H information filter. The filter established in Theorem 1 is equivalent to the following information form. Measurement update: ˆ 1 + S[1], 21 ˆx ˆx 1 + ˆP z S [1] ˆx Pseudo-measurement update: ˆx ˆP 1 + S[2] γ 2 I, 23 I + γ 2 I 1 S [2] ˆx. 24 Time update: P +1 A P A T + Q, 25 ˆx +1 A ˆx. 26 Proof. According to matrix inversion lemma, we have K P 1 CT C P 1 CT + R C T 1 R C T 1 CT R 1 + γ 2 I 1 CT R 1, then 15 can be rewritten as ˆx ˆx γ 2 I 1 z S [1] + S[2] ˆx 1 I + γ 2 I 1 S [2] ˆx γ 2 I 1 z S [1] ˆx 1. On the other hand, by substituting 22 into 24, we have ˆx I + γ 2 I 1 S [2] ˆx 1 + ˆP z S [1] ˆx 1 I + γ 2 I 1 S [2] ˆx 1 + I + γ 2 I 1 S [2] ˆP z S [1] ˆx 1. It is easy to obtain the following equation 1 ˆ I + γ 2 I 1 S [2] ˆ I S [2] γ 2 I 1 S [2] I + ˆP S [2] ˆ + γ 2 I, so 1 + γ 2 I I + γ 2 I 1 S [2] ˆP, which means 15 is equivalent to 22 and 24. Equation 13 is obtained immediately by combining 21 and 23. H IV. DISTRIBUTED H -CONSENSUS FILTERING WITH PSEUDO-MEASUREMENT In this section, we will develop a distributed H - consensus filtering based on the results presented in Section III. Consider a sensor networ whose topology is represented by an undirected graph G V, E, A of order r with the set of nodes V {1, 2,, r}, the set of edges E V V, and the adjacency matrix A [a ij ] r r with nonnegative adjacency elements a ij. An edge of G is denoted by an unordered pair i, j. The adjacency elements associated with the edges are positive, i.e., a ij > 0 i, j E. Node j is called a neighbor of node i if i, j E and j i. The neighbor set of node i is denoted by N i. Assume G is strongly connected. Assume the measurement of sensor node [ i is] described yi, by the linear model 2. Denote ȳy i,, Ci, 0

4 152 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 [ ] [ ] Ci, vi,, v H i, ε, we can get the following augmented measurement equation of sensor i, i ȳy i, C i, x + v i,, 27 where C i, is the measurement matrix of sensor i, H sgnx T, v i, N0, R i,, ε i, N0, σ2 i,. And let R i, diag{r i,, σi, 2 }. Define Y vec{ȳy 1,,,ȳy r, }, V vec{ v 1,,, v r, }, C vec{ C 1,,, C r, } and R diag{ R 1,,, R r, }. Then we have the augmented global measurement Denoting z 1 r r i1 C T i, Y C x + V. 28 R 1 i,ȳy i, 1 r r i1 C i, R 1 i, y i,, 29 S 1 r r i1 where S[1] 1 r 1 r r i1 σ 2 C T i, 1 R C [1] [2] i, i, : S + S, 30 r i1 CT i, R 1 i, C i, and S[2] i, HT H, we can obtain a local filter for each sensor node, which has the same performance as the centralized filter presented in Theorem 1 by using the augmented global measurement 28. Suppose every node of the networ applies the following filter ˆ i, [1] i, 1 + S, ˆx i, ˆx i, 1 + ˆP i, z i, ˆx i, ˆP 1 [2] i, + S γ 2 I, I i, + γ 2 I P i,+1 A P i, A T + rq, ˆx i,+1 A ˆx i,, S [1] ˆx i, 1, 1 S[2] ˆx i,, where P i,0 1 rp 0 1, then the local and central state estimates for all nodes are the same, i.e., ˆx i, ˆx. Proof. The proof is a direct combination of Corollary 1 and the result in [11]. In Corollary 2, the average measurement z and the average [1] inverse covariance matrix S can be computed approximately on each node using the following high-pass consensus filter proposed in [11] { q i β j N i q j q i + u j u i, β > 0, ξ i q i + u i. 31 It has been verified that, if the networ is strongly connected, ξ i 1 r r i1 u i as t for any i V. S[2] can be approximated on each node by Ŝ[2] i, σ 2 i, ĤT i,ĥi,, where Ĥ i, sgnˆx i, T. The distributed H filtering can be summarized as the following Algorithm 1. V. ILLUSTRATIVE EXAMPLE In this section, we will illustrate effectiveness of the method proposed in Section IV. Without loss of generality, consider a sensor networ with 6 nodes as shown in Fig. 1, whose topology is represented by an undirected graph G V, E, A with the set of nodes V {1, 2, 3, 4, 5, 6}, the set of edges E {1, 1, 1, 2, 1, 3, 2, 2, 2, 3, 2, 4, 2, 5, 3, 3, 3, 5, 3, 6, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6}, and the adjacency matrix Fig. 1. A Topological structure of the sensor networ. Here, we attempt to estimate a 10-sparse signal sequence {x } in R 256 and assume the sequence has a constant support set. The dynamics of {x } can be described by the following linear system { x i + w x +1 i i, i Supp{x 0 }, 32 0, otherwise, where {w i} is a zero-mean Gaussian white noise sequence with standard deviation For system 1, we have A I 256 and Q 10 4 I 256. Both the index i Supp{x } and the value of x i are unnown. The initial sparse signal x 0 is generated by creating the support set as i U int [1, 256] and setting the value as x 0 i N0, 1. For sensor node i, the sensing matrix C i, R consists of entries sampled according to N0, Then, we have the global measurement y cent vec{y 1,, y 6 } R 72

5 YU et al.: DISTRIBUTED SPARSE SIGNAL ESTIMATION IN SENSOR NETWORKS USING H -CONSENSUS FILTERING 153 TABLE I NMSEs WHEN 80 Algorithms KF CSKF-1 Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Errors and the global measurement matrix C cent vec{c 1,, C 6 } R This type of matrix lie C cent has been shown to satisfy the restricted isometry property with overwhelg probability. The measurement noise v i, N0, R i,, where R i, 10 4 I 12, and denote R cent diag{r 1,, R 6 } R Set filter initial states as ˆx i,0 1 0 i 1,, 6, and filter parameters as β 0.2, λ 1, γ 10, σ i, , and N 80. Now we are ready to design the distributed H filters for system 38. The centralized Kalman filtering and CSKF-1 algorithm proposed in [6] are implemented respectively using y cent, C cent and R cent, and we will compare their performances with that of Algorithm 1. The following normalized mean squared error NMSE is employed to evaluate the performance, e x ˆx 2 2 x 2. 2 The results are presented in Figs In Fig. 2, the first figure shows the actual signal x 80, the second figure gives ˆx 80 from centralized Kalman filtering, the third figure shows ˆx 80 from centralized CSKF-1 algorithm. Fig. 3 gives the estimates of x 80 from sensor nodes using Algorithm 1. It is obvious that all nodes give satisfactory estimates of the actual sparse signal. Fig. 4 presents the normalized mean squared errors, which indicates that all node filters are stable and reach a consensus on the estimates of sparse signals. Moreover, there exist smaller steady errors by using Algorithm 1 than centralized Kalman filtering and CSKF-1 algorithm, and the errors when 80 are presented in Table I. All of these results demonstrate effectiveness of the distributed filter presented in this paper. algorithm was established. A numerical example verified effectiveness of the proposed method. However, the algorithm presented in this paper is only suitable to deal with the timevarying sparse signal with an invariant or slowly changing support set, and the more general methods need to combine a support set estimator with the filter. The decomposition of sensing matrix needs further research as well. These problems will be discussed in future wor. Fig. 3. Estimates of x 80 using Algorithm 1. Fig. 2. Actual signal x 80 and its estimates ˆx 80 using centralized KF and CSKF-1. VI. CONCLUSION In this paper, the problem of distributed sparse signal estimation in sensor networs has been considered. The pseudomeasurement equation was interpreted as a 1-norm regularization term in the classic H performance index, and the l 1 -regularized H filtering method was proposed. By means of a high-pass consensus filter, the distributed H filtering Fig. 4. NMSEs of centralized KF, CSKF-1 and Algorithm 1. REFERENCES [1] Candes E J, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 522: [2] Candes E J, Romberg J, Tao T. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 2006, 598: [3] Candes E J, Wain M B. An introduction to compressive sampling. IEEE Signal Processing Magazine, 2008, 252: 21 30

6 154 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 [4] Vaswani D. Kalman filtered compressed sensing. In: Proceedings of the 15th IEEE International Conference on Image Processing. San Diego, USA: IEEE, [5] Charles A S, Rozell C J. Dynamic filtering of sparse signals using reweighted l 1. In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing. Vancouver, Canada: IEEE, [6] Carmi A, Gurfil P, Kanevsy D. Methods for sparse signal recovery using Kalman filtering with embedded pseudo-measurement norms and quasi-norms. IEEE Transactions on Signal Processing, 2010, 584: [7] Kanevsy D, Carmi A, Horesh L. Kalman filtering for compressed sensing. In: Proceedings of the 13th Conference on Information Fusion FUSION. Edinburgh, UK: IEEE, [8] Wan Yi-Ming, Dong Wei, Ye Hao. Distributed H filtering with consensus strategies in sensor networs: considering consensus tracing error. Acat Automatica Sinica, 2012, 387: in Chinese Yisha Liu Lecturer at the School of Information Science and Technology, Dalian Maritime University. She received her B. S. and Ph. D. degrees in control theory and engineering from Dalian University of Technology in 2005 and 2011, respectively. Her research interest covers sensor networs and intelligent mobile robots. Wei Wang Professor at the School of Control Science and Engineering, Dalian University of Technology. He obtained his Bachelor, Master and Ph. D. degrees in industrial automation from Northeastern University in 1982, 1986 and 1988, respectively. His research interest covers adaptive control, predictive control, robotics, and computer integrated manufacturing systems. Corresponding author of this paper. [9] Wang Shuai, Yang Wen, Shi Hong-Bo. Consensus-based filtering algorithm with pacet-dropping. Acat Automatica Sinica, 2010, 3612: in Chinese [10] Feng Xiao-Liang, Wen Cheng-Lin, Liu Wei-Feng, Li Xiao-Fang, Xu Li-Zhong. Sequential fusion finite horizon H filtering for Multisenor System. Acat Automatica Sinica, 2013, 399: in Chinese [11] Olfati-Saber R. Distributed Kalman filter in sensor networs. In: Proceedings of the 46th IEEE Conference on Decision and Control. Los Angeles, New Orleans, LA: IEEE, [12] Olfati-Saber R. Kalman-consensus filter: optimality, stability, and performance. In: Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai, China: IEEE, [13] Shen B, Wang Z D, Hung Y S, Chesi G. Distributed H filtering for polynomial nonlinear stochastic systems in sensor networs. IEEE Transactions on Industrial Electronics, 2011, 585: [14] Ugrinovsii V. Distributed robust filtering with H consensus of estimates. Automatica, 2011, 471: 1 13 [15] Hassibi B, Sayed A H, Kailath T. Indefinite Quadratic Estimation and Control: a Unified Approach to H 2 and H Theories. Philadelphia: SIAM, Haiyang Yu Ph. D. candidate at the School of Control Science and Engineering, Dalian University of Technology. He received his B. S. degree in mathematics from Liaoning Normal University in 2006, and M.S. degree in applied mathematics from Dalian University of Technology in His research interest covers distributed estimation in sensor networs, multi-source information fusion and multi-robot system.