A Distributed Implementation of Steady-State Kalman Filter
aa r X i v : . [ ee ss . S Y ] J a n A Distributed Implementation of Steady-StateKalman Filter
Xu Yang, Jiaqi Yan, Yilin Mo ∗ , and Keyou You Abstract —This paper studies the distributed state estimationin sensor network, where m sensors are deployed to infer the n -dimensional state of a linear time-invariant (LTI) Gaussiansystem. By a lossless decomposition of optimal steady-stateKalman filter, we show that the problem of distributed estimationcan be reformulated as synchronization of homogeneous linearsystems. Based on such decomposition, a distributed estimatoris proposed, where each sensor node runs a local filter usingonly its own measurement and fuses the local estimate of eachnode with a consensus algorithm. We show that the average ofthe estimate from all sensors coincides with the optimal Kalmanestimate. Numerical examples are provided in the end to illustratethe performance of the proposed scheme. I. I
NTRODUCTION
The past decades have witnessed remarkable research in-terests in multi-sensor networked systems. As one of itsimportant focuses, distributed estimation has been widely stud-ied in various applications including robot formation control,environment monitoring, spacecraft navigation (see [1]–[5]for examples). Compared with the centralized architecture, itprovides better robustness, flexibility and reliability [6].One of the most fundamental problems in distributed esti-mation would be the distributed realization of Kalman filter[7]. The solution generally consists of two parts: local mea-surement processing and neighboring information exchange.Recent advances in cooperative control diversify the design ofdistributed estimator. For example, in an early work [8], theauthors suggest a fusion algorithm for two-sensor networks,where local estimate of the first sensor is considered as apseudo measurement of the second one. Due to its ease ofimplementation, this approach has then inspired the sequentialfusion in multi-sensor networks [9], [10], where the multiplenodes repeatedly perform the two-sensor fusion in a sequentialmanner. As the result of serial operation, these algorithmsrequire special communication topology which should besequentially connected as a ring/chain. Gossip algorithm isfound as popular in the fusion process of distributed estimatorsas well [11], [12]. At every round of iteration, the sensor, eitherdeterministically or randomly, selects one node in its neigh-borhood, with which its local information is fused. Recently,Olfati-Saber et. al [13] introduce the consensus algorithms intodistributed estimation and propose Kalman-Consensus Filter(KCF) where the average consensus on local estimates isperformed. Since then, various consensus-based Kalman filterhas been proposed in literature [14]–[16]. Other popular fusion
The authors are with Department of Automation, BNRist, Tsinghua Uni-versity. Emails: [email protected], [email protected],[email protected],[email protected] protocols include diffusion processes ( [17], [18]) and moving-horizon estimation ( [19], [20]).In this paper, we investigate the same problem of estimatingthe state of a LTI Gaussian system using multiple sensors, andaim to provide a new design of distributed estimator, in thesense that data fusion task is distributed over multiple nodesand local solutions of sensor nodes cooperatively recover theKalman estimate.To this end, a lossless decomposition of steady-state Kalmanfilter is proposed in this paper. In order to approach thecentralized benchmark, each sensor node locally estimatesthis decomposition and seeks for the synchronization amongthemselves to get the best performance. Note by doing so, wereformulate the problem of distributed state estimation to thatof linear system synchronization. We, hence, are able to lever-age the methodologies from latter field to propose solutions forformer problem. To be specific, in the synchronization of linearsystems, the dynamics of each agent is described by a LTIsystem. Each agent updates its state by local communicationswith neighbors so that all agents asymptotically reach anagreement. Over the past years, lots of research efforts havebeen devoted to this area in different communication networkswith different control laws (see [21]–[23] for examples).These elegant results offer solid foundations to our designof distributed estimators. Specifically, this work resorts to theprotocol in [21]. The main steps of our estimator are describedas below:1) (Local measurement processing) In order to extract theinformation from local measurements, each sensor performs alocal filter and obtains the local estimate on system state basedon its own measurement.2) (Neighboring information fusion) For the reason ofachieving global convergence, the sensor makes further esti-mation on the local estimates of all the others and transmit it tothe neighbors. A consensus protocol, with subtle modificationon the algorithm proposed in [21], is introduced in fusionprocess to guarantee local nodes cooperatively recover theoptimal Kalman filter.
Notations : For vectors v i ∈ R m i , the vector (cid:2) v T , . . . , v TN (cid:3) T is defined by col { v , . . . , v N } . For any matrix A , we denote ρ ( A ) as its spectral radius. Moreover, A ⊗ B indicates theKronecker product of matrices A and B .II. P ROBLEM F ORMULATION
In this paper, we consider the LTI system as given below: x ( k + 1) = Ax ( k ) + w ( k ) , (1) where x ( k ) ∈ R n is the system state, w ( k ) ∼ N (0 , Q ) isindependent and identically distributed (i.i.d) Gaussian noisewith zero mean and covariance matrix Q ≥ . The initialstate x (0) is also assumed to be Gaussian with zero mean andcovariance matrix Σ > , and is independent from the processnoise { w ( k ) } .Suppose a sensor network consisting of m sensors is mon-itoring the above system. The measurement from each sensor i ∈ { , , ..., m } is given by : y i ( k ) = C i x ( k ) + v i ( k ) , (2)where y i ( k ) ∈ R is the measurement of sensor i , C i is a non-zero n -dimensional row vector, and v i ( k ) ∈ R is the Gaussianmeasurement noise.By stacking the measurement equation, one gets y ( k ) = Cx ( k ) + v ( k ) , (3)where y ( k ) , y ( k ) y ( k ) ... y m ( k ) , C , C C ... C m , v ( k ) , v ( k ) v ( k ) ... v m ( k ) , (4)and v ( k ) is zero-mean i.i.d. Gaussian noise with covariance R ≥ and is independent from w ( k ) and x (0) .Without loss of generality, throughout this paper, we assumethat ( A, C ) is observable. Otherwise a Kalman decompositioncan be used to remove unobservable state. On the other hand, ( A, C i ) may not necessarily be observable, i.e., a single sensormay not be able to observe the whole state space.Note that if all measurements are collected by a single fu-sion center, the following centralized Kalman filter is optimalfor state estimation purpose: ˆ x ( k | k −
1) = A ˆ x ( k − ,P ( k | k −
1) = AP ( k − A T + Q,K ( k ) = P ( k | k − C T (cid:2) CP ( k | k − C T + R (cid:3) − , ˆ x ( k ) = ˆ x ( k | k −
1) + K ( k ) [ y ( k ) − C ˆ x ( k | k − ,P ( k ) = [ I − K ( k ) C ] P ( k | k − , (5)where the recursion starts from initial conditions ˆ x (0) = 0 and P (0) = Σ . Since ( A, C ) is observable, it is well-known that the errorcovariance and Kalman gain will converge to P = lim k →∞ P ( k | k − ,K = P C T (cid:0) CP C T + R (cid:1) − . (6)Since the operation of a typical sensor network lasts for anextended period of time, we assume that the Kalman filter isin the steady state, or equivalently Σ = P , which results in a The results in this paper can be readily generalized to cases where eachsensor outputs a vector measurement. steady-state Kalman filter with fixed gain . Accordingly, theoptimal Kalman estimate is computed recursively as ˆ x ( k + 1) = A ˆ x ( k ) + K ( y ( k ) − C ˆ x ( k ))= ( A − KCA )ˆ x ( k ) + Ky ( k ) . (7)It is clear that optimal estimate (7) requires the informationfrom all sensors. However in a distributed framework, eachsensor is only capable of communicating with immediateneighbors, which renders the centralized solution impractical.Therefore, this paper is devoted to approaching the optimalKalman estimate in a distributed fashion.III. D ECOMPOSITION OF K ALMAN F ILTER
To address the above problem, this section provides adecomposition of Kalman filter, which will be leveraged inthe next section to create a distributed estimation algorithm.Throughout the rest of this paper, we make the followingassumptions:
Assumption 1. A − KCA has n distinct eigenvalues; and2) A − KCA and A do not share any eigenvalues.A. Local decomposition of Kalman filter Assume the eigenvalues of A − KCA are λ , λ , ..., λ n .Since they are distinct, there exists a non-singular matrix V ,such that A − KCA = V Λ V − , (8)where Λ , diag { λ , λ , ..., λ n } . For simplicity, let us denotethe Kalman gain as K = [ K , · · · , K m ] with K i being the i thcolumn of K . Then suppose each sensor i perform the belowfilter based on its local measurement: ˆ ξ i ( k + 1) = Λ ˆ ξ i ( k ) + n y i ( k + 1) , (9)where n ∈ R n is a vector of all ones. The optimal Kalmanfilter can be decomposed as a weighted sum of local estimates,as stated below: Lemma 1 ( [24]) . Suppose Assumption 1 holds. The optimalKalman estimate (7) can be recovered from the local estimates ˆ ξ i ( k ) , i = 1 , , ..., m as ˆ x ( k ) = m X i =1 F i ˆ ξ i ( k ) , (10) where F i = V diag( V − K i ) and diag( V − K i ) is an n × n diagonal matrix with j th diagonal entry equaling the j th entryof vector V − K i . Now let us recall the local filter (9). We note the systemmatrix A may have unstable modes which cause y i ( k ) to beunbounded. To make a feasible implementation of Kalmanfilter, we instead rewrite (9) using the stable residual z i ( k ) (aswill be proved in Lemma 2). That is, z i ( k ) = y i ( k + 1) − β T ˆ ξ i ( k ) , ˆ ξ i ( k + 1) = S ˆ ξ i ( k ) + n z i ( k ) , (11) Notice that even if Σ = P , the Kalman estimate converges to the steady-state Kalman filter, i.e., the steady-state estimator is asymptotically optimal. where S , Λ + n β T , (12)such that β = [ β , · · · , β n ] T solves the following equation: n X i =1 β i ( A − λ i I ) − = I. (13)We further denote matrix G i as G i , C i A ( A − λ I ) − ... C i A ( A − λ n I ) − , (14)which is well defined as A does not share any eigenvalueswith Λ . The validity of the above filter is then proved in thefollowing lemma: Lemma 2.
Consider local filter (11) . The following statementshold at ant instant:1) ˆ ξ i ( k ) is a stable estimate of G i x ( k ) ;2) The local filter (9) is equivalent to (11) ;3) z i ( k ) is stable, i.e., the covariance of z i ( k ) is alwaysbounded.Proof. For any i ∈ V , we denote e i ( k ) , G i x ( k ) − ˆ ξ i ( k ) . (15)One infers from (9) that e i ( k + 1)= G i x ( k + 1) − ˆ ξ i ( k + 1)= G i Ax ( k ) + G i w ( k ) − Λ ˆ ξ i ( k ) − n y i ( k + 1)= G i Ax ( k ) + G i w ( k ) − Λ ˆ ξ i ( k ) − n [ C i Ax ( k ) + C i w ( k ) + v i ( k + 1)]= ( G i − n C i ) Ax ( k ) − Λ ˆ ξ i ( k ) + ( G i − n C i ) w ( k ) − n v i ( k + 1) . (16)By the definition of G i , one knows ( G i − n C i ) A = Λ G i . (17)Therefore, e i ( k + 1) = Λ G i x ( k ) − Λ ˆ ξ i ( k ) + ( G i − n C i ) w ( k ) − n v i ( k + 1)= Λ e i ( k ) + ( G i − n C i ) w ( k ) − n v i ( k + 1) . (18)Hence, e i ( k ) is stable at any time, which proves statement 1).On the other hand, it holds from (13) that β T G i = C i A n X i =1 β i ( A − λ i I ) − = C i A. (19) One thus has z i ( k )= y i ( k + 1) − β T ˆ ξ i ( k )= y i ( k + 1) − β T ( G i x ( k ) − e i ( k ))= C i ( Ax ( k ) + w ( k )) + v i ( k + 1) − C i Ax ( k ) + β T e i ( k )= C i w ( k ) + v i ( k + 1) + β T e i ( k ) , (20)indicating that z i ( k ) is stable. Moreover, it can be calculatedthat S ˆ ξ i ( k ) + n z i ( k )=(Λ + n β T ) ˆ ξ i ( k ) + n [ y i ( k + 1) − β T ˆ ξ i ( k )]=Λ ˆ ξ i ( k ) + n y i ( k + 1) . (21)This completes the proof.To simplify notations, we define the following aggregatedmatrices: ˜ S , I m ⊗ S, ˜ L i , e i ⊗ n , ˜ L , [ ˜ L , ˜ L , · · · , ˜ L m ] , (22)where I m is an m -dimensional identity matrix and e i is the i th canonical basis vector in R m . We thus collect (10) and(11) in matrix form as: ˆ ξ ( k + 1) ... ˆ ξ m ( k + 1) = ˜ S ˆ ξ ( k ) ... ˆ ξ m ( k ) + ˜ L z ( k ) ... z m ( k ) , ˆ x ( k ) = F ˆ ξ ( k ) ... ˆ ξ m ( k ) . (23)where F , [ F , F , · · · , F m ] .IV. L OCAL I MPLEMENTATION OF K ALMAN FILTER
In this part, we provide distributed solutions for state esti-mation, where each sensor node performs measurement pro-cessing and data fusion using the information obtained fromlocally connected neighbors, such that the optimal Kalmanestimate can be cooperatively recovered.In this work, a weighted undirected graph G = {V , E , A} is used to model the interaction among nodes, where V = { , , ..., m } is the set of sensors, E ⊂ V × V is the setof edges, and A = [ a ij ] is the weighted adjacency matrix.It is assumed a ij ≥ and a ij = a ji , ∀ i, j ∈ V . An edgebetween sensors i and j is denoted by e ij ∈ E , indicatingthese two agents can communicate directly with each other.Note e ij ∈ E if and only a ij > . By denoting a degreematrix as D , diag (deg , . . . , deg N ) with deg i = P Nj =1 a ij , the Laplacian matrix of G is defined as L G , D − A . Inthis paper, a connected network is considered. We thereforecan arrange the eigenvalues of Laplacian matrix as µ <µ ≤ · · · ≤ µ m . We first propose solutions in the case where n ≥ m , namelythe number of states is no less than the number of sensor.Instead of fusing the local estimates at a center as proposed in[24], we aim to design a local fusion algorithm at each sensorside, such that the information is fused in the distributed wayover multiple nodes.
1) Description of the distributed estimator:
In light of (10),the recovery of optimal Kalman filter requires local estimatesfrom all sensors. Therefore, in order to achieve global con-vergence, let each sensor node i make further estimation on m ˆ ξ j ( k ) , which is denoted by η i,j ( k ) . As a result, it keeps onemore variable storing its estimate on the local estimates fromall the others, as defined below: η i ( k ) , η i, ( k ) ... η i,m ( k ) ∈ R mn . (24)By defining communication state as ∆ i ( k ) , ˜Γ η i ( k ) , where ˜Γ = I m ⊗ Γ and Γ is a design parameter to be given later, weare now ready to present the main algorithm. Suppose eachnode i is initialized with ˆ x i (0) = 0 and η i (0) = 0 . At anyinstant k > , its update is outlined in Algorithm 1. Algorithm 1
Distributed estimation algorithm ( n ≥ m )1: Using the latest measurement from sensor i , compute thelocal residual and update the local estimate by (11).2: Fuse neighboring information with the consensus algorithmas η i ( k + 1) = ˜ Sη i ( k ) + ˜ L i z i ( k ) + ˜ B m X j =1 a ij (∆ j ( k ) − ∆ i ( k )) , (25)where ˜ B , I m ⊗ n .3: Update the estimate on system state as: ˘ x i ( k + 1) = mF η i ( k + 1) . (26)4: Transmit the new state ∆ i ( k + 1) to neighbors.A few remarks are given regarding the proposed algorithm.Firstly, each agent updates its local estimate with the latestmeasurement. After that, the agent modifies its opinion oneveryone’s local estimates as in Step 2. Note in (25), we resortto the consensus protocol in [21] to facilitate the agreementamong communication states. As to be shown later, the perfor-mance convergence can be guaranteed by this procedure sincethe observability of plant is ensured through interconnections.In Step 3, state estimation is made, forcing local estimatesmoving towards the optimal one.
2) Performance analysis:
This part is devoted to the per-formance analysis of Algorithm 1. We shall first provide thebelow theorem:
Theorem 1.
With Algorithm 1, the average of local estimatesfrom all sensor coincides with the optimal Kalman estimate atany instant. That is, m m X i =1 ˘ x i ( k ) = ˆ x ( k ) , ∀ k ≥ . (27) Proof.
Summing (25) over all i = 1 , , ..., m yields m X i =1 η i ( k + 1) = ˜ S m X i =1 η i ( k ) + m X i =1 ˜ L i z i ( k ) , (28)where we use the fact that a ij = a ji for any i, j ∈ V . Compareit with (9), we conclude the following relation for any instant k and any j ∈ V : ˆ ξ j ( k ) = m X i =1 η i,j ( k ) . (29)Thereby, the below equation is satisfied at any k ≥ : m m X i =1 ˘ x i ( k ) = m X i =1 F η i ( k ) = m X i =1 m X j =1 F j η i,j ( k )= m X j =1 F j h m X i =1 η i,j ( k ) i = m X j =1 F j ˆ ξ j ( k ) = ˆ x ( k ) . (30)This completes the proof.V. N UMERICAL E XAMPLE
In this section, we present some numerical examples toverify the theoretical results obtained in previous sections.Consider the case where four sensors cooperatively estimatethe system state. The system parameters are listed below: A = (cid:20) . . (cid:21) , C = (cid:20) − (cid:21) T ,Q = 0 . I , R = 4 I . (31)With the eigenvalue decomposition of A − KCA , we have
Λ = (cid:20) . . (cid:21) . (32)Suppose the topology of these four sensors is a ring withweight for each edge. The Laplacian matrix is thus: L G = − − − − − − − − . (33)Therefore, the second smallest and the largest eigenvalues of L G respectively are µ = 2 , µ = 4 . In order to satisfy theconstraint Q j (cid:12)(cid:12) λ uj ( A ) (cid:12)(cid:12) < ζ − ≤ µ /µ m − µ /µ m , we choose ζ =0 . .We set the initial state x (0) ∼ N (0 , I ) and the initial localestimate ˘ x i (0) = 0 for each sensor i ∈ { , , , } . It can beseen that the mean squared local estimate error e i ( k ) becomesstable after a few steps (see Fig. 1 and Fig. 2).VI. C ONCLUSION
In this paper, the problem of distributed state estimationhas been studied for a LTI Gaussian system. We investigateboth cases where m > n and m ≤ n , and propose distributedestimators for both cases to introduce low communication cost.The solutions are proved to be globally optimal, in the sensethat the average of local estimates from all sensors coincides . . . . Time E s ti m a t ee rr o r o f s t a t e x KFs1s2s3s4Fig. 1: Average mean square estimation error of state x undercentralized KF and local estimators in 10000 experiments. . . Time E s ti m a t ee rr o r o f s t a t e x KFs1s2s3s4Fig. 2: Average mean square estimation error of state x undercentralized KF and local estimators in 10000 experiments.with the optimal Kalman estimate. As the consequence of ourmajor merit, which reformulates the problem of distributedestimation to that of linear system synchronization, the resultsin this paper lay a solid foundation for future works to developmore flexible distributed estimators in different scenarios.R EFERENCES[1] M. V. Subbotin and R. S. Smith, “Design of distributed decentralizedestimators for formations with fixed and stochastic communicationtopologies,”
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