A Lyapunov-based distributed consensus filter for a class of nonlinear stochastic systems

Automatica 86 (2017) 53–62

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A Lyapunov-based distributed consensus filter for a class of nonlinear stochastic systems✩ Ahmadreza Jenabzadeh, Behrouz Safarinejadian School of Electrical and Electronic Engineering, Shiraz University of Technology, Modarres Blvd., P.O. Box: 71555-313, Shiraz, Iran

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Article history: Received 29 April 2016 Received in revised form 4 March 2017 Accepted 1 August 2017

Keywords: Consensus filter Nonlinear stochastic systems Sensor networks State estimation Target tracking

a b s t r a c t This paper considers the distributed state estimation problem for nonlinear stochastic systems over sensor networks. It is assumed that the nonlinear functions are bounded in the pseudo Lipschitz condition. Based on the stochastic Lyapunov stability theory, a distributed consensus filter (DCF) is proposed for both continuous and discrete nonlinear stochastic systems for each node in a sensor network. It will be shown that the estimation errors of the proposed filters are exponentially ultimately bounded in the sense of mean square in terms of linear matrix inequality (LMI). Furthermore, a criterion is presented to optimize the filter gains based on minimizing the upper bound of mean-square error. Numerical examples are used to verify the theoretical results. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In recent decades, the problem of distributed state estimation (DSE) has received great attention for its successful applications in different areas including environmental monitoring, surveillance, cooperative control of multi-agent systems, target tracking and so on (Olfati-Saber, Fax, & Murray, 2007; Xie, Choi, Kar, & Poor, 2012). The essential principles of DSE algorithms in sensor networks are state estimation at every node and reaching consensus based on the estimated states of each node and its neighboring nodes. Here the word ‘‘consensus’’ means that each sensor node uses distributed filters that can agree on an estimated value with their neighbors. These types of algorithms are called distributed consensus algorithms. In recent decades, many distributed consensus algorithms were introduced and applied (Farina, Ferrari-Trecate, & Scattolini, 2010; Olfati-Saber, 2005, 2007, 2009; Zhu, Chen, Li, Yang, & Guan, 2013). Distributed consensus algorithms can be classified into four groups: consensus on state estimation, consensus on innovations, consensus on information and H∞ consensus. The first group has consensus on state estimation, in which estimates are averaged to reach a consensus (Açıkmeşe, Mandić, & Speyer, 2014; Farina et al., 2010; Olfati-Saber, 2007, 2009; Zhu et al., 2013). Some of the first consensus algorithms on state estimation were given ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Bert Tanner under the direction of Editor Christos G. Cassandras. E-mail addresses: [email protected] (A. Jenabzadeh), [email protected] (B. Safarinejadian). http://dx.doi.org/10.1016/j.automatica.2017.08.005 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

in Olfati-Saber (2007, 2009). Actually, by adding a consensus term to the Kalman filter, consensus filters in discrete and continuous forms, which are called Kalman-consensus filters, were introduced in Olfati-Saber (2007). The error covariance of the Kalman-consensus filter was not optimal in a discrete manner, which could cause unacceptable estimation errors. Thus, an optimal consensus filter was proposed in Olfati-Saber (2009) and its stability was investigated. In the second group, a consensus is performed on local innovations (Li & Jia, 2011, 2012; OlfatiSaber, 2005, 2007). In Olfati-Saber (2005), the first consensus filter based on consensus on innovations has been proposed. A modified version of this consensus filter which can be applied in a sensor network with different observation matrices was introduced in Olfati-Saber (2007). In papers Li and Jia (2011, 2012), consensus on innovations was used to design consensus filters for jump Markov systems and discrete-time nonlinear systems with non-Gaussian noise. In the third group, consensus occurs on the inverse of the state estimation covariance matrix or information matrix that was firstly applied in a distributed state estimation problem (Battistelli, Chisci, Morrocchi, & Papi, 2011). Later, a novel consensus filter has been proposed to study the distributed target tracking problem over a sensor network (Battistelli, Chisci, Fantacci, Farina, & Graziano, 2013). This filter was extended to be used in a tracking problem for a maneuvering target (Battistelli, Chisci, Fantacci, Farina, & Graziano, 2015a). More recently, a consensus filter based on the unscented Kalman filter and consensus on information was presented for systems with sensor saturation and state saturation (Li, Wei, & Han, 2014). Finally, the fourth group of consensus filters is H∞ consensus which was originally introduced in Shen, Wang, and Hung (2010). The main reason

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A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

to apply H∞ consensus is this fact that the practical systems are often along with parameter uncertainties and disturbances. Consequently, we cannot use ordinary consensus filters. Therefore, the H∞ consensus has been recently developed (Han, Wei, Song, & Li, 2015; Ugrinovskii, 2014). Target tracking is one of the fundamental problems in sensor networks in both theory and application (Morbidi & Mariottini, 2013; Olfati-Saber & Jalalkamali, 2012; Ou, Du, & Li, 2012; Ou, Gu, Wang, & Dong, 2015; Zhu et al., 2013). In Olfati-Saber and Jalalkamali (2012), the authors have solved the problem of tracking control for mobile sensors with linear dynamics to estimate the states of a linear target and track that linear target based on a flocking manner. In this work, a distributed Kalman filter is proposed to estimate the states of the target. This filter is developed for tracking a target with linear dynamics in heterogeneous sensor networks (Zhu et al., 2013). In Morbidi and Mariottini (2013), a team of unmanned aerial vehicles are considered as mobile sensors and a distributed estimation and control algorithm is suggested for them to track a target with linear dynamics. In Ou et al. (2012), a distributed controller has been presented to solve the cooperative control problem of mobile sensors with nonlinear dynamics. The proposed controller makes mobile sensors converge to a desired trajectory. In Ou et al. (2015), the problem of finite-time tracking control of multiple nonholonomic mobile robots subject to external disturbances has been solved. An observer is presented to estimate the disturbance and a finite time controller is designed for each robot to track the target with nonholonomic dynamics. In this work, the target’s position is assumed to be available. The targets in the aforementioned target tracking problems have continuous dynamics and their states are considered to be available or have linear dynamics. But, these assumptions actually do not hold in practical environments because many practical targets such as unmanned aerial, ground, or underwater vehicles and satellites have continuous nonlinear dynamics and their states must be estimated by sensor networks. Thus, it is necessary to pay much attention to the design of DSE algorithms for continuous nonlinear systems. Some algorithms have also been proposed for state estimation of targets with nonlinear dynamics (Battistelli, Chisci, Mugnai, Farina, & Graziano, 2015b; Hu & Hu, 2010; Li, Wei, Han, & Liu, 2016). In Hu and Hu (2010), a nonlinear convergent filter has been presented to estimate the target’s states in a sensor network. The target’s dynamics has been described by a continuous-time linear system whose input is generated by another linear system. In Battistelli et al. (2015b), a hybrid consensus filter is presented with a combination of consensus on information, consensus on measurements and extended Kalman filter algorithm. The stability analysis of the proposed filter is limited to linear systems. In Li et al. (2016), a nonlinear consensus filter has been suggested by employing a consensus approach and unscented Kalman filter. This filter had bounded estimation error in the sense of mean-square and could estimate the states of a target with discrete-time nonlinear dynamics. It should be noted that the proposed algorithms in Battistelli et al. (2015b) and Li et al. (2016) have obtained for state estimation of the targets with discrete-time nonlinear dynamics and cannot be implemented for state estimation of continuous-time nonlinear systems. Furthermore, although Hu and Hu (2010) has proposed a continuous-time consensus filter, the target has a special structure in which most of the targets are not classified. These limitations motivate us for this study. In this paper, by using the proposed estimator in Xie and Khargonekar (2012) and consensus techniques on state estimation, a DCF for estimating the states of continuous nonlinear systems is proposed. In fact, Xie and Khargonekar (2012) suggested an estimator for estimating states and parameters of a class of continuous nonlinear stochastic system based on Lyapunov stability theory

with a suboptimal gain. Additionally, the discrete-time version of proposed DCF is presented for implementation application. Therefore, the main contributions of this paper are as follows:

• A novel DCF is presented to estimate the states of a continuous nonlinear stochastic system. The discrete-time version of the proposed DCF is also introduced. • The exponentially ultimately boundedness of the estimation errors of proposed DCFs is proved and suboptimal gains are obtained for both continuous and discrete DCF by minimizing the upper bound of the estimation error. The remainder of this paper is organized as follows. Section 2 introduces a DCF for continuous nonlinear system and analyzes its convergence and optimality. In Section 3, a discrete-time DCF is presented based on the discrete version of Lyapunov theory and a suboptimal gain is obtained. The proposed DCFs performance is studied with numerical examples both for continuous and discrete systems in Section 4. Finally, the conclusions are drawn in Section 5. Notation and graph theory. λmax (.) and λmin (.) denote the biggest and the smallest eigenvalues, respectively. ⊗ represents the Kronecker product. E[.] denotes the expectation operator. C 2,1 denotes the family of all nonnegative functions V (x(t), t) that are continuously twice differentiable in x and once differentiable in t. IM denotes an M × M identity matrix. The sensor nodes of the network are communicated over an undirected graph G = (v, ε, A), where v = {1, 2, . . . , N } is the sensor node set, ε ∈ v × v = {(i, j) : i, j ∈ v} is the communication link set and A = [aij ] ∈ RN ×N is the adjacent matrix. If nodes i and j are connected, then the node i is the neighbor of node j and aij = aji > 0. The Laplacian matrix for graph G is defined as L = D − A, ∑ in which D is a diagonal matrix with the diagonal elements di = j∈Ni aij . The eigenvalues of a Laplacian matrix can be ordered as λ1 (L) ≤ λ2 (L) ≤ . . . ≤ λN (L) in which the second smallest eigenvalue, λ2 (L) is called the algebraic connectivity of the network. Ni = {j ∈ v : (i, j) ∈ ε, j ̸ = i} denotes the set of neighbors of node i. In this paper, the assumption is that no node is connected with itself, i.e. aii = 0 ; 1 ≤ i ≤ N. If there is a link between nodes i and j, the corresponding element in the adjacent matrix will be 1. G is called a connected graph if and only if there is at least one path between every two arbitrary nodes. It is a critical point that an undirected graph is connected if and only if its algebraic connectivity is positive: λ2 (L) > 0. 2. Continuous-time distributed consensus filter (CDCF) Consider the continuous-time nonlinear system with the following dynamic equations: x˙ (t) = f (x(t)) + Bw (t), yi (t) = hi (x(t), t) + Di vi (t),

x(t) ∈ RM i = 1, . . . , N

(1) (2)

where w (t) and vi (t) are white noises with covariances Q (t) and Ri (t), respectively. x(t) is the state vector, yi (t) is the ith sensor measurement vector. f (.) and hi (.) are nonlinear functions. It is noteworthy that B can be a function of x(t). The problem is to estimate the states of the nonlinear system (1) by providing a novel distributed estimation algorithm in a sensor network. As mentioned in the introduction section, the proposed CDCF is concluded from Xie and Khargonekar (2012). In fact, Xie and Khargonekar (2012) has employed the observers in Tarn and Rasis (1976), Yaz and Azemi (1993), Cho and Rajamani (1997) and introduced an estimator that could estimate the states and unknown parameters of a class of nonlinear stochastic systems. This reference has used the structure in Tarn and Rasis (1976) and sufficient conditions for estimation error boundedness and the optimal filter gain have

A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

been obtained based on LMI using Cho and Rajamani (1997) and Yaz and Azemi (1993). The observer in Tarn and Rasis (1976) had the following simple structure: x˙ˆ (t) = f (xˆ (t)) + K (y(t) − h(xˆ (t)))

(3)

where f (.) is the known nonlinear term in Eq. (1), K is the observer gain, h(.) and y(t) are the terms in Eq. (2) without the subscript i. It should be noted that the observer (3) is not a distributed observer and cannot be applied to sensor networks. Thus, by adding a consensus term to the observer (3), a CDCF is proposed for sensor networks. The dynamic equation of this filter for every node in the sensor network is given as follows: x˙ˆ i (t) = f (xˆ i (t)) + Ki (yi (t) − hi (xˆ i (t))) ∑ + Ti Pi−1 [ˆxj (t) − xˆ i (t)], i = 1, . . . , N

(4)

j∈Ni

where xˆ i (t) is the estimation of x(t), Ki ∈ RM ×S is the filter gain matrix, Ti = ϕi × IM > 0 is the consensus matrix and Pi ∈ RM ×M is the filter matrix of node i. The gain Ki and the matrix Pi formulas should be provided for the CDCF. Obtaining these parameters and stability analysis of this filter will be given in Section 2.1. 2.1. Stability analysis of the CDCF Before stability analysis, a couple of assumptions should be mentioned. Assumption 1. There are positive constants γ , γ1 , γ2 , γ3 , γ4 and constant matrices Ai , Ci so that for every vector x, we have: (1) Boundedness condition: ∥B∥F ≤ γ . (2) The following pseudo Lipschitz conditions

∥f (x + δ ) − f (x) − Ai δ∥ ≤ γ1 ∥δ∥ + γ2 ∥h(x + δ ) − h(x) − Ci δ∥ ≤ γ3 ∥δ∥ + γ4

√

where ∥.∥ is the Euclidean vector norm and ∥B∥F = tr(BT B) is the Frobenius norm. Various constants such as γ , γ1 , . . . , γ4 should be chosen such that the deviations or the upper bounds are made as small as possible (Yaz & Azemi, 1993). Assumption 2. There are a matrix Pi > 0 and vector Ki that satisfy the following two conditions: Qi =: Pi (Ai − Ki Ci ) + (Ai − Ki Ci )T Pi + (γ1 + γ2 )Pi Pi

+ (γ3 + γ4 )Pi Ki KiT Pi + (γ1 + γ3 )IM < 0 2ϕi λ2 (L) + |λmax (Qi )| > 0.

To obtain the CDCF parameters including Pi > 0 and Ki by using Theorem 1, it is necessary that Assumptions 1–3 are satisfied. Since it is assumed that the sensor network is connected throughout this paper, it is only necessary to solve the inequality (5) in Assumption 2. Therefore, we assume that in inequality (5), the matrix Qi is such that Qi < −α I where α > 0 is a given number. To solve this inequality, the LMI technique is used. In the following, the LMI for obtaining CDCF parameters in case of bounded estimation error is presented. The LMI related to the boundedness: The matrix inequality Qi < −α I is converted matrix √ to the following √ ( ) inequality by using Lemma A.2,

P1

∗ ∗

(γ1 + γ2 )Pi −I 0

(γ3 + γ4 )Pi Ki 0 −I

< 0 where

P1 = Pi (Ai − Ki Ci ) + (Ai − Ki Ci )T Pi + (γ1 + γ3 + α )I, Pi > 0 and α > 0. The ‘*’ symbols denote the symmetric entries of the matrix. Supposing Xi = Pi Ki , the following LMI problem is created:

⎛

P2

⎝∗ ∗

√

(γ1 + γ2 )Pi −I 0

⎞

√

(γ3 + γ4 )Xi ⎠<0 0 −I

(7)

where P2 = Pi Ai − Xi Ci + ATi Pi − CiT XiT + (γ1 + γ3 + α )I. In the following, the normal gain procedure is summarized as follows. Algorithm 1. The algorithm to compute normal gain Step 1: Obtain positive constants γ , γ1 , . . . , γ4 and constant matrices Ai , Ci such that the conditions in Assumption 1 are satisfied. Step 2: Choose a positive value for α > 0. Step 3: Compute matrices Ki and Pi by solving the LMI (7) assuming Pi > 0 by using the YALMIP toolbox in MATLAB. 2.2. Suboptimal gain of the CDCF In this section, an optimality criterion is presented to improve the CDCF performance. With the Lyapunov function (A.4) and Assumption 3, it can be obtained V (e(t)) =

N ∑

eTi (t)Pi ei (t) ≥

i=1

N ∑

λmin (Pi ) ∥ei (t)∥2

i=1

= λmin (Pi ) ∥e(t)∥2 .

(8)

Taking the expectation of both sides of Eq. (8) and using inequality (A.9) yield,

[ λmin (Pi )E ∥e(t)∥

[

] 2

= λmin (Pi )E

N ∑

] ∥ei (t)∥2

i=1

(5) (6)

Remark 1. Inequality (5) implies that (Ai , Ci ) must be detectable. Furthermore, according to the positivity of |λmax (Qi )| and ϕi , the inequality (6) holds when λ2 (L) is positive and λ2 (L) is positive when the sensor network graph is connected. Assumption 3. All sensor nodes are identical in every aspect and the same gain is designed for the distributed consensus filter of all of the nodes. Theorem 1. Consider a sensor network with a connected topology for estimating the states of a nonlinear system with Eqs. (1)– (2). Under Assumptions 1–3, a CDCF with the dynamic equation (4) is obtained for every sensor node in the sensor network in which estimation error ei (t) = x(t) − xˆ i (t) is exponentially ultimately bounded in mean square. Proof. See Appendix.

55

≤ k3 exp(−k1 t) + k2 /k1 ⇒ E[∥e(t)∥2 ] [ N ] ∑ k3 exp(−k1 t) + k2 /k1 2 ∥ei (t)∥ ≤ . =E λmin (Pi )

(9)

i=1

Eq. (9) provides an upper bound for the E[∥e(t)∥2 ]. Since ∥e(t)∥2 is the summation of ∥ei (t)∥2 elements, then the reduction of the upper bound of the E[∥e(t)∥2 ] leads to a suboptimal gain for the CDCF. Therefore, our goal is to reduce the upper bound of the E[∥e(t)∥2 ]. In the upper bound, there is an exponential term that k /k goes to zero as t → ∞. Thus, by minimizing λ 2 (P1 ) , a suboptimal min i gain is obtained. Since Assumption 3 implies an identical gain for all sensor network nodes, then the (γ2 + γ4 + q1 ) is identical for all CDCF. ∑N Therefore, the equation k2 in (A.8) is converted to k2 = = N√ (γ2 + γ4 + q1 ). By using i=1 (γ2 + γ4 + q1 ) tr(AT A) and tr(AB) ≤ the facts tr(AB) = tr(BA), ∥A∥F = tr(A)tr(B) for semi definite positive matrices, the ( ( ) following) inequality holds: k2 = N tr (BBT + Ki Di DTi KiT )Pi + γ2 + γ4 ≤ ( ( ) ) 2¯ ˜ N tr γ 2 Pi + DTi KiT Pi Ki Di + γ2 + γ4 ≤ N( ( ) γ k + k) where T T 4) ¯k = tr (P ) + (γ2 +γ ˜ and k = tr Di Ki Pi Ki Di . Since we assume γ2

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A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

that there is the given parameter α > 0 such that Qi < −α I. Then, it implies that |λ(Qi )| > α . Considering the aforementioned equations and the principle that |λ(Qi )| > α , one obtains 2N (Pi )k2 (Pi ) k2 ˆ = λ (P )(2λϕmax ≤ (2ϕ λγ (L) × (k) × λλmax(P λ (P )k λ (L)+|λ (Q )|) +α) ) min

i

1

min

where kˆ = k¯ +

k˜

i

γ2

max

i 2

i

i 2

min

i

. For the given parameters λ2 (L), α , ϕi , N and γ ,

we can achieve an optimal upper bound for λ ˜

kˆ or k¯ + γk2 and the term

k2 min (Pi )k1

by minimizing

λmax (Pi ) . The LMI technique is used to solve λmin (Pi )

ˆ two matrices Z , W > 0 are this optimal problem. To minimize k, defined so that Pi < Z , DTi KiT Pi Ki Di /γ 2 < W . For minimizing duced as

(10)

λmax (Pi ) , two positive constants λ1 λmin (Pi )

and λ2 are intro-

λ2 I < Pi < λ1 I .

3. Discrete-time distributed consensus filter Consider the discrete-time nonlinear stochastic system xk+1 = f (xk ) + Bwk yik = hik (xk ) + Di vki ,

min {tr(Z ) + tr(W )} , min λ1 , max λ2

(

−W s.t. Pi − Z < 0, Xi Di /γ Pi < λ1 I , λ2 I < Pi .

DTi XiT /γ

)

−Pi

(12)

Algorithm 2. The algorithm to compute suboptimal gain Step 1: Obtain positive constants γ , γ1 , γ2 , γ3 , γ4 and constant matrices Ai , Ci such that conditions in Assumption 1 are satisfied. Step 2: Choose a positive value for α > 0. Step 3: Compute matrix Pi by solving the following LMIs assuming Pi > 0 and Xi ( = Pi K√ i by minimizing √ λ1 and)maximizing λ2 : P2

∗ ∗

(γ1 + γ2 )Pi −I 0

(γ3 + γ4 )Xi 0 −I

where wk and v are white noises with covariances nw and nv , respectively. xk is the state vector, and yik is the ith sensor measurement vector. f (.) and hik (.) are nonlinear functions. The goal is to estimate the states of the nonlinear system (14) by providing a novel discrete-time distributed consensus filter (DDCF). The DDCF has a form as follows: xˆ ik+1 = f (xˆ ik ) + Ki yik − hik (xˆ ik )

)

(

+ Ti

∑

[ˆxjk − xˆ ik ], i = 1, . . . , N

where xˆ ik is the estimation of xk . Ki and Ti > 0 are filter and consensus gain of node i, respectively. The estimation error of filter (16) is exponentially ultimately bounded in mean square. The stability analysis of filter (16) is as follows.

< 0.

Let eik = xk − xˆ ik be the estimation error of the DDCF of node i, then the error dynamics is given as: eik+1 = f (xk ) − f (xˆ ik ) + Bwk − Ki hik (xk ) − hik (xˆ ik )

(

− Ki Di vki − Ti

(17)

j∈Ni

Define the error vector as ek = [(e1k )T , . . . , (eNk )T ]T and the block diagonal matrix P = diag {P1 , . . . , PN }. We show that the estimation error of the DDCF is bounded.∑ To this goal, a Lyapunov function is N i T i defined as Vk (ek ) = eTk Pek = i=1 (ek ) Pi (ek ) where Pi is a positive definite matrix. The derivative of the Lyapunov function was used for the CDCF. However, for discrete-time nonlinear systems, a difference-like function is used defined by

Remark 2. The suboptimal gain is obtained when some parameters such as λ2 (L), α , ϕi , N are assumed as given parameters. Here, we compute the upper bound of estimation error e(t) and consider the effect of these parameters √ on this bound. From inequality

eik+1 = (Ai − Ki Ci ) eik + f (xk ) − f (xˆ ik ) − Ai eik

DTi XiT /γ −Pi

< 0.

k exp(−k t)+k /k

3 1 2 1 . As t → ∞, the (9), we have E[∥e(t)∥] ≤ λmin (Pi ) expectation of estimation error converges exponentially into the ball

L1 =

≤

⎧ ⎪ ⎪ ⎨

(18)

Before the calculation of (18), it is needed to simplify (17). By adding and subtracting Ai eik and Ci eik , Eq. (17) is converted to

[

]

[ ] − Ki hik (xk ) − hik (xˆ ik ) − Ci eik + Bwk − Ki Di vki ∑ j ( ) [ek − eik ] := A0,i eik + pk,i − Ki qk,i + Ti j∈Ni

+ Bwk − Ki Di vki + Ti

∑

[˜ek ].

(19)

j∈Ni

Using (19) and calculating (18) yields

e(t) ∈ RM ×N | E[∥e(t)∥]

⎪ ⎪ ⎩    √

)

∑ j [ˆxk − xˆ ik ].

∆Vk = E [Vk+1 |ek ] − Vk .

minimizing tr(Z ) + tr(W ): Pi − Z < 0,

(16)

j∈Ni

Step 4: Substitute the computed Pi in step 3 in the following LMIs and solve it to compute matrix Ki(assuming Z , W ) > 0 by −W Xi Di /γ

(15)

3.1. Stability analysis of the DDCF

<0

By solving the LMIs (7) and (12), the CDCF with a suboptimal gain is obtained. In the following, the suboptimal gain optimization procedure is summarized as follows.

Pi < λ1 I , λ2 I < Pi ,

i = 1, . . . , N

i k

(11)

These optimal problems can be solved by the YALMIP toolbox in Matlab. The LMI related to computing a suboptimal upper bound: By minimizing tr(Z ) + tr(W ), λ1 and maximizing λ2 , the inequalities (10) and (11) are converted to the following LMIs:

(14)

( ) λmax (Pi )N C˜ + γ2 + γ4

⎫ ⎪ ⎪ ⎬

λmin (Pi ) (2ϕi λ2 (L) + |λmax (Qi )|) ⎪ ⎪ ⎭

∆Vk =

)

E (eik )T A0,i

(

)T (

Pi A0,i eik

)

i=1

(13)

where C˜ = tr (BBT + Ki Di DTi KiT )Pi . From the inequality (13) it is concluded that when the parameters λ2 (L), α and ϕi are larger and the parameter N is smaller, the upper bound of the estimation error is smaller.

(

N ( ∑

( )T ( )T + 2(eik )T A0,i Pi pk,i − 2(eik )T A0,i Pi Ki qk,i − 2(pk,i )T Pi Ki qk,i + (pk,i )T Pi pk,i ) + (qk,i )T KiT Pi Ki qk,i |ek +

N ∑ [

E (Bwk )T Pi (Bwk ) + (Ki Di vki )T Pi (Ki Di vki )

i=1

]

A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62 N ∑

+

⎞ ( )T

E ⎝2Ti A˜

∑ [˜ek ]

Pi ×

j∈Ni

⎞ ∑ ∑ + Ti2 [˜ek ]T Pi × [˜ek ]|ek ⎠ − Vk . j∈Ni

(20)

j∈Ni

for the term 2(ek ) A0,i

Pi pk,i and

)T

)T

η

( T

(qk,i )T KiT PKi qk,i ,

( )T

E ⎝2Ti A˜

∑ [˜ek ]

Pi ×

j∈Ni

+ Ti2

η > 0,

(25)

j∈Ni

⎛

i=1

− 2(eik ) A0,i Pi Ki qk,i ≤ η(eik ) A0,i Pi A0,i eik

( T

j∈Ni

Here, it is needed to simplify the following term: N ∑

where A˜ = A0,i eik + pk,i − Ki qk,i . Using similar inequalities as (A.7) ( )T i T

+

∑ ∑ ∑ [˜ek ] + Ti2 [˜ek ]T × Pi [˜ek ]|ek ⎠ .

×

j∈Ni

i=1

1

57

⎛

∑

⎞ ∑ [˜ek ]T × Pi [˜ek ]|ek ⎠ .

j∈Ni

j∈Ni

(26)

(21) Extending Eq. (26) gives

then Eq. (20) is converted to N ∑

N

∆Vk ≤

∑ ( E

(eik )T

ψ

eik

T

− 2(pk,i ) Pi Ki qk,i

+ (pk,i )T Pi pk,i + 1 +

1

)

η

j∈Ni

(qk,i )T KiT Pi Ki qk,i |ek

⎞

)

E (Bwk ) Pi (Bwk ) + T

v

v

(Ki Di ki )T Pi (Ki Di ki )

]

+ γ2 +

N ∑

j∈Ni

⎛

N

∑ ∑ ( )T ( )T [˜ek ]|ek ⎠ = E ⎝2Ti eik A0,i Pi

× Pi

i=1

∑ ∑ ( )T ( )T × [˜ek ] + 2Ti pk,i Pi × [˜ek ] − 2Ti qk,i KiT Pi

i=1

)

∑ ∑ [˜ek ] + Ti2 [˜ek ]T

Pi ×

j∈Ni

N ∑ ( [

+

( )T

E ⎝2Ti A˜

i=1

i=1

(

⎛

j∈Ni

⎛

⎞ N ∑ ∑ ∑ ∑ × [˜ek ]|ek ⎠ + E ⎝Ti2 [˜ek ]T × Pi [˜ek ]|ek ⎠ .

j∈Ni

i=1

j∈Ni

⎞

( )T ∑ [˜ek ] E ⎝2Ti A˜ Pi ×

⎛

j∈Ni

i=1

j∈Ni

(27)

j∈Ni

⎞ + Ti2

∑ ∑ [˜ek ]T Pi × [˜ek ]|ek ⎠ j∈Ni

(22)

)T (

)T

where ψ = (1 + η) A0,i Pi A0,i − Pi + (γ1 + γ2 ) A0,i Pi Pi A0,i + γ1 IM . Furthermore, the pseudo Lipschitz conditions in Assumption 1 imply that

(

)

(

)

(

(

2 pk,i

ε

Pi ×

1 ∑

µ )T

∑

( )T ( )T [˜ek ] ≤ µ eik A0,i Pi A0,i eik

T

[˜ek ] Pi

j∈Ni

∑ [˜ek ], j∈Ni

∑ ( )T Pi × [˜ek ]|eik ≤ σ pk,i Pi pk,i

1 ∑

[˜ek ]T Pi

+

T

(23)

σ j∈Ni ( )T

− 2 qk,i

(qk,i )T KiT Pi Ki qk,i , ε > 0

∑ [˜ek ], j∈Ni

KiT Pi

×

∑ ( )T [˜ek ] ≤ β qk,i KiT Pi Ki qk,i j∈Ni

E (Bwk )T Pi (Bwk ) ≤ γ 2 λmax (Pi )nw

1 ∑

+

and also from the statistic properties of wk and vki , one obtains

[

)T

j∈Ni

− 2(pk,i ) Pi Ki qk,i ≤ ε (pk,i ) Pi pk,i 1

A0,i

j∈Ni

(

T

+

( )T (

2 eik

+

  )2 (pk,i ) Pi pk,i ≤ λmax (Pi ) γ1 eik  + γ2 (  2 ) ≤ 2λmax (Pi ) γ12 eik  + γ22 , )2 (   (qk,i )T KiT Pi Ki qk,i ≤ λmax (KiT Pi Ki ) γ3 eik  + γ4 (  2 ) ≤ 2λmax (KiT Pi Ki ) γ32 eik  + γ42 , T

Using similar inequality as (21) for the first term on the right-hand side of the (27), one gets

j∈Ni

β

∑ [˜ek ]. [˜ek ]T Pi

j∈Ni

(28)

j∈Ni

]

Furthermore, by using the pseudo Lipschitz conditions in Assump-

E (Ki Di vki )T Pi (Ki Di vki ) ≤ ∥Di ∥F λmax (Ki Pi Ki )nv i .

[

]

(24)

k

Substituting inequalities (23) and (24) into (22) yields

∆Vk ≤

N (( ∑

tion 1, one can obtain

(

2 pk,i

[

)T

j∈Ni

 2

(eik )T ψ eik + 2(1 + ε ) λmax (Pi )γ12 eik 

+

i=1

)[

(

]  2 1 1 λmax (KiT Pi Ki )γ32 eik  + λmax (Pi )γ22 + 2 1 + + η ε N ]) ) ∑ + λmax (KiT Pi Ki )γ42 |ek + (γ 2 λmax (Pi )nw i=1

+ ∥Di ∥F λmax (Ki Pi Ki )nvi + γ2 ) +

( (  2 )) ∑ [˜ek ] ≤ σ 2λmax (Pi ) γ12 eik  + γ22

Pi ×

N ∑

k

i=1

⎛ ( )T

E ⎝2Ti A˜

Pi

1 ∑

[˜ek ]T Pi

σ j∈Ni ( )T

− 2 qk,i

∑ [˜ek ], j∈Ni

KiT Pi

×

∑ [˜ek ] ≤ j∈N

( i  2 )) β 2λ γ32 eik  + γ42 ∑ 1 ∑ + [˜ek ]T Pi [˜ek ]. β (

T max (Ki Pi Ki )

j∈Ni

j∈Ni

(29)

58

A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

:= −k1 Vk (ek ) + k2

Substituting inequalities (28) and (29) into (27) yields N ∑

⎛ ( )T

E ⎝2Ti A˜

Pi ×

∑

where k1 and k2 are defined as k1

[˜ek ]

≤ Ti

k

orem A.1 to (17) for the mean-square boundedness. Now, we summarize the above arguments in the following theorem.

j∈Ni

N ( ( )T ( ) ∑ T A0,i Pi A0,i eik E µ eik

Theorem 2. Consider a sensor network for estimating the states of a nonlinear system with Eqs. (14)– (15). If there exist positive numbers σ , η, µ, α , ε , β, ς , ξ and matrices Pi > 0 and Ki such that inequalities (31)–(34) hold, then under Assumptions 1 and 3, a DDCF with the dynamic equation (16) is obtained for every sensor node in the sensor network in which estimation error eik = xk − xˆ ik is exponentially bounded in mean square.

i=1

(

(

))

 2 + σ 2λmax (Pi ) γ12 eik  + γ22 )) (  2 ( + β 2λmax (KiT Pi Ki ) γ32 eik  + γ42 ⎛ ⎞ ⎞ N ( )∑ ∑ ∑ + E ⎝Ti T˜i [˜ek ]T Pi [˜ek ]⎠ |ek ⎠ j∈Ni

i=1

j∈Ni

where T˜ = σ + µ + β + Ti . Assume that the consensus gain Ti is the same for all of the nodes. Then, by using graph theory, we have 1

N ∑

1

1

⎞ ( ) ( )∑ ∑ E ⎝Ti T˜ [˜ek ]T Pi [˜ek ]|ek ⎠ ≤ Ti T˜ ⎛

j∈Ni

i=1

λmax (Pi ) × × λmax ((L ⊗ IN ) (L ⊗ IN )) Vk (ek ). λmin (Pi )

Q + 2(1 + ε + σ Ti )γ12 λmax (Pi ) + 2(β˜ 1 )γ32 λmax (KiT Pi Ki )

N

∑

(eik )T Qeik +

i=1

∑(

2(1 + ε + σ Ti )

i=1

[ [  2 ]  2 ] × λmax (Pi )γ12 eik  + 2(β˜ 1 ) λmax (KiT Pi Ki )γ32 eik  [ ] [ ] + 2(1 + ε + σ Ti ) λmax (Pi )γ22 + 2(β˜ 1 ) λmax (KiT Pi Ki )γ42 ) + γ 2 λmax (Pi )nw + ∥Di ∥F λmax (Ki Pi Ki )nvi + γ2 k ( ) λ (P ) max i × λmax ((L ⊗ IN ) (L ⊗ IN )) Vk (ek ) + Ti T˜ λmin (Pi ) ( )T ( ) ( )T where Q = (1 + η + Ti µ) A0,i Pi A0,i − Pi + (γ1 + γ2 ) A0,i ( ) Pi Pi A0,i + γ1 IM and β˜ 1 = 1 + η1 + 1ε + β Ti . Now if the following conditions are satisfied, (1 + η + Ti µ) A0,i

(

)T (

)

Pi A0,i − Pi

( )T ( ) + (γ1 + γ2 ) A0,i Pi Pi A0,i + (ξ + ζ )I + γ1 IM < −α I , α>0 2(1 + ε + σ Ti )γ

2 1 Pi

< ξ I,

ξ >0

2(β˜ 1 )γ32 KiT Pi Ki < ζ I ,

ζ >0 ( ) λ (P ) α max i − Ti T˜ × 0< λmax (Pi ) λmin (Pi ) × λmax ((L ⊗ IN ) (L ⊗ IN )) < 1. Then N ( ) λ (P ) ∑  2 max i ∆Vk ≤ − α eik  + Ti T˜ × λmin (Pi )

( )T ( ) < (1 + η + Ti µ) A0,i Pi A0,i − Pi ( )T ( ) + (γ1 + γ2 ) A0,i Pi Pi A0,i + (ξ + ζ )I + γ1 IM < −α I . (36)

(30)

From (25) and (26)–(30), we can obtain

∆Vk ≤

Proof. From Theorem (2), we conclude 0 < k1 < 1 since it is assumed that (34) holds. Inequalities (32) and (33) are equivalent to the inequalities 2(1 + ε + σ Ti )γ12 λmax (Pi ) < ξ and 2(β˜ 1 )γ32 λmax (KiT Pi Ki ) < ζ , respectively. Hence it follows from (31) that

j∈Ni

N

α λmax (Pi )

∑N × λmax ((L ⊗ IN ) (L ⊗ IN )) and k2 = i=1 (2(1 + ε + ] [ ] σ Ti ) λmax (Pi )γ22 + 2(β˜ 1 ) × λmax (KiT Pi Ki )γ42 + γ 2 λmax (Pi )nw + ∥Di ∥F λmax (Ki Pi Ki )nvi + γ2 ). This implies that we may apply The-

⎞ ∑ ∑ [˜ek ]T × Pi [˜ek ]|ek ⎠ + Ti2 j∈Ni

=

λmax (Pi ) λmin (P[i )

j∈Ni

i=1

(35)

( ) − Ti T˜ ×

Here, from (36) it is concluded that (35) holds and Theorem A.1 can be applied for the error system (17). Therefore, it is proved that the error estimation of the DDCF (16) is exponentially bounded in mean square. LMI for the boundedness: Inequalities (31)–(33) can be transformed into LMIs. Let Xi = KiT Pi , Then

⎛

P3

⎝∗ ∗

a1 ATi Pi − CiT Xi − Pi 0

(

(

2(1 + ε + σ Ti )γ12 Pi < ξ I ,

( −ζ ∗

a3 X i −Pi

)

<0

√

√

(1 + η + Ti µ), a2 = (γ1 + γ2 ), a3 = 2(β˜ 1 ). Moreover, for implementing inequality (34), two positive constants λ1 and λ2 are introduced as

λ2 I < Pi < λ1 I (31)

such that

(32)

0<

(33)

(37)

where P3 = (γ1 + ξ + ζ + α )I − Pi , Pi >√0, α > 0, and a1 =

(38)

( ) λ α 1 − Ti T˜ × × λmax ((L ⊗ IN ) (L ⊗ IN )) < 1. λ1 λ2

(39)

By solving the LMIs (37) and (38), we can obtain the DDCF gain Ki . (34)

3.2. Suboptimal gain of the DDCF To compute the suboptimal gain, an optimality criterion is used to improve the filter performance. Notice that the inequality (31) implies that Pi > α I which yields

i=1

× λmax ((L ⊗ IN ) (L ⊗ IN )) Vk (ek ) + k2 ( ( ) λ (P ) −α max i = + Ti T˜ × λmax (Pi ) λmin (Pi ) ) × λmax ((L ⊗ IN ) (L ⊗ IN )) Vk (ek ) + k2

)⎞

a2 ATi Pi − CiT Xi ⎠<0 0 −I

)

Vk (ek ) = eTk Pek =

N ∑

(eik )T Pi (eik )

i=1 N ∑  2 ≥ α eik  = α ∥ek ∥2 . i=1

(40)

A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

59

Taking the expectation of both sides of Eq. (40) and using inequality (A.2) in Appendix yields,

α E[∥ek ∥2 ] ≤ (1 − k1 )k V (0) + k2 /k1 (1 − k1 )k V (0) + k2 /k1 . ⇒ E[∥ek ∥2 ] ≤ α

(41)

2 The Eq. (41) creates an upper bound for the 2 mean of ∥ek ∥ . Since 2 i ∥ek ∥ is composed of the summation of ek  elements, the reduction of the upper bound of the mean of ∥ek ∥2 leads to the reduction of the upper bound of the estimation error of the DDCF. According to these explanations, our goal is to reduce the upper bound of the mean of ∥ek ∥2 . In the upper bound, there is the term (1 − k1 )k that k /k goes to zero as k → ∞. Thus, the bound 2 α 1 should be as small as k2 /k1 possible. In fact, by minimizing α , a suboptimal filter is created. k /k First, the equation 2 α 1 is rewritten as

k2

α k1

(

N ( ∑ i=1

(

1

1

)

] [ + 2 1 + + + β Ti λmax (KiT Pi Ki )γ42 + η ε )) 2 ∥ ∥ γ λmax (Pi )nw + Di F λmax (Ki Pi Ki )nvi /α k1 k

(42)

k /k

Since we assume all parameters of 2 α 1 except λmin (Pi ), λmax (Pi ), and λmax (KiT Pi Ki ) are given, they do not have effectiveness in these optimal problems. Therefore, a suboptimal criterion to minimize k2 /k1 is minimizing λmax (Pi ), λmax (KiT Pi Ki ) and maximizing λmin (Pi ). α k /k For minimizing 2 α 1 , the matrix Z > 0 is introduced as KiT Pi Ki < Z .

(43)

This optimal problem can be solved by the YALMIP toolbox in Matlab software. The LMI related to computing a suboptimal gain of the DDCF: By minimizing λ1 and maximizing λ2 , the inequalities (38) and (43) are converted to the following LMIs: max λ2 , min λ1 , s.t. λ2 I < Pi < λ1 I , Z < λ1 I ,

(

−Z XiT

Xi −Pi

)

< 0.

Example 1. Consider the nonlinear continuous system and measurement equation of each sensor node as follows:

{

(44)

By solving the LMIs (37) and (44), the DDCF with a suboptimal gain is obtained. Remark 3. In the design of the suboptimal gain, some parameters such as σ , η, µ, α , ε , β , Ti are considered as given parameters. Here, we investigate the effect of these parameters on the upper bound of the √ estimation error ek . From inequality (41), we have

yi (t) = 4x1 (t) + 0.5 sin(4x1 (t)) + 2x2 (t)

E[∥ek ∥] ≤

k2 /k1

α

.

(45)

From the inequalities (42) and (45), it is concluded that when parameters η, µ and α are larger and parameter Ti is smaller, the upper bound of estimation error is smaller. It should be noted that we cannot investigate the effect of parameters σ , ε and β because k /k they are both in numerator and in denominator of 2 α 1 . 4. Numerical examples In this section, in order to investigate the performance of the CDCF and the DDCF, two examples are presented.

(46)

where ω ˜ = 0.1ω(t). w(t) and vi (t) are independent white noises with variances 0.1 and 0.01, respectively. The purpose is to estimate the states of the system (46) using the CDCF in a sensor network. The sensor network considered for this purpose is shown in Fig. 1. The graph of this sensor network is connected. Consequently, λ2 (L) is positive and condition (6) is satisfied. The parameters considered for the CDCF and Assumptions 1, 2 are ( ) −0.7 0.5

1

= (4 2), γ = 0.1414, γ1 = 0.2, γ2 = 0, γ3 = 0, γ4 = 1, α = 0.01, ϕi = 0.25 and Di = 0.1. To obtain the normal and suboptimal filter gain, we use Algorithms ( 1 and ) 0.0506 2. The computed normal and suboptimal gains are Ki = 0.1813 ( ) 0.0098 and Ki = 0.0101 , respectively. By applying these gains, the sensor A =

−1 , C

network simulations are implemented. The simulations are done in Matlab/Simulink and the results are shown after averaging 100 samples. Fig. 2 shows the estimated states and estimation errors of x1 (t) of the CDCF with normal gain for every sensor node or agent. The CDCF with normal gain is successful in reaching consensus in a short time. Although, the estimation errors are negligible, we can improve the estimate of system states using the suboptimal filter gain. Fig. 3 shows the estimation errors of the CDCF with normal and suboptimal gains from 100 to 200 s. It is clear that the estimation errors of the CDCF with suboptimal gain are less than those of the CDCF with normal gain and this verifies the desired performance of the optimizing method. Example 2. Consider the following discrete-time system:

(

(1−k )k V (0)+k /k

1 2 1 E[∥ek ∥] ≤ . As k → ∞, the expectation of the α estimation error has the following upper bound:

√

x˙ 1 (t) = −0.7x1 (t) + x2 (t) − 0.8(1 + 0.1 sin(x1 (t))) + ω ˜ x˙ 2 (t) = 0.5x1 (t) − x2 (t) + 0.2 × 0.8 cos(x2 (t)) + ω ˜

+ 0.1vi (t), i = 1, . . . , N

] [ 2(1 + ε + σ Ti ) λmax (Pi )γ22 + γ2

=

Fig. 1. Sensor network topology with N = 4 sensor nodes.

x1 (k + 1) x2 (k + 1)

)

( =

0.7x1 (k) − (1 + 0.1 sin(x1 (k)))x2 (k) x2 (k)

)

+ Bwk y (k) = 4x1 (k) + 0.5 sin(4x1 (k)) + v i (k), i = 1, . . ., N i

where wk : N(0, 0.01), v i (k):N(0, 0.02) and B =

(47)

() 1 0

. The aim

is to estimate the states of the system (47) using the DDCF in a sensor network described in Fig. 1. The parameters ( ) considered for 0.7 0

−1

, C = (4 0), 1 γ1 = 0.1414, γ2 = 0, γ3 = 0, γ4 = 1, α = 0.1 and Ti = 0.1. At

the DDCF and Assumptions 1, 2 are A =

first, the normal filter gain is computed by solving LMIs (37) and (38). For simplicity, the parameters λ1 and λ2 are previously chosen such that ( ) (39) holds. The computed normal filter gain is Ki = 0.2693 −0.0933

. By applying this gain, the sensor network simulations

are implemented. Fig. 4 shows the true and estimated states of x1 (k) and estimation error e1 (k) of the DDCF with normal gain for every agent. It can be seen from Fig. 4 that the DDCF with normal

60

A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

Fig. 2. The estimated states and estimation errors of x1 (t) of the CDCF with normal gain for every agent.

Fig. 4. The estimated states and estimation errors of x1 (k) of the DDCF with normal gain for every agent.

Fig. 5. Estimation errors of the DDCF with normal and suboptimal gains from 250 to 300 time steps.

Fig. 3. Estimation errors of the CDCF with normal and suboptimal gains from 100 to 200 s.

gain has a suitable performance in state estimation. But like the CDCF one can have a lower estimation error using suboptimal gain. To obtain suboptimal gain LMI, (44) is solved where the computed Pi of LMIs( (37) and (38) ) is substituted. The suboptimal filter gain is Ki =

0.5184 × 10−5 0.4900 × 10−5

. Fig. 5 illustrates the estimation errors of

the DDCF with normal and suboptimal gain for every agent from 250 to 300 time steps. It is clear that the DDCF with suboptimal gain has a lower estimation error than the DDCF with normal gain. 5. Conclusions In this paper, Lyapunov-based distributed consensus filters are proposed for both continuous and discrete stochastic nonlinear systems. The sufficient conditions for the ultimate boundedness of the estimation error in the proposed filters are also presented. Furthermore, a suboptimal filter gain is introduced for both continuous and discrete cases. The provided numerical examples show the promising performance of the CDCF and the DDCF. Appendix. Proof of Theorem 1 Definition A.1 (Zakai, 1967). Consider the stochastic differential equation: dx(t) = f (x(t))dt + g(x(t))dω(t).

(A.1)

x(t) is said to be exponentially ultimately bounded in mean square if [there exist ] positive constants h1 , h2 , h3 such that for all t ≥ 0, E ∥x(t)∥2 ≤ h1 exp(−h2 t) + h3 .

Lemma A.1 (Zakai, 1967). Consider the stochastic differential (A.1). Assume that there exists a function V (x, t) ∈ C 2,1 and positive constants h1 , k1 , k2 such that h1 ∥x(t)∥2 ≤ V (x, t) and ℓV (x, t) ≤ −k1 V (x, t) + k2 . Furthermore, suppose that E [V (x0 , 0)] < ∞. In this case, x(t) is called exponentially ultimately bounded in mean square. Also, it can be said that: E [V (x, t)] ≤ (E [V (x0 , 0)] − k2 /k1 ) exp(−k1 t) + k2 /k1 . Lemma A.2 (Boyd, El Ghaoui, Feron, and Balakrishnan, ( 1994 )Schur X X Complement). For a given symmetric matrix X = X11 X12 , the 12

22

following conditions are equivalent: (a) X < 0, (b) X11 < 0, X22 − −1 T −1 T X12 < 0 and (c) X22 < 0, X11 − X12 X22 X12 < 0. X11 X12

Theorem A.1 (Xie and Khargonekar, 2012). Consider a discrete-time stochastic process xk . Assume that there exist a stochastic function Vk (xk ) and numbers 0 < k1 ≤ 1, k2 , k3 > 0 such that k3 ∥xk ∥2 ≤ Vk (xk ) and E [Vk+1 (xk+1 )|xk ] − Vk (xk ) ≤ −k1 Vk (xk ) + k2 , Then the Lyapunov function Vk (xk ) and the stochastic process xk are exponentially bounded in mean square. More specifically, we have k3 E ∥xk ∥2 ≤ (1 − k1 )k V (0) + k2 /k1 .

[

]

(A.2)

Proof of Theorem 1. The proof of Theorem 1 will be provided in the Ito form (Mao, 2007). The Ito form of Eqs. (1), (2) and (4) are as follows: dx(t) = f (x(t))dt + BdW (t), dzi (t) = hi (x(t))dt ∑ + Di dVi (t) and dxˆ i (t) = f (xˆ i (t))dt + Ki (dzi (t) − hi (xˆ i (t))dt) + Ti Pi−1 j∈N [ˆxj (t) − i xˆ i (t)]dt ∫where W (t) and Vi (t) are standard Brownian motions and t zi (t) = 0 yi (s)ds, zi (0) = 0. Let ei (t) = x(t) − xˆ i (t) be the error of each node filter. Then

˜ dei (t) = dx(t) − dxˆ i (t) = f˜ (x(t), xˆ i (t))dt + B(t)d w ˜ (t). Define the error vector as e(t) = [ ,..., stochastic Lyapunov function as follows: eT1 (t)

V (e(t)) =

N ∑ i=1

eTi (t)Pi ei (t).

(A.3)

eTN (t) T .

] Consider a

(A.4)

A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

61

The differential generator ℓ of V with respect to (A.3) can be computed by

According to Assumption 2, constants k1 and k2 are positive. By using Lemma A.1, we have

N ( ] ∑ ∂ V [˜ ˆ f (x(t) , x (t)) ℓV (e(t)) = i ∂ eTi (t) i=1 N ( )) ∑ 1 ( + tr B˜ B˜ T × 2Pi = 2eTi (t)Pi f˜ (x(t), xˆ i (t))

E[V (e(t))] ≤ E[V (e(0))] exp(−k1 t)

2

+ =

1 2

N ∑

+ k2 /k1 (1 − exp(−k1 t)) : ≤ k3 exp(−k1 t) + k2 /k1 .

Since E[∥e(t)∥2 ] ≤ λ (P ) , consequently, it is proven from min i Lemma A.1 that the estimation error is exponentially ultimately bounded in mean square. E[V (e(t))]

i=1 T

(

tr (BB +

Ki Di DTi KiT )2Pi

2eTi (t)Pi f˜ (x(t), xˆ i (t)) +

i=1

)

) References

N ∑

q1 .

(A.5)

i=1

After adding and reducing Ai ei (t) and Ci ei (t) to Eq. (A.5), it is converted to

ℓV (e(t)) =

N ∑

2eTi (t)Pi (Ai ei (t) + [f (x(t)) − f (xˆ i (t))

i=1 [ ] − Ai ei (t)] − Ki (hi (x(t)) − hi (xˆ i (t))) − Ci ei (t) ∑ − Ki Ci ei (t) − Ti Pi−1 [ˆxj (t) − xˆ i (t)] + q1 ) j∈Ni

=

N ∑

eTi (t) Pi (Ai − Ki Ci ) + (Ai − Ki Ci )T Pi ei (t)

[

]

i=1

+

N ∑

2eTi (t)Pi f (x(t)) − f (xˆ i (t)) − Ai ei (t)

[

]

i=1

−

N ∑

2eTi (t)Pi Ki (hi (x(t)) − hi (xˆ i (t))) − Ci ei (t)

[

]

i=1

−2

N ∑

Ti eTi (t)

i=1

N ∑ ∑ [ˆxj (t) − xˆ i (t)] + q1 . j∈Ni

(A.6)

i=1

⏐

⏐

By using the inequalities 2 ∥x∥ ∥y∥ ≤ xT x + yT y, ⏐xT y⏐ ≤ ∥x∥ ∥y∥ and Assumption 1, the second and third terms in Eq. (A.6) are simplified as follows: 2eTi (t)Pi f (x(t)) − f (xˆ i (t)) − Ai ei (t)

[

]

  ≤ 2 ∥Pi ei (t)∥ f (x(t)) − f (xˆ i (t)) − Ai ei (t) ≤ 2 ∥Pi ei (t)∥ (γ1 ∥ei (t)∥ + γ2 ) ≤ eTi (t) ((γ1 + γ2 )Pi Pi + γ1 IM ) ei (t) + γ2 , [ ] − 2eTi (t)Pi Ki hi (x(t)) − hi (xˆ i (t)) − Ci ei (t)  T   ≤ 2 Ki Pi ei (t) hi (x(t)) − hi (xˆ i (t)) − Ci ei (t)   ≤ 2 KiT Pi ei (t) (γ3 ∥ei (t)∥ + γ4 ) ( ) ≤ eTi (t) (γ3 + γ4 )Pi Ki KiT Pi + γ3 IM ei (t) + γ4 .

(A.7)

By substituting Eqs. (A.7) in Eqs. (A.6), xˆ j (t) − xˆ i (t) = ei (t) − ej (t) and 2, the following inequality is obtained: ℓV (e(t)) ≤ ∑N Assumption T T i + (γ3 + i=1 ei (t)[Pi (Ai − Ki Ci ) + (Ai − Ki Ci ) Pi + (γ1 + γ2 )Pi P ∑ γ4 )Pi Ki KiT Pi + (γ1 + γ3 )IM ]ei (t) − 2ϕi eT (t)(L ⊗ IN )e(t) + Ni=1 (γ2 + ∑ γ4 + q1 ) ≤ Ni=1 eTi (t)Qi ei (t) − 2ϕi λ2 (L)eT (t)e(t) + k2 . Using matrix ∑N T T properties yields ℓV (e(t)) ≤ i=1 ei (t)Qi ei (t) − 2ϕi λ2 (L)e (t)e(t) + k2 ≤ λmax (Qi )eT (t)e(t) − 2ϕi λ2 (L)eT (t)e(t) + k2 ≤ −k1 V1 (e(t)) + k2 where constants k1 and k2 are defined as

k1 =

2ϕi λ2 (L) + |λmax (Qi )|

λmax (Pi )

, k2 =

(A.9)

N ∑

(γ2 + γ4 + q1 ).

i=1

(A.8)

Açıkmeşe, Behçet, Mandić, Milan, & Speyer, Jason L. (2014). Decentralized observers with consensus filters for distributed discrete-time linear systems. Automatica, 50(4), 1037–1052. Battistelli, Giorgio, Chisci, Luigi, Fantacci, Claudio, Farina, Alfonso, & Graziano, Antonio (2013). Consensus CPHD filter for distributed multitarget tracking. IEEE Journal of Selected Topics in Signal Processing, 7(3), 508–520. Battistelli, Giorgio, Chisci, Luigi, Fantacci, Claudio, Farina, Alfonso, & Graziano, Antonio (2015a). Consensus-based multiple-model Bayesian filtering for distributed tracking. IET Radar, Sonar and Navigation, 9(4), 401–410. Battistelli, Giorgio, Chisci, Luigi, Morrocchi, Stefano, & Papi, Francesco (2011). An information-theoretic approach to distributed state estimation. IFAC Proceedings Volumes, 44(1), 12477–12482. Battistelli, Giorgio, Chisci, Luigi, Mugnai, Giovanni, Farina, Alfonso, & Graziano, Antonio (2015b). Consensus-based linear and nonlinear filtering. IEEE Transactions on Automatic Control, 60(5), 1410–1415. Boyd, Stephen, El Ghaoui, Laurent, Feron, Eric, & Balakrishnan, Venkataramanan (1994). Linear matrix inequalities in system and control theory. SIAM. Cho, Young Man, & Rajamani, Rajesh (1997). A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Transactions on Automatic Control, 42(4), 534–537. Farina, Marcello, Ferrari-Trecate, Giancarlo, & Scattolini, Riccardo (2010). Distributed moving horizon estimation for linear constrained systems. IEEE Transactions on Automatic Control, 55(11), 2462–2475. Han, Fei, Wei, Guoliang, Song, Yan, & Li, Wangyan (2015). Distributed H∞ -consensus filtering for piecewise discrete-time linear systems. Journal of the Franklin Institute, 352(5), 2029–2046. Hu, Jiangping, & Hu, Xiaoming (2010). Nonlinear filtering in target tracking using cooperative mobile sensors. Automatica, 46(12), 2041–2046. Li, W., & Jia, Y. (2011). Consensus-based distributed information filter for a class of jump Markov systems. IET Control Theory & Applications, 5(10), 1214–1222. Li, Wenling, & Jia, Yingmin (2012). Distributed consensus filtering for discretetime nonlinear systems with non-Gaussian noise. Signal Processing, 92(10), 2464–2470. Li, Wangyan, Wei, Guoliang, & Han, Fei (2014). Consensus-based unscented Kalman filter for sensor networks with sensor saturations. In 2014 International conference on mechatronics and control. (ICMC), (pp. 1220–1225). IEEE. Li, Wangyan, Wei, Guoliang, Han, Fei, & Liu, Yurong (2016). Weighted average consensus-based unscented Kalman filtering. IEEE Transactions on Cybernetics, 46(2), 558–567. Mao, Xuerong (2007). Stochastic differential equations and applications. Elsevier. Morbidi, Fabio, & Mariottini, Gian Luca (2013). Active target tracking and cooperative localization for teams of aerial vehicles. IEEE Transactions on Control Systems Technology, 21(5), 1694–1707. Olfati-Saber, Reza (2005). Distributed Kalman filter with embedded consensus filters. In 44th IEEE conference on decision and control, 2005 and 2005 European control conference. CDC-ECC’05, (pp. 8179–8184). IEEE. Olfati-Saber, Reza (2007). Distributed Kalman filtering for sensor networks. In 2007 46th IEEE conference on decision and control (pp. 5492–5498). IEEE. Olfati-Saber, Reza (2009). Kalman-consensus filter: Optimality, stability, and performance. In Proceedings of the 48th IEEE conference on decision and control, 2009 held jointly with the 2009 28th Chinese control conference. CDC/CCC 2009, (pp. 7036–7042). IEEE. Olfati-Saber, Reza, Fax, J Alex, & Murray, Richard M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215–233. Olfati-Saber, Reza, & Jalalkamali, Parisa (2012). Coupled distributed estimation and control for mobile sensor networks. IEEE Transactions on Automatic Control, 57(10), 2609–2614. Ou, Meiying, Du, Haibo, & Li, Shihua (2012). Finite-time tracking control of multiple nonholonomic mobile robots. Journal of the Franklin Institute, 349(9), 2834–2860. Ou, Meiying, Gu, Shengwei, Wang, Xianbing, & Dong, Kexiu (2015). Finite-time tracking control of multiple nonholonomic mobile robots with external disturbances. Kybernetika, 51(6), 1049–1067.

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A. Jenabzadeh, B. Safarinejadian / Automatica 86 (2017) 53–62

Shen, Bo, Wang, Zidong, & Hung, Yeung Sam (2010). Distributed H∞ -consensus filtering in sensor networks with multiple missing measurements: the finitehorizon case. Automatica, 46(10), 1682–1688. Tarn, Tzyh-Jong, & Rasis, Yona (1976). Observers for nonlinear stochastic systems. IEEE Transactions on Automatic Control, 21(4), 441–448. Ugrinovskii, Valery (2014). Gain-scheduled synchronization of parameter varying systems via relative H∞ -consensus with application to synchronization of uncertain bilinear systems. Automatica, 50(11), 2880–2887. Xie, Le, Choi, Dae-Hyun, Kar, Soummya, & Poor, H Vincent (2012). Fully distributed state estimation for wide-area monitoring systems. IEEE Transactions on Smart Grid, 3(3), 1154–1169. Xie, Li, & Khargonekar, Pramod P. (2012). Lyapunov-based adaptive state estimation for a class of nonlinear stochastic systems. Automatica, 48(7), 1423–1431. Yaz, E., & Azemi, A. (1993). Observer design for discrete and continuous nonlinear stochastic systems. International Journal of Systems Science, 24(12), 2289–2302. Zakai, Moshe (1967). On the ultimate boundedness of moments associated with solutions of stochastic differential equations. SIAM Journal on Control, 5(4), 588–593. Zhu, Shanying, Chen, Cailian, Li, Wenshuang, Yang, Bo, & Guan, Xinping (2013). Distributed optimal consensus filter for target tracking in heterogeneous sensor networks. IEEE Transactions on Cybernetics, 43(6), 1963–1976.

Ahmadreza Jenabzadeh received the M.Sc. degree in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in 2013. He is currently pursuing the Ph.D. degree in electrical engineering at Shiraz University of Technology, Shiraz, Iran. His research multiagent systems, cooperative control, and nonlinear control.

Behrouz Safarinejadian received the B.S. and M.S. degrees from Shiraz University, Shiraz, Iran, in 2002 and 2005, respectively, and the Ph.D. degree from Amirkabir University of Technology, Tehran, Iran, in 2009. Since 2009, he has been with the Faculty of Electrical and Electronic Engineering, Shiraz University of Technology, Shiraz, Iran. His research interests include multi-agent systems, control systems theory, estimation theory, statistical signal processing, sensor networks and fault detection.