Event-based recursive filtering for time-delayed stochastic nonlinear systems with missing measurements

Author’s Accepted Manuscript Event-based recursive filtering for time-delayed stochastic nonlinear systems with missing measurements Jingyang Mao, Derui Ding, Yan Song, Yurong Liu, Fuad E. Alsaadi www.elsevier.com/locate/sigpro

PII: DOI: Reference:

S0165-1684(16)30346-2 http://dx.doi.org/10.1016/j.sigpro.2016.12.004 SIGPRO6332

To appear in: Signal Processing Received date: 17 October 2016 Revised date: 2 December 2016 Accepted date: 6 December 2016 Cite this article as: Jingyang Mao, Derui Ding, Yan Song, Yurong Liu and Fuad E. Alsaadi, Event-based recursive filtering for time-delayed stochastic nonlinear systems with missing measurements, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2016.12.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Event-Based Recursive Filtering for Time-delayed Stochastic Nonlinear Systems with Missing Measurements Jingyang Mao, Derui Ding ∗ , Yan Song, Yurong Liu, and Fuad E. Alsaadi

Abstract—In this paper, the recursive filtering is investigated for a type of time-delayed stochastic nonlinear systems with event-based communication protocols and missing measurements. A varying condition threshold in event triggering protocols is introduced to better match up with the system dynamic performance. The purpose of this paper is to develop an easyimplemented recursive algorithm with consideration of linearization errors, time-delays, packet losses as well as adopted communication protocols. By carrying out some elaborate mathematical operation, the desired parameters can be derived by means of the solutions to two Riccati-like difference equations. Such parameters can effectively suppress the trace of filtering error covariances and therefore the developed filtering algorithm is suboptimal. Finally, an illustrative example is presented to show the effectiveness of the developed filtering algorithm.

Keywords: Time-delayed stochastic nonlinear systems; Recursive filtering; Event-triggering protocols; Varying thresholds. I. I NTRODUCTION As we all knew, time-delays are ubiquitous for many practical systems [8], [13], [21], [25], [38], [39], like the belt transmission systems, heat-conduction systems as well as chemical reaction processes. In addition, the units of plants are usually integrated via communication networks, and therefore the information exchange could be subject to delays due to environmental disturbances or network access This work was supported in part by the Royal Society of the UK, the National Natural Science Foundation of China under Grants 61573246 and 61460254, the Shanghai Rising-Star Program of China under Grant 16QA1403000, and the Program for Capability Construction of Shanghai Provincial Universities under Grant 15550502500. J. Mao, D. Ding and Y. Song are with the Shanghai Key Lab of Modern Optical System, Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China. Y. Liu is with the Department of Mathematics, Yangzhou University, Yangzhou 225002, P. R. China. He is also with the Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. F. E. Alsaadi is with the Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. ∗ Corresponding author. E-mail address: [email protected] (D. Ding).

collision. Furthermore, state delays exist in various real-world systems and there are chiefly two reasons for such timedelayed phenomena. The first is that state delays exist naturally in the system, such as the slow process of chemical reactions and the heating of furnace [12]. Another is influenced by the aging or network congestion of the controller and the generator components. If it cannot be adequately considered, the performance of plant systems or estimation errors might be poor, or even instable and oscillated. In current years, several ways have been put forward to deal with the state estimation of stochastic nonlinear systems. Such ways include, but are not limited to, the Kalman filtering for special nonlinear systems [29], the extended Kalman filtering (EKF) [19], [24], the particle filtering [28], the unscented Kalman filtering (UKF) [17], the moving horizon estimation [31], the cubature Kalman filtering (CKF) [2], and so on. Among them, owing to its easy-to-compute character (i.e. the first-order linearization and the iterative computation), EKF achieves an ever-increasing application in various engineering systems, such as engineering electrical electronic [5], automation control [1], telecommunications [3], instruments instrumentation [11], engineering aerospace [27]. For instance, it has been utilized with success in [37] to estimate the charge state in battery management system modeled by time-series data. It’s worth noting that EKF algorithm has been successfully adopted to carry out the filtering issues of systems with timedelayed measurements. For example, the algorithm has been used in [34] to analyze the gene regulatory network from gene time-series data. In [30], the multiple-input multiple-output radio propagation path has been detected and tracked from multidimensional channel sounding measurements in view of EKF algorithms. Furthermore, by adopting limited and delayed measurements, it has been implemented to estimate the state of controlled membrane bioreactor systems in [26]. In nonline-of-sight (NLOS) environments [18], the updating step in the EKF algorithm has been modified to reduce the NLOS errors. It deserves to be mentioned that, in comparison with the case of delayed measurements, there are not so much research

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about that with state delays. The main reason could be that the impact of state delays are transformed as part of measurement delays in much research to simplify the performance analysis. Conspicuously, it is of great realistic significance to carry out more deeper research about state-delays. As the application of information technology, data missing is often encountered in various practical systems [10], [14], [15], [23], [40] owing to bandwidth constraints. Furthermore, that improving the communication efficiency and reliability and reducing the communication burden is the pressing need which leads to an ever-increasing attention on event-triggering communication protocols [6], [7], [16], [21], [22], [33], [35]. In an event-triggered strategy with a preset event generator, measurement outputs are not allowed to be transmitted to the estimator until a certain event condition is violated. With the help of the generator in charge of event-triggered transmission, a lot of energy will be saved and the age of components could be prolonged. For example, in the multi-agent system [9], a self-triggered consensus algorithm has been proposed to determine when the agent updates its control input and transmits the average state to its neighbors. Up to now, the wildly used event-triggered conditions involved state-dependent ones [20] and state-independent ones [4] (or absolute triggered ones). In many conditions, the threshold that governs the communication burden is a known constant or a designed constant. However, such a set cannot effectively govern the communication data in initial stage of systems and therefore does not possess the adjustment ability of communication burden between the transitory stage and the stable stage. Thus, it constitutes another motivation of our discussed investigation. As is mentioned above, the main attention of this paper focus on the recursive filtering for time-delayed stochastic nonlinear systems under event-triggered protocol with varying threshold. The challenges faced in investigation are as follows: 1) how to develop a new filtering structure to cater to statedelayed stochastic nonlinear systems? 2) how to design an effective event-triggered condition to coordinate the communication burden of different time stages? 3) how to develop a concise algorithm to guarantee the desired estimation performance? In order to obtain the satisfied answer to considered challenges, an effective and concise iterative approach is elaborated to handle the filtering problem of state-delayed stochastic nonlinear systems with missing measurements. The main contribution of this paper can be concluded as follows: (1) the considered system is subject to state delays and the data transmission with missing measurements is governed by event-triggered protocols; (2) the state-independent condition with varying threshold is put forward to coordinate the communication burden; (3) an iterative approach is developed to guarantee the suboptimality of filtering performance.

2

The rest of paper is organized as follows. Section II briefly introduces the problem under consideration and draw the adopted protocol with varying threshold. In section III, the evolutions of one-step prediction error covariance and filtering error covariance are obtained for the discussed model in light of the linearization. In addition, the filtering gain is designed to minimizing the upper bound of the filtering error covariance. An illustrative example is presented in Section IV to show the effectiveness of the proposed algorithm. The paper is concluded in Section V. Notation The notation used throughout the paper is standard except where otherwise stated. R n and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n × m real matrices. I and 0 represent the identity matrix and the zero matrix of compatible dimension. The notation X ≥ Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X −Y is positive semi-definite (respectively, positive definite). For a matrix P , P T and P −1 represent its transpose and inverse in the order given. E{x} stands for the expectation of stochastic variable x. Matrices, if their dimensions are not expressly stated, are assumed to be compatible for algebraic operations. II. P ROBLEM FORMULATION AND PRELIMINARIES Consider a discrete-time nonlinear system with time-delays on finite-horizon [0, N ]: xk+1 = f (xk ) + g(xk−τ ) + Bwk ,

(1)

with measurements modeled by yk = h(xk ) + Dvk ,

(2)

where k ∈ N is the discrete time, N = {0, 1, ...}. x k ∈ Rnx represents the state vector which cannot be observed straightway, yk ∈ Rny represents the measurement output. The process noise wk ∈ Rnw and the measurement noise v k ∈ Rny are the additive noises sequences which are uncorrelated in k with zero-mean and unity covariance, severally. B and D are known matrices with compatible dimensions. τ is a known positive integer. The nonlinear functions f (x k ) : Rnx → Rnx , g(xk ) : Rnx → Rnx and h(xk ) : Rnx → Rny are assumed to be known and analytic everywhere. In this paper, the event-triggered communication protocol is adopted in order to reduce network burden. Specially, the following event generator function is utilized (yk , σk ) = (yk − ysi )T (yk − ysi ) − σk ,

(3)

where σk = δ1 e−δ2 k + δ3 , δ1 , δ2 and δ3 are three predefined scalars, and y ks is the transmitted information at the latest event instant. For such a communication protocol, if (yk , σk ) > 0, then the current measurement y k will be

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transmitted to a filer via networks. Obviously, the sequence of event triggered instants 0 ≤ s 0 ≤ s1 ≤ · · · ≤ si · · · is determined iteratively by si+1 = inf{k|(yk , σk ) ≥ 0, k > si }. Furthermore, considering the phenomenon of packet loss, the data y˜k received by a filter with zero-order holder (ZOH) is of the form y˜k = αsi ysi ,

k ∈ {si , si + 1, · · · , si+1 − 1},

(4)

where the stochastic variable α si , which governs the missing communication, takes values on 0 or 1. It is a Bernoulli distributed white sequence with the undermentioned probabilities Prob{αsi = 0} = 1 − α ¯,

Prob{αsi = 1} = α ¯,

where α ¯ ∈ [0, 1) is a known constant. In this paper, the Kalman-type recursive filter over [si , si+1 − 1] to be figured is of the following structure:  x ˆk+1|k = f (ˆ xk|k ) + g(ˆ xk−τ |k−τ ) (5a)   x ˆk+1|k+1 = xˆk+1|k + Kk+1 y˜k+1 − α ¯ h(ˆ xk+1|k ) (5b) where x ˆk|k ∈ Rnx is the state estimate of xk at time step ¯0 , x ˆk+1|k ∈ Rnx is the one-step prediction, k with x ˜0|0 = x Kk+1 is the filter parameter to be solved. Remark 1: The proposed event generator function (3), which means that the measurement can be transferred only when this significant rule is violated, can improve the efficiency of resource utilization and prolong the components’ life. Obviously, the varying threshold can effectively adjust the communication burden in different dynamical stages. Similar scheme has been utilized in [41] to address the output feedback control problem. The predefined scalars δ 1 and δ2 guarantee the accuracy of estimation in the first few steps. In addition, we employ δ3 to insure the minimum triggered times according to different cases, which guarantees the accuracy of estimation in the long run. To an utmost degree, almost all measurements would be transferred as δ 1 e−δ2 k + δ3 tends to 0, and the strategy reduces to conventional time-triggered transmission. In addition, the term on state-delays, g(ˆ x k−τ |k−τ ), is naturally added in filter (5a) to improve the filtering performance. Note that, the missing on xˆ k−τ +1|k−τ +1 , · · · , x ˆk−1|k−1 can effectively reduce the computation complexity. Let us denote the one-step prediction error and the filtering ˆk+1|k and x ˜k+1|k+1 := xk+1 − error as x ˜k+1|k := xk+1 − x x ˆk+1|k+1 in the order given. The aim of this paper is to design a Kalman-type filter of the structure (5a) and (5b) so that the following requirements are met: T1) For the adopted communication protocol and the phenomenon of packet loss, an upper bound for the filtering error covariance is supposed to be derived, namely, there exists a

sequence of positive-definite matrices Σ k+1|k+1 (0 ≤ k ≤ N ) that satisfies ˆk+1|k+1 )(xk+1 − x ˆk+1|k+1 )T } ≤ Σk+1|k+1 (6) E{(xk+1 − x T2) Design the filter parameter K k+1 to minimize the upper bound Σ k|k through a recursive scheme. III. M AIN RESULTS In this section, to dispose the nonlinear filtering problem with state-delays and missing measurements, the linearization with first-order approximation is implemented to facilitate the following discussion. Then, with the help of stochastic analysis and some common matrix inequalities, the one-step prediction error covariance and the filtering error covariance are derived step by step to design the algorithm on finite horizon. In addition, the filtering gain is calculated to minimize the upper bound of the filtering error covariance. In the first place, subtracting (5a) from (1) results in x ˜k+1|k =xk+1 − x ˆk+1|k =f (xk ) + g(xk−τ ) + Bwk

(7)

− f (ˆ xk|k ) − g(ˆ xk−τ |k−τ ) With the help of the Taylor series expansion to f (x k ) around x ˆk|k , we have f (xk ) = f (ˆ xk|k ) + Ak x ˜k|k + o(|˜ xk|k |)

(8)

 ∂f (xk )  Ak = ∂xk xk =ˆxk|k

where

and o(|˜ xk|k |) stands for the first-order term of the Taylor series expansion. In addition, this high-order term can be changed into the undermentioned style and handled more easily: ˜k|k o(|˜ xk|k |) = Ck ℵ1,k Lk x

(9)

where Ck is a matrix that depends on the problem, L k is adopted to adjust the filtering with a further extent of freedom, and ℵ1,k is an unknown discrete-time matrix that represents the error of linearization of model that meets ℵ1,k ℵT1,k ≤ I

(10)

And so do g(xk−τ ) ˜k−τ |k−τ + o(|˜ xk−τ |k−τ |) = g(ˆ xk−τ |k−τ ) + Ek−τ x where Ek−τ

(11)

 ∂g(xk−τ )  = ∂xk−τ xk−τ =ˆxk−τ |k−τ

and o(|˜ xk−τ |k−τ |) = Fk−τ ℵ2,k−τ Lk−τ x˜k−τ |k−τ

(12)

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Here, Fk−τ is a matrix that is dependent on the problem, and ℵ2,k−τ is an unidentified discrete-time matrix that accounts the error of linearization of model that fulfills

positive constant γ, if γ −1 I − EXE T > 0, then the following inequality holds

ℵ2,k−τ ℵT2,k−τ ≤ I

≤ A(X −1 − γE T E)−1 AT + γ −1 HH T

(13)

Similar to functions f and g, linearizing h(x k+1 ) around x ˆk+1|k leads to h(xk+1 ) = h(ˆ xk+1|k ) + Gk+1 x˜k+1|k + o(|˜ xk+1|k |) where Gk+1 =

(A + HF E)X(A + HF E)T

(14)

 ∂h(xk+1 )  ∂xk+1 xk+1 =ˆxk+1|k (15)

(16)

It follows from (7) to (9) that x ˜k+1|k =(Ak + Ck ℵ1,k Lk )˜ xk|k + g(xk−τ ) + Bwk − g(ˆ xk−τ |k−τ ) xk|k + Bwk = (Ak + Ck ℵ1,k Lk )˜

∀X ≤ Y = Y T

φk (y) ≥ ξk (y)

Hk+1 is a problem-dependent matrix, and ℵ 3,k+1 is an unknown discrete-time matrix that plays a role of error of linearizations of the model satisfying ℵ3,k+1 ℵT3,k+1 ≤ I

ξk (Y ) ≥ ξk (X), and

and ˜k+1|k o(|˜ xk+1|k |) = Hk+1 ℵ3,k+1 Lk+1 x

Lemma 3: [32] For the given matrix X = X T > 0, if the matrix value functions ξ k (X) = ξkT (X) ∈ Rn×n and φk (x) = T φk (x) ∈ Rn×n over [0, N ] satisfy

(17)

then the solutions M k and Nk to the following difference equations Mk+1 = ξ(Mk ),

Nk+1 = φ(Mk ),

M 0 = N0 > 0

satisfy M k ≤ Nk . Now, we have the following theorems on the prediction error covariance and the filtering error covariance. Theorem 1: For addressed system (1) with Kalman-type recursive filtering (5a)-(5b), the one-step prediction error covariance Pk+1|k can be recursively calculated as Pk+1|k = (Ak + Ck ℵ1,k Lk )Pk|k (Ak + Ck ℵ1,k Lk )T

+ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )˜ xk−τ |k−τ According to (4) and (5b), the filtering error can be expressed as:

+ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )Pk−τ |k−τ × Ek−τ + Fk−τ ℵ2,k−τ Lk−τ T + (Ak + Ck ℵ1,k Lk )E{˜ xk|k x˜Tk−τ |k−τ }

x ˜k+1|k+1

× (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T

= xk+1 − x ˆk+1|k+1 = xk+1 − x ˆk+1|k − Kk+1 [˜ yk+1 − α ¯ h(ˆ xk+1|k )]  =x ˜k+1|k − Kk+1 y˜k+1 + αsi yk+1  − αsi yk+1 − αh(ˆ ¯ xk+1|k )

+ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )E{˜ xk−τ |k−τ x ˜Tk|k } × (Ak + Ck ℵ1,k Lk )T + BB T (18)

= [I − αsi Kk+1 (Gk+1 + Hk+1 ℵ3,k+1 Lk+1 )]˜ xk+1|k − αsi Kk+1 (ysi − yk+1 ) − (αsi − α ¯ )Kk+1 h(ˆ xk+1|k ) − αsi Kk+1 Dvk+1 Before proceeding further, let us introduce the following lemmas which will be helpful in subsequent developments. Lemma 1: For any two vectors x, y ∈ R n , the following inequality holds T

(19)

T

T

xy + yx ≤ εxx + ε

−1

yy

T

where ε > 0 is a scalar. Lemma 2: [36] Assume that the appropriate dimensions’ matrices A, H, E and F satisfying F F T ≤ I is given. For a symmetric positive definite matrix X and an arbitrary

Proof: Considering (8), (9), (11), (12) and (17), we have Pk+1|k =E{˜ xk+1|k x ˜Tk+1|k }  xk|k =E [(Ak + Ck ℵ1,k Lk )˜ + (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )˜ xk−τ |k−τ + Bwk ] xk|k × [(Ak + Ck ℵ1,k Lk )˜ + (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )˜ xk−τ |k−τ + Bwk ]T =(Ak + Ck ℵ1,k Lk )Pk|k (Ak + Ck ℵ1,k Lk )T + (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )Pk−τ |k−τ × (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T + (Ak + Ck ℵ1,k Lk )E{˜ xk|k x ˜Tk−τ |k−τ } × (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T + (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )E × {˜ xk−τ |k−τ x ˜Tk|k }(Ak + Ck ℵ1,k Lk )T + BB T



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which is identical to (19). Theorem 2: For addressed system (1) with Kalman-type recursive filter (5a)-(5b), the filtering error covariance P k+1|k+1 is given by Pk+1|k+1  = I − αsi Kk+1 (Gk+1 + Hk+1 ℵ3,  × Pk+1|k I − αsi Kk+1 (Gk+1 T + Hk+1 ℵ3, k+1 Lk+1 )



k+1 Lk+1 )

T +α ¯ Kk+1 E{(ysi − yk+1 )(ysi − yk+1 )T }Kk+1 T + (¯ α−α ¯ 2 )Kk+1 E{h(ˆ xk+1|k )hT (ˆ xk+1|k )}Kk+1 T +α ¯ Kk+1 DDT Kk+1

−α ¯ [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3, × −

E{vk } = 0, E{vk vkT } = 1. It follows from above formulas that (21) can be rewritten as (20). The proof is now complete. Up to now, we have proposed the prediction error covariance and the filtering error covariance. In what follows, we need to get the filter gain while guaranteeing the upper bound constraint. Theorem 3: Note covariance matrices of the one-step prediction error and the filtering error in (19) and (20). Assume that (10), (13) and (16) are true. For arbitrary positive constants γ1,k , γ2,k−τ , γ3,k+1 , ε1 , ε2 and ε3 , and the given initial condition Σ0|0 = P0|0 > 0, if, for all 0 ≤ k ≤ N , the subsequent two discrete-time Riccati-like difference equations: Σk+1|k

k+1 Lk+1 )]

T −1 T = (1 + ε1 )Ak (Σ−1 Ak k|k − γ1,k Lk Lk )

T

T E{˜ xk+1|k (ysi − yk+1 ) }Kk+1 α ¯ Kk+1 E{(ysi − yk+1 )˜ xTk+1|k }

−1 −1 + (1 + ε1 )γ1,k Ck CkT + (1 + ε−1 1 )Ek−τ (Σk−τ |k−τ (22) T − γ2,k−τ LTk−τ Lk−τ )−1 Ek−τ

T × [I − αK ¯ k+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]T Kk+1 T + (¯ α−α ¯ 2 )Kk+1 E{(ysi − yk+1 )hT (ˆ xk+1|k )}Kk+1 T + (¯ α−α ¯ 2 )Kk+1 E{h(ˆ xk+1|k )(ysi − yk+1 )T )}Kk+1 (20)

−1 T T + (1 + α ¯ ε2 )γ3,k+1 Kk+1 Hk+1 Hk+1 Kk+1

= E{˜ xk+1|k+1 x ˜Tk+1|k+1 }

+

= M Pk+1|k M T T + αK ¯ k+1 E{(ysi − yk+1 )(ysi − yk+1 )T }Kk+1 T + (¯ α−α ¯ 2 )Kk+1 E{h(ˆ xk+1|k )hT (ˆ xk+1|k )}Kk+1

+ (21)

T + αK ¯ k+1 DDT Kk+1

− M12 − M13 − M14 + M23 + M24 + M34 T T T T T T − M12 − M13 − M14 + M23 + M24 + M34

where M = I − αsi Kk+1 (Gk+1 + Hk+1 ℵ3,

k+1 Lk+1 )

T ¯ M E{˜ xk+1|k (ysi − yk+1 )T }Kk+1 M12 = α T M13 = M E{˜ xk+1|k (αsi − α)h ¯ T (ˆ xk+1|k )}Kk+1 T T α ¯ M E{˜ xk+1|k vk+1 }DT Kk+1

T M23 = Kk+1 E{αsi (αsi − α)(y ¯ si − yk+1 )hT (ˆ xk+1|k )}Kk+1 T T M24 = α ¯ Kk+1 E{(ysi − yk+1 )vk+1 }DT Kk+1 T T α)h(ˆ ¯ xk+1|k )vk+1 }DT Kk+1

+

+α ¯ ε−1 2 2

Prob{αsi = 0} = 1 − α ¯ , Prob{αsi = 1} = α ¯ one has E{(αsi − α ¯ )2 } = α ¯−α ¯2 .

As vk is the additive noise with zero-mean and unity covariance, and is mutually uncorrelated in k, so we have

(23)

T σk [¯ α + (¯ α−α ¯ )ε−1 3 ]Kk+1 Kk+1 T (¯ α−α ¯ )(1 + ε3 )Kk+1 ψk+1|k Kk+1 T α ¯ Kk+1 DDT Kk+1 2

with ψk+1|k = h(ˆ xk+1|k )hT (ˆ xk+1|k ) have positive definite solutions Σk+1|k and Σk+1|k+1 , and the following three constrains −1 I − Lk Σk|k LTk > 0 γ1,k −1 γ2,k−τ I −1 γ3,k+1 I

(24)

− Lk−τ Σk−τ |k−τ LTk−τ > − Lk+1 Σk+1|k LTk+1 > 0

0

(25) (26)

are satisfied, then with the filter gain K k+1 given by T −1 ¯ (1 + α ¯ ε2 )(Σ−1 Kk+1 =α k+1|k − γ3,k+1 Lk+1 Lk+1 )

¯ 2 (1 + α × GTk+1 α ¯ ε2 )Gk+1 (Σ−1 k+1|k

− γ3,k+1 LTk+1 Lk+1 )−1 GTk+1

(27)

−1 T + (1 + α ¯ ε2 )γ3,k+1 Hk+1 Hk+1

+ σk [¯ α+α ¯ ε−1 α−α ¯ 2 )ε−1 2 + (¯ 3 ]

Recalling

E{αsi − α ¯ } = 0,

Σk+1|k+1 − γ3,k+1 LTk+1 Lk+1 )−1 (I − αK ¯ k+1 Gk+1 )T

Pk+1|k+1

M34 = Kk+1 E{αsi (αsi −

and = (1 + αε ¯ 2 )(I − α ¯ Kk+1 Gk+1 )(Σ−1 k+1|k

Proof: Noting(18), we have

M14 =

−1 T T + (1 + ε−1 1 )γ2,k−τ Fk−τ Fk−τ + BB

+ (¯ α−α ¯ 2 )(1 + ε3 )ψk+1|k + α ¯ DDT

−1

,

the matrix Σk+1|k+1 is an upper bound for P k+1|k+1 , namely, Pk+1|k+1 ≤ Σk+1|k+1 . Proof: In the first place, on the basis of (19) and (20),rewrite the one-step prediction error covariance P k+1|k and the filtering error covariance P k+1|k+1 as the functions of P k|k and

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6

Pk+1|k as follows:

Similarly, we have xTk+1|k } −α ¯ Kk+1 E{(ysi − yk+1 )˜

Pk+1|k

× [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]T

=φ(Pk|k )

−α ¯ [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]

=(Ak + Ck ℵ1,k Lk )Pk|k (Ak + Ck ℵ1,k Lk )T

T × E{˜ xk+1|k (ysi − yk+1 )T }Kk+1

+ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )Pk−τ |k−τ

≤α ¯ ε2 [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]Pk+1|k

× (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T

× [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]T

+ (Ak + Ck ℵ1,k Lk )E{˜ xk|k x ˜Tk−τ |k−τ }

T T +α ¯ ε−1 2 Kk+1 E{(ysi − yk+1 )(ysi − yk+1 ) }Kk+1

× (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T

and

+ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )E{˜ xk−τ |k−τ x ˜Tk|k }

T (¯ α−α ¯2 )Kk+1 E{(ysi − yk+1 )hT (ˆ xk+1|k )}Kk+1

× (Ak + Ck ℵ1,k Lk )T + BB T

T + (¯ α−α ¯ 2 )Kk+1 E{h(ˆ xk+1|k )(ysi − yk+1 )T )}Kk+1 T ≤(¯ α−α ¯2 )ε3 Kk+1 E{h(ˆ xk+1|k )hT (ˆ xk+1|k )}Kk+1

Pk+1|k+1

+ (¯ α−α ¯ 2 )ε−1 3 Kk+1 E{(ysi − yk+1 )

=φ(Pk+1|k ) = [I − αsi Kk+1 (Gk+1 + Hk+1 ℵ3,

T × (ysi − yk+1 )T )}Kk+1

k+1 Lk+1 )]Pk+1|k

× [I − αsi Kk+1 (Gk+1 + Hk+1 ℵ3,

k+1 Lk+1 )]

(31)

T

So, we get

T +α ¯ Kk+1 E{(ysi − yk+1 )(ysi − yk+1 )T }Kk+1

Pk+1|k+1

T + (¯ α−α ¯2 )Kk+1 E{h(ˆ xk+1 )hT (ˆ xk+1 )}Kk+1

≤ (1 + αε ¯ 2 )[I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]

T +α ¯ Kk+1 DDT Kk+1

−α ¯ [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3, × − × + +

(30)

¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]T × Pk+1|k [I − α

k+1 Lk+1 )]

+ [¯ α+α ¯ ε−1 α−α ¯2 )ε−1 2 + (¯ 3 ]Kk+1

T

T E{˜ xk+1|k (ysi − yk+1 ) }Kk+1 α ¯ Kk+1 E{(ysi − yk+1 )˜ xTk+1|k }

T × E{(ysi − yk+1 )(ysi − yk+1 )T }Kk+1 T

T [I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )] Kk+1 T (¯ α−α ¯2 )Kk+1 E{(ysi − yk+1 )hT (ˆ xk+1 )}Kk+1 T (¯ α−α ¯2 )Kk+1 E{h(ˆ xk+1 )(ysi − yk+1 )T )}Kk+1

After that, it is feasible to verify that Lemma 3 is satisfied. With the help of Lemma 1, we have

In light of (3), we have (ysi − yk+1 )(ysi − yk+1 )T ≤ σk I

(33)

¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]T × Pk+1|k [I − α (28)

≤ε1 (Ak + Ck ℵ1,k Lk )Pk|k (Ak + Ck ℵ1,k Lk )T

T + σk [¯ α+α ¯ ε−1 α−α ¯ 2 )ε−1 2 + (¯ 3 ]Kk+1 Kk+1

(34)

+ (¯ α−α ¯ 2 )(1 + ε3 )Kk+1 T × E{h(xk+1|k )hT (xk+1|k )}Kk+1

T + ε−1 1 (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ ) Pk−τ |k−τ

T +α ¯ Kk+1 DDT Kk+1

× (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T So Pk+1|k

× Pk−τ |k−τ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T + BB T

T ×α ¯ Kk+1 DDT Kk+1

≤ (1 + αε ¯ 2 )[I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]

+ (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )E{˜ xk−τ |k−τ x˜Tk|k }

T + (1 + ε−1 1 )(Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )

T × E{h(ˆ xk+1|k )hT (ˆ xk+1|k )}Kk+1

Pk+1|k+1

× (Ek−τ + Fk−τ ℵ2,k−τ Lk−τ )T

≤ (1 + ε1 )(Ak + Ck ℵ1,k Lk )Pk|k (Ak + Ck ℵ1,k Lk )T

+ (¯ α−α ¯ 2 )(1 + ε3 )Kk+1

As a result, we have

(Ak + Ck ℵ1,k Lk )E{˜ xk|k x˜Tk−τ |k−τ }

× (Ak + Ck ℵ1,k Lk )T

(32)

(29)

With consideration of equation (5a) and the assumption that nonlinear functions f (x k ) : Rnx → Rnx , g(xk ) : Rnx → Rnx and h(xk ) : Rnx → Rny are known and analytic everywhere and τ is a known positive integer, we can deduce that E{h(xk+1 )hT (xk+1 )} is calculable and rewrite it as the following equation ψk+1|k = E{h(ˆ xk+1|k )hT (ˆ xk+1|k )}

(35)

REVISION

7

Subsequently, we can infer that

Based on the above equation, the optimal filtering gain Kk+1 can be determined as

Pk+1|k+1

T −1 T ¯ (1 + α ¯ ε2 )(Σ−1 Gk+1 Kk+1 = α k+1|k − γ3,k+1 Lk+1 Lk+1 )

−1 2 T × α ¯ (1 + α ¯ ε2 )Gk+1 (Σk+1|k − γ3,k+1 Lk+1 Lk+1 )−1

≤ (1 + α ¯ ε2 )[I − α ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )] ¯ Kk+1 (Gk+1 + Hk+1 ℵ3,k Lk+1 )]T × Pk+1|k [I − α T + σk [¯ α+α ¯ ε−1 α−α ¯ 2 )ε−1 2 + (¯ 3 ]Kk+1 Kk+1

(36)

−1 T × GTk+1 + (1 + α ¯ ε2 )γ3,k+1 Hk+1 Hk+1

T + (¯ α−α ¯ 2 )(1 + ε3 )Kk+1 ψk+1|k Kk+1

+ σk [¯ α+α ¯ ε−1 α−α ¯ 2 )ε−1 2 + (¯ 3 ]

T +α ¯ Kk+1 DDT Kk+1

T + (¯ α−α ¯ 2 )(1 + ε3 )ψk+1|k + αDD ¯

Then, on the basis of (22) and (23), we continue to rewrite Σk+1|k and Σk+1|k+1 as the function of Σ k|k and Σk+1|k as follows: Σk+1|k =ϕ(Σk|k ) T −1 T =(1 + ε1 )Ak (Σ−1 Ak k|k − γ1,k Lk Lk ) −1 + (1 + ε1 )γ1,k Ck CkT + (1 + ε−1 1 )Ek−τ

× +

(37)

T −1 T (Σ−1 Ek−τ k−τ |k−τ − γ2,k−τ Lk−τ Lk−τ ) −1 −1 T T (1 + ε1 )γ2,k−τ Fk−τ Fk−τ + BB

and Σk+1|k+1 =ϕ(Σk+1|k ) =(1 + αε ¯ 2 )(I − α ¯ Kk+1 Gk+1 )(Σ−1 k+1|k − γ3,k+1 LTk+1 Lk+1 )−1 (I − αK ¯ k+1 Gk+1 )T −1 T T + (1 + α ¯ ε2 )γ3,k+1 Kk+1 Hk+1 Hk+1 Kk+1 T + σk [¯ α+α ¯ ε−1 α−α ¯ 2 )ε−1 2 + (¯ 3 ]Kk+1 Kk+1 T + (¯ α−α ¯ 2 )(1 + ε3 )Kk+1 ψk+1|k Kk+1 T +α ¯ Kk+1 DDT Kk+1

(38) where ψk+1|k is defined in (35). In terms of (29), (36)-(38), Lemma 2 and Lemma 3, we have Pk+1|k+1 ≤ Σk+1|k+1 Next, we are ready to show that the filtering gain given by (27) is optimal in the sense that it minimizes the upper bound Σ k+1|k+1 . Taking the partial derivative of Σ k+1|k+1 with respect to Kk+1 and letting the derivative be zero, we have ∂Σk+1|k+1 ∂Kk+1 = − 2(1 + αε ¯ 2 )(I − α ¯ Kk+1 Gk+1 ) T −1 α ¯ GTk+1 × (Σ−1 k+1|k − γ3,k+1 Lk+1 Lk+1 )

+ 2Kk+1 σk [¯ α+α ¯ ε−1 α−α ¯ 2 )ε−1 2 + (¯ 3 ] −1 T + 2Kk+1 (1 + α ¯ ε2 )γ3,k+1 Hk+1 Hk+1 T + 2Kk+1 (¯ α−α ¯ 2 )(1 + ε3 )Kk+1 ψk+1|k Kk+1

+ 2Kk+1 α ¯ DDT = 0

−1

which is as same as (27). It is clear that the filtering gain given by (27) is optimal that minimizes the upper bound Σ k+1|k+1 for the filtering error covariance. This completes the proof. Remark 2: Note that, it is almost impossible to get the exact value of the filtering error covariance P k+1|k+1 which is ascribable to the consideration of the linearization errors, event-triggering protocols and time-delays. In addition, in comparison with the augmented approach adopted in [42], the utilized inequalities (28) and (29) prominently reduce the calculation burden and improve its real-time. Remark 3: The iterative algorithm with the form of two Riccati-like difference equations is established by minimizing the upper bound for the filtering error covariance. Note that the schoolbook matrix algebra has a multiplication-time complexity bounded by O(nmp), where the inputs are an n × mmatrix and an m × p-matrix, and the output is a n × p-matrix. In addition, the complexity of matrix inversion is bounded by O(n3 ) for an n × n-matrix. Taking these into consideration, we can obtain that the whole computational complexity of this algorithm is O(nx 3 ), which depends on the dimension of state variables. IV. A NUMERICAL EXAMPLE In this section, a numerical example is adopted to illustrate the effectiveness of the the event-based recursive filtering algorithm developed in this paper for discrete time-delayed stochastic nonlinear system with missing measurements. Consider the target system (1) with τ = 1 and other parameters given as follows:

0.72x2k − 0.6x1k x2k f (xk ) = 0.42 sin(x1k x2k ) + 0.6x2k

0.12x1k−τ + 0.15 cos(x1k−τ x2k−τ ) g(xk−τ ) = 0.24x2k−τ − 0.15x1k−τ x2k−τ h(xk ) = 2x1k + 5x2k B = [0.01 0.03]T , D = 0.05 Denote xk = [x1k x2k ]T , and formulate the first-order expansion term coefficient of f (x k ) as Ak =

−0.6ˆ x2k 0.42 cos(ˆ x1k x ˆ2k )ˆ x2k

0.72 − 0.6ˆ x1k 1 2 1 0.42 cos(ˆ xk x ˆk )ˆ xk + 0.6

REVISION

8

with the high-order expansion term C k = diag{0.1, 0.2} and set the first-order expansion term coefficient of g(x k−τ ) as

ek11 −τ ek12 −τ Ek−τ = ek21 −τ ek22 −τ

1.6

States and their estimation

1.2

x1k−τ xˆ2k−τ )ˆ x2k−τ , ek12 −τ = with ek11 −τ = 0.12 − 0.15 sin(ˆ −0.15 − 0.15 sin(ˆ x1k−τ xˆ2k−τ )ˆ x1k−τ , ek21 −τ = −0.15ˆ x2k−τ , 1 xk−τ , and the high-order expansion ek22 −τ = 0.24 − 0.15ˆ

V. C ONCLUSIONS In this paper, the recursive filtering problem has been deliberated for a type of discrete time-delayed stochastic nonlinear systems with event-based measurement transmissions

1 0.8 0.6 0.4 0.2 0 −0.2

0

10

20

30

40 50 Time (k)

60

70

80

90

Fig. 1. The states and their estimations.

2.5 Measurement outputs Received data Measurement outputs and received data

term Fk−τ = diag{0.1, 0.25} and give value to the firstorder expansion term coefficient of h(x k+1 ) and its high-order expansion term as G k+1 = Hk+1 = 0.15. The threshold of the event generator function (3) is modeled with σk = δ1 e−δ2 k + δ3 , where δ1 = 0.01, δ2 = 0.05, δ3 = 0.08, and the probability of successful communication is chosen as α ¯ = 0.9. In the simulation, the initial conditions of estimations are set as x−1 = [−0.4 0.3]T , x0 = [1.6 − 0.2]T , T x ˆ−1|−1 = xˆ0|0 = [0 0] , Σ0|0 = 25I2 . Other parameters are designed as L k = Lk−τ = Lk+1 = 0.01I2 , γ1,k = 0.002, γ2,k−τ = 0.003, γ3,k+1 = 0.004, 1 = 0.4, 2 = 0.35, 3 = 0.3. Let MSE1 represents the mean squared error(MSE) to estimate the former state 2 N  of x like (1/N ) k=1 [1 0][xk − x ˆk|k ] , and, analogously, 2 N  ˆk|k ] , MSE2 of the latter state as (1/N ) k=1 [0 1][xk − x where N is the amount of samples. According to (22), (23) and (27) in Theorem 3, we can get the upper bound of the filtering error covariance and filtering gain at each time step through recursive calculation. Consequently, the stated problem can be figured out via the addressed filter structure (5a) and (5b). The simulation results are presented in Figs.1-5, where Fig.1 depicts the states x1k and x2k and their estimations and confirms that the estimations of system states perform well. Then, Fig.2 shows the measurement outputs and the corresponding real received signals, from which we can infer that the efficiency of transmission will be largely improved. After that, Fig.3 plots points where the event-triggering condition is violated. In addition, Fig.4 demonstrates ratios of network burden to original ones. What’s more, we can infer from Fig.5 the trace of the upper bounds Σ 1k|k 1 and Σ2k|k and the MSE for the states x1k and x2k , which substantiate that the upper bounds keep above the MSE. It is confirmed that the proposed filtering could improve transmission efficiency and carry out the estimation favorably in terms of missing measurements and event-based measurement transmissions for time-delayed stochastic nonlinear systems.

x1,k Estimation of x1,k x2,k Estimation of x2,k

1.4

2

1.5

1

0.5

0

−0.5

−1

0

10

20

30

40 50 Time (k)

60

70

80

90

70

80

90

Fig. 2. Measurement outputs and received data.

0

10

20

Fig. 3. The triggered time.

30

40 50 Time (k)

60

REVISION

9

50

[2]

45

Network burden ratio (%)

40 35

[3]

30 25

[4]

20 15 10

[5]

5 0

50

0

350

300

250 200 150 Experiment No. (s)

100

400

[6]

Fig. 4. Ratios of network burden to original ones. [7] 6

[8]

Log(MSE and their upper bound)

4

[9]

2

0

M SE1 Upper bound of M SE1 M SE2 Upper bound of M SE2

−2

[10]

[11]

−4

−6

[12] −8

0

10

20

30

40 50 Time (k)

60

70

80

90

[13] Fig. 5. MSE and their upper bounds.

and missing measurements. A novel form of event-based condition threshold has been suitably drawn where the current measurement is transmitted to filter through network only when the designed communication protocol is transgressed in the event generator. Then, the random occurrence of missing measurements has been described by a set of random variables which obey Bernoulli distribution. In addition, we have properly designed a Kalman-type recursive filtering so as to derive the upper bound of filtering error covariance and minimize it at each time step. Finally, an illustrative simulation example has been carried out to demonstrate the effectiveness and applicability of the proposed algorithm. Further research topics include the extension of our results to more communication protocol problems with various communication constraints.

[14]

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[17]

[18]

[19]

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