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Energyefficient filtering algorithm for a class of industrial sensor network systems with packet dropouts, timevarying delay, and multiplicative noises
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 162 (2019)
Abstract
In this paper, for the purpose of improving the energy efficiency of the industrial sensor networks, we investigated the eventbased H_{∞} filtering problem for a class of discretetime nonlinear sensor network systems with timevarying delay, packet dropout, and multiplicative noises. Instead of traditional timetriggered communication mechanism, the eventtriggered strategy is adopted in industrial sensor network, which could not only reduce the transmission frequency of the sensor measurement output, but also guarantee the prescribed filtering performance, if only the threshold in the eventtriggered function is chosen suitably. The timevarying delay characteristic of systems is considered with the eventtriggered strategy, which has seldom been studied due to the complexity of timevarying delay and eventtriggered strategy. The most common networkinduced phenomenon of packet dropout in industrial sensor network is described. The purpose is to design a filter satisfying exponentially stable and H_{∞} indexes. The main result is that sufficient conditions are established, guaranteeing our proposed filter satisfying filtering performance constraints, and the parameters of filter could be got through the derived linear matrix inequality (LMI), if only it is feasible. At last, the filtering approach is demonstrated by a simulation.
Introduction
Over the past decades, the H_{∞} filtering technique has attracted considerable research attention and fruitful results have appeared, see for example [1–13] and the references therein. This is mainly due to the following two reasons. Firstly, in a lot of practical engineering, it is hard to get the probabilistic information of disturbance and the H_{∞} technique could well deal with this kind of noise signals. Secondly, no matter how precise the system model is, there is also some error between the physical plant and its model. And the robustness of the H_{∞} filtering approach may tolerate such error in system model. From the above analysis, we could find that investigating the H_{∞} filtering technique has not only theoretically importance but also engineering significance. As such, we will employ the H_{∞} approach to design the filter for a class of sensor network systems.
It is well known that the limited network channel bandwidth and limited power are significant factors constraining the performance of industrial sensor network systems [14–19]. In traditional timetriggered communication mechanism, the signal of sensor is transmitted to the filter or controller at every time, which does not consider the limited bandwidth of communication channel and therefore increases the burden of industrial sensor network channel. To avoid the unnecessary frequent communication and save limited energy, an effective method is adopting eventtriggered strategy [20–24], in which sensor measurement output is transmitted only when an eventtriggered condition is satisfied. If only the eventtriggered condition is suitably constructed, the transmission frequency of measurement will decrease while maintaining the prescribed filtering performance. During recent years, the eventtriggered communication mechanism has been successfully applied to controller design for various engineer systems, such as networked systems [25, 26] and multiagent systems [27–29]. Also, some results about eventbased filter design have appeared, see for example [30–34]. However, when it comes to the industrial sensor network systems, considering the inevitable networkinduced phenomena, the eventbased filter design approach has not been adequately investigated and still has many problems needed to be solved. Therefore, the eventtriggered communication mechanism will be adopted in the filtering problem for the proposed industrial sensor network systems.
Noting that, nonlinear control and filtering have attracted much interest [4, 35–41], due to the popular existence of nonlinearity in a lot of practical systems and its important effectiveness to systems. In [4], a sectorbounded approach is proposed to handle with a class of nonlinearities. It is pointed out that many plants may be modeled by systems with multiplicative noises and some characteristics of nonlinear systems can be closely approximately by models with multiplicative noises rather than by linearized models [42, 43]. Therefore, in this paper, the nonlinearity of addressed systems is described by a nonlinear function and statemultiplicative noises, which could better present the practical nonlinearity.
As a main source of system instability, timedelay widely exists in practical industrial sensor network systems and should be taken into the analysis process of systems. As such, the H_{∞} filtering for various timedelay plants has attracted much interest, see [35, 44–46] and the reference therein. For example, the robust filter is designed for systems with packet dropout and constant delay in [44]. In [35], a delaydependent H_{∞} filtering method is proposed for delay systems whose postpone is timevarying. Very recently, in [30], the eventtriggered strategy is adopted to address distributed H_{∞} filtering problem for industrial sensor networks with timeinvarying delay. Unfortunately, up to now, when eventtriggered communication is adopted, the relative investigation about eventbased H_{∞} filter design problem has seldom taken timevarying delay into account. Therefore, we will investigate the eventbased H_{∞} filtering problem for industrial sensor networks whose postpone is timevarying.
Summarizing the above discussions, the eventbased H_{∞} filtering problem will be investigated for a class of nonlinear industrial sensor network systems with packet dropouts, multiplicative noises and timevarying delay. The main contributions are highlighted as follows:
1. During the design of filter for a class of discretetime sensor network systems with timevarying delay, the eventtriggered communication mechanism is adopted.
2. A comprehensive model of nonlinear sensor network systems is proposed which subjects to packet dropouts, multiplicative noises, and timevarying delay.
3. Sufficient conditions are built which could ensure proposed filter and corresponding eventbased filtering algorithm is addressed.
Section 2 introduces the methods utilized for the energyefficient filter. In Section 3, the delay sensor network with packet dropouts and multiplicative noises is introduced. The results and discussions are given in Section 4, where sufficient condition is derived for the H_{∞} filter and the filtering method is addressed. A numerical example is given in Section 5. Finally, we conclude in Section 6.
Methods
In this paper, the energyefficient filter is designed based on Lyapunov theory method and linear matrix inequality method. The simulation experiment is based on the LMI toolbox of MATLAB R2014a.
Problem formulation and preliminaries
Here, the following discrete nonlinear sensor network system with timevarying delay and multiplicative noise is considered:
where \(x(k)\in {\mathbb {R}}^{n}\) represents the state vector, \(y(k) \in {\mathbb {R}}^{r}\) is sensor output, \(z(k) \in {\mathbb {R}}^{m}\) is the signal to be estimated, \(w(k)\in {\mathbb {R}}^{p}\) and \(v(k)\in {\mathbb {R}}^{q}\) are disturbance belonging to l_{2}[0,∞],f(·):R^{n}→R^{n} is nonlinear vector function, \(\tilde {w}_{i}(k)(i=1, 2,..., \alpha)\) and \(\tilde {v}_{j}(k)(i=1, 2,..., \beta)\) are zero mean Gaussian white noise with \({\mathbb E}\{\tilde {w}_{i}(k)\}=0, {\mathbb E}\{\tilde {w}^{2}_{i}(k)\}=1, {\mathbb E}\{\tilde {w}_{i}(k)\tilde {w}_{j}(k)\}=0(i\neq j), {\mathbb E}\{\tilde {v}_{j}(k)\}=0, {\mathbb E}\{\tilde {v}^{2}_{j}(k)\}=1, {\mathbb E}\{\tilde {v}_{i}(k)\tilde {v}_{j}(k)\}=0(i\neq j), {\mathbb E}\{\tilde {w}_{i}(k)\tilde {v}_{j}(k)\}=0\). The timevarying delay τ(k)∈[d_{m},d_{M}]. A, A_{i}, A_{d}, A_{dj}, B, C, L, andD are known, real matrices with appropriate dimensions.
f(x(k)) is assumed to satisfy the following condition:
where θ>0 is a known scalar and G is a known matrix.
Remark 1
As an essential characteristic for many practical networked systems, timedelay should be considered, due to it is a main source of system instability. Although, for the purpose of decreasing the difficulty of filter design, in many filter design algorithm, timedelay is assumed to be constant. But, the fact is that timedelay is almost timevariant. Therefore, it is more practical significant to design filter for network systems with timevarying delay.
Remark 2
The addressed system (1) is a comprehensive model for industrial sensor network systems which includes the multiple noises, nonlinearity, and timevarying delay. As far as we know, due to the complexity of the addressed system (1), the relevant research results are few. This motivates our research interest.
Different from traditional filter design, the eventtriggered strategy is considered, which could reduce communication frequency. As such, a event generator function g(·,·) is defined as follows:
where σ(k)=y(k_{i})−y(k) with y(k_{i}) being the measurement at the latest event time k_{i} and y(k) is the current measurement. δ∈[0,1] is the threshold. In practical engineering, δ can be determined on the basis of the filtering requirement. When a smaller filtering error is needed, δ is set to be smaller.
The current measurement y(k) of the sensor is transmitted if only the following condition
is met. Thus, the eventtriggered sequence 0≤k_{0}≤k_{1}≤···≤k_{i}≤··· is determined iteratively by
Remark 3
The eventtriggered strategy is adopted in the networked filter design for industrial sensor network. As is well known, in timetriggered communication mechanism, the measurement output of sensor is transmitted by network communication channel with limited bandwidth at every sampling time, even though the measurement output changes slightly in the next instant, which increases the burden of network channel and wastes a lot of source of industrial sensor network. However, in eventtriggered communication mechanism, only when the designed condition is met, then measurement signal of sensor is transmitted. And a suitable threshold in the event generator function could not only reduce the measurement communication frequency but also make sure prescribed filtering performance.
As is well known, the measurement of sensor transmitted by network may encounter packet dropouts. When the phenomenon of packet dropouts is considered, the real measurement obtained by filter can be depicted as
Here, stochastic variable α(k_{i}) is employed to govern the phenomenon of packet dropouts in industrial sensor network. It is assumed to be Bernoullidistributed white sequence with
For system (1), construct the following filter:
where \(x_{f}(k)\in {\mathbb {R}}^{n}\) is the estimate of the state \(x(k), z_{f}(k) \in {\mathbb {R}}^{m}\) represents the estimate of z(k), and A_{f},B_{f}, and C_{f} is the filter gain matrix to be designed.
By letting \(\eta (k) = [x^{T}(k) \quad e^{T}(k)]^{T}, \tilde {z}(k)=z(k)z_{f}(k), e(k)=x(k)x_{f}(k), \bar {w}=[w^{T}(k) \quad v^{T}(k)]^{T}, h(\eta (k))=[f^{T}(x(k))\ f^{T}(x(k))]^{T}\), and \(\tilde {\alpha }(k)=\alpha (k)\bar {\alpha }\), we could get the augmented system:
where,
Definition 1
[13]: The augmented system (8) with \(\bar {w}(k)=0\) is exponentially meansquare if there exist constant ε>0 and 0<κ<1 thus
Our aim is to design a filter satisfying the following requirements: (Q1) the filtering error system (8) is exponentially meansquare stable, and (Q2) under the zero initial condition, for given scalar γ>0, filtering error \(\tilde {z}(k)\) satisfies
for all nonzero \(\bar {w}(k)\).
Results and discussions
The main results and some discussions are presented in this section.
Analysis of H _{∞} performance
First of all, we introduce the following lemma.
Lemma 1
(Schur complement) Given constant matrices S_{1},S_{2}, and S_{3}, where \(S_{1}=S_{1}^{T}\) and \(0<S_{2}=S_{2}^{T}\), then \(S_{1}+S_{3}^{T} S_{2}^{1}S_{3}<0\) if and only if
Theorem 1
:Consider the sensor network system(1) and let the filter parameters A_{f},B_{f}, and C_{f} be given. Thus, the filtering error system(8) with \(\bar {w}(k)=0\) is exponentially stable in meansquare, if there exist positive definite matrixes P>0,Q>0 and positive constant scalars ε_{1}, satisfying
where
Proof
: Choose the following Lyapunov function
where
□
Then, according to (8) with \(\bar {w}(k)=0\), there is
Next, it can be derived that
and
Let
It follows from (13)–(15) that
where
Moreover, if follows from (2) that
Furthermore, it follows from (16) and (17) that
Considering the eventtriggered condition (3), we have
According to Theorem 1, we have Φ_{1}<0. Thus, for all \(\zeta (k)\neq 0, {\mathbb E}\{\Delta V(k)\}\leq {\mathbb E}\{\zeta ^{T}(k)\tilde {\Phi }_{1}\zeta (k)\}<0\). Furthermore, similar to [13], system (8) can be proved to be exponentially meansquare stable. The proof is complete.
Then, the H_{∞} index will be analyzed.
Theorem 2
: Let A_{f},B_{f}, and C_{f} and γ be given. Then, system(8) is exponentially stable in the meansquare and satisfies the H_{∞} performance constraint (9) for any nonzero \(\bar {w}(k)\)under zero initial condition, if there exist matrices P>0,Q>0 and positive constant scalar ε_{1} satisfying
where
\(\bar {D}=[0 \ \ \ D]\).
Proof
: It is clear that (20) implies (11). From Theorem 1, system (8) is exponentially stable. □
Then, we will analysis the H_{∞} performance.
where
To handle with H_{∞} performance, the following index is introduced:
where n is a nonnegative integer.
Under the zero initial condition, we have
According to Theorem 2, we have Φ_{2}<0,J(n)<0. When n→∞, there is
The proof is complete.
Eventbased H _{∞} filter design
Here, the H_{∞} filtering algorithm will be solved in Theorem 3.
Theorem 3
Let the disturbance attention level γ>0 be given. Then, for sensor network system (1) and filter (7), the H_{∞} performance constraints (9) and exponential stability are guaranteed, if there exist positive matrices P>0,Q>0, and ε_{1}>0 and matrices X and C_{f} satisfying
where
Furthermore, if (P, Q, X, C_{f},ε_{1}) is a feasible solution of (25), then the filter matrices (A_{f},B_{f},C_{f}) could be obtained by means of matrices X and C_{f}, where
Proof
: Rewrite Φ_{2} as follows:
where
□
According to Lemma 1, (27) is equivalent to
Moreover, rewrite the parameters in (8):
Thus, (28) is equivalent to (25). Then, from Lemma 2, we obtain (9), and system (8) is exponentially stable. The proof is complete.
Remark 4
The sufficient conditions guaranteeing the eventbased filter satisfy Q1 and Q2 are proposed in Theorem 2. The design problem of desired filter is addressed in Theorem 3. It is easy to find that all the relevant information is contained in the LMI, such as system parameters, nonlinearity, and the threshold of eventtriggered function.
Numerical simulations
The system (1) is as follows:
f(k,x(k)) and disturbance w(k)andv(k) are chosen as
where x_{i}(i=1,2,3) denotes the ith element of the system state x(k). Then, the constraint (2) can be met with
The initial value of state is x(0)=[0.3 0.25 −0.5]^{T}.The initial value of state estimation is \(\hat {x}(0)=[0\quad 0\quad 0]^{T}\). The probability of stochastic variable α(k) is taken as \(\bar {\alpha }=0.9\). Delay is d_{M}=3,d_{m}=1. Choose the event threshold δ=0.3. The disturbance attenuation level is γ=0.95.
The filter parameters can be obtained as follows:
Figures 1, 2, 3, 4, 5, 6, and 7 show the simulation results. When setting the threshold δ=0.3, the results are described in Figs. 1, 2, 3, and 4. Figure 1 depicts the state variables x_{3}(k) and its estimate \(\hat {x}_{3}(k)\), and Fig. 2 plots the output z(k) and its estimation \(\hat {z}(k)\), whereas the estimation error \(z(k)\hat {z}(k)\) is shown in Fig. 3. Eventtriggered times are plotted in Fig. 4, whereas one represents the times that eventtriggered condition is satisfied and sensor signal is transmitted and zero represents times that eventtriggered condition is not satisfied. It follows from Fig. 4 that the eventtriggered communication mechanism can reduce the transmission frequency of the measurement output, which is energy efficient. According to Figs. 1, 2, and 3, it is easy to find that the proposed filter can estimate the state of the system well, and the energyefficient filtering strategy has satisfying filtering performance. Next, we will compare the eventtriggered mechanism with the timetriggered mechanism. When setting the threshold δ=0, e.g., the timetriggered mechanism, the corresponding results are depicted in Figs. 5, 6, and 7. Corresponding to Figs. 1, 2, and 3, Fig. 5 describes x_{3}(k) and its estimate \(\hat {x}_{3}(k)\), and Fig. 6 plots z(k) and its estimation \(\hat {z}(k)\), whereas the estimation error \(z(k)\hat {z}(k)\) is shown in Fig. 7. Compared with the simulation results between δ=0 and δ=0.3, we conclude that, with suitable threshold δ, the eventtriggered mechanism could reduce the network burden while ensuring certain system performance. The results confirm the proposed filter design method which could well achieve the desired filtering requirement.
Conclusions
In this paper, based on the eventtriggered mechanism, we have designed the energy efficiency H_{∞} filter for a class of industrial sensor network system with timevarying delay, packet dropouts, and multiplicative noises. The eventtriggered communication mechanism is adopted to improve energy efficiency. It could not only reduce the transmission frequency of the measurement output, but also guarantee the prescribed filtering performance. The timevarying delay is considered with eventtriggered strategy, which has seldom been studied. Sufficient conditions are found through stochastic analysis technique. The filter parameters could be obtained by solving the certain LMI. Finally, the simulation confirms the proposed method.
Availability of data and materials
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Abbreviations
 LMI:

Linear matrix inequality
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Funding
This work was supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20180467), the Research Startup Funds of Nanjing University of Science and Technology, and the Alexander von Humboldt Foundation.
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Authors’ contributions
ML carried out the literature analysis and raised and refined the proposed issue in this paper. Meanwhile, she gave the mathematical description of the proposed issue. HL analyzed and designed the filter. JZ verified the analysis and design of the filter by simulation experiments. BZD and YMB checked, reviewed the manuscript, and gave valuable suggestions on the structure of the paper. All authors have read approved the final manuscript.
Authors’ information
Hui Li received his BSc degree in Electrical Engineering and Automation in 2010 from Yangtzue University, JingZhou, China, his MSc degree in Control Theory and Control Engineering in 2013 from University of ShangHai for Science and Technology, ShangHai, China. From April 2016, he studied in Nanjing University of Science and Technology for PhD degree. His current research fields include networked filtering systems, multimedia big data processing, and sensor network systems.
Ming Lyu was born in Taizhou, China, in June 1980. She received her BSc degree in Automatic Control in 2002, her MSc degree in Automatic Control in 2004, and her PhD degree in Control Theory and Control Engineering in 2007, all from Nanjing University of Science and Technology, Nanjing, China. She is currently a research fellow in the School of Automation, Nanjing University of Science and Technology, Nanjing, China. Her current research interests include filtering, networked systems, multimedia big data processing, and sensors network systems.
Jie Zhang received his BSc degree in Automatic Control in 2002, his MSc degree in Automatic Control in 2004, and his PhD degree in Control Theory and Control Engineering in 2011, all from Nanjing University of Science and Technology, Nanjing, China. From April 2013 to March 2014, he was an Academic Visitor in the Department of Information Systems and Computing, Brunel University, UK. He is currently an associate research fellow in the School of Automation, Nanjing University of Science and Technology. His current research interests include stochastic systems, networked systems, wireless sensor network systems, and neural networks.
Yuming Bo received the Ph.D. degree in control theory and control engineering from Nanjing University of Science and Technology, Nanjing, China, in 2005. His research interests are focused on filtering and system optimization.
Baozhu Du received the B.S. in Information and Computing Science, and M.S. degree in Operational Research and Cybernetics from Northeastern University, Shenyang, Liaoning Province, China, in 2003 and 2006, respectively. She obtained the Ph.D. degree in Mechanical Engineering from The University of Hong Kong in 2010. Her current research interests include stability analysis and robust control/filter theory of timedelay systems, positive systems, Markovian jump systems, and networked control systems.
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Li, H., Lyu, M., Du, B. et al. Energyefficient filtering algorithm for a class of industrial sensor network systems with packet dropouts, timevarying delay, and multiplicative noises. J Wireless Com Network 2019, 162 (2019). https://doi.org/10.1186/s1363801914825
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Keywords
 Energy efficiency
 Eventtriggered communication mechanism
 Timevarying delay
 Industrial sensor network system
 H _{∞} filtering
 Multiplicative noises
 Packet dropouts