Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter Rui Wang a, Yanxiao Li b, Hui Sun a,n, Zengqiang Chen c a

College of Information Engineering and Automation, Haihang Building, South campus, Civil Aviation University of China, Tianjin 300300, China Innovation Center, China Electronics Technology Avionics Co., Ltd., Chengdu, China c Computer and Control Engineering College, Nankai University, Tianjin 300071, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 30 September 2016 Received in revised form 1 May 2017 Accepted 27 June 2017

The modern civil aircrafts use air ventilation pressurized cabins subject to the limited space. In order to monitor multiple contaminants and overcome the hypersensitivity of the single sensor, the paper constructs an output correction integrated sensor configuration using sensors with different measurement theories after comparing to other two different configurations. This proposed configuration works as a node in the contaminant distributed wireless sensor monitoring network. The corresponding measurement error models of integrated sensors are also proposed by using the Kalman consensus filter to estimate states and conduct data fusion in order to regulate the single sensor measurement results. The paper develops the sufficient proof of the Kalman consensus filter stability when considering the system and the observation noises and compares the mean estimation and the mean consensus errors between Kalman consensus filter and local Kalman filter. The numerical example analyses show the effectiveness of the algorithm. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Distributed wireless sensor networks Integrated sensors Kalman consensus filter Stability

1. Introduction Modern civil aviation aircrafts use atmospheric ventilated pressurized cabin to ensure the flight safety of passengers and crew members. It is well known that the safe and comfortable cabin environment is an important factor for passengers to choose airlines, and is also the health guarantee for crew members who work for a long time in cabins. With the continuous development of aviation industry, civil aviation puts forward higher requirements on the cabin environment quality. Many countries and regions have clearly defined the limits of relevant cabin environmental parameters in the corresponding civil aviation standards [1–3]. However, these standards are only for temperature, pressure, humidity and other basic environmental parameters. Smoke is the only prescribed contaminant indicator, but only works as a supplementary means for fire monitoring. However, in reality, due to the narrow space of aircraft cabin and crowded passengers, it is necessary to detect the contaminants immediately and accurately in order to protect the passengers and crew members from direct damages during long-range flight. In particular, this paper can provide a new method for new aircraft airworthiness certification of cabin environment. In recent years, many countries have carried out the relevant n

Corresponding author.

experimental works on aircraft cabin contaminants. Literature [4] tested contaminant levels for 81 different types of aircrafts from three airlines and compared the test results with those from civil buildings. Literature [5] randomly selected 16 flights of a certain commonly used flight types and measured the variations of volatile organic compounds and carbon dioxide throughout the whole flight process. The above experimental results demonstrated that various types of contaminants can be detected in the aircraft cabin, most of which are lower than the limits of current environmental standards for civil buildings. Only a small number of contaminants exceed the limits. Other works include the protection against contaminant in cabin [6] and the improvement of cabin air quality [7]. When monitoring contaminants in the cabin, due to the excessive sensitivity of local individual sensors, a large error in the measurement results will lead to spurious alarms when noises exist. Therefore, it is necessary to design a stable and accurate parameter estimation model. When the serious warning occurs during actual flight, the pilot must operate aircraft to land in other airports in the vicinity immediately. Spurious alarm will seriously affect the normal flight schedule and cause heavy human and financial resources losses. Increasing the threshold of alarm can reduce the spurious alarm rate, however, it has a serious potential safety hazard. Therefore, the priority concern of the design aims at increasing the accuracy of the sensor measurement. Today, the most mature application field of integrated sensor is aircraft integrated navigation system as presented in [8]. A new integrated

http://dx.doi.org/10.1016/j.isatra.2017.06.027 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

2

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

system is designed to perform micro-environmental monitoring by taking the advantages of the wireless sensor network in [9]. In hi-tech environment, a strict surveillance system often utilizes a group of distributed sensors that obtain information from the local targets and then uses the distributed Kalman filter (DKF) for data fusion [10]. The distributed sensors are often organized as networked control systems. The output-feedback communication and event-triggered strategy for the distributed networked systems are proposed in [11]. In networked control systems, the time delay is an essential problem that influences the stability. Literature [12] considered the Markov network-induced delay and then investigated the finite-time event-triggered H1 boundedness for network-based Markovian jump nonlinear system. And Literature [13] considered the networked control systems with random time delay and given a tuning method of optimal fuzzy PID controller. Consensus strategy is a kind of the important method in the DKF which can make the estimation value of each sensor consistently approach the true value. Olfati-Saber proposed the classical Kalman consensus filter (KCF) in [14]. In order to eliminate the estimation error among the local node and its neighbors, the consistency term is added in the state estimation in Kalman filter, and the corresponding stability analysis for continuous-time condition was provided. Literature [15] introduced a scalable suboptimal KCF algorithm, and then gave the stability proof for discrete-time condition. In [16], a new energy efficient DKF is proposed based on the classical KCF and the site percolation model which is a kind of random graph. In the stability analysis of all the above algorithms, the system noises and the observation noises are ignored, finally the algorithms prove that the estimation values of each sensor asymptotically converge to the true values. However, in practical applications, the measured dynamic systems and sensors are inevitably affected by noises. And computer simulation results also show that there is a certain deviation between the estimated value and the true value for each sensor. From the energy optimization point of view, the Literature [17] proposed an event-triggered KCF algorithm and considered the effect of noise about algorithm stability. Recently, sensor scheduling became an effective methodology in the field of distributed filtering. The deterministic scheduling and stochastic scheduling have been investigated [18–20]. These scheduling strategies provide an enlarged set of feasible activation solutions, therefore, they can improve the estimation performance and reduce power assumption. Another important methodology is the pinning control approach, which has been applied both in the field of distributed filtering and the complex networks [21–24]. By this approach, only a small fraction of sensors need to be measured and controlled, and the network could still be controlled. There are two contributions for this paper. One is to provide a distributed integrated sensor network architecture applied on monitoring cabin contaminants in this paper. For each type of contaminants, two sensors with different principles are integrated in a single measurement node called the integrated sensor. These nodes are connected to a distributed network and use KCF for data fusion. This network finally outputs the accurate contaminant concentration and the evaluation index of the environmental condition. On the other hand, the paper considers the interference of noise working condition. In general, under noises condition, the classical KCF cannot converge to the true state value asymptotically. By using Lyapunov-based approach, we present a sufficient condition for the stability analysis of the distributed integrated monitoring network in order to ensure that the estimation error of the classical KCF is exponentially bounded in mean square under the noise interference condition. The rest of this paper is organized as follows. The complete distributed integrated cabin contaminant monitoring network is established in Section 2. Section 3 gives the abstract mathematical

model of integrated sensor structure and uses the classical KCF for contaminant monitoring. A sufficient condition for ensuring the stochastic stability of the proposed algorithm is given in Section 4. Section 5 illustrates a numerical example and analyses in detail. Conclusions are finally summarized in Section 6.

2. Distributed integrated cabin contaminant monitoring network 2.1. The architecture of cabin contaminant monitoring network Although there are ubiquitous contaminants in the aircraft cabin, current civil aviation regulations and design standards have not covered these into scope of study. Referring to existing measurement data and building indoor environmental standards, the following categories of contaminants are considered. The corresponding limits and measurement methods are shown in Table 1. During the actual pollutants tests in cabin, the over sensitive single sensor often leads to spurious alarm. In order to overcome the limitation of single sensor, modern industry has widely used the distributed multi-sensor information fusion technology. This technology can optimize the various data provided by the spatially distributed multi-sensors in order to monitor parameters accurately. The proposed cabin pollutants monitoring network installs sensors on the typical locations of the cabin, including the top, bottom, windows, seats, etc. The wireless sensor network sketch for monitoring the cabin contaminants is shown in Fig. 1. The specific deployment architecture of distributed sensor network in cabin can refer to Ref. [18]. As Fig. 1 shown, G = (U, E, A) represents the topological structure diagram of the monitoring network. U = {v1, v2, … vn} is the set of sensor nodes, and E ⊂ U × U is the set of information exchange edges between them. It is called ‘connected’ that two nodes communicate directly with each other. Ni = {vj ∈ U |(vi, vj ) ∈ E} is the set of neighborhood nodes for node i . The number of neighbors of node i is called degree and is denoted by di = Ni . The degree matrix D is a diagonal matrix with element di. F = [aij ] is the adjacent matrix of G where aij = 1 if and only if node i is connected with node j, and aij = 0, otherwise. L ¼ D - F is the Laplacian matrix of G. If the Laplacian matrix L has a nonzero eigenvalue, the undirected graph is connected. Table 1 The classification of cabin contaminants. Contaminants

Unit

Limits

Measurement methods

Carbon Dioxide

ppm

r 5000

Carbon Monoxide

ppm

r 10

Ozone

ppm

9750 m r 0.25 8230 m r 0.10 PM2.5 100 r 99 r 1 (24 h)

Non-dispersive Infrared Absorption Spectrometry Non-dispersive Infrared Absorption Spectrometry Chemiluminescence VIS-Spectrometer (623 nm)

Inhalable Particulate μg/m3 Acetone ppm Acetaldehyde Formaldehyde

ppm

Methylbenzene

ppm

Dichloromethane Endotoxin

ppm ng/m3

ASTM D6699 GC/FID HPLC, UV detector, 245 nm HPLC, UV detector, 365 nm r 0.08 GC/FID HPLC, UV detector, 365 nm r 40 ISO 16017-1 ISO 16000-6 r 0.86 (24 h) GC/FID LALM, Limulus amebocyte lysate r 4.5 (8 h) 9  170 (peak method value)

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Fig. 1. The schematic diagram of the distributed wireless sensor network for minitoring cabin contaminants.

2.2. Comparisons of multiple-sensor integrated measurement structures In order to overcome the instability of a single sensor, the multiple-sensor structures are introduced in the monitoring network shown as in Fig. 1. For a certain contaminant, sensors with different measuring principles are deployed in every node areas in order to make up the weakness of a single senor. Then the KCF is used to get the real monitoring results. The comparison among the following structures of sensor combinations are taken where subscript i indicates the number of the sensor. The direct fusion structure is shown as in Fig. 2. The local KCF carries out the data fusion for the state estimation based on the primary sensor measurement values, ZPi , and secondary sensor measurement values, ZSi , and outputs the final estimation of contaminant concentration ZOi . The corresponding air quality index (AQI) can be obtained by the monitor. The feedback compensation structure is shown as in Fig. 3. The state Zi is defined as the difference between the primary and the secondary sensor measurements, ZPi and ZSi. Then conduct the ^ error estimation by KCF and use the output Zi to amend the system dynamics. So the primary sensor measurement ZPi can be used as the final estimation of contaminant concentration. The output compensation structure is shown as in Fig. 4. Compared to the feedback compensation structure, output compensation ^ structure method uses the KCF to get the optimal error estimation Zi

Fig. 2. The direct fusion structure for a node.

Fig. 3. The feedback compensation structure for a node.

Fig. 4. The output compensation structure for a node.

3

coming from the error, Zi, between the primary and secondary ^ sensor measurements. That estimation Zi is used to correct the measurements of the primary sensor and then we can get the suboptimal estimation of actual contaminant concentration ZOi . Theoretically, if the system and the measurement equations can describe the sensors dynamics accurately, the above-mentioned three kinds of structures are the same in nature and they will get the correct results. However, from the practical engineering point of view, the real estimation results could not be the same. In the direct fusion structure, the system equation of KCF is the dynamic equation of the cabin contaminants. And it is always the complicated nonlinear time-varying equations, which will cause a large error even if nonlinear Kalman filter algorithm is employed. While in the feedback and the output compensation structures, the system equations of KCF are the error dynamic equations obtained by a first-order approximation according to the sensor characteristics. The error magnitude for the primary sensors are close to the secondary sensors, then the mature linear Kalman filtering algorithm can be used to get the accurate estimation. Both of these two structures estimate the errors among the sensor measurements, then KCF is used to correct the output results. The error states of the feedback compensation structure have been corrected, while those of the output compensation structure are not. Therefore, the feedback compensation structure can reflect the dynamics of system errors more accurately [8]. However, as Literature [8] analyzed, in reality, sensors in the feedback compensation structure are coupled together. Therefore, once one sensor fails, the wrong signal will broadcast a wrong message to other sensors immediately. Eventhough the fault will be detected in a short time, the other sensor dynamics has already been affected. Therefore, the fault will affect the normal operation of the whole system. On the contrary, in the output compensation structure, if a fault occurs in one sensor, the difference between the primary and secondary sensors will be larger, in addition, affects the output to some extent. However, when the fault is detected, the faulty sensor will be cut off immediately, then the whole system can still work properly with higher reliability. From the practical application point of view, although the feedback compensation structure is more accurate than the output compensation structure. However, the latter one is more reliable and owns stronger fault-tolerant capability than the feedback structure. Therefore, we use the output compensation structure for cabin contaminants monitoring network design in order to guarantee the absolute safety for passengers and crew members.

3. The cabin contaminants monitoring algorithm based on KCF It is well known that in the sensor network, each sensor can get close but not exactly consistent measurement results under the influence of many random factors. The traditional solution is setting up a data fusion center, which conducts the weighted fusion for all the sensor's measurements. However, the amount of data in center fusion scheme is substantially large. What's more, the entire system will collapse once the data fusion center fails, which leads to a low reliability. Cabin pollutants monitoring requires a high real-time performance and accuracy and provides the pollutants concentration information. Moreover, the outbreak of pollutants must be monitored immediately by the monitoring networks. So the KCF is used to accomplish the accurate measurements and improve the reliability of the system. In this algorithm, each sensor node only needs to exchange information with several adjacent sensor nodes instead of all the others, and then the consistency estimation of global data can be obtained.

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

3.2. Cabin contaminant evaluation algorithm for the distributed integrated sensor networks

3.1. The measurement error model for integrated sensors This paper proposes the general model of pollutant monitoring algorithm. As the output correction structure as shown in Fig. 2, the measurement errors for the primary and secondary sensors are fused with the local Kalman filter. So the error dynamics equations of the integrated sensors should be established. According to the statistical analysis of the actual system performance and the error analysis of the sensor, the error model of each sensor can be obtained. For note simplicity, using superscript þ indicates the updated value of any variable. The error dynamics equation of the primary sensor is shown as Eq. (1).

xP+ = AP xP + BPwP

(1-1)

ZPi = HPixP + vPi

(1–2)

The error dynamics equation of the secondary sensor is shown as Equation 2.

x S+ = AS x S + BSwS

(2-1)

ZSi = HSix S + vSi

(2-2)

where xP ∈ R mP , xS ∈ R mS are the state vectors of the primary and the secondary sensor, respectively. And ZPi ∈ R p, ZSi ∈ R s are the output vectors of the primary and the secondary sensor, respectively. (AP , BP , HPi ) and (AS , BS , HSi ) are system matrices with appropriate dimensions. wP , vPi, wS , vSi are independent zero-mean white Gaussian noises with covariance matrices Q and R ij satisfying

It is well known that in the sensor network, each sensor can obtain approximately but not exactly consistent measurement results under the influence of multiple random factors. As proposed before, the traditional solution is to set up a data fusion center, which conducts the weighted fusion for all the sensor's measurements. However, the data traffic in center fusion scheme is extensively large and busy. What's more, the entire system will collapse once the data fusion center fails, which refers to a low reliability. Cabin pollutants monitoring requires a high real-time reliable performance and accuracy and provides the pollutants concentration. The KCF is used to achieve the accurate measurements and improve the reliability of the system. In the proposed algorithm of this paper, each sensor node only needs to exchange information with several adjacent sensor nodes instead of all the others, and the noises are considered. Then the consistency estimation of global data can be obtained with the noise consideration. We consider the following distributed Kalman consensus filter at sensor i,

⎧ x^ = x¯ + K (Z − H ⋅x¯ ) + AC ⋅ ¯ j − x¯ i ) i i i i i i ∑ (x ⎪ i j ∈ Ni ⎨ ⎪ + ⎩ x¯ i = Ax^i

(5)

where Ki and Ci represent the Kalman gain matrix and the consensus gain matrix, respectively. x^i and x¯ i are the estimation value and the prediction value of system state x, separately. They are defined by taking the expectations with respect to the measurement and process noises vi(k) and w(k) as follows:

E[w (k )w (l)T ] = Q (k )δkl

x^i(k ) = E[x(k ) Zi(k ), j ∈ Ni ]

(6)

E[vi(k )vj(l)T ] = R ij(k )δkl

x¯ i(k ) = E[x(k ) Zi(k − 1), j ∈ Ni ]

(7)

where δkl = 1 if k = l , and δkl = 0 , otherwise. For simplicity, the notation R ii is shortly denoted by R i when i = j . According to the actual performance and error analysis of sensors, the overall error equation of the integrated sensor can be derived by the Equations (1) and (2). Since it is hard to guarantee that the sampling times of the primary sensor are consistent with that of the secondary sensor, the measurement noise in the secondary sensor can be ignored in order to simplify the system observation equations in the alignment time. Suppose the measurement noise of integrated sensor is the same as that of the primary sensor. The error equation of the integrated sensor is shown as follows.

⎡ x + ⎤ ⎡ A 0 ⎤⎡ x ⎤ ⎡ B 0 ⎤⎡ w ⎤ P P ⎥⎢ P ⎥ + ⎢ P ⎥⎢ P ⎥ x = ⎢ +⎥ = ⎢ ⎢⎣ x S ⎥⎦ ⎣ 0 AS ⎦⎣ x S ⎦ ⎣ 0 BS ⎦⎣ wS ⎦

(3-1)

Zi = ZPi − ZSi

(3-2)

+

For the convenience of the following analysis, Equation (3) is expressed as the following unified form.

x+ = Ax + Bw

(4-1)

Zi = Hix + vi

(4-2)

where x = col{xP , xS} and w = col{wP , wS}. vi is the measurement noise of the primary sensor, that is, vi = vPi . And Hi is the measuring matrix of the integrated sensor obtained by the principle of sensor measurement error and engineering applications.

Then we define the state estimation error ei and the state prediction error e¯i of node i as follows:

ei = x^ i − x

(8)

e¯i = x¯ i − x

(9)

The corresponding estimation error covariance and prediction error covariance are Pij and Mij, respectively. These two parameters are defined as follows:

Pij = E[ei⋅ej T ]

(10)

Mij = E[e¯i⋅e¯j T ]

(11)

For simplicity, the notations Pij and Mij are shortly denoted by Pi and Mi when i = j . To design the Kalman gain Ki and the consensus gain Ci, we first consider the optimal method provided in Literatures [20,21]. By setting the activation probability in Lemma 1 of Literature [21], the optimal Kalman gain Ki* which minimizes the estimation error covariance Pi is obtained by

⎧ ⎫ ⎪ ⎪ Ki* = A⎨ Pi + Ci ∑ ⎡⎣ Pji − Pi⎤⎦⎬HiT Mi−1, ⎪ ⎪ j ∈ Ni ⎩ ⎭ Mi = HiPiHiT + R i, i = 1, 2, …, n.

(12)

Notice that this optimal Kalman gain Ki* in Eq. (12) contains the parameter Pji which can cause a large computational burden when the sensor networks have large scale by the following lemma.

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Lemma 1. [15]: Computational complexity of solving the Riccati equation in (12) by updating the Pji is O(n2). And as a subsystem of the aircraft environmental control system, the cabin contaminant monitoring system is integrated with other systems in the center computer which has limited computing resource. On the other hands, Lemma 1 also shows that the update of the optimal Kalman gain Ki* is not scalable in the number of sensors. In order to get an efficient and scalable filter for contaminant monitoring, we consider the suboptimal Kalman gain matrix Ki. As mentioned in Literatures [15,17,20], the suboptimal filter can be obtained by setting the consensus gain Ci to zero in (12). And the complete suboptimal KCF can be formulated as follows,

⎧ x^ = x¯ + K (Z − H ⋅x¯ ) + C ⋅ ¯ j − x¯ i ) i i i i i i ∑ (x ⎪ i j ∈ Ni ⎪ ⎪ T T −1 ⎪ Ki = PiHi (R i + HiPiHi ) ⎨ T T ⎪ Mi = FiPiFi + KiR iKi ⎪ + T T ⎪ Pi = AMiA + BQB ⎪ ¯+ ⎩ x = Ax^ i

i

i

i

Ui = HiT R i−1Hi After that, send message (u i , Ui, x¯ i ) to neighbors. (4) Receive message from all neighbors and conduct the following data fusions.

yi =

∑ uj j ∈ Ji

Si =

∑ Uj j ∈ Ji

(5) Update the covariance matrix Mi of state estimation error and conduct state estimation.

Mi = (Pi−1 + Si )−1 r=

θ ‖Pi‖ + 1

x^i = x¯ i + Mi(yi − Six¯ i ) + rPi ∑ (x¯ j − x¯ i ) j ∈ Ni

(13)

where Fi ¼ I - KiHi and I denotes the identity matrix. According the Literature [17], we define a weighted measurement yi = HiT R i−1Zi and information matrix Si = HiT R i−1Hi , then the filter can be denoted by the following form,

⎧ x^ = x¯ + M (y − S x¯ ) + C ¯ j − x¯ i ) i i i i i i ∑ (x ⎪ i j ∈ Ni ⎪ ⎪ −1 −1 ⎪ Mi = (Pi + Si ) ⎨ ⎪ Ci = γPi ⎪ + T T ⎪ Pi = AMiA + BQB ⎪ x¯ + = Ax^ ⎩

5

where θ is the system sampling time and the Frobenius matrix norm is used for the calculation of variable r. (6) Output the cabin pollutants concentration estimation ZOi .

^ ZOi = ZPi + Zi = ZPi + Hix^i (7) Real-time assessment for different pollutant concentrations If the concentration of any kind of pollutant is greater than that of the corresponding limit value, ZOil ≥ Zbl , then alarm. else, give the air quality index AQI = ∑l

αl(Zbl − ZOil ) where ‖Zbl‖

Zbi is

the limit value of certain pollutant l, and αi is the weighted value of the corresponding pollutant. (8) Update the estimation gain matrix Pi and the state prediction x¯ i .

(14)

The above filter is called the Kalman consensus information filter. It uses the covariance matrix of state prediction error Pi and state estimation error Mi to correct the state prediction x¯ k = E (xk|Zk − 1), and then collects the estimation of the error states. As mentioned in Literature [13], the data fusion effect of this algorithm is sub-optimal. However, it has a good scalability, which can be applied to large scale distributed monitoring networks. In this paper, the Kalman consensus information filter is applied to the proposed distributed integrated monitoring network, and the following iterative algorithm is constructed for the cabin pollutant monitoring network. Algorithm 1:

Pi+ = AMiAT

x¯ i+ = Ax^i (9) Return Step (2) and execute the loop. In our monitoring algorithm, the network communication occurs only between neighboring sensor nodes, which can effectively reduce the algorithm computational complexity and the energy consumption of sensors. More importantly, the problem of low reliability in centralized Kalman filter can be avoided. In Step (5) of the algorithm, the consistency term of the state estimation can make all the sensor nodes converge to the same estimation value. That means the sensor nodes combine their own measurement values with the information of neighbor nodes to achieve global consensus estimations.

(1) Initialize each sensor parameters.

Pi = P0

x¯ i = x(0) (2) Obtain primary sensor measurements ZPi and secondary sensor measurements ZSi , then calculate each sensor measurements error.

Zi = ZPi − ZSi (3) Obtain the covariance matrices Ri for measurement noises, and then calculate the information vector ui and information matrix Ui of node i .

ui = HiT R i−1Zi

4. Stability analysis of distributed integrated sensor networks subject to noises Although KCF proposed in Literature [13] can estimate the differences between the primary and the secondary contaminant sensor, it does not present stability analysis of that filter with the Gaussian white noises in system dynamics. However, in practical applications, the sensors in the cabin might have zero drift generated by the environments such as temperature, humidity, etc. Therefore, the paper considers the noise and concludes that all estimators asymptotically reach a consensus on state estimate. In fact, the noises will make estimations deviate from actual states,

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

and all the estimators cannot provide completely consensus results. Inspired by Literature [15], we develop the stability analysis with considering state and input noises in this section. When considering the general KCF form, substituting Mi into Pi+ and x^i into x¯ i+ in Eq. (5) yields a more compact form written as

⎧^+ ^ ^ ^ ^ ⎪ xi = Axi + AKi(Zi − Hixi ) + ACi ∑ (xj − xi ) j ∈ Ni ⎪ ⎨ −1 T T ⎪ Ki = PiHi (HiPiHi + Ri ) ⎪ + T T T T ⎩ Pi = (A − AKiHi )Pi(A − AKiHi ) + AKiRiKi A + BQB

Lemma 4. [26, Lemma 3.3]: Under Assumption 3, there is a real number εi > 0 (i ¼ 1, 2, …, N) such that.

{

}

E viT KiT AT (Pi+)−1AKivi + w T BT (Pi+)−1Bw ≤ εi

(22)

The main results of the stability analysis are presented as follows.

(15)

Two key preliminary assumptions and three lemmas are used for the formal stability proof. Assumption 1. The wireless sensor networks graph G is connected. Assumption 2. (A, Hi) is uniformly observable. Assumption 3. There are positive scalars a¯ , b¯ , h¯ , q̲ , r̲ , p̲ and p¯ such that the filter parameter matrices are bounded as follows.

Theorem 1. For discrete-time linear time-varying systems (4) and the KCF (5). Under Assumptions 1 to 3, the prediction error is exponentially bounded in mean square and bounded with probability 1 provided that the initial prediction error is bounded. Proof: According to the Eqs. (8) and (11), then define the prediction T error e = ⎡⎣ e1T , … eNT ⎤⎦ and the corresponding block diagonal matrix P = diag{P1, … PN}. The following Lyapunov function is constructed, N

V (e) = eT P −1e =

∑ eiT P −1ei .

(23)

i=1

As p̲ i I ≤ Pi ≤ p¯i I described in Assumption 2, we have

‖A‖ ≤ a¯ , ‖B‖ ≤ b¯ , ‖Hi‖ ≤ h¯ ,

1 1 ‖e‖2 ≤ V (e) ≤ ‖e‖2 p¯ p̲

Q ≥ qI̲ , Ri ≥ rI̲ , p̲ i I ≤ Pi ≤ p¯i I .

(16)

where I is the identity matrix. Remark 1. It is worth to mentioning that, according to Corollary 5.2 in [25], in this paper, Assumption 2 is a sufficient condition to guarantee the last condition in Assumption 3. And the first five conditions in Assumption 3 hold for real world systems. Lemma 2. [26, Lemma 2.1]: Assume there is a stochastic process Vk(ξk ) as well as real numbers v̲ , v¯ , μ > 0 and 0 < α ≤ 1 such that.

(24)

where p¯ = max {p¯1, … , p¯N } and p̲ = min {p̲ , … , p̲ }. Inequality 1 N (24) meets the first condition in Lemma 1. To further prove that the process e is exponentially bounded, the expectation of the Lyapunov function in next time instant E{V +(e+)} must be considered. According to the definition of ei , the prediction error dynamics of ei , ei+ is shown below,

ei+ = x¯ i+ − x+ =A(I − KiHi )ei + ACi ∑ (ej − ei ) − Bw + AKivi . (25)

j ∈ Ni

Then substitute (25) into (23), we have

v̲ ‖ξk‖2 ≤ Vk(ξk ) ≤ v¯‖ξk‖2

(17)

N

V +(e+) =

and

∑ (ei+)T (P +)−1eiT i=1

{

}

E Vk(ξk ) ξk − 1 ≤ (1 − α )Vk − 1(ξk − 1) + μ

(18)

N

= ∑ eiT (A − AKiHi )T (Pi+)−1(A − AKiHi )ei i=1

Then the stochastic process is exponentially bounded in mean square, i.e. we have

{

E ‖ξk‖2

} ≤ vv¯̲ E{ ‖ξ ‖ }(1 − α) 0

2

k

+

μ v̲

N

+2 ∑ qiT CiT AT (Pi+)−1(A − AKiHi )ei

k

∑ (1 − α )i i=1

i=1 N

(19)

and the stochastic process is bounded with probability 1. In the following analysis, Lemma 1 will be used as the boundary condition for the estimation process of KCF. And for each sensor node, the boundaries of these parameters are held as described by Lemmas 3 and 4.

+ ∑ qiT CiT AT (Pi+)−1ACiqi i=1 N i=1 N

∑ wT BT (Pi+)−1Bw i=1

N

− ∑ viT KiT AT (Pi+)−1Bw − ∑ w T BT (Pi+)−1(AKi )vi i=1

Lemma 3. [26, Lemma 3.1]: Under Assumption 3, there is a real number 0 < κi < 1 (i ¼ 1, 2, …, N) such that.

N

+ ∑ viT (AKi )T (Pi+)−1(AKi )vi +

i=1

N

−2 ∑ eiT (A − AKiHi )T (Pi+)−1Bw i=1 N

T

(A − AKiHi )

(Pi+)−1(A

− AKiHi ) ≤ (1 −

κi )Pi−1

(20)

i=1 N

where

⎡ ⎤−1 q̲ ⎥ κi = 1 − ⎢ 1 + ⎢⎣ p¯i (a¯ + a¯ ⋅p¯i h¯ / r̲ )2 ⎥⎦

+2 ∑ eiT (A − AKiHi )T (Pi+)−1AKivi N

−2 ∑ qiT CiT AT (Pi+)−1Bw + 2 ∑ qiT CiT AT (Pi+)−1AKivi i=1

(21)

i=1

(26)

where qi = ∑j ∈ N (ej − ei ). i Take the expectation on both sides of (26) and then use Lemmas 1, 2, 3, we can get

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

N

E{V +(e+)} ≤

0 < κ̲ − σ 2p¯ λ max(LT L )⋅( p̲ −1 − κ̲ p̲ −1) < 1

N

∑ (1 − κi)eiT Pi−1ei + ∑ εi i=1

i=1

7

(38)

N

N

∑ εi > 0

+2 ∑ qiT CiT AT (Pi+)−1(A−AKiHi )ei

(39)

i=1

i=1 N

+ ∑ qiT CiT AT (Pi+)−1ACiqi

(27)

i=1

Condition σ < σ * makes the quadratic inequality (36) hold, where

κ̲ p¯ −1

In order to transfer the third term and fourth term of (27) to an easy handling form, define the consensus gain Ci as follows

σ* =

Ci = σ[(A − AKiHi )T (Pi+)−1A]−1Λi

Then the inequality (39) obviously exists according to the result of Lemma 3. Therefore, the prediction error ei is exponentially bounded in mean square and bounded with probability 1.

(28)

where Λi = (A − AKiHi )T (Pi+)−1(A − AKiHi ). Multiplying Ci from right by the inverse term in Eq. (28), the following equation can be easily yielded

CiT AT (Pi+)−1(A − AKiHi ) = σΛi

(29)

T

λ max(L L )( p̲ −1 − κ̲ p̲ −1)

(40)

5. Performance simulations of the cabin contaminant monitoring network

Then substituting Λi into Ci yields

Ci = σA−1(A − AKiHi )

(30)

(A − AKiHi )−1ACi = σ

(31)

further substitute (31) into (29), then yield,

CiT AT (Pi+)−1ACi = σ 2Λi

(32)

Using Eqs. (29) and (32), the third and fourth term on the right hand side of (27) can be expressed by N

N

2 ∑ qiT CiT AT (Pi+)−1(A − AKiHi )ei + i=1

∑ qiT CiT AT (Pi+)−1ACiqi i=1

N

N

=2σ ∑ qiT Λi ei + σ 2 ∑ qiT Λi qi i=1

(33)

i=1

Note that Λ = diag{Λ1, … ΛN } and q = col{q1, … qN }. Define L as the Laplacian matrix of the network graph and e = col{e1, … eN }, then q can be expressed as q = − L⋅e with L = L ⊗ I . The right hand side of Eq. (33) can be further amplified as follows N

N

5.1. Simulation model of the distributed integrated sensor network The simulations of the algorithm are conducted before the practical deployment of the aircraft cabin contaminant monitoring network. The simulation model of the monitoring network is shown in Fig. 5. The primary and the secondary sensor outputs are the measurement values, ZPi and ZSi, affected by their respective system noise and measurement noise, Wi and Vi, and the difference of these values is the measurement error Zi. The local KCF also measures the error Zj from its neighboring nodes, and then out^ puts the measurement error estimation Zi . The primary sensor's ^ measurement, ZPi, can be corrected by that estimation Zi , and ZOi is the final result. The simulation experiments employ Monte Carlo method to carry out a large number of independent repeated experiments. The performance of the monitoring network is analyzed by the statistical mean of the parameters. Inspired by Literature [14], the performance indicators are defined as follows. Mean Estimation Error: N

MEEk =

∑i = 1 (eiT, kei, k ) N

(41)

2σ ∑ qiT Λi ei + σ 2 ∑ qiT Λi qi i=1

ei, k = ZO, k − ZOi, k

i=1 T

≤ − 2σλ min(Λ)⋅e Le + σ 2λ max(Λ)⋅λ max(LT L )⋅eT e

Mean Consensus Error:

≤ σ 2λ max(Λ)⋅λ max(LT L )⋅eT e

(34) N

where λ min and λ max denote the minimum and the maximum eigenvalue of the matrix. Formulate the above terms, then yield N

E{V +(e+)} ≤

(42)

MCEk =

∑i = 1 (δiT, kδi, k ) N

(43)

N

∑ (1 − κi)eiT Pi−1ei + ∑ εi + σ 2λ max(Λ)⋅λ max(LT L)⋅eT e i=1

i=1 N

≤[1 − ε + σ 2p¯ λ max(Λ)⋅λ max(LT L )]eT P −1e +

∑ εi

(35)

i=1

where κ̲ = min {κ1, … κN}. As shown in Lemma 2, this inequality is obviously shown below

λ max(Λ) ≤

1 − κ̲ p̲

(36)

Substitute (36) into (35), then yield N

E{V +(e+)} ≤ [1 − κ̲ + σ 2p¯ λ max(LT L )⋅( p̲ −1 − κ̲ p̲ −1)]eP −1e +

∑ εi i=1

(37)

To satisfy the condition of Lemma 1, we must guarantee that

Fig. 5. The simulation model of the monitoring network.

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

Fig. 6. The topological graph of monitoring network. Fig. 7. Mean estimation errors. N

δi, k = ZOi, k −

∑i = 1 ZOi, k (44)

N

where k represents the simulation instant, and N is the total number of nodes. The mathematical model of the primary and secondary sensors as shown in Fig. 5 can be obtained from theoretical analysis of the measurement error or from the system identification of the actual measurement values. It should be noted that, due to the limitation of system identification precision and the performance of embedded equipment, the system and measurement equations of KCF are only approximations of the real integrated sensors. 5.2. Analyses of simulation results Consider the following monitoring network which is connected with 10 sensor nodes and the topological graph is shown in Fig. 6. And the Laplacian matrix is

⎡ 5 ⎢ ⎢ −1 ⎢ −1 ⎢ −1 ⎢ ⎢ −1 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ −1 ⎢ ⎣ 0

−1 3 −1 0 0 0 0 0 0 −1

−1 −1 5 0 −1 0 −1 0 0 −1

−1 0 0 4 0 0 −1 −1 0 −1

−1 0 −1 0 4 −1 0 −1 0 0

0 0 0 0 −1 2 0 −1 0 0

0 0 −1 −1 0 0 3 0 −1 0

0 0 0 −1 −1 −1 0 5 −1 −1

−1 0 0 0 0 0 −1 −1 3 0

0⎤ ⎥ −1⎥ −1⎥ −1⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ −1⎥ 0⎥ ⎥ 4⎦

sampling instant which satisfies Assumption 1. The above system holds the following inequalities which satisfies Assumption 2.

‖A‖ ≤ a¯ = 1.002, ‖B‖ ≤ b¯ = 0.5, ‖Hi‖ ≤ h¯ = 1, Q ≥ q̲ I = 10I , R ≥ r̲I = 100I , 0.0114I = p̲ I ≤ P ≤ p¯ I = 10I .

(46)

According to Eq. (21), we can get κ̲ = 0.4515, which meets the precondition in Lemma 2. And by the definition of L , we can get λ max(LT L ) = 7.1992. Then use Eq. (40) to get the critical value σ * = 0.7417 of the consensus parameter. In order to verify the effectiveness of the proposed algorithm, two algorithms are employed and compared for the monitoring network, 1000 independent Monte Carlo simulations are carried out for the two algorithms. The performance indexes of (41) and (43) are shown in Figs. 7 and 8 below. Due to the lack of information about the neighboring nodes, the overall effect generated by the local Kalman filter is relatively poor. As shown in Fig. 7, the estimation error of local Kalman filter is higher than that of the proposed algorithm and converges more slowly. More importantly, the consensus error of local Kalman filter is much bigger than that of the proposed algorithm as shown in Fig. 8, that is, the local Kalman filter cannot estimate the value of

The measurement error dynamics of the integrated sensor is

⎡ 1 −0.02⎤ ⎡ 0.5 0 ⎤ x+ = ⎢ ⎥x + ⎢⎣ ⎥w ⎣ 0.02 1 ⎦ 0 0.5⎦

(45-1)

⎡ 1 0⎤ ⎡ 1 0⎤ Zi = ⎢ x+⎢ v ⎣ 0 1⎥⎦ i ⎣ 0 1⎥⎦

(45-2)

We assume that the initial measurement error is x 0 = (8, 12)T , and the initial prediction error matrix of KCF is P0 = 10I2 . The system and the measurement noises are the independent Gaussian white noise with covariance 10 and 100i, respectively, where i is the node index. The sampling period of system is 10 ms. The stability analysis is presented as follows. First, verify two assumptions in Section 4. Ak is obviously singular in every

Fig. 8. Mean consensus error.

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i

R. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

9

Technology Project (MHRD20150220), the National Natural Science Foundation of China (61403395, U1533201), the Fundamental Research Funds for the Central Universities-Civil Aviation University of China (3122017003) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. The authors would like to extend our sincere gratitude to Dr. Zongli Lin from University of Virginia, for his instructive advices and suggestions on this paper.

References

Fig. 9. The rates of spurious alarm.

each sensor more accurately. And this also demonstrates that the single sensor is prone to produce spurious alarms. While the MEE and MCE of the proposed algorithm can converge to the stable value immediately. Due to the system and the measurement noises existing in the simulation process, the estimation of each node cannot converge to the true value completely. Meanwhile, there are some errors in the estimation values of each node, but these errors converge quickly to a finite value. Compared to the traditional local Kalman filter, the proposed algorithm has a higher estimation precision, and the measurement value of each node tends to be consistent. Fig. 9 provides the statistical analysis of the spurious alarm generated by simulation using two algorithms. This figure demonstrates that the rate of spurious alarms by using KCF is less than that by using local Kalman filter for the individual node. Furthermore, the rates of spurious alarm of each node tend to stable and are relatively close to each other generated by the KCF algorithm. This also indicates that KCF for the integrated output compensation sensor structure can effectively reduce the rates of the spurious alarm to overcome the limitation of the local single sensor.

6. Conclusions This paper deploys the distributed wireless sensor network to the typical location of the aircraft cabin and uses the integrated sensor with output compensation structure to overcome the spurious alarm caused by the single sensor. The general mathematical model of the measurement error used in the integrated sensor is also given. Through theoretical and simulations analyses, it proves that the proposed algorithm can make the measurements of sensors converge more precisely. On the other hand, the paper proposes the sufficient condition for the stability analysis of KCF with considering system and measurement noises. Furthermore, the simulation results also demonstrate that the error of the proposed algorithm is bounded and more efficiently converges. In the near future, the corresponding experimental validation for this algorithm will be conducted in the cabin.

Acknowledgements This work was supported by the Civil Aviation Science and

[1] Civil Aviation Administration of China. China Civil Aviation Regulations -25-R4[S]. 2011. [2] Federal Aviation Administration. Federal Aviation Regulations (FAR) part 25 airworthiness standards: transport category airplanes-128 [S]. 2014. [3] European Aviation Safety Agency. Joint Airworthiness Requirements (JAR) part 25 large aeroplanes [S]. 2014. [4] Spengler JD, Vallarino J, Mcneely E, et al. In-flight/onboard monitoring: ACER’s component for ASHRAE 1262, part 2[R]. Boston: MA National Air Transportation Center of Excellence for Research in the Intermodal Transport Environment (RITE), 2012. [5] Jun GUAN, Xudong YANG, Chao WANG, et al. Measurement and analyses on volatile organic compounds in aircraft cabins. Heat Vent Air Cond 2013;43(12):80–4. [6] Farag AM, Sayed MMA and Abbas W. Air curtain CFD model for protection commercial aircraft cabin passengers from Sarin (GB) [C]. In: Proceedings of the IEEE Aerospace Conference, Montana, USA: 1-12. doi: http://dx.doi.org/10.1109/AERO. 2016.7500523; 2016. [7] Farag AM and Khalil EE Numerical analysis and optimization of different ventilation systems for commercial aircraft cabins[C]. In: Proceedings of the IEEE Aerospace Conference, Montana, USA, 2015: 1-12. doi: http://dx.doi.org/10.1109/AERO.2015. 7119230. [8] BIAN Hongwei, LI An, TAN Fangjun, et al. Application of Modern Information Fusion Technology in Integrated Navigation... Beijing: China, National Defense Industry Press; 2010. [9] Cao Xianghui, Chen Jiming, Zhang Yan, et al. Development of an integrated wireless sensor network micro-environmental monitoring system. ISA Trans 2008;47:247– 55. [10] MAHMOUD MS, KHALID HM. Distributed Kalman filtering: a bibliographic review. IET Control Theory Appl 2013;7(4):1–6. http://dx.doi.org/10.1049/iet-cta.2012.0732. [11] Mahmoud MS, Sabih M, Elshafei M. Event-triggered output feedback control for distributed networked systems. ISA Trans 2016;60:294–302. [12] Zhang Honglu, Cheng Jun, Wang Hailing, et al. Robust finite-time event-triggered H1boundedness for network-based Markovian jump nonlinear systems. ISA Trans 2016;63:32–8. [13] Pan Indranil, Das Saptarshi, Gupta Amitava. Tuning of an optimal fuzzy PID controller with stochastic algorithms for networked control systems with random time delay. ISA Trans 2011;50:28–36. [14] OLFATI-SABER R. Distributed Kalman filtering for sensor networks [C]. In: Proceedings of the IEEE Conference on Decision and Control, New Orleans, USA: p. 5492–5498; 2007. [15] OLFATI-SABER R. Kalman consensus filter: optimality, stability, and performance [C]. Joint In: Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China: p. 7036–7042; 2009. [16] Chao SUN, YANG Chunxi, Sha FAN. , et al.. Design of distributed consensus Kalman filter based on energy optimization [C]. In: Proceedings of the Chinese Control Conference, Nanjing, China: p. 7134–7138; 2014. [17] Wengling LI, Yingmin JIA, Junping DU. Event-triggered Kalman consensus filter over sensor networks. IET Control Theory Appl 2016;10(1):103–10. http://dx.doi.org/ 10.1049/iet-cta.2015.0508. [18] KUNTZE N, RUDOLPH C, HANKA O, et al.. A resilient and distributed cabin network architecture [C]. IEEE/AIAA In: Proceedings of the 33rd Digital Avionics Systems Conference, Colorado, USA, 2014: 4A4-1  4A4-10. [19] Yang W, SHI HB. Power allocation scheme for distributed filtering over wireless sensor network. IET Control Theory Appl 2015;9(3):410–7. [20] Yang W, Chen GR, Wang XF, Shi L. Stochastic sensor activation for distributed state estimation over a sensor network. Automatica 2014;50:2070–6. [21] Yang W, Yang C, Shi HB, Shi L, Chen GR. Stochastic link activation for distributed filtering under sensor power constraint. Automatica 2017;75:109–18. [22] Yu WW, Chen GR, Wang ZD, Yang W. Distributed consensus filtering in sensor networks. IEEE Trans Syst, Man Cybern, Part B: Cybern 2009;39(6):1568–77. [23] Chen GR. Pinning control and controllability of complex dynamical networks. Int J Autom Comput 2017;14(1):1–9. [24] Chen GR. Pinning control and synchronization on complex dynamical networks. Int J Control, Autom Syst 2014;12(2):221–30. [25] Andersen B, Moore J. Detectability and stabilisability of time-varying discrete-time linear systems. Siam J Control Optim 1981;19(1):20–32. http://dx.doi.org/10.1137/ 0319002. [26] KONRAD R, STEFAN G, ENGIN Y, et al. Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans Autom Control 1994;44(4):714–28. http://dx.doi. org/10.1109/9.754809.

Please cite this article as: Wang R, et al. Analyses of integrated aircraft cabin contaminant monitoring network based on Kalman consensus filter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.027i