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A Dynamic Event-Triggered Transmission Scheme for Distributed Set-Membership Estimation Over Wireless Sensor Networks Xiaohua Ge, Member, IEEE, Qing-Long Han , Senior Member, IEEE, and Zidong Wang , Fellow, IEEE

Abstract—This paper is concerned with the distributed set-membership estimation for a discrete-time linear time-varying system over a resource-constrained wireless sensor network under the influence of unknown-but-bounded (UBB) process and measurement noise. Sensors collaborate among themselves by exchanging local measurements with only neighboring sensors in their sensing ranges. First, a new dynamic event-triggered transmission scheme (ETS) is developed to schedule the transmission of each sensor’s local measurement. In contrast with the majority of existing static ETSs, the newly proposed dynamic ETS can result in larger average interevent times and thus less totally released data packets. Second, a criterion for designing desired event-triggered set-membership estimators is derived such that the system’s true state always resides in each sensor’s bounding ellipsoidal estimation set regardless of the simultaneous presence of UBB process and measurement noise. Third, a recursive convex optimization algorithm is presented to determine optimal ellipsoids as well as the estimator gain parameters and the event triggering weighting matrix parameter. Furthermore, the proposed dynamic ETS is applied to address the distributed set-membership estimation problem for a discrete-time linear time-varying system with a nonlinearity satisfying a sector constraint. Finally, an illustrative example is given to show the effectiveness and advantage of the developed approach. Index Terms—Distributed set-membership estimation, dynamic event-triggered transmission scheme (ETS), recursive convex optimization, unknown-but-bounded (UBB) noise, wireless sensor networks (WSNs).

I. I NTRODUCTION IRELESS sensor networks (WSNs) have become an emerging technology that can be used for a wide range of application areas, such as environment (detection of toxic chemicals in a contaminated environment), health (personal wellness determination), military (battlefield surveillance), and home (appliances monitoring). A WSN is composed of a large

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Manuscript received April 3, 2017; revised August 10, 2017; accepted October 29, 2017. This work was supported by the Australian Research Council Discovery Project under Grant DP160103567. This paper was recommended by Associate Editor R. Selmic. (Corresponding author: Qing-Long Han.) X. Ge and Q.-L. Han are with the School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia (e-mail: [email protected]). Z. Wang is with the Department of Computer Science, Brunel University London, Uxbridge UB8 3PH, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2017.2769722

number of spatially deployed smart sensor nodes which are linked together through wireless channels and possess data sensing, processing, and communicating components [1]. A key feature of the WSN is that these smart sensor nodes collaborate among themselves in accordance with an interaction topology so as to achieve a common detecting, tracking or monitoring objective. In most practical systems, noise is unavoidable during information sensing and transmission such as measurement noise and communication noise, and often causes the degradation of the system performance [2]. Therefore, a fundamental problem in WSNs is to develop an effective distributed estimation algorithm on each sensor to estimate an unavailable signal of interest through a disturbed target plant and/or noisy measurements. Depending on different types of noise signals, two prevalent methods, i.e., distributed Kalman filtering and distributed H∞ estimation, have been intensively studied in the literature. Motivated by some existing consensus algorithms of multiagent systems, consensus strategies were applied in [3] to deal with the coupled distributed estimation and motion control problem for mobile sensor networks. This was achieved by introducing a consensus term in a distributed Kalmanconsensus filter to reduce the disagreements of estimates among the sensors. In contrast, the problem of distributed Kalman filtering and smoothing was addressed in [4] based on diffusion strategies, where sensor information is diffused across the network through a sequence of Kalman iterations and data aggregation. Whereas, diffusion solutions focused on the recursive minimization of cost functions. Alternatively, some optimization approaches were adopted to deal with distributed Kalman filtering problems. For example, some distributed Kalman filtering and smoothing algorithms, which offered any-time minimum mean-square error optimal state estimations, were provided in [5] by the alternation direction of multipliers. In [6], based on the Kalman filter, the problem of distributed state estimation over WSNs was investigated under intermittent and random data packet dropouts. The theoretical analysis of minimizing the estimation error via searching an optimal consensus gain for a fixed network was provided. Combining the consensus and optimization methods, the problem of distributed optimal consensus filtering problem was generalized in [7] to heterogeneous sensor networks where two types of sensors with different processing abilities were involved. Readers are also referred to [8] for a classification

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of different approaches and methods for solving distributed Kalman filtering problems. Using an H∞ norm of the transfer function from a disturbance vector to an output vector as a performance index for disturbance attenuation, the problem of distributed H∞ estimation has received increasing attention in the last years and a number of results have been reported in the literature. To mention a few, in [9], a round-Robin type communication protocol was proposed in the context of robust distributed H∞ estimation to regulate information transmissions between sensing nodes. In [10], a stochastic sampled-data approach was developed to solve distributed H∞ estimation in sensor networks where each sensor’s measurement output was sampled under a randomly switched mechanism between two different values. In [11]–[16], some typical network-induced phenomena such as data packet dropouts, link failures, redundant channels, and switching topologies were incorporated in the proposed distributed H∞ estimation frameworks for various systems. Note that the celebrated Kalman filtering method requires an accurate model of the plant under consideration and a priori stochastic knowledge of the noise statistics being assumed. If these requirements are not met, Kalman filters may lead to poor performance [17]. In contrast, the H∞ estimation method is based on an assumption that the noise signal is energy bounded rather than random in the form of Gaussian noise. Nonetheless, H∞ estimators are designed to provide a bound for the worst-case estimation error of the system. Besides, under either a Kalman filtering framework or an H∞ estimation framework, the designed estimation algorithm only achieves a pointwise estimation, i.e., at each instant of time a filter or an estimator calculates a point estimation of the signal of interest which is often a single vector. This cannot guarantee that the signal of interest is always included in some reliable confidence region at each instant of time because the estimation has no hard bounds [18], [19]. In many realworld applications such as multiple marine vessels formation control [20], [21], each vessel usually should not enter the collision area of neighboring vessels so as to form a specific formation. Due to unpredictable environmental changes, such as wind, waves, and ocean currents, however, vessels are impossible to remain an exact formation pattern at each instant of time but can definitely be brought into individual safe formation regions in accordance with each vessel’s moving. In this case, even though a specific vessel slows down so as not to collide with the vessel in front of it, a region-based estimation algorithm can guarantee a reliable estimation as the vehicle at a lower speed is still in the safe region [22]. Another example is a power system where an estimation algorithm should provide a confidence region which always contains the true state of the power system for safety and reliability purposes [23]. This motivates the development of set-membership estimation (referred to also as ellipsoidal or set-valued estimation) methods. The key idea of a set-membership estimation is to provide a bounding ellipsoidal set enclosing all possible state estimations in the state space, which is guaranteed to contain the true state of the system by assuming unknown-but-bounded (UBB) noise signals instead of accurate statistical knowledge

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of the noise. During the past two decades, the set-membership estimation problems have attracted considerable interest for single sensor systems (see [17]–[19], [23], [24] and references therein). However, for systems over WSNs, the distributed set-membership estimation issue has not yet gained adequate research attention in the literature. It remains challenging to tackle a distributed set-membership estimation problem over a WSN such that even in the presence of UBB process noise and measurement noise, each sensor can provide a reliable ellipsoidal estimation of the system’s state based on not only its own measurement but also its neighboring sensors’ measurements. This gives rise to the first motivation of this paper. Notice that most of the existing distributed filtering or estimation algorithms rely on an implicit assumption of consecutive or periodic exchanges of information among sensors, which means that each sensor’s local measurement is broadcast to its neighboring sensors through a communication network at consecutive instants of time or regularly with equidistant sampling periods (see [2], [12]–[14], [25]). However, practical sensors are powered by finite battery and have restricted energy for performing a consecutive or periodic broadcasting [26]–[28]. Thus, from a resource conservation perspective, it is extravagant for individual sensors to consecutively or periodically communicate with neighboring sensors, especially when there is a little fluctuation between two consecutive broadcasts [29]. Additionally, consecutive or periodic exchanges of information among sensors will inevitably occupy more bandwidth and worse still will lead to severe network congestions, which serve as the main cause of data packet dropouts and transmission delays. For these reasons, it is significant to develop an efficient transmission scheme to reduce the continual use of limited communication resources in WSNs. Recently, in [30], an energy-efficient distributed H∞ filtering algorithm was developed by reducing each sensor’s measurement size. It has been shown that with the aid of reduced data packet size, the energy consumption on individual sensors can be alleviated. On the other hand, event-triggered transmission schemes (ETSs) have been emerging to mitigate unnecessary occupancy of resources while preserving certain system performance (see [22], [29], [31]–[36] and references therein). A critical concern of an event-triggered scheme is to reduce the frequency of a sensor’s data transmission through a network medium. This is done by defining a series of “events” on sensors and checking an event triggering condition at each sampling instant of time, thereby discerning when each sensor’s data should be transmitted to its neighboring sensors. As a result, an objective control or estimation task is only executed after the occurrence of an “event.” Compared with many results on event-triggered control published in [37]–[39], little effort has been made to the investigation on distributed event-triggered filtering or estimation over WSNs. For example, in [40], a co-design algorithm of distributed H∞ filters and event-triggered communication schemes was derived for continuous-time linear systems over sensor networks in the presence of communication delays. In [41], a distributed recursive filtering problem was investigated for discrete time-varying systems by developing an

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event-based communication mechanism. In [42] and [43], a distributed event-based set-membership filtering problem was addressed for discrete-time nonlinear systems subject to sensor saturations. In [44], a partial information-based ETS was developed to solve the problem of distributed set-membership filtering for a class of discrete time-varying systems with UBB noise. It is noteworthy that the distributed event-triggered filtering or estimation schemes aforementioned and most existing distributed event-triggered schemes surveyed in [28] and [45] can be categorized as static ETSs as the threshold (or threshold parameter) is a fixed scalar all the time. Generally, a static event triggering condition has the following form: f (ei ) − δi > 0 or f (ei ) − σi g(yi ) > 0, where f and g stand for two class K∞ functions; ei is the measurement error between the current measurement and the last transmitted measurement; δi represents a measurement-independent threshold; and σi g(yi ) stands for a measurement-dependent threshold with yi denoting either the current measurement or the last transmitted measurement and σi > 0 denoting a prescribed threshold parameter. Under the static ETS, an event is triggered once f (ei ) surpasses the static threshold δi or σi g(yi ). Intuitively, one would expect that if the threshold δi or σi g(yi ) can be dynamically adjusted such that f (ei ) should not easily exceed the static threshold at each checking instant of time, the total number of released events would be further decreased than the one by using a static ETS. Drawing intuition from this idea, such a static triggering strategy, although widely investigated in the literature, represents a conservative solution that may still lead to unnecessary data transmissions and broadcasts. It is thus the second motivation of this paper to develop an efficiently dynamic ETS for distributed estimation over WSNs. In this paper, we will address distributed set-membership estimation problems for both a discrete-time linear timevarying system and a discrete-time linear time-varying system with a nonlinearity satisfying a sector constraint in the presence of UBB process noise and measurement noise. The systems are observed by resource-constrained WSNs which consist of a group of spatially distributed smart sensors communicating among themselves via a wireless network medium. Each sensor has a capability to sense measurement from the plant, process the sensed measurement and broadcast the processed data to its all underlying neighbors in accordance with a prescribed interaction topology. We summarize the main contributions as follows. 1) A new dynamic ETS will be developed to determine when each sensor’s measurement should be transmitted to its neighbors. In contrast to a static ETS, an offset variable generated by an auxiliary system model will be introduced in the threshold. It will be analytically proved that the auxiliary variable is non-negative and thus the next event release instant generated by the dynamic ETS is no less than the one generated by a static ETS. In this sense, the total number of released events computed by the dynamic ETS will be no more than the one by a static ETS.

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2) Delicate event-triggered distributed set-membership estimators will be constructed by taking the influence of UBB process and measurement noise into account. Each estimator is based on aggregated local measurements triggered at local event instants of neighboring sensors. 3) A design criterion in terms of a set of recursive linear matrix inequalities will be derived to ensure the existence of desired distributed set-membership estimators under the dynamic ETS. 4) An efficient recursive convex optimization algorithm will be proposed. The event-triggered distributed setmembership estimation problem is formulated as finding a group of confidence state estimation ellipsoids for individual sensors such that the plant’s true state always resides in each sensor’s bounding ellipsoidal estimation set regardless of UBB process noise and measurement noise. The presented recursive algorithm can determine the event release time sequence and solve out the estimator gain matrix sequences and the event triggering weighting matrix sequences as well as matrix sequences for optimal ellipsoids. The rest of this paper is organized as follows. Section II formulates the event-triggered distributed set-membership estimation problem for a discrete-time linear time-varying system subject to UBB process and measurement noise under a dynamic ETS. Main results are presented in Section III where an estimator design criterion under the dynamic ETS is derived. Section III also includes a recursive convex optimization algorithm and design results for discrete-time linear time-varying systems with sector-bounded nonlinearities in the presence of UBB process noise and measurement noise. Section IV presents an illustrative example and Section V draws the conclusion. II. P ROBLEM F ORMULATION A. Notation 1) Interaction Topology: Let V = {1, 2, . . . , N} denote an index set of nodes, E ⊆ V × V denote an edge set of paired nodes and A = [aij ] ∈ RN×N denote a weighted adjacency matrix. A weighted directed graph G = (V, E, A) represents an interaction topology of N interacting nodes. For any i, j ∈ V, an adjacency element aij > 0 in A represents a positive weighting of the edge between two adjacent nodes, which means that node i can collect information from node j (otherwise, aij = 0 if no information link from node j to node i exists). The set of neighbors of node i ∈ V plus the node itself are denoted by Ni = {j ∈ V : (i, j) ∈ E}. An element of Ni is called a neighbor of node i. 2) Ellipsoid: An ellipsoid is described by E {a : a = b + Ec, c ≤ 1}, where b ∈ Rn is the center and E ∈ Rn×m with rank(E) = m ≤ n is the shape matrix of the ellipsoid. Let E be a lower triangular matrix, where every element on the diagonal is positive. By a Cholesky factorization, we have that P = EET > 0 and cT c = (a − b)T P−1 (a − b) ≤ 1. Hence, an alternative description of the ellipsoid can be given by E {a : (a − b)T P−1 (a − b) ≤ 1}. The “size” of the



ellipsoid is a function of the squared shape matrix P and can be measured by means of trace(P) [17]. Rn stands for the n-dimensional Euclidean space. · denotes the induced matrix 2-norm or the Euclidean vector norm as appropriate. colN {·} stands for a column vector with N blocks. diagN {·} represents a diagonal matrix with N blocks. N denotes the set of nonnegative integers. I is an identity matrix with an appropriate dimension. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. Other notations in this paper are quite standard. B. Description of Plant The plant to be observed is described by a discrete-time linear time-varying system of the following form: x(k + 1) = A(k)x(k) + F(k)w(k), x(0) = x0

(1)

Rnx

where x(k) ∈ is the state vector of the plant; x0 is a given initial condition; A(k) and F(k) are real-valued time-varying matrices with appropriate dimensions; and w(k) ∈ Rnw is the UBB process noise and is confined to a specified ellipsoid Wk w(k) : wT (k)R−1 (k)w(k) ≤ 1 (2) where R(k) = RT (k) > 0 is a real-valued time-varying matrix with an appropriate dimension. C. Output Measurement Model on Each Sensor At time k, sensor i’s output measurement model for system (1) is given by yi (k) = Ci (k)x(k) + Di (k)vi (k), ∀ i ∈ V

(3)

where yi (k) ∈ Rny is the measurement output vector received by sensor node i from the plant; Ci (k) and Di (k) are realvalued time-varying matrices with appropriate dimensions; and vi (k) ∈ Rnv is the UBB measurement noise and is confined to the following ellipsoid: Vki vi (k) : vTi (k)Q−1 (4) i (k)vi (k) ≤ 1 where Qi (k) = QTi (k) > 0 is a real-valued time-varying matrix with an appropriate dimension. Remark 1: In robotics applications, the distribution of the process and sensor noise is generally multimodal and imprecisely known [18]. Besides, in some practical situations, the noise signal may not be stochastic in the form of Gaussian noise. Rather, the noise possesses a deterministic characteristic and is unknown but can be bounded by either amplitude or frequency. For example, the noise during the audio design is usually one of a set of narrow-band signals with frequency spectrum in the range of 20 Hz to 20 KHz. Thus, it is reasonable to make assumptions of UBB process noise and measurement noise in (2) and (4). D. New Dynamic Event-Triggered Transmission Scheme It is noteworthy that actual data transmissions typically take place over some advanced digital networks, such as wireless local-area networks (802.11) and cellular data networks (3G), where communication resources such as bandwidth may be

Fig. 1. Schematic of distributed estimation over a WSN consisting of a group of smart sensors.

scarce due to various design and implementation considerations [46]. Therefore, from a resource conservation perspective, it is improvident to broadcast and transmit each sensor’s measurement to its neighbors at every sampling time. On the other hand, there is no need to update an estimator when there is a little fluctuation between two consecutive samples of the measurement output signal. Thus, it is of great necessity to develop an effective transmission scheme to reduce the frequency of sensors’ broadcasts and transmissions, and thus to achieve better resource efficiency. In essence, it is equivalent to scheduling sensors’ data transmissions at every sampling instant of time. To make a concept of a “dynamic ETS” clear, we first introduce its counterpart of a “static ETS,” which has been widely studied in [29], [31]–[34], [36], [40]–[44] and references therein. The plant of the form (1) is observed and measured by a group of smart sensors, as shown in Fig. 1. The interaction topology of these smart sensors is modeled by G. At time k, it is assumed that the processed measurement y˜ i (k) = yi (k) − Ci (k)ˆxi (k) and a time-stamp k are encapsulated into a data packet (k, y˜ i (k)), where xˆ i (k) ∈ Rnx is an estimation of the plant’s state x(k) calculated by estimator i. For smart sensor i, the data packet (k, y˜ i (k)) will be shared and exchanged among its neighboring sensors. However, when this data packet should be broadcast and transmitted to sensor i’s neighbors is determined by an eventbased data packet scheduler (DPS). For example, Fig. 2(a) shows the schematic of a static event-based DPS. An event is triggered or released, which means that the data packet (k, y˜ i (k)) is transmitted to sensor i’s neighbors. Thus, after sensor i measuring the output signal from the plant, DPS i will check the data packet (k, y˜ i (k)) to discern whether an event should be released. As a result, sensors’ data are scheduled at each time step such that the transmission frequency can be reduced. For sensor i, denote an event release time sequence by {tki | tki ∈ N; k ∈ N}, where t0i = 0, t1i , t2i , . . ., are monotonically increasing event release instants satisfying {t0i , t1i , t2i , . . .} ⊆ N. It is assumed that tki also denotes the instant when y˜ i (tki ) is successfully arrived at the estimator side. The key idea of an ETS is to identify the event release time sequence {tki }. First, the following static ETS, shown in Fig. 2(a), is considered,


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ETS, the event release instants on sensor i can be determined recursively as follows: i = inf k > tki | θi li (k) > δi (k) (7) dynamic ETS : tk+1 k∈N

where li (k) is defined in (6); δi (k) is an auxiliary offset variable satisfying the following dynamics: δi (k + 1) = ρi δi (k) − li (k), δi (0) = δi,0

with δi,0 ≥ 0 representing the initial condition of the auxiliary system. In (7) and (8), θi and ρi are prescribed scalars satisfying

(a)

0 < ρi < 1, θi ≥ 1/ρi .

(b) Fig. 2. Typical architecture of a smart sensor equipping a DPS under. (a) Static ETS. (b) Dynamic ETS.

through which DPS i recursively determines the event release instants on sensor i: i static ETS : tk+1 = inf k > tki | li (k) > 0 (5) k∈N

where

i T i li (k) = hTi (k)i (k)h ii(k) − σi y˜ i tk i (k)˜yi tk hi (k) = y˜ i (k) − y˜ i tk

(8)

(6)

with σi ∈ [0, 1) representing a prescribed threshold parameter and i (k) = iT (k) > 0 denoting a weighting matrix sequence to be determined. Storer i in Fig. 2 always sends the newest data packets to estimator i. At time k, DPS i checks whether the current data packet (k, y˜ i (k)) satisfies the event triggering i is calcondition li (k) > 0. If yes, the event release instant tk+1 culated by (5) and the current data packet is transmitted to estimator i and broadcast to its neighbors. Otherwise, the current data packet is discarded and the last transmitted data packet (tki , y˜ i (tki )) which is reserved in storer i is utilized instead. From (5), one can see that the static ETS involves a constant threshold parameter σi . Thus, once sensor i’s measurement is received by DPS i, whether or not the current data packet (k, y˜ i (k)) should be broadcast depends only on the threshold parameter σi , which is often preset by the designer or operating engineer. However, this may result in the unfavorable circumstance that some data packets containing useless measurement information of the system are excessively triggered. Intuitively, if the designer or operating engineer can dynamically adjust the threshold parameter σi such that the violation of li (k) ≤ 0 occurs not so often, then the total number of triggered events by using the static ETS (5) can be further reduced. Bearing this in mind, we next propose a dynamic ETS, as shown in Fig. 2(b), which is based on an offset variable generated by an auxiliary system model. More specifically, by using a dynamic

(9)

Remark 2: Compared with the static ETS (5), the introduction of δi (k) is a key ingredient of the dynamic ETS (7), which is partially motivated by the internal-clock-based dynamic triggering mechanism proposed in [47] for event-triggered control systems in the continuous-time context. Whereas, the dynamic triggering mechanism in [47] required a continual monitoring of the system’s state and was associated with only a single sensor node. In contrast, the dynamic ETS (7) is based on measurements at discretized instants of time and can be performed in a fully distributed fashion, i.e., each sensor only utilizes its local measurement information to trigger its data transmission. In the meanwhile, the auxiliary system of the form (8) is delicately constructed such that the ETS (7) can dynamically regulate the interevent time (i.e., the execution time distance i − tki ). Note that δi (k) can between two consecutive events tk+1 be deemed as an estimation of the signal li (k) and the estimator has a linear dynamics described by (8). In this sense, the value of δi (k) can be adjusted real-time in accordance with the current measurement y˜ i (k) and its last transmitted measurement y˜ i (tki ). We now present the following lemma, which shows an important feature of the auxiliary offset variable δi (k). Lemma 1: For prescribed scalars ρi and θi satisfying (9), the auxiliary offset variable δi (k) ≥ 0

(10)

holds for all k ∈ N. Proof: Under the dynamic ETS (7), no event is triggered for i i i i i all k ∈ ∞ k=0 ϒk = N, where ϒk {tk , tk + 1, . . . , tk+1 − 1} denotes the sampling-interval-like subset between two coni . Thus, we have secutive event release instants tki and tk+1 that δi (k) − θi li (k) ≥ 0

(11)

which leads to 1 δi (k). θi On the other hand, (8) indicates that li (k) ≤

li (k) = ρi δi (k) − δi (k + 1).

(12)

(13)

In light of (9), combining (12) and (13) yields

1 1 2 δi (k) ≥ ρi − δi (k − 1) δi (k + 1) ≥ ρi − θi θi k+1

1 ≥ · · · ≥ ρi − δi (0) ≥ 0. (14) θi This completes the proof.



It is shown in Lemma 1 that the auxiliary time-varying offset δi (k) is nonnegative. As a consequence, the triggering condition li (k) > δi (k)/θi > 0 in the dynamic ETS (7) is much stringent than li (k) > 0 in the static ETS (5). In this sense, it is expected that for a previously released instant tki , the next event release instant generated by the dynamic ETS (7) will be no less than that generated by the static ETS (5). To proceed with, we present the following lemma, which establishes the relationship between the next event release instants generated by (5) and (7). Lemma 2: Given prescribed scalars ρi and θi satisfying (9), and a previously released instant tki , let sik+1 be the next event i release instant computed by the static ETS (5) and dk+1 be the next event release instant computed by the dynamic ETS (7). i . Then we have that sik+1 ≤ dk+1 Proof: The proof is performed by contradiction. Let sik+1 > i dk+1 . From (5), it can be seen that i ≤ 0. (15) li dk+1 On the other hand, from (7), one has i i − δi dk+1 > 0. θi li dk+1

(16)

Recalling that δi (k) ≥ 0 for all k ∈ N, we have that i ) > (1/θ )δ (d i ) ≥ 0 from (16), which contrali (dk+1 i i k+1 dicts (15). This completes the proof. Remark 3: Lemma 2 indicates that, for a previously released instant tki , the interevent time by using the dynamic ETS (7) is larger than or is equal to the one by applying the static ETS (5), which gives a theoretical proof that the total number of released events computed by the dynamic ETS (7) will be no more than the one by the static ETS (5). For a quantitative illustration on this point, an example will be presented in Section IV. However, it should be pointed out that Lemma 2 does not provide anything on further event release instants due i ). to the fact that y˜ i (sik+1 ) = y˜ i (dk+1 E. Event-Triggered Distributed Set-Membership Estimators We are interested in designing a group of set-membership j estimators based on triggered local measurement y˜ j (t ˜ ) of kj neighboring sensors. For each i ∈ V, consider that the smart sensor i runs an estimator of the following form:

j xˆ i (k + 1) = Aˆ i (k)ˆxi (k) + Bˆ i (k) (17) aij y˜ j t ˜ j∈Ni

kj

for all k ∈ ϒki , where xˆ i (k) ∈ Rnx is an estimation of the plant’s j state; y˜ j (t ˜ ) are the latest transmitted local measurements kj

received from sensor i’s neighbors with k˜ j arg mink˜ {k − t ˜ | k j k > t ˜ , k˜ ∈ N}; Aˆ i (k) and Bˆ i (k) are estimator gain matrix k sequences to be designed; and the initial condition of estimator i is given as xˆ i,0 which belongs to a given ellipsoid T −1 (18) x0 − xˆ i,0 ≤ βi X0i x0 : x0 − xˆ i,0 Ui,0 j

T > 0 is a known real-valued matrix and where Ui,0 = Ui,0 βi > 0 is a scaling parameter of the ellipsoid.

F. Problem to be Addressed Generally, when a conventional distributed filtering or estimation problem is considered in a WSN setting, each sensor (or estimator) is required to compute a local pointwise estimation (denoted as xˆ i (k) ∈ Rnx ) with regard to the plant’s state x(k) at each time step. Since xˆ i (k) is only a single vector and has no hard bounds, there is no guarantee that the state distributions of x(k) can be included in some confidence region. However, in many real-world applications, such as military missile guidance, the target requires 100% confidence to be estimated [19], [24]. In other words, the future state of the target should be brought into a confidence estimation region which always contains the true state of the target, especially when there exist unpredictable environmental changes in the monitoring and tracking area. For this purpose, we will establish a distributed set-membership estimation framework for the system under consideration. To proceed with, the definition of distributed set-membership estimation is presented. Definition 1: The system of the form (1) and (3) subject to UBB process noise w(k) ∈ Wk and UBB measurement noise vi (k) ∈ Vki is said to achieve the distributed set-membership estimation on sensor i, ∀ i ∈ V, if the system’s future state i x(k + 1) always resides in a state estimation ellipsoid Xk+1 i computed by sensor i. Furthermore, the ellipsoid Xk+1 provides a set of state estimations in state space containing the true state of the system and has the following form: i x(k + 1) : eTi (k + 1)Ui−1 (k + 1)ei (k + 1) ≤ βi Xk+1 (19) for k ∈ N, where ei (k + 1) = x(k + 1) − xˆ i (k + 1) denotes an estimation error vector on sensor i and Ui (k + 1) = UiT (k + 1) > 0 represents a real-valued time-varying matrix. The event-triggered distributed set-membership estimation problem to be addressed is now stated as: for prescribed scalars σi ∈ [0, 1), βi > 0, ρi and θi satisfying (9), and UBB w(k) ∈ Wk and vi (k) ∈ Vki , i ∈ V, find an appropriate realvalued matrix sequence Ui (k + 1) and estimator i’s one-step ahead state xˆ i (k + 1) under the dynamic ETS (7) such that the system’s one-step ahead state x(k + 1) always resides in the i . confidence state estimation ellipsoid Xk+1 If the above problem is solvable, the actual estimation of the system’s state on each sensor at each time step is a set in state space rather than a single vector computed by the traditional distributed filtering or estimation algorithms. Therefore, the proposed distributed set-membership estimation algorithm guarantees that the set of all possible values of the system’s state can be included in an ellipsoidal estimation region. Remark 4: Note that one of the main challenging issues in designing set-membership estimation algorithms is to reduce the conservatism of the ellipsoidal estimation sets. This can be achieved by finding tight upper and lower bounds for the ellipsoidal estimation sets. From (19), it can be observed that by choosing different values of βi , one can obtain various upper and lower bounds for the estimation error. In other words, the introduction of the scaling parameter βi enables us to derive a favorable error bound and thus increases the flexible dimensions in the solution space for the set-membership


estimation problem, which will be verified through an example in Section IV. III. M AIN R ESULTS A. Design Criterion Under the Dynamic ETS To facilitate subsequent development, we introduce the following notations: e˜ (k) = colN {ei (k)}, x˜ (k) = colN {x(k)} xˆ (k) = colN xˆ i (k) , w(k) ˜ = colN {w(k)} ˜ {v v˜ (k) = colN i (k)}, h(k) = colN {hi (k)}

˜ = colN {σi }, (k) ˜ ˜ = colN {θi },

= diagN {i (k)} 1 α˜ = colN {αi }, β˜ = diagN βi2

i=1

βi + 1 (k)N + 2 (k)N + 3 (k)N +

(22)

j∈Ni

and (k) = [p,q (k)]5×5 is a time-varying sparse symmetric block matrix with its nonzero entries given by N

1

ei (k + 1) = A(k)x(k) + F(k)w(k) − Aˆ i (k)ˆxi (k) aij y˜ j (k) − hj (k) − Bˆ i (k)

˜ ˆ ˜ ˜ ˆ ˜ A(k) − A(k) xˆ (k), β˜ A(k) − B(k)A C(k) L(k), ˜ ˆ ˜ ˆ F(k), −B(k)A D(k), B(k)A

1,1 (k) = −

Thus, there exists a vector αi satisfying αi ≤ 1 such that

In light of (1), (17), and (22), calculating the one-step ahead state estimation error ei (k + 1) yields

In the following, we present a theorem, which provides a sufficient condition on the existence of desired distributed estimators in the form of (17) and state estimation ellipsoids i Xk+1 such that the event-triggered distributed set-membership estimation problem is solved under the dynamic ETS (7). Theorem 1: For prescribed scalars σi ∈ [0, 1), βi > 0, ρi and θi satisfying (9), w(k) ∈ Wk and vi (k) ∈ Vki , i ∈ V, if there exist real-valued matrix sequences Ui (k + 1) > 0, i (k) > 0, Aˆ i (k), Bˆ i (k), and scalar sequences m (k) > 0, m = 1, 2, 3, such that ˜ + 1) (k) −U(k ≤ 0, ∀ k ∈ N (20) ∗ (k)

(k) =

−1 ei,0 ≤ βi . Then, at time k, it is assumed that the that eTi,0 Ui,0 plant’s state x(k) belongs to its state estimation ellipsoid Xki and satisfies eTi (k)Ui−1 (k)ei (k) ≤ βi . Next, we shall prove that eTi (k + 1)Ui−1 (k + 1)ei (k + 1) ≤ βi holds. Recall that (x(k) − xˆ i (k))T Ui−1 (k)(xi (k) − xˆ i (k)) ≤ βi at time k. By virtue of Schur complement [48], the inequality is equivalent to βi−1 (x(k)− xˆ i (k))(x(k)− xˆ i (k))T ≤ Ui (k). Furthermore, by a Cholesky factorization, one has Ui (k) = Li (k)LiT (k), where Li (k) is a lower triangular matrix and every element −1/2 −1 Li (k)(x(k) − on the diagonal is positive. Setting αi = βi xˆ i (k)), it is easy to verify that T αiT αi = βi−1 x(k) − xˆ i (k) Ui−1 (k) xi (k) − xˆ i (k) ≤ 1. (21)

x(k) = xˆ i (k) + βi2 Li (k)αi .

˜ ˜ U(k) = diagN {Ui (k)}, L(k) = diagN {Li (k)} ˜ ˜ = diagN {Qi (k)} R(k) = diagN {R(k)}, Q(k) ˜ ˜ = diagN {F(k)} A(k) = diagN {A(k)}, F(k) ˜ ˜ C(k) = diagN {Ci (k)}, D(k) = diagN {Di (k)} ˆ ˆ A(k) = diagN Aˆ i (k) , B(k) = diagN Bˆ i (k) .

where

7

N

δi (k)

i=1

˜ L(k) ˜

˜ (k) ˜ β˜ C(k) ˜ 2,2 (k) = − 3 (k)I + L˜ T (k)C˜ T (k)β˜ T T T T ˜

˜ (k) ˜ D(k) ˜ 2,4 (k) = L˜ (k)C˜ (k)β˜ T T T ˜ ˜

˜ (k), ˜ 2,5 (k) = −L˜ (k)C (k)β˜ 3,3 (k) = − 1 (k)R˜ −1 (k) ˜ −1 (k) + D ˜

˜ (k) ˜ D(k) ˜ T (k) ˜ 4,4 (k) = − 2 (k)Q ˜

˜ (k), ˜ ˜

˜ − I (k) ˜ ˜ T (k) 5,5 (k) = 4,5 (k) = −D then the plant’s one-step ahead state x(k + 1) is guaranteed i by to always reside in its state estimation ellipsoid Xk+1 implementing the dynamic ETS (7) and estimators (17). Proof: The proof is performed by using a mathematical induction method. First, from (18), it is straightforward to have

1 = A(k) − Aˆ i (k) xˆ i (k) + βi2 A(k)Li (k)αi + F(k)w(k) 1 − Bˆ i (k) aij βj2 Cj (k)Lj (k)αj j∈Ni

− Bˆ i (k)

aij Dj (k)vj (k) − hj (k) .

(23)

j∈Ni

Letting ψ(k) = [1, α˜ T , w˜ T (k), v˜ T (k), h˜ T (k)]T , from (23), one has that e˜ (k + 1) = (k)ψ(k). Hence, the quadratic one-step ahead estimation error constraint eTi (k + 1)Ui−1 (k + 1)ei (k + 1) ≤ βi in (19) can be rewritten as ˜ −1 (k + 1) (k) + ψ(k) ≤ 0 ψ T (k) T (k)U (24) N where = diag{− i=1 βi , 0, 0, 0, 0}. From (2), (4), and αi ≤ 1, we have that ψ T (k)1 (k)ψ(k) ≥ 0, ψ T (k)2 (k)ψ(k) ≥ 0, and ψ T (k) 3 ψ(k) ≥ 0, where 1 (k) = diag{N, 0, −R˜ −1 (k), 0, 0}, ˜ −1 (k), 0}, and 3 = diag{N, 2 (k) = diag{N, 0, 0, −Q −I, 0, 0, 0}. On the other hand, it is derived from (6) and (11) that δi (k)− θi {hTi (k)i (k)hi (k)−σi [˜yi (k)−hi (k)]T i (k)[˜yi (k)−hi (k)]} ≥ 0, which can be rewritten as ψ T (k)(k)ψ(k) ≥ 0, where (k) = [p,q (k)]5×5 is a 5-by-5 time-varying sparse symmetric block matrix with its nonzero entries given by 1,1 (k) = N i=1 δi (k), 2,2 (k) = 2,2 (k) + 3 (k)I, 2,4 (k) = 2,4 (k), 2,5 (k) = ˜ −1 (k), 4,5 (k) = 4,5 (k), 2,5 (k), 4,4 (k) = 4,4 (k)+ 2 (k)Q and 5,5 (k) = 5,5 (k). Applying S-procedure [48], inequality (24) holds if there exist positive scalar sequences m (k), m = 1, 2, 3, such that ˜ −1 (k + 1) (k) + + 1 (k)1 (k)

T (k)U + 2 (k)2 (k) + 3 (k)3 + (k) ≤ 0.

(25)

By Schur complement, (20) can be deduced from (25). This completes the proof.


Remark 5: By Theorem 1, the proposed event-triggered distributed set-membership estimation problem can be converted into the feasibility problem of a set of recursive linear matrix inequalities (20), through which one can solve out the estimator gain matrix sequences Aˆ i (k) and Bˆ i (k), and the event triggering weighting matrix sequence i (k), as well as deteri containing the true state mine state estimation ellipsoids Xk+1 of the plant even when the UBB process noise and measurement noise are present. Thus, Theorem 1 establishes a criterion for designing desired set-membership estimators in the form of (17), the dynamic ETS of the form (7), and confidence state i defined in (19) for the plant’s future estimation ellipsoids Xk+1 state x(k + 1). B. Recursive Convex Optimization Algorithm Under the Dynamic ETS Note that even though the principle of determining the state estimation ellipsoids is outlined in Theorem 1, it does not provide an optimal state estimation ellipsoid on each sensor. In what follows, by applying a convex optimization approach, the proposed event-triggered distributed setmembership estimation problem is transformed into the following optimization problem (OP) such that some optimal ellipsoids can be derived: ˜ + 1) min trace U(k ˜ + 1) (k) −U(k s.t. ≤0 (26) ∗ (k) ˜ + 1), (k), and (k) are defined in Theorem 1. where U(k ˜ ˜ + 1), (k), Remark 6: Notice that (20) is linear to U(k ˆ ˆ and m (k), m = 1, 2, 3. Hence, OP (26) can be A(k), B(k), efficiently solved by some existing semidefinite programming via an interior-point approach [49]. Note that in (26) the trace ˜ + 1) is minimized at each time step in an effort to of U(k find the smallest state estimation ellipsoids. In this sense, the estimation performance of the system on each sensor can be optimized at each time step by recursively solving the OP (26). Based on the OP (26), we are in a position to present a recursive distributed algorithm, i.e., Algorithm 1, which generates the event release time sequence {tki }, solves out the gain matrix sequences for estimators (17), the weighting matrix sequences for the dynamic ETS (7), and matrix sequences Ui (k + 1), as well as determines real-time optimal state estimation ellipsoids for each sensor. C. Design Criterion Under the Static ETS In (7), when δi (k) ≡ 0, the dynamic ETS reduces to a static ETS of the form (5). The following theorem presents a criterion for designing admissible distributed estimators (17) i such that the proposed and state estimation ellipsoids Xk+1 event-triggered distributed set-membership estimation problem is solved under the static ETS (5). The proof is similar to that of Theorem 1, thus omitted. Theorem 2: For given scalars σi ∈ [0, 1) and βi > 0, and w(k) ∈ Wk and vi (k) ∈ Vki , i ∈ V, if there exist real-valued matrix sequences Ui (k + 1) > 0, i (k) > 0, Aˆ i (k), Bˆ i (k), and


Algorithm 1 (i).

(ii). (iii). (iv).

Choose initial conditions x0 , xˆ i,0 , δi,0 , t0i , and suitable R(k), Qi (k) and Ui,0 such that (2), (4) and (18) hold. Let a vector αi satisfy αi ≤ 1. Set the maximum time step M. Find Li,0 by Ui,0 = T . Set k = 0, ti = ti , xˆ (k) = xˆ , δ (k) = δ Li,0 Li,0 i,0 i i,0 and Li (k) = k 0 i Li,0 ; Solve the OP (26) and determine Ui (k + 1), Aˆ i (k), Bˆ i (k) and i (k). Find Li (k + 1) such that Ui (k + 1) = Li (k + 1)LiT (k + 1); i Calculate δi (k + 1) by (8). Determine tk+1 by checking whether (k, y˜ i (k)) satisfies dynamic ETS (7) on each sensor; Compute xˆ i (k + 1) by (17). Use xˆ i (k + 1) as the center of the i to calculate Xi (k + 1) = xˆ i (k + 1) + estimation ellipsoid Xk+1 1

(v).

βi2 Li (k + 1)αi . If k = M, go to (v); otherwise, set k = k + 1, go to (ii); Output matrix sequences {Aˆ i (k)}, {Bˆ i (k)}, {i (k)} and {Ui (k + 1)}; the event release time sequence {tki }; and the ellipsoidal state estimation set {Xi (k + 1)}. Exit.

scalar sequences m (k) > 0, m = 1, 2, 3 such that ˜ + 1) (k) −U(k ≤ 0, ∀ k ∈ N ˜ ∗ (k)

(27)

˜ ˜ p,q (k)]5×5 is a time-varying sparse symmetwhere (k) = [ ric block matrix with its nonzero entries given by ˜ 1,1 (k) = −

N

βi + 1 (k)N + 2 (k)N + 3 (k)N

i=1

˜ L(k) ˜ 2,2 (k) = − 3 (k)I + L˜ T (k)C˜ T (k)β˜ T

˜ (k) ˜ β˜ C(k) ˜ ˜ 2,4 (k) = L˜ T (k)C˜ T (k)β˜ T

˜ (k) ˜ D(k) ˜ T T T ˜ 2,5 (k) = −L˜ (k)C˜ (k)β˜

˜ (k), ˜ ˜ 3,3 (k) = − 1 (k)R˜ −1 (k) ˜ −1 (k) + D ˜ 4,4 (k) = − 2 (k)Q ˜ (k) ˜ D(k) ˜ T (k)

˜ T ˜ (k), ˜ ˜ 5,5 (k) =

˜ 4,5 (k) = −D ˜ − I (k) ˜ ˜ (k)

then the plant’s one-step ahead state x(k + 1) is guaranteed i by to always reside in its state estimation ellipsoid Xk+1 implementing the static ETS (5) and estimators (17). Remark 7: Setting σi ≡ 0 in (5) or δi (k) = σi ≡ 0 in (7), i = one has li (k) > 0 for all k ∈ N, which means that tk+1 i tk +1. Then, the proposed static ETS and dynamic ETS reduce to a conventional periodic transmission scheme (PTS). The corresponding results on this subject can be easily deduced from Theorems 1 and 2, thus are omitted for brevity. D. Design Criterion for Discrete-Time Linear Time-Varying System With Nonlinearity Satisfying Sector Constraint In this section, we apply the dynamic ETS (7) to address the problem of distributed set-membership estimation for a discrete-time linear time-varying system with a nonlinearity satisfying a sector constraint in the presence of UBB process and measurement noise. Consider the plant described by x(k + 1) = A(k)x(k) + B(k)f (x(k)) + F(k)w(k)

(28)

where x(k), w(k), A(k), and F(k) are defined in system (1); B(k) is a real-valued matrix sequence with an appropriate dimension; the nonlinear vector-valued function f (·) : Rnx → Rnx is assumed to be continuous and satisfies f (0) = 0 and the sector-bounded condition ˜ − z)] ≤ 0 [f (x) − f (z) − G(x − z)]T [f (x) − f (z) − G(x

(29)


9

˜ are real-valued matrices of for all x, z ∈ Rnx , where G and G ˜ is a symmetric positive appropriate dimensions, and G − G definite matrix. The distributed set-membership estimators are of the form

j aij y˜ j t ˜ + B(k)f xˆ i (k) xˆ i (k + 1) = Aˆ i (k)ˆxi (k) + Bˆ i (k) j∈Ni

kj

(30) ϒki .

for all k ∈ For convenience of development, denote f˜i = f (x(k)) − f xˆ i (k) , fˇ = colN f˜i

˜ ¯ = diagN {G}, G ˇ = diagN G ˜ B(k) = diagN {B(k)}, G

Fig. 3.

and other notations are the same as the ones in Theorem 1. We now state and establish the following theorem. Theorem 3: For given scalars σi ∈ [0, 1), βi > 0, ρi and θi satisfying (9), w(k) ∈ Wk and vi (k) ∈ Vki , i ∈ V, if there exist real-valued matrix sequences Ui (k + 1) > 0, i (k) > 0, Aˆ i (k), Bˆ i (k), and scalar sequences m (k) > 0, m = 1, 2, 3, 4, such that ˇ ˜ + 1) (k) −U(k ≤0 (31) ˇ ∗ (k) for all k ∈ N, where ˜ ˆ ˜ ˜ ˇ ˆ ˜

(k) = A(k) − A(k) xˆ (k), β˜ A(k) − B(k)A C(k) L(k) ˜ ˆ ˜ ˆ ˜ F(k), −B(k)A D(k), B(k)A, B(k) ˇ ˇ p,q (k)]6×6 is a time-varying sparse symmetric and (k) = [ block matrix with its nonzero entries given by ˇ 2,4 (k) = 2,4 (k), ˇ 2,5 (k) = 2,5 (k) ˇ 1,1 (k) = 1,1 (k), ˇ 4,4 (k) = 4,4 (k), ˇ 4,5 (k) = 4,5 (k) ˇ 3,3 (k) = 3,3 (k), ˇ 6,6 (k) = − 4 (k)I ˇ 5,5 (k) = 5,5 (k), (k) ¯ TG ˇ β˜ L(k) ˇ 2,2 (k) = 2,2 (k) − 4 L˜ T (k)β˜ T G ˜ 2 (k) ¯T +G ˇT ˇ 2,6 (k) = 4 L˜ T (k)β˜ T G 2 then the plant’s one-step ahead state x(k + 1) is guaranteed i by the to always reside in its state estimation ellipsoid Xk+1 dynamic ETS (7) and estimators (30). ˇ Proof: Let ψ(k) = [1, α˜ T , w˜ T (k), v˜ T (k), h˜ T (k), fˇ T ]T . T Noting that ei (k+1)Ui−1 (k+1)ei (k+1) ≤ βi , it is not difficult to obtain that ˇ T (k)U ˇ ˇ ψ(k) ˇ ˜ −1 (k + 1) (k) ψˇ T (k) + ≤0 (32) N ˇ = diag{− i=1 βi , 0, 0, 0, 0, 0}. where From (29), we have T ˇ e(k) ≤ 0. ¯ e(k) fˇ − G˜ (33) fˇ − G˜ ˜ α˜ into (33) yields Substituting e˜ (k) = β˜ L(k) ˇ ψˇ T (k)(k)ψ(k) ≥ 0,

(34)

where (k) = [p,q (k)]6×6 is time-varying sparse symmetric block matrices with nonzero entries given by 2,2 (k) =

Physical structure of the CSTR.

¯ TG ˇ β˜ L(k), ¯T + ˜ 2,6 (k) = (1/2)L˜ T (k)β˜ T (G −(1/2)L˜ T (k)β˜ T G T ˇ ), and 6,6 (k) = −I. The rest of the proof is similar to G the counterpart in the proof of Theorem 1. Remark 8: With the help of S-procedure and Schur complement, the newly developed event-triggered distributed setmembership estimation approach can deal with several issues, such as time-varying system parameter variations, sectorbounded nonlinearities, UBB noise, and event-triggered data transmission in a unified framework for practical systems over WSNs, as shown in Theorem 3. While it should be mentioned that, the above approach can be further developed to deal with some complex yet important phenomena such as sensor saturations in [19], [42], and [43], where a saturation function satisfied a sector condition, and signal quantization in [25], where static logarithmic quantizers were studied. Interested readers can incorporate these parts by the similar modeling and analysis procedures into the corresponding results on distributed set-membership estimation. We omit these results due to the page limit. IV. I LLUSTRATIVE E XAMPLE In this section, the developed event-triggered setmembership estimation approach is applied to an industrial nonisothermal continuous stirred tank reactor (CSTR), where chemical species A reacts to form species B [50]. The reactor inflow contains only the educt A in low concentration CA0 . The reactor outflow represents the desired product B mixed with A. Fig. 3 demonstrates a simple physical structure of the CSTR, where CA is the output concentration of the educt A; CB is the output concentration of the desired product B within the reactor; T denotes the reactor temperature; and Tc is the cooling medium temperature. The state matrix of the discretized and linearized state-space model of the CSTR near the operating point is borrowed from [51] and [52] and given by (1) x (k + 1) 0.9719 −0.0013 x(1) (k) = −0.0340 0.8628 x(2) (k + 1) x(2) (k) where x(1) (k) denotes the output concentration of the educt A; and x(2) (k) represents the reactor temperature. In practice, the process noise may stem from poisoning of the reaction



and/or from fouling of the cooling coils. In addition, due to unpredictable environmental changes, parameters of the CSTR system may be affected to some extent, thus the system matrix parameter of the CSTR should be time-varying to reflect parameter variations from time to time. Hence, in this example, we consider the CSTR model has the form of (1) with its time-varying parameter matrix sequences given by

0.9719 −0.0013 A(k) = −0.0340 0.8628 + 0.2 sin(k) 0.1 + 0.1 sin(k) F(k) = . 0.3

To enhance the reliability and to improve the estimation performance of the estimation system, a network of five smart sensors, i.e., V = {1, 2, . . . , 5}, are deployed to cooperatively monitor the reactor temperature. Each sensor only broadcasts and shares its measurement and local estimation to its neighboring sensors. The interaction topology of these sensors is depicted in Fig. 3, where the adjacency matrix of the topology is selected as a binary matrix, whose element is either 1 (when sensor i can receive information from sensor j) or 0 (otherwise). The measurement model is subject to measurement noise vi (k) and has the form of (3) with parameter matrices given by Ci (k) = [0, 1 + 0.1i − 0.1 sin(k)] and Di (k) = 1/i for any i ∈ V. The initial output concentration of the educt A is selected as 5.1 mol/l and the initial reactor temperature is selected as 130 ◦ C. The initial conditions of estimators are chosen as xˆ 10 = [5.8, 128]T , xˆ 20 = [4.6, 132]T , xˆ 30 = [6.0, 131]T , xˆ 40 = [4.2, 129]T , xˆ 50 = [5.0, 133]T . Set U0 = diag2 {40, 40}, βi = 1, R(k) = 0.3, and Qi (k) = 0.1. A. Simulation Objectives 1) To design a confidence state estimation ellipsoid for each sensor such that at each time step the CSTR’s true states (the output concentration of the educt A and the reactor temperature) always reside in the estimation ellipsoids of sensors regardless of the UBB process noise and measurement noise. The developed dynamic ETS (7) will be applied to determine when each sensor’s current measurement should be broadcast to its all underlying neighboring sensors so as to save limited communication resources to some extent. 2) To conduct a comparative study of the effects of different transmission schemes on estimation performance and data transmission performance of each sensor through the wireless network medium. More specifically, to provide a quantitative comparison of how many data packets are actually transmitted on each sensor, we consider the same problem setting under three different transmission scenarios. S1: A traditional PTS is implemented to derive the distributed set-membership estimator design results. In this case, each sensor will broadcast and transmit its current measurement yi (k) to its neighboring sensors at each time step k ∈ N. This case can be achieved by setting δi (k) ≡ 0 and σi ≡ 0 in (7) or by letting σi ≡ 0 in (5).

Fig. 4. True state of the CSTR x(k) = [x(1) (k), x(2) (k)]T , its estimation (1) (2) xˆ i (k) = [ˆxi (k), xˆ i (k)]T on sensor i, ∀ i ∈ V, the upper bound and the lower bound of the state in the case of the dynamic ETS.

S2: The static ETS is employed to derive the corresponding results. In this case, whether or not each sensor should broadcast and transmit its current measurement to its neighbors is determined by the static ETS presented in (5). The threshold parameters in (5) are selected as σ1 = 0.11, σ2 = 0.12, σ3 = 0.08, σ4 = 0.08, and σ5 = 0.06. S3: The newly proposed dynamic ETS is utilized instead. In this case, when each sensor’s current measurement should be broadcast to its all underlying neighboring sensors is decided by the proposed dynamic ETS in (7). Let ρi = 0.12, θi = 10, δ10 = 0.4, δ20 = 0.35, δ30 = 0.3, δ40 = 0.25, and δ50 = 0.15. B. Effectiveness of the Newly Developed Event-Triggered Set-Membership Estimation Approach By letting the simulation run for 60 s and solving Algorithm 1, it is found that the proposed distributed setmembership estimation problem is solved under the dynamic ETS. Fig. 4 shows that, in the dynamic ETS case, the true value of the output concentration of the educt A and the true reactor temperature always reside between the upper bound and lower bound which are determined by calculat(1/2) Li (k)αi , where αi is chosen as a unit ing x(k) = xˆ i (k) + βi ball. In other words, Fig. 4 verifies that the true states of the CSTR always belong to the estimated region computed by each sensor, and thus the proposed distributed set-membership estimation approach can provide an ellipsoidal state estimation for the CSTR system.


11

Fig. 6. Event release instants on sensor i, ∀ i ∈ V, in different cases of the static ETS and the dynamic ETS.

Fig. 5. Number of transmitted data packets on sensor i, ∀ i ∈ V, in different cases of the PTS, the static ETS and the dynamic ETS.

C. Comparison of Data Packet Transmission Ratios Between Different Transmission Schemes To facilitate comparison analysis, we define a data packet transmission ratio (DPTR) on each sensor as a transmission i i /N i ), where N i = (Natdp performance index by Jdptr tsdp atdp denotes the number of actually transmitted data packets on i denotes the number of totally sampled-data sensor i and Ntsdp packets on sensor i over the simulation time window. Applying the corresponding results developed in the preceding section, it is found that the proposed distributed set-membership estimation problem is solvable under either the PTS or the static ETS. Note that within the simulation time range [0, 60 s), there are totally 60 sampled-data packets on each sensor. In the PES case, all the 60 data packets on each sensor are transmitted, i = 100% on each senwhich means that the DPTR is Jdptr sor. However, implementing the static ETS (5), we find that the DPTR is 60% on sensor 1, 61.67% on sensor 2, 61.67% on sensor 3, 66.67% on sensor 4, and 66.67% on sensor 5, respectively. In contrast, employing the dynamic ETS (7), the DPTR on each sensor through wireless transmission chan1 2 = 21.67%, Jdptr = 20%, nels is significantly reduced to Jdptr 3 5 4 Jdptr = 23.33%, Jdptr = 23.33%, and Jdptr = 28.33%, respectively. A detailed comparison of the number of transmitted data packets on each sensor in different cases is provided in Fig. 5. Therefore, it can be concluded that the dynamic ETS is more efficient than the PTS and the static ETS in reducing the frequency of data transmission on sensors through WSNs, thus has more potential to alleviate the continual occupancy of limited network resources. The superiority of the developed dynamic ETS over the PTS and the static ETS is verified. We also provide Fig. 6 which shows the specific event release instants on each sensor node in the static ETS case and the dynamic ETS case. It is shown in Fig. 6 that the average interevent times generated by the dynamic ETS are always larger than that generated by the static ETS. D. Comparison of Ellipsoidal Estimation Performance Between Different Transmission Schemes Define a quadratic estimation error on each sensor as an i = eTi (k + 1) ellipsoidal estimation performance index by Ek+1

i Fig. 7. Quadratic estimation errors Ek+1 on sensor i, ∀ i ∈ V in different cases of the PTS, the static ETS and the dynamic ETS.

Ui−1 (k + 1)ei (k + 1). The simulation results between difference transmission schemes are then analyzed by calculating the quadratic estimation errors on sensors. Fig. 7 demonstrates the quadratic estimation errors on sensors between different i ≤ βi = 1 transmission schemes. It can be observed that Ek+1 holds for each sensor in the PTS case, the static ETS case and the dynamic ETS case, respectively, which also validates that the true states of the CSTR always reside in sensors’ state estii , i ∈ V. It should be also pointed that mation ellipsoids Xk+1 in this example, the PTS and the static ETS provide approximately the same quadratic estimation errors, while the dynamic ETS does not provide larger quadratic estimation errors than those by applying the PTS and the static ETS even though less data packets are transmitted through the wireless network medium to compute local estimation sets. E. Comparison of the Ellipsoidal Estimation Set Bounds With Different βi Finally, in the dynamic ETS case, we compute the error between the upper bound and the lower bound of the ellipsoidal estimation set each sensor node by choosing different values of the ellipsoid scaling parameter βi . For brevity, we only provide Fig. 8 which illustrates the bound errors on sensor 4 in difference cases of βi . It is shown that by adjusting the values of βi for sensors, one may obtain different upper and lower bounds for the ellipsoidal estimation sets. Therefore, Fig. 8 enables one to properly select βi to obtain a desirable 4 , which means that the introducstate estimation ellipsoid Xk+1 tion of the scaling parameter βi in (19) increases the flexible



(1)

(2)

Fig. 8. Bound errors of e4 (k) = [e4 (k), e4 (k)]T under different values of βi in the case of the dynamic ETS.

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Xiaohua Ge (M’18) received the B.Eng. degree in electronic and information engineering from Nanchang Hangkong University, Nanchang, China, in 2008, the M.Eng. degree in control theory and control engineering from Hangzhou Dianzi University, Hangzhou, China, in 2011, and the Ph.D. degree in computer engineering from Central Queensland University, Rockhampton, QLD, Australia, in 2014. He was a Research Assistant with the Centre for Intelligent and Networked Systems, Central Queensland University, from 2011 to 2013 and a Research Fellow, in 2014. From 2015 to 2016, he was a Research Fellow with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. In 2017, he joined the Swinburne University of Technology, Melbourne, VIC, Australia, where he is currently a Lecturer with the School of Software and Electrical Engineering. His current research interests include distributed networked control systems, multiagent systems, and sensor networks.

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Qing-Long Han (M’09–SM’13) received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Sc. and Ph.D. degrees in control engineering and electrical engineering from the East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From 1997 to 1998, he was a Post-Doctoral Researcher Fellow with the Laboratoire d’Automatique et d’Informatique Industielle (currently, Laboratoire d’Informatique et d’Automatique pour les Systémes), École Supérieure d’Ing’enieurs de Poitiers (currently, École Nationale Supérieure d’Ingé nieurs de Poitiers), Université de Poitiers, France. From 1999 to 2001, he was a Research Assistant Professor with the Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, Edwardsville, IL, USA. From 2001 to 2014, he was a Laureate Professor, an Associate Dean of research and innovation with the Higher Education Division, and the Founding Director of the Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton, QLD, Australia. From 2014 to 2016, he was the Deputy Dean of Research with the Griffith Sciences, and a Professor with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. In 2016, he joined the Swinburne University of Technology, Melbourne, VIC, Australia, where he is currently the Pro Vice-Chancellor of research quality and a Distinguished Professor. In 2010, he was appointed as a Chang Jiang (Yangtze River) Scholar Chair Professor by the Ministry of Education, Beijing, China. His current research interests include networked control systems, neural networks, time-delay systems, multiagent systems, and complex dynamical systems. Prof. Han is one of the World’s Most Influential Scientific Minds: 2014–2016, and the Highly Cited Researcher Award in Engineering by Thomson Reuters. He is an Associate Editor of a number of international journals, including the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS, the IEEE T RANSACTIONS ON I NDUSTRIAL I NFORMATICS, the IEEE T RANSACTIONS ON C YBERNETICS, and Information Sciences.

Zidong Wang (SM’03–F’14) was born in Jiangsu, China, in 1966. He received the B.Sc. degree in mathematics from Suzhou University, Suzhou, China, in 1986, and the M.Sc. degree in applied mathematics and the Ph.D. degree in electrical engineering from the Nanjing University of Science and Technology, Nanjing, China, in 1990 and 1994, respectively. He is currently a Professor of dynamical systems and computing with the Department of Information Systems and Computing, Brunel University London, Uxbridge, U.K. From 1990 to 2002, he held teaching and research appointments in universities in China, Germany, and the U.K. He has published over 300 papers in refereed international journals. His current research interests include dynamical systems, signal processing, bioinformatics, and control theory and applications. Prof. Wang was a recipient of the Alexander von Humboldt Research Fellowship of Germany, the JSPS Research Fellowship of Japan, and the William Mong Visiting Research Fellowship of Hong Kong. He serves (or has served) as the Editor-in-Chief of Neurocomputing, and an Associate Editor for 12 international journals, including the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY, the IEEE T RANSACTIONS ON N EURAL N ETWORKS, the IEEE T RANSACTIONS ON S IGNAL P ROCESSING, and the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART C: A PPLICATIONS AND R EVIEWS. He is a fellow of the Royal Statistical Society, and a member of program committee for several international conferences.