On Stochastic Sensor Network Scheduling for Multiple Processes

Transcription

1 On Stochastic Sensor Network Scheduing for Mutipe Processes Duo Han, Junfeng Wu, Yiin Mo, Lihua Xie arxiv:6.08v [cs.sy 8 Dec 06 Abstract We consider the probem of mutipe sensor scheduing for remote state estimation of mutipe process over a shared ink. In this probem, a set of sensors monitor mutuay independent dynamica systems in parae but ony one sensor can access the shared channe at each time to transmit the data packet to the estimator. We propose a stochastic eventbased sensor scheduing in which each sensor makes transmission decisions based on both channe accessibiity and distributed event-triggering conditions. The corresponding minimum mean squared error MMSE) estimator is expicity given. Considering information patterns accessed by sensor scheduers, time-based ones can be treated as a specia case of the proposed one. By utiizing reatime information, the proposed schedue outperforms the time-based ones in terms of the estimation quaity. Resorting to soving an Markov decision process MDP) probem with average cost criterion, we can find optima parameters for the proposed schedue. As for practica use, a greedy agorithm is devised for parameter design, which has rather ow computationa compexity. We aso provide a method to quantify the performance gap between the schedue optimized via MDP and any other schedues. I. INTRODUCTION Sensor scheduing is crucia for remote state estimation in cyber-physica systems CPS). Typicay, the main task for sensor scheduing is to improve estimation quaity subject to communication constraints [ [8. In this paper we focus on the probem of bandwidth constrained sensor scheduing for state estimation. To be specific, the distributed sensors in charge of different monitoring tasks are sharing a common channe for data transmission. At each time instant, ony one sensor can access the communication channe. We consider a sensor schedue deciding which sensor is abe to access the channe at each time in order to optimize the overa state estimation quaity. The probem of singe sensor scheduing subject to imited transmission rate has been studied in [, [5, [7, [9. Generay speaking, if a sensor uses ony prior knowedge of systems, an optima poicy for transmission is very ikey to transmit the data packets periodicay [3, [5, [9. This kind of poicies are referred to as time-based schedues. When the This research is partiay funded by the Repubic of Singapore Nationa Research Foundation NRF) through a grant to the Berkeey Education Aiance for Research in Singapore BEARS) for the Singapore-Berkeey Buiding Efficiency and Sustainabiity in the Tropics SinBerBEST) Program. BEARS has been estabished by the University of Caifornia, Berkeey as a center for inteectua exceence in research and education in Singapore. : Schoo of Eectrica and Eectronic Engineering, Nanyang Technoogica University, Singapore, E-mai: dhanaa,ymo,ehxie@ntu.edu.sg. : ACCESS Linnaeus Center, Schoo of Eectrica Engineering, KTH Roya Institute of Technoogy, Stockhom, Sweden. E-mai: junfengw@kth.se. sensor makes transmission decisions based on reatime innovations, which generay has better estimation performance then periodic ones if propery designed but induces higher computationa compexity, the transmission is ikey to be random [, [7, [8. This kind of poicies are referred as event-based schedues. Weerakkody et a. [0 extended [7 by considering mutipe sensors monitoring one process, but without any constraint on channe accessibiity. Compared with the singe system case, not enough research efforts have been put in the case of different sensors monitoring different systems, which is widey encountered in practice. A simpe motivating exampe is the underground petroeum storage using WireessHART technoogy in [. Underground sat caverns are often used for crude oi storage. Brine and crude oi fowing in both directions are measured by sensors and reported to the contro center through a gateway device. The contro center aims to maintain a certain pressure inside the caverns. Obviousy, each sensor competes with others for the gateway access to achieve their own goa. Therefore, a schedue for optimizing an objective function taking the benefits of a sensors into account is desirabe. A few preiminary works have considered this case. Savage and La Scaa [ studied mutipe sensor scheduing for a set of scaar Gauss-Markov systems over a finite horizon. They considered the optimaity in terms of termina estimation error covariance. An optima poicy is to schedue a the transmissions in the end of the horizon. However, the termina error covariance metric is ony suitabe for finite horizon scheduing probems. To study an infinite horizon scheduing, Shi et a. [3 adopted a metric of averaged estimation error, studied two muti-dimensiona systems over an infinite horizon and proposed an expicit optima periodic sensor schedue. The concusion in [3 benefits from the mutua excusiveness of two sensors. When the number of sensors is beyond three, cosed-form optima scheduing is formidabe to obtain. Another simiar probem where one sensor is schedued to monitor mutipe processes was studied in [4. The probem is same with the mutipe sensor scheduing probem in nature. The authors proposed agorithms to search schedues such that the error covariance of each system is bounded by some constant matrix. Unike the singe-sensor case, muti-sensor event-based scheduing is not fuy investigated to push the imit of performance. A preiminary work by Han et a. [5 improved [3 by designing an event-based schedue which depends on the importance of a sensor s measurements. In this note we consider an event-based sensor scheduing design for a set of sensors. At each time every sensor makes a transmission decision based on both channe accessibiity

2 by sensing the carrier and the importance of its own data by checking some criteria. Ony when the channe is accessibe and the data is considered as being sufficienty important, the sensor wi transmit the data packet. The sensor scheduing studied in this note is restricted to a kind of stochastic eventbased sensor scheduing since it can maintain Gaussianity of estimation process and bypass the noninear probem induced by the event-triggering mechanism, e.g., [, [8. Compared with time-based schedues, the proposed event-based schedue dynamicay assigns the communication resource according to the needs of sensors. From operating principe point of view, the time- and event-based sensor schedues are anaogous to time division mutipe access TDMA) and carrier sense mutipe access with coision avoidance CSMA/CA) medium access contro MAC) in communication networks, respectivey. The main contributions are summarized as foows: ) We first propose an event-based scheduing infrastructure with network cooperation and sef event-triggering mechanism. Any time-based schedues can be treated as a specia case of the framework. ) Based on the underying schedue, we derive the minimum mean squared error MMSE) estimator and anayze the communication behaviour of each sensor. 3) We mode an Markov decision process MDP) probem with average cost criterion to seek the optima parameters for the cass of proposed stochastic schedues. 4) For computationa simpicity, we aso propose a greedy parameter design method. Moreover, we are abe to anayticay quantify the performance any schedue by showing a ower bound of the optima cost. Notations: Z + is the set of positive integers. S n + S n ++) is the set of n by n positive semi-definite definite) matrices. Tr[ denotes the trace of a matrix. X / denotes the square root of X S n +. For functions f with appropriate domains, f 0 X) := X, and f t X) := f f t X) ). x[i represents the ith entry of the vector x. For a matrix X, Xi, j) represents the entry on the ith row and jth coumn of X. For x R, x is the argest integer that is not arger than x and x is the smaest one that is not ess than x. Denote the set or sequence {x i } k i=j = {x j, x j+,..., x k }, j k and if j > k then {x i } k i=j =. A. System Mode II. PROBLEM SETUP Consider the foowing n mutuay independent inear timeinvariant LTI) systems, which are monitored by n sensors respectivey: x i k + ) = A i x i k) + ω i k), y i k) = C i x i k) + ν i k), i S a) b) where S := {,..., n} is the index set of the n processes or sensors, x i k) R ni is the state of the ith process at time k and y i k) R mi is the measurement obtained by the ith sensor at time k. For shorthand denote s i as the ith sensor. The system noise ω i k) s, the measurement noise ν i k) s and the initia system state x i 0) are mutuay independent zeromean Gaussian random variabes with covariances Q i > 0, R i > 0, and Π i 0, respectivey. Assume that A i, C i ) is detectabe. Furthermore, we assume that A i is unstabe ike [3 for two reasons: ) unstabe systems bring out stabiity issues rather than stabe ones do; ) most process estimation tasks wi become unpredictabe if eft unattended too ong. Each sensor measures its corresponding system state and generates a oca estimate first. More specificay, at each time s i ocay runs a Kaman fiter to compute the minimum mean square error MMSE) estimate of x i k), i.e., ˆx i,oca k) := E[x i k) {y i )} k =0. The corresponding estimation error and error covariance are ɛ i,oca k) := x i k) ˆx i,oca k), ) [ P i,oca k) := E ɛ i,oca k)ɛ i,oca k)) {y i )} k =0. 3) The quantities ˆx i,oca k) and P i,oca k) can be obtained through a standard Kaman fiter [6. It is we known that P i,oca k) converges exponentiay fast to P i S ni + which is the soution of a discrete agebraic Riccati equation [6. Since we consider a probem over the infinite horizon in the seque, we ignore the transient behaviour and make a standing assumption that P i,oca k) = P i, k Z +. For the purpose of state estimation, a sensors transmit their own oca estimates to the remote estimator over a shared channe. In this work we consider a bandwidth-imited sensor network which aows one sensor to access the channe each time instant. In other words, ony a singe sensor can send its estimate over the shared channe at each time instant whie the others sti keep their oca copies. B. Scheduing Probem The probem of interest is how to efficienty schedue the transmission of sensors at each time in terms of some performance metric. We first define some mathematica notations. Denote the transmission indicator for s i at time k as a binary variabe γ i k), i.e., γ i k) = means s i sends data and vice versa. A sensor schedue θ is defined as a sequence of γ i k), i.e., θ := {γ i k)},k Z+. Moreover, we define a coection of the information sets I i,θ k) s for i S at the estimator side as I i,θ k) := {γ i 0)ˆx i,oca 0),..., γ i k)ˆx i,oca k)}. 4) Due to the mutua independency among the systems, the estimator computes the estimate of x i k) as ˆx i k) := E[x i k) I i,θ k) with the corresponding error covariance P i k) := E[x i k) ˆx i k))x i k) ˆx i k)) I i,θ k). Note that γ i k), ˆx i k) and P i k) are functions of θ though we do not expicity show that in the notations. Simiar to [3, we use the overa average expected estimation error covariance as a performance metric, i.e., Jθ) := im sup T T Tr [ T E [P i k)). 5) k=0 Let J i θ) denote the individua expected estimation error covariance correspondingy. The probem of interest is formay

3 3 k k+ Awaiting Phase Transmission Phase Sensor Sensor Sensor 3 Fig. : Iustration of the awaiting phase and transmission phase for three sensors between time k and k +. stated as: minimize θ Jθ) subject to γ i k) =. 6) Time-based schedues refer to that θ is a function of time ony. In that case, γ i k) is independent of ˆx i,oca k). The optima scheduing soution turns out to be a periodic TDMAike schedue ike [3. The main advantage of this type of schedue is its simpicity. From 4), however, we can see that when γ i k) = 0, I i,θ k) contains no information on ˆx i,oca k) and thus the estimator gains nothing about the system state from s i at time k when γ i k) = 0. To outperform the time-based schedue, we study a type of event-based schedue meaning that γ i k) is a function of the rea-time estimates or measurements which contain information on the underying system state. Both the sensor and the estimator know how γ i k) at each time is determined based on some triggering rues. Hence, even with γ i k) = 0, the estimator can sti infer some information about the system. Therefore, the event-triggering mechanism improves the estimation performance by providing extra information. C. Stochastic Event-based Sensor Schedue Mimicking the protoco of Carrier Sense Mutipe Access with Coision Avoidance CSMA/CA) widey used in wireess sensor networks, we propose a distributed event-based sensor schedue to enhance the overa estimation performance compared with the time-based schedues. Now we introduce the infrastructure. There are two phases for each sensor in each transmission frame: the awaiting phase and the transmission phase. Since the packet transmission time is often much arger than the uncertainties in communication networks such as the propagation deays, it is safe to assume the transmission time is much arger than the awaiting time during each epoch. Any sensor istens to the channe carrier for a short period before it sends anything. If the channe is occupied during the awaiting phase, then the sensor hods its data; otherwise, it sends the packet to the estimator in the transmission phase. The ends of awaiting phase of a sensors are made different in order to avoid coision see Fig. ). In other words, the sensors form a queue qk) with different queueing time in the awaiting phase. For exampe, the queue s,..., s n ) means s i has higher priority to access the channe than s i+. The queue qk) can be time varying or invariant and denote Q to be the set of possibe priority queues, i.e., a permutations of s,..., s n. The idea of event-based scheduing behind is as foows. If the data of s i contains itte innovative information, which can be checked by some criteria introduced ater, then s i wi be unikey to transmit the data and s i+ in the queue can take the chance to use the channe. The queue impemented on the top of carrier sensing is usefu here to resove the confict when more than one sensor want to transmit. In other words, the transmission decision of s i depends on the importance of oca data and the channe accessibiity. Let us first define two binary indicators before formay proposing the schedue: the data importance indicator η i k) and the channe accessibiity indicator µ i k). Various eventtriggering criteria for determining the importance of a singe measurement have been investigated in [, [7, [0. In the spirit of [7, [0, we design a stochastic event-triggering rue to check the importance of the data. The dominant feature of the stochastic event-triggering mechanism is to maintain the Gaussianity of the estimation process and bypass the noninear probem such as [. We define η i k) based on the difference between the MMSE estimate under θ at oca sensors and the prediction at the estimator side, i.e., ɛ i k) := ˆx i,oca k) A iˆx i k ), and the corresponding covariance as Σ i k) := E [ ɛ i k)) ɛ i k) I i,θ k ). 7) Define the mapping φ i z, Π) : R ni S ni + [0, as φ i z, Π) := exp ) z Π z. Now we define the data importance indicator η i k) as foows: {, if ξi k) > φ η i k) := i ɛ i k), α i k)σ i k)), 8) 0, otherwise. where ξ i k) U[0, is an i.i.d. auxiiary random variabe, and α i k) is a tunabe parameter which refects the importance of the data packet. For smaer α i k), η i k) is more ikey to be. Namey, for a more important sensor, we woud ike to pick a smaer α i k) to ensure it can transmit more data. The tuning parameter {α i k) 0},k Z+ and the queue parameter {qk)} k Z+, depending on the historica arriva pattern of a sensors, are designed and broadcasted by the centra estimator. The estimator usuay has advantageous resources compared with sensors, i.e., stronger computation capabiity and arger energy storage. The design procedure wi be discussed in Section IV. Let µ i k) denote the channe accessibiity indicator, i.e., { i, if µ i k) := j= γ jk) = 0. 9) 0, otherwise Now we are ready to propose an event-based schedue θ + := {γ i k)},k Z+ and denote Θ + to be the set of We abuse the notation X to represent the inverse of X S n i ++ or the Moore-Penrose pseudo-inverse of a singuar X S n i +. Simiary, we use detx) to denote the determinant of a square matrix X or its pseudodeterminant if X is singuar. For concise presentation, by reabeing the sensors we assume the queue qk) = s,..., s n) by defaut. More discussion on the design of the optima queue is shown in Section IV.

4 4 a event-based schedues in 0) subject to a possibe {α i k)},k Z+ and {qk)} k Z+ : { µi k)η γ i k) := i k), if i n µ i k), otherwise. 0) Notice that γ i k) depends on ɛ i k) and Σ i k), and ɛ i k + ) and Σ i k+) depends on γ i k). Thus θ + cannot be determined offine. Aso note that if s n with the ast priority detects that the channe is ide, it sends the data without checking 8). The condition 8) impies that if the prediction error is sma, η i k) is very ikey to be 0 which prevents unecessary transmission. In the subsequent sections, we anayze the estimation performance using such a schedue and design the parameters {α i k) 0},k Z+ and the queue parameters {qk)} k Z+ optimay by formuating optimization probems. III. OPTIMAL FILTERING After proposing the stochastic event-based schedue θ +, we are keen in deriving the MMSE estimator under θ + and quantifying the estimation performance of θ +. Moreover, from the viewpoint of communication, we present the transmission probabiity of each sensor. To faciitate derivations, we define the foowing operators for X S ni + and α R: h i X) := A i XA i + Q i, ) g i X, α) := α + α A ixa i + h i P i ) P i ), ) t i X, α) := + α P i + α + α h ix). 3) The foowing emma on the properties of the innovation and estimation error is usefu for proving the main resut whose partia proof can be found in [7. Let the incrementa innovation for s i be denoted as δ i k) := ˆx i,oca k) A iˆx i,oca k ). 4) Lemma Given ɛ i,oca k), δ i k) defined in ) and 4), the foowing statistica properties hod: i) δ i k) is zero mean Gaussian distributed, i.e., δ i k) N 0, h i P i ) P i ). ii) E [ δ i k)δ i j)) = 0 for any k j. iii) E [ ɛ i,oca k)δ i k 0 )) = 0 for any k 0 k. iv) E [ˆx i,oca k)ɛ i,oca k )) = 0 for any k k and E [ˆx i,oca k)δ i k )) = 0 for any k > k. The foowing emma on Bayes inference is presented before showing the main resut. Lemma Suppose z R ni is a Gaussian random variabe with z N 0, Z) with respect to the Lebesgue measure on Z := {Z / x : x R ni } and ξ is uniformy distributed over [0,. The foowing statements hod: i) The occurring probabiity of the foowing event is Pr {ξ φ i z, Π)} = deti + ZΠ ) /. ii) The conditiona pdf of z is fz ξ φ i z, Π)) N 0, Z + Π ) ). Proof: First we have [ Pr {ξ φ i z, Π)} = E exp ) z Π z π) r/ = exp ) detz) / z Zz exp ) z Π z dz Z = deti + ZΠ ), 5) where r = rankz). Statement ii) is a direct resut of the Bayes theorem. For notationa simpification, denote γ i k) := γ i k), µ i k) := µ i k). Furthermore, denote the eave duration of s i as τ i k) := min{k k 0 : γ i k 0 ) =, k 0 k}. Now we are ready to present the main resut. Theorem Under the proposed schedue θ + given in 0), the MMSE state estimate for each process is given by { ˆxi,oca k), if γ ˆx i k) = i k) = A iˆx i k ), if γ i k) = 0, 6) and the corresponding estimation error covariance is P i, if γ i k) = P i k) = h i P i k )), if µ i k) = 0. 7) t i P i k ), α i k)), otherwise Moreover, Σ i k) in 7) is given by Σ i k) = h i P i k )) P i. 8) Proof: When γ i k) =, from [6 we know that E [x i k) ˆx i,oca k), I i,θ k ) = ˆx i,oca k), and P i k) = P i. When µ i k) = 0 γ i k) must be 0) which impies the channe is occupied by another sensor, the estimator can ony do prediction on the estimate of x i k), i.e. ˆx i k) = A iˆx i k ), P i k) = h i P i k )). 9) The two cases above are easy to anayze. Next we consider the remaining case, i.e., when µ i k) γ i k) =. The foowing equation is easy to verify and usefu for subsequent derivations. x i k) = A iˆx i k ) + ɛ i,oca k) + ɛ i k). 0) Without oss of generaity, assume that γ i k) = or µ i j) = 0 for any j [k τ i k), k, and µ i k+) γ i k+) = occurs at time k +. Moreover, the foowing recursive equation aways hods: ɛ i k) = A i ɛ i k ) + δ i k). ) Aso from 9) and 0) we have x i k + ) = A τik+) i ˆx i,oca k τ i k + )) + ɛ i,oca k + ) + ɛ i k + ) = A τik+) i ˆx i,oca k τ i k + )) + ɛ i,oca k + ) + τik) j=0 Aj i δ ik + j), where the ast two terms on the RHS are mutuay independent in view of Lemma iii). Due to Lemma i) and the fact [6 that fɛ i,oca k + ) I i,θ k + )) = fɛ i,oca k + )) N 0, P i ), we have E [x i k + ) I i,θ k + ) = A iˆx i k). Therefore, from 0) we know that for j [k τ i k), k + the equaity hods: P i k) = P i + Λ i k), )

5 5 where Λ i k) := E [ ɛ i k)) ɛ i k) I i,θ k). Hence we have Λ i k) = P i k) P i. Then together with ) we can concude that fɛ i k+) I i,θ k)) N 0, A i P i k) P i )A i +h i P i ) P i ) which proves 8). Then from Lemma, we have fɛ i k + ) I i,θ k + )) N 0, g i P i k) P i, α i k))). 3) Thus, from ) and 3) we have other words, compared with the time-based schedues, even if s i does not transmit anything, the estimator is sti ikey to obtain an estimate better than a pure prediction. Denote as r i k) the rank of Σ i k) which can be computed offine from 8). The probabiity of kinds of events during transmission can be computed as foows. For concise notation, denote α i k) := α i k)/ + α i k)) 5) β i k) := α i k) rik)/, βi k) := β i k). 6) fx i k + ) I i,θ k + )) N A iˆx i k), P i + g i P i k) P i, α i k))). 4) Notice that ˆx i k+) is actuay A τik+) i ˆx i,oca k τ i k+)) which is the same with the predicted estimate but with a smaer error covariance. Consequenty, no matter what µ i k + ) and γ i k + ) are at time k +, the mutua independence of the three terms in 0) and the recursion in ) hod. Therefore, the conditiona pdf of ɛ i k + ) can be computed in a simiar fashion as 3), which is zero mean Gaussian distributed. Thus ˆx i k + ) is sti a predicted estimate, i.e., A iˆx i k+), with the corresponding error covariance dependent on µ i k + ) and γ i k + ). Recursivey, we can verify 6) and 7) which competes the proof. The resut in 7) shows that if s i is ide due to other competitors, i.e., µ i k) = 0, then the estimator simpy runs a prediction. If s i decides to hod its packet even if it has the access to the channe, i.e., µ i k) γ i k) =, the estimator updates the estimate using the information encoded by 8). Using the foowing emma, we can see the estimation performance under µ i k) γ i k) = is ower bounded by that under γ i k) = and upper bounded by that under µ i k) = 0. Lemma 3 The foowing statements hod for any i S: i). For any, Z + with <, h i P i) h i P i). ii). For any Z +, Tr [ [ P i < Tr hi P i ) Tr [ h i P i). iii). For any X which is a convex combination of {h j i [ P i)} j=0 and any [ α 0, X t ix, α) and Tr t i X, α) Tr t i, X, α) for <. Proof: From [8, Lemma A., we can prove that P i h i P i ). Since h i ) is affine, we can take h i ) on both sides and draw the concusion i). Next we show the strictiveness in ii), i.e., Tr [ [ P i < Tr hi P i ). Assuming P i = h i P i ), then we can find that Q = 0 which contradicts the assumption Q > 0, or C i = 0 which contradicts the fact A i, C i ) is detectabe. Thus P i h i P i ). From i) we immediatey have Tr [ [ P i Tr hi P i ) Tr [ h i P i). Moreover, we know that { P i h i P i ) Tr [ [ P P i h i P i ), P i S ni i Tr hi P i ). + The ast statement is a direct resut of iii). From Lemma 3iii), for any reaization of P i k ) we know that P i t i P i k ), α i k)) h i P i k )). In Theorem The foowing statements hod true: Pr {η i k) = } = β i k), Pr {µ i k) = Ξ i k)} = Pr {γ i k) = Ξ i k)} = where Ξ i k) := {η i k),..., η k)}. {, if i = i j= β jk), if i >, { n j= β jk), if i = n β i k) i j= β jk), if i < n, Proof: From Lemma i), we have Pr {ξ i k) < φ i ɛ i k), α i k)σ i k))} = [α i k)/ + α i k)) rik)/. For ii), it is easy to see that Pr {µ k) = Ξ i k)} = 0 for s. From 9) and 0), we know that Pr {µ i k) = Ξ i k)} = i j= Pr {γ jk) = Ξ i k)} and Pr {γ i k) = Ξ i k)} = Pr {η i k) = 0} Ξ i k)) Pr {µ i k) = Ξ i k)}. After some cacuation, omitted for ongevity, we obtain ii) and iii). Remark Since Σ i k) in 8) can be written as j κ jh j i P i) P i where j κ j =, we can guarantee Σ i k) to be fu rank as ong as h j i P i) P i > 0, j hods. A sufficient condition is that A i has fu rank. Since we have known how the error covariance of each process is updated under different conditions of {γ i k), µ i k), η i k)} and the probabiity of each outcome in Theorem, we can next investigate how to optimay set the event-triggers for each sensor. IV. PARAMETER DESIGN In this section, we aim to design the optima parameters for the cass of stochastic event-based schedues based on the resuts in Section III. It turns out that finding the optima schedue is computationay chaenging. Thus, we propose a greedy agorithm to achieve suboptima schedues. Moreover, we anayticay quantify the performance gap between the optima and suboptima schedues. We first show that the proposed schedue performs at east as good as any time-based schedue. Theorem 3 There exists a nonempty subset of Θ + from which θ + is at east as good as any time-based schedue θ does, i.e., {θ + Θ + : Jθ + ) Jθ ), θ }.

6 6 The proof is straightforward since any time-based schedue is a specia case of θ + in 0). For exampe, if s i is deterministicay schedued to transmit at time k, then one can set α i k) = 0 and α j k) = for j i. Thus by optimizing the parameters in the schedue θ +, one can aways find a schedue in Θ + performs as we as any time-based schedue can. A. Optima Parameter Design by Soving an MDP Probem Now we are keen on the optima θ + Θ + of minimizing 5). By formuating an Markov decision process MDP) probem with average cost criterion, we can design the optima time-varying parameters {α i k)},k Z+ and the queue {qk)} k Z+. First define the state space V to be the set of a possibe P k),..., P n k)), where P i k) beongs to C i which is the set of any convex combination of h j i P i), j, i.e., C i := { j λ jh j i P i): j λ j = }. The action is a tupe of b := u, q) U Q where U := {u R n : 0 u[i, i S}. The compact action space is denoted as B, which is identica for each state in V. Denote the set K := {v, b) : v V, b B} which is a Bore space. The transition aw Q v, b) with v, b) K is a stochastic kerne on V given K, which can be obtained from Theorem and Theorem. The cost function cv, b) is thus defined as the sum of the trace of each matrix in v. We thus mode an MDP denoted as Ω := V, B, Q, ), c, )). A decision at time k is a mapping dk) : V B. A poicy ζ for Ω is a sequence of decision rues. We define the average expected cost of the poicy ζ per unit time by g ζ v) = im sup T T E ζ,v [ T rvk), bk)), 7) k=0 where v V is the initia state, and the expectation is taken based on the stochastic process {vk), bk)} uniquey determined by the poicy ζ and v. The target for an MDP probem with average cost criterion is to search an optima stationary poicy ζ for Ω such that the average cost is minimized, i.e., g ζ v) g ζ v), v V. It is easy to see that finding the optima poicy ζ for Ω is equivaent to searching the optima θ + Θ + of minimizing 5). Numerous iterature have studied the optimaity conditions for a poicy of an MDP probem with average cost criterion in Bore spaces such as [9. Though some researchers have attempted to sove the MDP probem in Bore spaces ike [0, obtaining the optima cost and the optima poicy in a computationay efficient fashion, however, is generay chanenging. The technique of discretizing the state space and rebuiding the stochastic kerne from the origina mode is a popuar approach to approximatey sove the MDP with Bore spaces. Unfortunatey, for soving the discrete MDP, the computationa burden is high. For exampe, the cassica poicy iteration agorithm for soving an MDP requires the computationa effort of V d B d + 3 V d mutipications/iterations [?, Chapter 8, where V d represents the cardinaity of the discretized state set and B d the cardinaity of the discretized action set. The number of discretized state space is growing at east exponentiay with the number of the sensors in order to keep the same grid size. B. Greedy Agorithm The optima stochastic schedue is formidabe to obtain in practice. Instead, we propose a suboptima greedy schedue which minimizes the next-step trace of tota expected error covariance. At each time k, the estimator broadcasts the priority queue qk) and the event-triggering parameters {α i k)} to a sensors which minimizes the one-step cost function: [ J greedy k) = Tr E [P i k). 8) Unike searching the optima priority queue for the optima schedue via enumeration, we find a simpe rue for the optima priority queue for the greedy schedue if r i k) = r j k), i j. An exampe is that the system matrices of severa identica vehices have fu rank, i.e., ranka i ) = ranka j ) = n i = n j, i, j S. Theorem 4 If r i k) = r j k) for any i, j S, the queue qk) = {s ϕ,..., s ϕm,..., s ϕn }, where ϕ m S, is optima for the greedy schedue if and ony if for each m < n the foowing inequaity hods: Tr [ h ϕm P ϕm k )) P ϕm Tr [ h ϕm+ P ϕm+ k )) P ϕm+. 9) Proof: Note that the condition 9), which appies to each adjacent pair in the queue, is a tota order on the set S. Therefore, if we prove ony if, then if is automaticay proved. Next we wi prove ony if by contradiction. Assume q + k) to be optima and there exists some adjacent pair in q + k), without oss of generaity, i.e., s i, s i+ ) with Tr [ h i P i k )) P i < Tr [ hi+ P i+ k )) P i+ 30) If we can show that by swapping s i and s i+ in q + k) we can construct a better schedue q k) than q + k), we wi finish the proof by contradiction. We wi discuss two cases: I) s i, s i+ ) is not the ast pair in q + k); II) s i, s i+ ) is the ast pair. Denote the quantities β i k),β i+ k), α i k), α i+ k) defined in 5) and 6) under q + k) q k)) as β + i,β+ i+, α+ i, α+ i+ β i,β i+, α i, α i+ ). We wi omit the time index k or k if it is cear from the context. Case I. Note that the next-step error covariance of sensors before s i in the queue are not affected by the swapping prodedure. In order not to disturb the next-step error covariance of sensors behind s 3, based on Theorem iii) we et β + i β+ i+ = β i β i+ := c where c [0, is a constant such that the expected error covariances of a sensors except s i, s i+ under q +, q are the same. From Theorem we have

7 7 J greedy under q +, q are given as J + greedy = Tr P i + h i+ P i+ ) + α + i ) + [ α i h ip i ) P i ) ) +c + α + + i ) )h i+ P i+ ) P i+ ) J greedy = Tr P i+ + h i P i ) + α i+ ) +c + α + i+ ) )h i P i ) P i ) 3) [ α i+ h i+p i+ ) P i+ ) ). 3) Denote α + i, α i+) to be the minimizer of 3) and 3) and J +, J to be the corresponding costs. If we can show that J = J + J = Tr [ h i P i ) P i [λ + c + α + i ) α + i ) α i+) ) + ) + c + α i+) ) α i+) ) α + i ) + ) 0 under 30), then the optimaity of q + is vioated. Next we wi show that J 0 for λ < and J 0 for λ > hods. First we need to know α + i and α i+). By taking the first derivative of J + greedy to be 0, we have an equation Φx) := + )x +4 λx + λc + = 0, x > 0 33) where λ := Tr[hi+Pi+) P i+ Tr[h ip i) P i derivatives of Φx), we know that x = λ +4. By taking first and second is a goba minimum on 0, + ), Φx) is decreasing on 0, λ +4 ) and Φx) is increasing on λ +4, + ). Since Φ0) < 0, there is ony one root for Φx) = 0, denoted by x. Therefore, we can concude that J + greedy is decreasing for α+ i 0, x and increasing for α + i x, + ). From β + i β+ i+ = β i β i+ = c [0, and β + i, β+ i+, β i, β i+ [0,, we have α+ i [c/,. To sum up, we have α + i = minmaxx, c / ), ). Simiary, α i+) = minmaxx +, c / ), ) where x + is the positive soution to the foowing equation: + )x +4 λx + λc + = 0, x > 0 34) where λ = λ. From 33) we have dx dλ = + c + x ) + + )x ) x ) +4 + c + ) 0. 35) λ Since x is a nondecreasing function of λ, there is λ = + )/ + c + ) > such that α + i = for a λ > λ. When λ > λ, i.e., α + i =, based on 3), 3), 34) we can derive that J = Tr [ + h P ) P α i+) ) λ α i+) ) 0. Simiary, when λ < /λ, J 0. Now we consider the case of /λ λ λ. Denote J gλ) = and its first and second derivatives are Tr[h ip i) P i g λ) and g λ). Then we have g λ) = + c + α + i ) α + i ) α g λ) = + λ i+) ) + λ α i+) ) d α i+) d/λ) α+ i ) 36) ) d α + i dλ 37) If we can prove that gλ) is convex for /λ λ <, i.e., g λ) 0. Since g/λ ) < 0 and g) = 0, we have that gλ) < 0 for /λ λ <. Moreover, if we can prove that g/λ) = gλ)/λ, we have gλ) > 0 for < λ < λ. Namey, we can show that J = 0 has a unique root at λ =. First we prove the convexity. From 33) we have + dλ = c + α + i ) + + ) α + i ) α + i ) +4 d α + i d α i+) d/λ) = Then from 37) we have + c + + c + α i+) ) + ) λ + ) α i+) ) α i+) ) +4 + c + /λ g λ) = κ[c + α + i α i+) ) + α + i λ α i+) ) 38) + c +4 λ α i+) ) + α + i ) + ) 39) + α + i α i+) ) +4 α i+) ) λ α + i ) ) 40) + c + λ α i+) )+ α + i )+ ), 4) where κ is some positive term. Now we wi show that the terms in 38)-4) are a positive. We first prove α + i λ α i+) ) in 38) is positive. Since there is ony one root for 33), the fact that α + i λ α i+) > 0 is equivaent to that Φλ α i+) ) < 0. We have that Φλ α i+) ) = + )λ α i+) ) +4 λλ α i+) ) + λc + = λ α i+) ) + + λ λ) + ) [ α i+) λ + ) We can easiy show that Λλ) = λ + λ) + ) λ + ) λ + λ)+). ) is a decreasing function of λ for λ > 0. Since λ <, we have the minimum of Λλ) at λ =. By using L Hospita s rue, we have im λ =. Since α i+), we have Φλ α i+) ) < 0 and thus α + i λ α i+) ) in 38) is positive. Now we prove λ α i+) ) + α + i ) + ) in 39) is positive. From 35) and λ <, we have that α i+) > α+ i. From 33) and 34), we have + )λ α i+) ) +4 α + i ) +4 ) = λ α i+) ) + α + i ) + ) > 0.

8 8 Due to λ α i+) ) +4 > α + i ) +4 > 0 and α + i > λ α i+), we have λ α i+) ) + > α + i ) +. It is easy to see that α i+) ) > λ α + i ). Due to λ α i+) ) +4 > α + i ) +4 > 0 and α i+) > α+ i and λ <, we have λ α i+) )+ > α + i )+. Thus we can show that g λ) 0 and gλ) is convex for /λ λ <. From 3)-34), it is easy to show that g/λ) = gλ)/λ. Hence we compete the proof of the fact that J = 0 has a unique root at λ =. Case II. The ast pair s n, s n ) needs specia attention because the ast sensor does not check its own data importance. Since the transmission of the ast sensor ony depends on its preceding sensor, we have 3)3) rewritten into J + greedy = Tr P n + h n P n ) + α + n ) [ α + n h n P n ) P n ) h n P n ) P n ) ) 4) J greedy = Tr P n + h n P n ) + α n ) [ α n h n P n ) P n ) h n P n ) P n ) ) 43) In this case, we can see that c = 0 compared with 3)3). We can find the minimizer of J + greedy by taking the first derivative of J + greedy and we have the minimizer α+ n ) as foows: + { α n ) = if + λ >, α + n ) = + λ if 0 < + λ, where λ = Tr[h np n) P n Tr[h n P n ) P n minimizer α n for J greedy as foows: { α n = if α n = + Simiary, we have the + λ >, λ if 0 < + λ. Since α + n ), α n depend on the vaue of λ, we discuss J for different ranges of λ ike we did in Case I. When + λ +, we have J = Tr [ h n P n ) P n λ ) + λ + ) ) λ With some cacuations, we know that there is ony one soution to d J/dλ = 0. Moreover, we have d J/dλ < 0 at λ = + + and. Then we can concude that there are two roots or no root for d J/dλ = 0. If there are two roots r and r, it must be true that d J/dλ < 0 for λ < r or λ > r, and d J/dλ > 0 for r < λ < r. Since J < 0 at λ = + + and J > 0 at, there must be odd number of roots for J = 0. If there is no root for d J dλ = 0, then J is decreasing function which is impossibe. So there are two roots for d J dλ = 0. From the Roe s theorem, there is ony one root for J = 0. By inspection we have one root λ = and it is unique based on our previous reasoning. Therefore, we have that { J > 0 if + λ >, J 0 if λ > +. So if J > 0, we choose,. Otherwise, we choose,. When λ +, we have J = Tr [ [ ) / h n P n ) P n + + λ > 0. When λ +, we have J = Tr [ h n P n ) P n [ + > 0. ) + λ Thus in summary, we can concude that J > 0 if λ > and J 0 if λ. Hence the concusion that J < 0 for λ < and J > 0 for λ > hods for both Case I and Case II. Since the above proof appies to any adjacent pair in any feasibe schedue no matter it is optima or not), we can aways swap the pair ike s i, s i+ to improve the performance if a condition ike 30) hods. Thus we prove the ony if part by contradiction which competes the proof. After determining the optima qk), we can optimize J greedy k) in 8) in terms of {α i k)}. Based on Theorem and, we can expicity obtain a cost function which turns out to be a signomia of {α i k)}. The signomia programming SP) probem is widey studied in communication society [. Though it is not convex, there are many efficient agorithms for oca optimum or goba optimum see, e.g., [, [). The computationa efforts required for different signomia programming probems are iustrated with exampes in [. C. Optimaity Gap Notice that the performance gap between the suboptima schedues and the optima θ + Θ + is upperbounded by the gap between the suboptima performance and a ower bound for the optima performance. Next we discuss the upper bound of such performance gap to quantify the performance of any suboptima stochastic schedue. First we construct an artificia schedue whose performance is better than the optima θ + Θ +. We reax the origina constraint i γ ik) = by requiring the sum of the rates of a sensors to be ess than. In other words, we aow that channe can be used by mutipe sensors but their sum communication rate must be ess than. Denote Θ as the set of a schedues in 0) with forcing µ i k) to be for i n and a k. In other words, any absence of ˆx i,oca k) impies 8) for i n. Moreover, we aso impement the event-triggering mechanism for s n. By soving the foowing optimization probem: Probem minimize Jθ), subject to θ Θ E [γ i k), 44) it is not hard to see the soution to Probem is better than the optima θ + Θ + since any absence of each sensor now

9 9 conveys some information to the estimator. Whie for θ + Θ + the information is ost for the sensors whose priority is behind the designated one. Aso note that the queue is useess for θ Θ. Now α i k) soey depends on τ i k) and so does β i k). From 7) we know that J i θ) is determined by the underying stochastic process {τ i k)} k=. With a itte abuse of notations, we use β j i to denote the probabiity of Pr {η i k) = 0} for sensor i with τ i k) = j and α j i accordingy for i n. We can quantify the performance of a suboptima schedue by using the foowing resut. Proposition Suppose θ Θ + to be an optima soution of minimizing 5). For any schedue θ Θ + incuding any time-based schedues), define the optimaity gap to be θ ) := Jθ ) Jθ ). The foowing inequaity hods: θ ) Jθ # ) Jθ ). The schedue θ # is characterized by {βi }, Z + which is the soution to the foowing probem: minimize π 0 i,β i R subject to where P j i := j πi 0 Tr [ Pi 0 j + πi 0 π 0 i + π 0 i j=0 =0 πi 0, u=0 β j u j=0 =0 j βi, i S, =0 [ βi )Tr βi u ) /riu) h j i P j i) + β j i u=0 P j+ i, ) /rij )) i ) /rij ) ) h ip i ), i S. 45) Proof: In Probem, the transmission of each sensor is not affected by others and we can first consider such a subprobem for one sensor: minimize J i θ), subject to E [γ i k) ϱ i. 46) θ Θ The stochastic process {τ i k)} k= is a Markov chain with countabe state space M := {0,,,...}. The transition matrix T i is given by β j i, if j = 0 T i j, j ) = β j i, if j = j +, 0, otherwise where T i j, j ) := Pr {τ i k) = j τ i k) = j }. Though we can design β j i freey, we have to choose the set of {βj i } j=0 to guarantee that the state τ i k) = 0 is recurrent. Otherwise, the Markov chain {τ i k)} is transient and J i θ) wi go[ unbounded due to Lemma 3iii) and the fact that im j Tr h j i P i). Therefore, the stationary distribution π i = [πi 0, π i, π i,... must exist [3. Denote the corresponding estimation error covariance for τ i k) as P τik) i. From Theorem and i) we Schedues and LB θ offine θ greedy θ MDP LB J ) TABLE I: Performance comparison among different schedues. The ower bound of the optima J is aso isted as LB. have that Pi 0 = P i, P j i = tp j i, α j i ) and thus P j i in 45). Now the cost function becomes J i θ) = ϱ i Tr [ Pi 0 + max j=0 ϱ i [ j =0 β i )Tr P j+ i. Note that the impicit equaity constraint for Probem is j πj i =. Since the cost function J i θ) is a decreasing function of πi 0, we can transform it into the inequaity constraint and the optima soution must be reached ony when the equaity hods. Therefore, by soving the optimazition probem in the proposition we can obtain a ower bound for the optima cost and thus the optimaity gap bound. Note that the optimization probem is aso a signomia programming probem whose goba optimum can be numericay soved [, [. Remark For a sma network, we can enumerate the optima or suboptima schedues for different orders and choose the best order. However, it is formidabe to do this for a arge network. One heuristic technique is to assign a timeinvariant priority queue in parameter design according to the optima πi 0 in Proposition, i.e., etting the queue to be {s φ,..., s φ,..., s φn }, where φ S, with πφ 0 πφ 0 + since a arger πφ 0 usuay impies that error covariance of may grow faster than others. s φ V. EXAMPLE In this section, we use a simpe exampe to show the superiority of the proposed event-based schedue and compare event-based schedues with different parameter designs. To compare with the optima time-based schedue in [3 which is ony appicabe to the two-sensor case, we aso conduct [ simuations over [ two processes with the parameters: [ 0 A =, C 0 =, Q =, R 0 = and [ [ [. 3 0 A =, C 0 =, Q =, R 0 3 =. We compare the performance among the foowing schedues: θ offine : the optima time-based schedue [3 is {s, s, s, s, s, s,...} with the period 3. θ greedy : the suboptima schedue via the Greedy agorithm. θ MDP : the approximate optima event-based schedue via MDP approach. By discretizing the continuous state and action space and rebuiding the transition aw, we approximate the origina MDP probem by a finite state discrete MDP probem and sove it by vaue iteration. The number of state grids and action grids is 000 and 00, respectivey. We aso compute the ower bound for the optima eventbased schedue according to Proposition.

10 0 tr[pik) Priority Sensor i on duty Sensor Sensor a) Time b) Time c) Time Fig. : Reaization of the event-based schedue θ greedy : a) Tr [P i k) of each sensor, b) y-axis denotes the sensor index with the higher priority, c) y-axis denotes the on-duty sensor index. We summarize the resuts in Tabe I. For performance of each schedue, we repeatedy run 500 experiments and take their average. The estimation performance of θ offine is the worst among a. The two event-based schedues reduce the cost function Jθ offine ) by 43.8% and 40.4%, respectivey. The approximate soution to the MDP does not outperform the soutions given by the greedy agorithm in this case due to the discretization approximation. Moreover, the gap between Jθ greedy ) and LB is sma, impying that the suboptima schedue via the greedy agorithm is a good choice with ow computationa compexity compared with the MDP-based optima schedue. The computationa time for θ greedy at each time and the ower bound are 3.3s and 5.s, respectivey 3. We aso pot the reaization of θ greedy to iustrate the superiority of the event-based mechanism over the time-based one. At time k =, Sensor has the first priority of using the channe in Fig. b). This refects the vaidity of Theorem 4 from the fact Tr [h P k)) P > Tr [h P k)) P seen from Fig. a). In Fig. c), however, Sensor is schedued to transmit its data because the event-triggering mechanism of Sensor cassifies its data as unimportant and eaves the channe access to Sensor. The design of queue and eventtriggering thus aocates the channe access more efficienty than the offine schedue does. VI. CONCLUSION We have studied a stochastic sensor scheduing framework for sensor networks monitoring different processes. The sensors make transmission decisions based on both channe accessibiity and sef-triggering events depending on the reatime innovations. 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