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Distributed Dynamic State Estimation and LQGControl in Resource-Constrained Networks

Yasin Yılmaz, Member, IEEE, Mehmet Necip Kurt, and Xiaodong Wang, Fellow, IEEE

Abstract—In this paper, the discrete-time distributed dynamicstate estimation and Linear Quadratic Gaussian (LQG) controlproblems are analyzed for resource-constrained networked sys-tems. Following a holistic approach we provide a complete systemdesign for the signal processing, communication, and controltasks involved in the problems; and evaluate their performance.In the presence of a controller node and a number of sensornodes, the sensor nodes, in a resource-efficient way, reporttheir information entities to the controller node using an event-triggered sampling technique called level-crossing sampling. Wedemonstrate the performance gains due to level-crossing samplingover conventional time-triggered uniform sampling, as well as theadvantages of processing data locally before transmitting to thecontroller. In particular, it is shown that the proposed decentral-ized schemes with local processing and level-crossing samplingensure a very close approximation, with a bounded error, to theoptimum (centralized) estimation and control schemes, and as aresult yield order-2 asymptotic optimality. Moreover, non-idealcommunication between sensors and the controller is considered,and optimal modulation techniques are provided for differentchannel models. Simulation results are provided to support thepresented discussions.

Index Terms– Networked control systems, level-crossing sam-pling, distributed Kalman filter, LQG control, asymptotic opti-mality, unreliable communications.

I. INTRODUCTION

With the interplay between signal processing, communi-cations, and control, and the recent advancements in thesefields, it is now possible to control dynamic systems re-motely via networked control systems (NCS) [1], [2]. InNCS, sensors and controllers are geographically separated andthere is a communication system between them to exchangeinformation. They are widely used due to ease of installation,ease of maintenance, system flexibility, low complexity, andlow costs [1], [2]. However, traditional signal processing andcontrol techniques do not directly apply to NCS due to strictcommunication constraints.

In resource-constrained systems, the communication rateshould be reduced as much as possible while maintaininga satisfactory system performance such that resources areused in a smart way. Although conventional time-triggeredsystems have the advantage of ease of design and analysisbased on well-established theories, they typically suffer from

This work was supported in part by the U.S. National Science Foundation(NSF) under grants CNS-1737598, ECCS-1405327, in part by the U.S. Officeof Naval Research (ONR) under grant N000141410667, and in part by theSCEEE-17-03 grant.

Y. Yılmaz is with the Electrical Engineering Department, University ofSouth Florida, Tampa, FL 33620 (yasiny@usf.edu). M.N. Kurt and X. Wangare with the Electrical Engineering Department, Columbia University, NewYork, NY 10027 (m.n.kurt@columbia.edu, wangx@ee.columbia.edu).

inefficient use of resources, such as energy and communicationbandwidth. As a result, there is an increasing interest in theevent-triggered signal processing and control techniques overthe past years [3]. For instance, in a networked control system,when the system is at its steady state, periodic samplingand transmission between sensors and the controller wastesbandwidth, energy, and computation resources. On the otherhand, in event-triggered systems, actions depend on a set ofpredefined events, and information transmission takes placeonly when such events occur, see e.g., [4], [5]. With properlydefined events, event-triggered signal processing and controlis preferred for networked systems [6], [7].

A main task of a networked system is estimating the systemstate based on collected measurements over the network. Fora linear dynamic system, if the noise terms are Gaussian,the centralized Kalman filter is the optimal state estimatorin minimizing the mean squared error [8]. However, if thesystem is located in a geographically large area, then collectingand processing measurements at a single node is infeasibledue to communication constraints. Hence, distributed Kalmanfiltering has received significant attention in the literature, seee.g., [9] for a review. For a network in which only neighboringnodes are allowed to communicate with each other, the KalmanConsensus Filter (KCF) is proposed in [10] that combineslocal Kalman filters and a dynamic consensus algorithm forreducing disagreements on state estimates of the local Kalmanfilters. With the aim of reducing the communication rate in theKCF, event-triggered communication schemes are proposed in[11] and [12]. Furthermore, in [13], an event-based distributedKalman filter is proposed, where each node decides on sharingits local measurements with the other nodes depending on thelevel of deviation of the local measurements from the predictedones. It is also proposed in [14], [15] that at each time a sensorsends a quantized measurement innovation signal, which is thedifference between the actual measurement and the predictedmeasurement by the Kalman filter, to all other sensors, whichthen update their state estimates based on the broadcastedmessage.

The presence of a communication system in networked sys-tems brings some challenges such as limited data rate, quan-tization, and unreliable channels with data losses and com-munication delays, whose effects on the system performanceare extensively studied, see e.g., [16]–[23]. For instance, [16]examines the linear quadratic Gaussian control problem inNCS in the presence of medium access constraints and delays.Delay compensator is used together with a medium accesspolicy. In [17], communication channels have limited datarates. A quantization, coding and control scheme is designed

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to have the smallest possible data rate to achieve stability anddesired control performance. Moreover, the tradeoff betweenlinear quadratic cost and the data rate is shown. Authors of[18] considered distributed event-triggered control for NCS inthe presence of both data losses and communication delays.Maximum allowable number of successive data dropouts andtransmission delay deadlines are predicted locally. In [19],joint effect of delay and quantization on the stability of NCS isexamined. The relationship between choosing the quantizationparameters and communication delays is shown.

In [20], discrete-time LQG control problem is examinedover lossy networks for both observation and control packets.It is argued that if packets are acknowledged at the receivers ina TCP-like network, then separation principle in LQG controlholds and the optimal control input is a linear function ofthe estimated state provided that the arrival rates of packetsare above some critical levels. However, if the packets arenot acknowledged in a UDP-like network, then the separationprinciple does not hold and optimal control law is nonlinearin general. In [21], in a TCP-like network, acknowledgmentmessages are also subject to random data dropouts. In thiscase, the optimal control law is again nonlinear. It is alsoshown that with quantization, the separation principle doesnot hold. In [22], an unreliable channel that randomly erasestransmitted data is considered and a threshold-based rule isused as the event-triggering mechanism. Control performanceis measured by a linear-quadratic cost for a given eventthreshold level and the tradeoff between control performanceand communication cost is illustrated. Cyber-security is alsoa concern in NCS. For instance, in [23], random packetlosses due to interrupting jamming attacks are considered andconditions for stability of the system are provided.

The event-triggered paradigm is mainly used for nonuniformsampling in the signal processing applications [24]. In con-ventional uniform sampling, sampling times are periodic andpredetermined, in general, according to the highest expectedspectral frequency in the signal. Such high-frequency samplingis a waste of energy when the lower-frequency components aredominant in the signal. On the other hand, in event-triggeredsampling, sampling times are dynamically determined basedon the signal. Specifically, a sample is taken when a predefinedevent occurs in the signal (e.g., signal amplitude crosses apredetermined level). Consequently, event-triggered samplingencodes the signal in the sampling times, whereas in uniformsampling the signal is encoded in the sample amplitude. Inreal-time applications, the time-encoding feature of event-triggered sampling results in a significant advantage as thesampling times can be tracked using simple one-bit signaling[25].

In level-crossing sampling, sampling is triggered when thesignal amplitude crosses one of the predetermined levels,which are usually uniformly spaced. When the successivecrossing of the same level is ignored, the sampling mech-anism is called level-crossing sampling with hysteresis [3].In this paper, we use level-crossing sampling with hysteresisto transmit the local information entities at the sensor nodesto the controller. For simplicity, we drop the term hysteresisthroughout the paper.

In this paper, we consider the discrete-time dynamicstate estimation and LQG control problems in a resource-constrained networked system. Following a holistic approachwe provide a complete decentralized system design for thesignal processing, communication, and control tasks in anetwork of multiple sensor nodes and a controller. The sensornodes take several noisy observations of the time-varyingsystem states, and after processing observations they transmittheir local information to the controller in a resource-efficientway using level-crossing sampling. We show that the proposeddecentralized schemes closely approximate the optimum cen-tralized schemes, achieving strong (i.e., order-2) asymptoticoptimality under reliable communication channels. We thenconsider noisy communication channels between sensors andthe controller. For different channel models, we derive theoptimal modulation techniques. Through simulations, we il-lustrate the advantages of transmitting a finalized informationcompared to transmitting raw measurements to the controller,and also the advantages of using level-crossing sampling overconventional time-triggered sampling.

The remainder of the paper is organized as follows. The sys-tem model and dynamic state estimation are explained in Sec-tion II. Distributed state estimation for resource-constrainednetworked systems is examined in Section III. Non-ideal com-munications is considered and corresponding optimal modu-lation schemes are presented in Section IV. Distributed LQGcontrol problem for resource-constrained networked systemsis studied in Section V, which is followed by the numericalresults in Section VI. Finally, Section VII concludes the paper.In this paper, we represent vectors and matrices with boldfacesmall and capital letters, respectively, and all vectors arecolumn vectors.

II. SYSTEM DESCRIPTION

A. System Model

1

2

m1

1

2

m2

1

2 mK

node 1 node 2

node K

controller

Fig. 1. The considered networked control system.

Consider a networked control system with K nodes and acontroller, as shown in Fig. 1. At each time t ∈ N each nodek takes mk noisy measurements of n dynamic states, resultingin a total of m =

∑Kk=1mk measurements systemwide. The

controller, gathering some kind of local information from

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nodes, applies a control input to the system. Specifically, weassume the following linear state space model

xt+1 = Axt +But + vt

zkt = Ckxt +wkt ,

(1)

where xt ∈ Rn is the state vector; A is the state matrix; ut ∈Rp is the input, i.e., control vector; B is the input matrix; vt isthe additive white Gaussian system noise with the covariancematrix R1; and zkt ∈ Rmk , Ck, wk

t are the observation vector,observation matrix, and the additive white Gaussian noise withthe covariance matrix R2,k, respectively, at node k. We furtherassume that the noise vectors vt and wk

t are independent,hence we have E

[vtv

Tt′

]= δ(t − t′)R1, E

[wkt (wk

t′)T]

=δ(t− t′)R2,k, and E

[wkt v

Tt′

]= 0, ∀k, t, t′, where E[·] is the

expectation operator, and δ(t) is the Dirac delta function att = 0.

We assume A, B and Ck are given and satisfy theobservability and controllability conditions for our discrete-time linear time-invariant system. In Section III, the input utis assumed to be given, and in Section V, it is designed forLQG control. Instead of transmitting the raw observations zktfrom nodes to the controller, leveraging the information formof Kalman filter we propose to transmit local contributionsto the optimum estimator xt (Section III) and the optimumcontrol input ut (Section V). Since xt and ut typically havemuch smaller dynamic ranges and number of dimensions thanzkt (see Fig. 8), the proposed method turns out to be quiteadvantageous over the traditional methods (see Table I). Wealso show through simulations that the control input producedby the proposed method is closer to the optimum value thanthat produced by transmitting zkt (see Fig. 7).

B. Information Filter

We have a discrete-time linear dynamic system as describedin (1). Since the noise terms are Gaussian, Kalman filter is theoptimal state estimator [8]. The information filter, or inversecovariance filter, is algebraically equivalent to the Kalmanfilter. Instead of the estimates of the state covariance matrixand the state vector, it uses the following information matrixand information vector, respectively,

Y t|t′ , P−1t|t′ (2)

yt|t′ , P−1t|t′ xt|t′ = Y t|t′ xt|t′ , (3)

where P t|t′ and xt|t′ are the predicted (for t′ = t − 1) andupdated (for t′ = t) covariance matrix and state estimate in theKalman filter, respectively. The information matrix and vectorare predicted as

Y t|t−1 = (I − F t−1)Gt−1, (4)

yt|t−1 = (I − F t−1)A−Tyt−1|t−1 + ξt−1, (5)

and updated as

Y t|t = Y t|t−1 +

K∑k=1

CTkR−12,kCk, (6)

yt|t = yt|t−1 +

K∑k=1

CTkR−12,kz

kt , (7)

where Gt , A−TY t|tA

−1, (8)

F t , Gt

(Gt +R−11

)−1, (9)

ξt−1 , Y t|t−1But−1, (10)

and I is the identity matrix [26]. Denote the informationcontributions from node k with

Φk , CTkR−12,kCk, (11)

and φkt , CTkR−12,kz

kt . (12)

Then, from (6) and (7) we can write Y t|t = Y t|t−1 + Φ andyt|t = yt|t−1+φt, where Φ ,

∑Kk=1 Φk and φt ,

∑Kk=1 φ

kt

are the systemwide information contributions.In this study, for state estimation, we prefer to use the

information filter over the Kalman filter for several reasons.Firstly, the information filter is more convenient for a dis-tributed setting due to its simple update rules, given in (6) and(7). In particular, in the classical distributed implementation ofthe information filter, each node k reports its new informationcontribution φkt at each time t to the controller, which sumsthem to update the information vector yt|t−1. On the otherhand, in the Kalman filter each node k needs to report itsraw observation zkt at each time t, which is processed atthe controller before the update, hence causes propagation ofquantization/communication errors. Secondly, at each time t,the inverse of an n × n matrix is computed to obtain F t inthe information filter, whereas in the Kalman filter an m×mmatrix is inverted, where usually m n. However, we notethat the structure of the information filter requires the matricesA, R1, and R2,k to be invertible.

III. DISTRIBUTED STATE ESTIMATION FORRESOURCE-CONSTRAINED SYSTEMS

In resource-constrained systems with strict energy and band-width limitations, e.g., wireless sensor networks, nodes cannotexactly report their information contributions. Specifically,each node k needs to sample and quantize its informationentity before transmitting it to the controller. Since such asampling and quantization process induces information loss,which grows with further processing at the controller, weintend to perform the necessary information processing aslocally as possible before transmitting the information. To thisend, defining the shorthand subscript t , t|t we present thefollowing lemma.

Lemma 1. The systemwide optimum state estimate xt can bewritten as the sum of local contributions and a term relatedto the previous-time control inputs, i.e.,

xt = ϕt +

K∑k=1

xkt , (13)

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where ϕt is the input related term, defined in (23) and (18),and xkt is the local contribution from node k to xt.

Proof: From (5) and (7), we write

yt = ξt−1 + Ωt−1yt−1 +

K∑k=1

φkt , (14)

where Ωt−1 , (I − F t−1)A−T . (15)

Note that due to (10), ξt−1 is related to ut−1. We split yt intotwo parts as

yt = λt−1 + ζt (16)

where λt−1 is equal to the sum of all terms related to theprevious-time control inputs and ζt is the contribution frominformation entities. We split yt−1 as in (16), and rewrite (14)as follows:

yt = ξt−1 + Ωt−1(λt−2 + ζt−1) +

K∑k=1

φkt

= ξt−1 + Ωt−1λt−2︸ ︷︷ ︸λt−1

+ Ωt−1ζt−1 +

K∑k=1

φkt︸ ︷︷ ︸ζt

. (17)

Based on (17), we write the evolutions of the terms λt and ζtin time as follows:

λt = Ωtλt−1 + ξt, (18)

ζt = Ωt−1ζt−1 +

K∑k=1

φkt , (19)

where λ−1 = 0 and ζ0 = 0 are the initial values ofthese terms, respectively. Furthermore, from (19) we haveζ1 =

∑Kk=1 φ

k1 , ζ2 = Ω1

∑Kk=1 φ

k1 +

∑Kk=1 φ

k2 , and so on.

Defining Ψts , ΩtΩt−1 · · ·Ωs we write ζt as

ζt =

t∑s=1

Ψt−1s

K∑k=1

φks ,

=

K∑k=1

t∑s=1

Ψt−1s φks ,

=

K∑k=1

ykt , (20)

where Ψt−1t = I and ykt ,

∑ts=1 Ψt−1

s φks . Note that φks =CTkR−12,kz

ks is the local information of node k at time s, and

Ψt−1s = Ωt−1Ωt−2 · · ·Ωs is a function of A and F rt−1r=s,

where F r, given by (9), is a function of A, R1, and Y r.From (6), Y r is computed using Ck and R2,k. Since A,R1, R2,k, and Ck are known and time-independent, eachnode k can compute its local information entity ykt at eachtime t. Moreover, similar to (19), it can recursively update yktas

ykt = Ωt−1ykt−1 + φkt . (21)

The terms related to the previous inputs, i.e., λt canbe computed and updated by the controller node using (18).Hence, there is no need to transmit these information entities

from the nodes. Then, each node k can locally process φkt ,given by (12), and transmit ykt instead of transmitting φkt asin the standard distributed implementation of the informationfilter. In fact, to minimize the information loss at the controller,we can further process ykt locally and transmit a finalizedinformation entity. Specifically, from (3), (16), and (20), theglobal state estimate at time t is given by

xt = Y −1t

(λt−1 +

K∑k=1

ykt

)

= ϕt +

K∑k=1

xkt , (22)

where ϕt , Y−1t λt−1, (23)

and xkt , Y −1t ykt . Note that the matrix Y t is available to all

nodes and from (4) and (6), it can be recursively computed as

Y t = Ωt−1Y t−1A−1 +

K∑k=1

Φk. (24)

A. Information Transmission using Level-Crossing Sampling

As a result of Lemma 2, we propose that the nodes transmitxkt Kk=1 to the controller, which sums them and the termrelated to the previous-time inputs to obtain the optimal stateestimate xt. An event-triggered sampling technique, called thelevel-crossing sampling, can be used to accurately report xktin an energy- and bandwidth-efficient way. The level-crossingsampling procedure is quite simple: Firstly, a set of signallevels is selected to trigger sampling. Here we use a set withuniform spacing ∆, as shown in Fig. 2. As the signal isobserved sequentially, a new sample is taken when the signalcrosses a sampling level that is different than the most recentlycrossed one.

This method is, in fact, known as level-crossing samplingwith hysteresis in the literature. In the original level-crossingsampling procedure, a sample is taken every time a samplinglevel is crossed, even when the same level is crossed consecu-tively, which makes the procedure even simpler. However, thismethod does not serve our purpose of reporting the changes inthe local information entity (see Fig. 2). For simplicity, herewe drop the term hysteresis.

In this adaptive sampling scheme, the sampling times aredynamically determined by the signal, hence random, asopposed to the traditional time-triggered uniform sampling, inwhich samples are periodically taken in time. In the traditionalsampling, the time axis is partitioned uniformly, whereas inthe level-crossing sampling, we partition the magnitude axisuniformly, time being the dependent variable.

In particular, node k samples the ith entry xk,it of xkt at therandom sampling times τk,iq q given by

τk,iq , mint ∈ N : |xk,it − γ

k,iq−1∆| ≥ ∆

, (25)

where γk,iq−1 ∈ N is the sampling level in terms of ∆ that wasmost recently crossed. In other words, each node k runs n

5

−∆

∆

2∆

3∆

1 2 3 4 5 6 7 8 9 10 11 12 13τ k,i1 τ k,i2 τ k,i3 τ k,i4

t

xk,it

εk,i1

εk,i2

εk,i3

εk,i4

Fig. 2. The level-crossing sampling procedure. A new sample is taken whena sampling level that is different than the one at the last sampling time iscrossed. At each sampling time τk,iq , a discrete-time signal over/under shootsthe sampling level by εk,iq .

parallel level-crossing samplers for xk,it ni=1. Note that thethreshold ∆ in (25) determines the sampling rate. The smallerit is, the more frequently samples are taken in a nonuniformfashion. A small ∆ may also cause xk,it to cross multiplesampling levels at once, i.e., at time τk,iq the number of crossedsampling levels is given by

θk,iq ,

⌊|xk,iq − γ

k,iq−1∆|

∆

⌋≥ 1. (26)

Then, at each sampling time τk,iq node k transmits θk,iqrepetitions of the sign bit

δk,iq , sign(xk,iq − γk,iq−1∆) = sign(γk,iq − γ

k,iq−1), (27)

indicating the current sampling level

γk,iq = γk,iq−1 + δk,iq θk,iq . (28)

Here we use the shorthand notation xk,iq for the signal level,and γk,iq for the sampling level (in terms of ∆) at time τk,iq .

The controller, upon receiving δk,iq (and possible repetitions)at time τk,iq , updates its estimate for xk,it by δk,iq θk,iq ∆, i.e.,

˜xk,iq = ˜xk,iq−1 + δk,iq θk,iq ∆, ˜xk,i0 = 0, (29)

where ˜xk,iq−1 is the estimate of xk,it at the controller node duringthe time interval τk,iq−1 ≤ t < τk,iq . Since xt = ϕt+

∑Kk=1 x

kt ,

as stated in (13), in fact, the controller updates its estimates foreach entry of xt regardless of which of the nodes transmittingthe information bits. We define ˜xq = [˜x1q,

˜x2q, . . . ,˜xnq ] ,∑K

k=1˜xkq , where ˜xk,iq is the ith entry of ˜xkq . The controller

node, upon receiving the qth (in the global order) bit δiq fromnode kq , it performs the following update

˜xiq = ˜xiq−1 + δiq∆,˜xi0 = 0, (30)

and uses ˜xq as the estimate of∑Kk=1 x

kt until the next received

bit. If multiple bits arrive at the same time, then it processesthem in a random order. The proposed procedures at nodek and the controller are summarized in Algorithms 1 and 2,respectively.

Algorithm 1 The proposed procedure at node k1: Compute Y tt≥0 and Ωtt≥0 as in Fig. 3

2: Initialization: t← 0, y ← 0, γi ← 0, i = 1, . . . , n

3: while t ≥ 0 do4: t← t+ 1

5: y ← Ωt−1y + φt

6: x← Y −1t y

7: if |xi − γi∆| ≥ ∆, i = 1, . . . , n then8: θi ←

⌊|xi−γi∆|

∆

⌋9: Transmit δi = sign(xi − γi∆) to the controller (θi times)

10: γi ← γi + δiθi

11: end if12: end while

Algorithm 2 The proposed procedure at the controller1: Compute Y tt≥0 and Ωtt≥0 as in Fig. 3

2: Compute Y t|t−1t≥1 as in (4)

3: Initialization: t← 0, ˜xiq ← 0, i = 1, . . . , n, λ−1 ← 0

4: while t ≥ 0 do5: t← t+ 1

6: ξt−1 ← Y t|t−1But−1

7: λt−1 ← ξt−1 + Ωt−1λt−2

8: ϕt ← Y −1t λt−1

9: if δiq arrives, i = 1, . . . , n then10: ˜xiq ← ˜xiq−1 + δiq∆

11: end if12: ˜xt ← ϕt + ˜xq

13: end while

Note that lines 7-11 in Algorithm 1, and 9-11 in Algorithm2 are processed for each entry i in parallel. In Algorithm 1 andAlgorithm 2, the matrices Y t and Ωt are iteratively computedoffline following the flow chart in Fig. 3.

B. Performance Analysis

The optimum state estimate xt, is achievable only undera centralized setup, in which the controller has access to allobservations systemwide. In a distributed system with resourceconstraints, a decentralized state estimate ˜xt inevitably incursa nonzero performance gap. Given a specific decentralizedscheme, the lower and upper bounds on the difference xit− ˜xitdefine the 100% confidence interval for each entry xit. We hereanalyze both the non-asymptotic and asymptotic behaviors ofthis maximum-level, i.e., deterministic confidence interval, as ameasure of performance/reliability for the proposed decentral-ized scheme based on level-crossing sampling. Specifically, weshow that this interval is easily controllable through parameterselection, and remains bounded for all xit, i.e., xit− ˜xit = O(1)1

even if |xit| → ∞, yielding a strong type of asymptoticoptimality called order-2.

1O(·) is the big-O notation and O(1) denotes a constant.

6

Yt

Gt = A−TYtA−1

Ft = Gt(Gt + R−11 )−1

Ωt = (I − Ft)A−T

Yt+1 = ΩtYtA−1 +

K∑k=1

CTk R

−12,kCk

Fig. 3. Flow chart for iteratively computing Y t,Gt, F t, and Ωt. See (4)–(8)and (15) for the mathematical relations.

This strong type of asymptotic optimality provides a higherperformance standard than the conventional order-1 asymp-totic optimality, which necessitates the convergence of ˜xitto xit, i.e.,

˜xit

xit

= 1 + o(1)2 as |xit| → ∞. Note that inorder-1 asymptotic optimality, the deterministic confidenceinterval may become unbounded at a lower rate than xit,that is, xit − ˜xit = o(xit) as |xit| → ∞. Order-2 asymptoticoptimality is used as a benchmark for high performance inresource-constrained distributed systems, e.g., [25], [27]. In thefollowing theorem, we show that the proposed decentralizedscheme achieves this high-performance benchmark.

Theorem 1. In the decentralized state estimation schemebased on level-crossing sampling, given in Algorithms 1 and 2,each state estimate ˜xit is guaranteed to lie within K∆ boundof the optimum state estimate xit, i.e.,

|xit − ˜xit| < K∆, ∀i, t, (31)

where K is the number of nodes, ∆ is the level-crossing sam-pling threshold, xit and ˜xit are the ith entries of state estimatevectors at time t in the optimum scheme and the proposedscheme, respectively. As a result, it is order-2 asymptoticallyoptimum, i.e., xit − ˜xit = O(1) even if |xit| → ∞.

Proof: Due to the sampling rule in (25) the currentsignal level xk,it at each node k is within ∆ bound of thecurrent sampling level γk,iq ∆, that was most recently crossed.The controller keeps track of the current sampling level γk,iq ,given by (28), for each local signal xk,it as it is informed ofeach sampling level crossing through δk,iq , given by (27). Itestimates xk,it with ˜xk,it = γk,iq ∆, as shown in (29), hence|xk,it − ˜xk,it | < ∆,∀k, i, t. Let the ith entry of the vector ϕtbe denoted with ϕit. Due to (13), xit = ϕit +

∑Kk=1 x

k,it and

˜xit = ϕit +∑Kk=1

˜xk,it . Hence, we have

|xit − ˜xit| <K∑k=1

|xk,it − ˜xk,it | < K∆, ∀i, t,

proving (31). Then, the asymptotic optimality result followsfor fixed K and ∆.

Theorem 1 presents a strong result (i.e., the confidencebound K∆) that holds for any xit value (small/large, non-asymptotic/asymptotic), constituting an important advantage

2o(·) is the little-o notation and o(1) denotes a diminishing value.

over the conventional scheme based on conventional time-triggered sampling. In particular, in the scheme which uni-formly samples and quantizes xk,it , the confidence intervalincreases with range of xk,it , that is, the larger values xk,itgets, the larger the confidence interval is. Furthermore, insuch a scheme with a small number of quantization bits, theconfidence interval is, in general, much larger than K∆.

IV. NON-IDEAL COMMUNICATIONS

Existing works assume there exists a communication system(a pair of transmitter and receiver for some channel model),and analyze the effects of noisy communications in discrete-time. Similarly, we start our discussion with a discrete-timechannel model with error probability p, e.g., binary erasurechannel (BEC) and binary symmetric channel (BSC). In such amodel, a transmitted bit δk,iq is received as δk,iq at the controller,where δk,iq = δk,iq with probability 1− p and δk,iq 6= δk,iq (i.e.,δk,iq = 0 for BEC and δk,iq = −δk,iq for BSC) with probabilityp. When a bit error occurs (i.e., a bit is lost or flipped), thecontroller performs an erroneous update δk,iq ∆ with δk,iq 6=δk,iq .

If such errors happen infinitely often (i.o.), the discrepancy|xit − ˜xit| may grow unboundedly if the system is unstable,i.e., |xit| → ∞. If the system tends to be unstable in time forsome reason, it can be stopped and restarted to let the systemstates return to initial values, e.g., [29]. However, for a stablesystem, as long as bit error happens at the same rate p forboth δk,iq = 1 and δk,iq = −1, the stability of the system ispreserved as t→∞, as shown next.

Theorem 2. In a practical system with resource constraintsand a nonzero channel error probability p, the deterministicconfidence interval η for the proposed LCS-based state es-timate ˜xit (i.e., |xit − ˜xit| < η) is unbounded, i.e., η → ∞,as |xit| → ∞. Recall from Theorem 1 that for p = 0,η = K∆,∀t. Hence, as opposed to Theorem 1, for p > 0,order-2 asymptotic optimality does not hold. However, for astable system where |xit| <∞ as t→∞, our estimate is alsostable, i.e., |˜xit| <∞, for any p > 0.

Proof: From the Borel-Cantelli lemma,

P(bit error occurs i.o.) = 0 if∞∑t=1

p <∞,

which does not hold unless p → 0 at a rate at least as fastas 1/t, that is, p = O(1/t). In the channel coding theorem,p→ 0 only if the block length goes to infinity, i.e., the numberof bits θk,iq → ∞. Obviously, this is not the case here asthe local signal xk,it has only finite jumps. Moreover, due tothe resource constraints, we cannot transmit a large numberof bits per sample. As a result, P(bit error occurs i.o.) > 0.If |xit| → ∞, that means either the number of ∆ or −∆changes (i.e., the number of δk,iq = 1 or δk,iq = −1) goes toinfinity faster than the other. In this case one of the bit typesis lost infinitely more often than the other, and we cannotdeterministically bound the error |xit − ˜xit|.

For a stable system where |xit| <∞ ∀t, i, the number of ∆and −∆ changes will go to infinity as t→∞ at the same rate

7

such that the state estimate remains bounded. Hence, even if biterror occurs infinitely often, errors for δk,iq = 1 and δk,iq = −1will occur randomly at the same rate p, canceling each other,so the error will not accumulate to infinity.

In theory, the order-1 asymptotic optimality is possible, asshown next.

Theorem 3. If xit → ∞ at a rate faster than t, that is,xit = ω(t), the proposed LCS-based scheme achieves order-1asymptotic optimality, i.e.,

˜xit

xit

= 1 + o(1) as |xit| → ∞.

Proof: This is because˜xit

xit

= 1+˜xit−x

it

xit

, and from Theorem

2, ˜xit− xit may become unbounded as fast as t, i.e., ˜xit− xit =O(t), since there may occur an error of 2∆

∑Kk=1 θ

k,iq < ∞

at each time t. Hence,˜xit

xit

= 1 + o(1) if xit = ω(t).However, such an extreme case, where xit → ∞ at a rate

faster than t, is not of practical interest. In this section we con-sider improving the non-asymptotic performance by designingpractical (continuous-time) communication systems. We firstconsider the simplest case in which each node k reports eachith entry xk,it of its local vector xkt to the controller througha separate channel that is orthogonal to the other channelsin the network, resulting in Kn parallel channels in total. Inthis model, we identify the optimum modulation techniquesunder additive white Gaussian noise (AWGN) and fadingchannels. Then, we propose more bandwidth-efficient modelsvia multiple-access channels.

A. Parallel Channels for Nodes

1) AWGN Channels: Suppose each node k sends the wave-forms a s(t) and b s(t) for δk,iq = 1 and δk,iq = −1, respec-tively, through an AWGN channel. The controller, applying amatched filter and sampling uniformly, receives the followingdiscrete-time signal

rk,it = ck,it + nk,it , (32)

where ck,it is either a or b, and nk,it is the (zero-mean) whiteGaussian noise with variance σ2. Hence, rk,it ∼ N (ck,it , σ2),that is, the mean is a, b, and 0 for δk,iq = 1, δk,iq = −1, and notransmission, respectively, and the variance is the same in allcases. To minimize the probability of demodulation error, weshould separate a, b, and 0 as much as we can. Thus, underthe peak transmission power constraint max(a2, b2) ≤ P 2, theantipodal signaling a = −b = P is optimum, as shown in Fig.4.

2) Fading Channels: Under a fading channel model, thecontroller receives

rk,it = hk,it ck,it + nk,it , (33)

where the channel coefficients hk,it ,∀t, i, k are i.i.d. withthe distribution N (µ, ρ2) for Rician fading (µ 6= 0) andRayleigh fading (µ = 0). Let us first analyze the Rayleighfading case. The received signal rk,it is zero-mean Gaussianwith variance a2ρ2 + σ2, b2ρ2 + σ2, and σ2 for δk,iq = 1,δk,iq = −1, and no transmission, respectively (see Fig. 5). Thistime we should separate a2, b2, and 0 to minimize the errorprobability. Hence, an asymmetric constellation is optimum

-10 -8 -6 -4 -2 0 2 4 6 8 10

rk,it

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

p(r

k,i

t)

δk,iq = 1

No transmissionδk,iq = −1

δk,iq = −1 δ

k,iq = 0 δ

k,iq = 1

Fig. 4. Probability density function (pdf) of the signal rk,it , given in (32),received by the controller through an AWGN channel when the optimumantipodal signaling a = −b = 5 is used. In this example, the peak powerconstraint is P 2 = 25 and the variance is σ2 = 1. For maximum likelihood(ML) demodulation, the decision boundaries and the corresponding decisionsare shown with the vertical dotted lines and the boxed texts, respectively.The controller indeed makes such a decision at each time t. To minimize themodulation error, represented by the intersection of pdfs, signaling levels aand b are set apart as much as the peak power constraint allows.

under Rayleigh fading, as shown in Fig. 5. Under the samepeak transmission power constraint, either a2 or b2 should beP 2, say a2 = P 2. Then, the choice for b2 ∈ [0, P 2] is givenby the following theorem.

-10 -8 -6 -4 -2 0 2 4 6 8 10

rk,it

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

p(r

k,i

t)

δk,iq = 1

No transmissionδk,iq = −1

δk,iq = −1

−r1(b) r1(b)r2(b)−r2(b)

δk,iq = 0

δk,iq = −1

δk,iq = 1 δ

k,iq = 1

Fig. 5. Probability density function (pdf) of the signal rk,it , given in (33),received by the controller through a Rayleigh fading channel when theoptimum asymmetric signaling, a2 = 25 and b2 = 4, is used. In this example,the peak power constraint is P 2 = 25; the variances are σ2 = ρ2 = 1; andthe transmission probabilities are P(δk,iq = 1) = P(δk,iq = −1) = 0.23 andP(No transmission) = 0.54. For maximum likelihood (ML) demodulation,the decision boundaries ±r1(b),±r2(b), and the corresponding decisionsare shown with the vertical dotted lines and the boxed texts, respectively.The controller indeed makes such a decision at each time t. To minimize theexpected modulation error, represented by the intersection of pdfs in the wrongdecision regions, the signaling level b is determined based on the transmissionprobabilities, as shown in (34).

Theorem 4. Consider Rayleigh fading channels between thenodes and the controller [see (33)], a peak power constraintP 2, and maximum likelihood (ML) demodulation. Then, theoptimum signaling levels, in terms of minimizing the MSE perbit, for transmitting the LCS bits [see (27)] are given by aray =

8

±P and

bray = arg minb

P(δ = 1)

[4Φ

(r1(b)

α

)− 3Φ

(r2(b)

α

)]− P(δ = 0)Φ

(r2(b)

σ

)+ P(δ = −1)

[−4Φ

(r1(b)

β(b)

)+ Φ

(r2(b)

β(b)

)], (34)

where a and b are interchangeable; Φ(·) is the cumulativedistribution function (cdf) of the standard Gaussian distribu-tion; P(δ = 1), P(δ = −1), P(δ = 0) are a node’s prob-abilities of transmitting +1, −1, and nothing, respectively;α =

√a2ρ2 + σ2 and β(b) =

√b2ρ2 + σ2 are the standard

deviations of the signal received by the controller [see (33)]when +1 and −1 are transmitted, respectively; and

r1(b) = αβ(b)

√2 log(α/β(b))

α2 − β(b)2

r2(b) = β(b)σ

√2 log(β(b)/σ)

β(b)2 − σ2

(35)

are the decision boundary values for the received signal forML demodulation (see Fig. 5).

Proof: From (30), the MSE per bit is given by

E[(δ∆− δ∆)2] =

(2∆)2P(δ = −1, δ = 1) + ∆2P(δ = 0, δ = 1)

+ ∆2P(δ = −1, δ = 0) + ∆2P(δ = 1, δ = 0)

+ ∆2P(δ = 0, δ = −1) + (2∆)2P(δ = 1, δ = −1) (36)

where δ ∈ −1, 0, 1 is the transmitted bit, and δ ∈ −1, 0, 1is the received bit through the Rayleigh fading channel. Abovewe don’t show the indices for node, dimension, and time sincewe assume i.i.d. channels. As shown in Fig. 5, the MSE perbit can be written as

E[(δ∆− δ∆)2] =

(2∆)2 P (r2(b) ≤ |r| ≤ r1(b) | δ = 1) P(δ = 1)

+ ∆2 P(−r2(b) ≤ r ≤ r2(b) | δ = 1) P(δ = 1)

+ ∆2 P(r2(b) ≤ |r| ≤ r1(b) | δ = 0) P(δ = 0)

+ ∆2 P(|r| ≥ r1(b) | δ = 0) P(δ = 0)

+ ∆2 P(−r2(b) ≤ r ≤ r2(b) | δ = −1) P(δ = −1)

+ (2∆)2 P(|r| ≥ r1(b) | δ = −1) P(δ = −1). (37)

After some manipulations, it is straightforward to show that

E[(δ∆− δ∆)2] =

2∆2

P(δ = 1)

[4Φ

(r1(b)

α

)− 3Φ

(r2(b)

α

)− 1

2

]+ P(δ = 0)

[1− Φ

(r2(b)

σ

)]+ P(δ = −1)

[7

2− 4Φ

(r1(b)

β(b)

)+ Φ

(r2(b)

β(b)

)], (38)

b

P (δ = −1)

P (δ = 1)

1

0.8

0.6

0.40

0

1

0.2 0.2

2

0.40.6

3

00.8

4

1

5

0 0.1 0.2 0.3 0.4 0.5

P (δ = 1) = P (δ = −1)

0

1

2

3

4

5

b

1

0.8

0.6

P (δ = −1)

00

1

0.40.2

2

P (δ = 1)

b

0.4

3

0.20.6

4

0.8

5

01

0 0.1 0.2 0.3 0.4 0.5P (δ = 1) = P (δ = −1)

0

1

2

3

4

5

b

Fig. 6. The optimum signaling level b under Rayleigh fading channel vs.the transmission probabilities for the MSE per bit criterion [left figure - see(34)] and the MAE per bit criterion [right figure - see (39)]. P(δ = 1)and P(δ = −1) denote the probability of transmitting at the level a and b,respectively. The probability of no transmission is given by1 − P(δ = 1) − P(δ = −1). In practice, the case in which P(δ = 1) =

P(δ = −1) is of special interest since xk,it is neither increasing or decreasing.This case, represented by the red line, is shown specifically in the upper rightcorner in both figures.

minimizing which over b is equivalent to minimizing (34). Thedecision boundaries r1(b) and r2(b) are the intersection pointsat which the zero-mean Gaussian pdf with variance β(b)2

coincides with the zero-mean Gaussian pdfs with variancesα2 and σ2, respectively. Equating the pdf expressions at thesame point, for instance for r1(b),

e−r1(b)2/2α2

√2πα

=e−r1(b)

2/2β(b)2

√2πβ(b)

,

we get the closed-form solutions given in (35).

Fig. 6 shows the optimum signaling level b for differenttransmission probabilitites according to MSE per bit as inTheorem 4, and also mean absolute error (MAE) per bitE[|δ∆ − δ∆|]. For the MAE criterion, following the proofof Theorem 4 we can show that the optimum signaling levels

9

are given by aray = ±P and

bray = arg minb

P(δ = 1)

[2Φ

(r1(b)

α

)− Φ

(r2(b)

α

)]− P(δ = 0)Φ

(r2(b)

σ

)+ P(δ = −1)

[−2Φ

(r1(b)

β(b)

)+ Φ

(r2(b)

β(b)

)]. (39)

In Fig. 6, it is seen that as the probability P(δ = 1) oftransmitting the level a increases from 0 to 1, the optimumlevel for b, as expected, transitions from 5, which is the sameas a, to 0, which correpsonds to no transmission. In the MAEcase, the transition takes place later, and the transition regionis wider because the penalty for flipping a bit in this case ismilder (2 times the penalty for losing a bit) than that in theMSE case (4 times the penalty for losing a bit).

Note that the transmission probabilities are determined bythe sampling threshold ∆, as well as the signal to be sampledxk,it . In particular, larger ∆ causes smaller P(δ = 1) andP(δ = −1), and thus larger no transmission probability. Inpractice, by setting ∆ accordingly, we tend to attain moderatetransmission rates (e.g., 1 bit per dimension per unit time),which correspond to the transition regions in Fig. 6, hence noinformation loss due to the identical signaling levels.

It might make more sense to minimize the cumulativeMSE = E[‖xt− ˜xt‖22] instead of the MSE per bit. In this case,the optimum b value that minimizes the cumulative MSE canbe numerically found through offline simulations.

Since Rician fading includes AWGN and Rayleigh fadingas extreme cases with ρ2 = 0 and µ2 = 0, respectively, theoptimum signaling scheme under Rician fading is a functionof µ2 and ρ2.

Recall from (26) that node k transmits θk,iq repetitions of thesign bit δk,iq , where θk,iq denotes the number of level crossings.Instead of transmitting θk,iq times, one may choose to encodeθk,iq in time or frequency or amplitude based on availableresources. Specifically, assuming θk,iq < M , for all q, i, kand for some M > 0, which easily holds in practice, onecan uniformly partition the available time/frequency/amplitudeinterval into M subintervals, and transmit δk,iq in the θk,iq thsubinterval. Such an encoding enjoys a single transmission,hence higher resource efficiency, whereas multiple transmis-sions (i.e., θk,iq repetitions of δk,iq ) provide robustness againstthe demodulation errors. Note that encoding θk,iq in energylevels (i.e., pulse-amplitude modulation) works well onlyunder AWGN channels, but not fading channels due to themultiplicative noise. We next discuss using multiple-accesschannels.

B. Multiple-Access Channels

Since a received bit at the controller, regardless of itssource node, causes the same update (∆ or −∆), nodes cansynchronously transmit in multiple-access channels in a morebandwidth-efficient way than the previous setup with Knparallel channels.

Under fading channels with asymmetric signaling, samepolarity bits that correspond to the same state estimate, i.e.,

δk,iq = jk,q, i = 1, . . . , n, j = ±1, can be transmittedin the same multiple-access channel, resulting in 2n parallelchannels. Then, the controller receives

rijt = cj

Kijt∑

k=1

hk,ijt + nijt , (40)

where Kijt is the number of nodes transmitting bits at time

t regarding the state estimate i with polarity j, and theasymmetric transmission levels c+1 = a and c−1 = b.The update information carried by rijt for xit is Kij

t j∆,where Kij

t is unknown. Hence, the controller needs to esti-mate Kij

t from rijt . Defining the effective channel coefficienthijt ,

∑Kijt

k=1 hk,ijt we have hijt ∼ N (Kij

t µ,Kijt ρ

2) andthus rijt ∼ N (cjK

ijt µ, c

2jK

ijt ρ

2 + σ2). As a result, using themaximum likelihood (ML) criterion the controller can estimateKijt as

Kijt = arg min

Kijt ∈0,1,...,K

(rijt − cjKijt µ)2

c2jKijt ρ

2 + σ2+log(c2jK

ijt ρ

2+σ2),

(41)and update ˜xiq with Kij

t j∆ [cf. (30)].The optimum total update for ˜xit at time t is (Ki,+1

t −Ki,−1t )∆ and estimated at the controller as (Ki,+1

t −Ki,−1t )∆.

Note that in symmetric signaling (i.e., b = −a), we cantransmit all bits regarding the same input, that is, δk,iq k,q , ina single multiple-access channel without losing the necessaryand sufficient information Ki

t , Ki,+1t − Ki,−1

t . In otherwords, we can combine different polarity channels regardingthe same input, resulting in n parallel channels, decreasingthe number of required parallel channels from 2n (in theasymmetric case) to n. Specifically, the signal received by thecontroller at time t for xit is given by

rit = a

Ki,+1t∑k=1

hk,i,+1t −

Ki,−1t∑k=1

hk,i,−1t

+ nijt . (42)

Defining the effective channel coefficient hit ,∑Ki,+1t

k=1 hk,i,+1t −

∑Ki,−1t

k=1 hk,i,−1t we see thathit ∼ N (Ki

tµ,Kitρ

2) and thus rit ∼ N (aKitµ, a

2Kitρ

2 + σ2).Similar to (41), the controller finds the ML estimator as

Kit = arg min

Kit∈−K,...,K

(rit − aKitµ)2

a2Kitρ

2 + σ2+ log(a2Ki

tρ2 + σ2),

(43)and updates ˜xiq with Ki

t∆. Note that in (43), Kit ∈

−K, . . . ,K, whereas in (41) Kijt ∈ 0, 1, . . . ,K.

Similarly, under AWGN channels with symmetric signaling,we can transmit all the bits regarding the same input in a singlemultiple-access channel. In this simpler case, the controllerreceives

rit = aKit + nit, (44)

which is clearly distributed according to N (aKit , σ

2). Hence,the ML estimator is given by

Kit = arg min

Kit∈−K,...,K

(rit − aKit)

2

= arg minKi

t∈−K,...,K

∣∣∣∣Kit −

rita

∣∣∣∣ . (45)

10

That is, Kit is the closest Ki

t ∈ −K, . . . ,K to rita , which

we can write as

Kit = min

(max

(round

(rita

),−K

),K

). (46)

It is clearly seen that synchronous communication overmultiple-access channels provides significant bandwidth sav-ings (n or 2n vs. Kn parallel channels). The number of levelcrossings θk,iq can be reported to the controller by again eithermultiple transmissions or encoding in time/frequency/energy,as discussed in Section IV-A.

V. DISTRIBUTED LQG CONTROL FORRESOURCE-CONSTRAINED SYSTEMS

As a practical application of the proposed distributed stateestimation method in networked control systems, we considerthe finite-horizon linear quadratic Gaussian (LQG) controlproblem in which the following quadratic cost function JNis minimized,

JN , E

[xTNQ0xN +

N−1∑t=0

(xTt Q1xt + uTt Q2ut

)], (47)

where Q0, Q1, and Q2 are symmetric and positive semidef-inite matrices of appropriate dimensions. Our objective is tofind the optimal control strategy ut that minimizes the costfunction JN above.

Due to the separation principle [30, page 157] in the optimalLQG control (cf. Remark 1 at the end of this section), we firstobtain the optimum state estimate xt, and then the optimumcontrol vector

ut = −Ltxt, (48)

where Lt is the optimum feedback gain matrix

Lt = (BTSt+1B +Q2)−1BTSt+1A, (49)

and St is given by the recursive equation

St = ATSt+1A−ATSt+1BLt +Q1 (50)

for t = N − 1, . . . , 1 with the initial value SN = Q0. Notethat given the system matrices A,B, and the cost matricesQ0,Q1,Q2, using (49) and (50) the optimum feedback gainLt can be computed offline. Hence, in the optimal LQGcontrol only the state estimate xt is computed online.

Lemma 2. The systemwide optimum control input ut can bewritten as the sum of local contributions as follows

ut =

K∑k=1

ukt ,

where ukt is the local contribution from node k, given by

ukt = −Lt

(Y −1t

t−1∑s=1

Ψt−1s Y sA

−1Buks + xkt

). (51)

Proof: Using (13), (23) and (48), we can write

ut = −Lt

(ϕt +

K∑k=1

xkt

)

= −Lt

(Y −1t λt−1 +

K∑k=1

xkt

). (52)

The update rules for λt and ζt, given by (18) and (19),respectively, share the same structure. Thus, similar to (20),we can show that

λt =

t∑s=1

Ψts+1ξs (53)

where from (10), (4), (8) and the definition of Ωt in (15), wehave

ξt = ΩtY tA−1But. (54)

Combining (52), (53) and (54) we get

ut = −Lt

(Y −1t

t−1∑s=1

Ψt−1s Y sA

−1Bus +

K∑k=1

xkt

), (55)

Note that u1 =∑Kk=1−Ltx

k1 , and as a result, for all t, ut is

a function of∑K

k=1 xkt

t≥1

and globally known matrices.

Hence, (55) can be rewritten as

ut =

K∑k=1

−Lt

(Y −1t

t−1∑s=1

Ψt−1s Y sA

−1Buks + xkt

)

=

K∑k=1

ukt .

Note that each node can compute the matrix Lt using (49)and (50), as well as Y t,Ψ

ts,A,B. Hence, we propose that

each node k computes ukt as in (51), and reports it to thecontroller node using level-crossing sampling, as described inSection III-A for xkt . In particular, each node k runs p parallellevel-crossing samplers for uk,it

pi=1, where uk,it is the ith

entry of ukt . Let uk,iq denote the approximation of uk,it at thecontroller node during the time interval τk,iq ≤ t < τk,iq+1. Then,the controller computes the input vector ut = [u1t , . . . , u

pt ]T ,

where uit =∑Kk=1 u

k,iq , and applies it to the system.

Corollary 1. For each input uit, under ideal communications,the performance gap between the proposed decentralizedscheme and optimum centralized scheme is deterministicallybounded, i.e.,

|uit − uit| < K∆, ∀i, t, (56)

and yields order-2 asymptotic optimality, i.e., uit− uit = O(1)even if |uit| → ∞.

Proof of (56) is similar to the proof of Theorem 1. Moreover,since in a large system, in general, p n, this marks anotheradvantage of reporting ukt instead of φkt (or ykt or xkt ) in thedistributed LQG control.

Following a similar discussion to Theorem 2, it can beshown that as |uit| → ∞ the confidence interval |uit − uit| is

11

unbounded under non-ideal communication channels, henceorder-2 asymptotic optimality is not satisfied. However, for apractical scenario where |uit| < ∞ as t → ∞, we also have|uit| < ∞ for any p > 0. Similar to the arguments givenin Theorem 3, order-1 asymptotic optimality is possible, i.e.,uit

uit

= 1 + o(1) as |uit| → ∞, if uit → ∞ at a rate fasterthan t, that is, uit = ω(t). Since this condition is practicallyinfeasible, the modulation techniques presented in Section IVcan be used to improve the non-asymptotic performance.

Remark 1: Our discussion and results in this section arebased on the separation principle and system stability, whichare not guaranteed to hold in general, especially under noisycommunication channels. In particular, they require somespecial conditions, e.g., a minimum level of data rate anda maximum level of communication delay, as discussed inIntroduction in more detail. Hence, the performance shown inCorollary 1 is valid only if such conditions are satisfied.

Remark 2: Together with Theorem 2, the results presentedin Corollary 1 and the following discussions show that theproposed entire solution (i.e., decentralized state estimatorand LQG control scheme) is bounded-input bounded-output(BIBO) stable even under noisy channels as long as theoptimum centralized solution is BIBO stable.

VI. NUMERICAL RESULTS

In this section, we illustrate the advantages of the proposedschemes based on level-crossing sampling (LCS) throughnumerical results considering a practical electrical power grid.In particular, we examine the performance of the distributedLQG control scheme presented in Section V since it is apractical application that includes both state estimation andcontrol. Throughout this section we use the IEEE 57-busdata in MATPOWER [31] with n = 57 state variables,m = 80 measurements, p = 7 control inputs, and Q0, Q1,Q2, R1, R2,k equal to the identity matrices of appropriatesizes. In Fig. 7, in a system consisting of four nodes (i.e.,K = 4), in terms of mean squared error (MSE), givenby E[‖ut − ut‖22] =

∑pi=1 E[(uit − uit)

2], we compare theproposed scheme with the conventional schemes that reportzkt , φkt , ukt via uniform sampling (US) and quantization. Inthe literature, it is conventional that each node k transmits theraw measurement zkt or the information vector φkt , and thecontrol center computes the Kalman filter or the informationfilter, respectively. In that sense, a scheme that transmits thelocal control vector ukt based on (51) is not conventional. Itis conventional in terms of transmission method if it usesuniform sampling and quantization. For fair comparison, inthe proposed scheme, on average a single bit is transmittedper dimension per unit time, and in the other schemes basedon uniform sampling at each time a single quantization bit istransmitted per dimension.

A. MSE vs. Time

As shown in the top figure in Fig. 7, the “fully” conventionalscheme that transmits φkt via uniform sampling suffers a hugeperformance gap due to the insufficient one-bit representationsof the wide range of φkt values (see Fig. 8). We see in Fig.

0 10 20 30 40 50 600

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70 80 90 100

t

0

1

2

3

4

5

6

7

MSE

=E

[ku

t!

~utk

2 2]

#10 -3

US-zUS-uLCS

Fig. 7. Mean squared error vs. time for the proposed scheme based on level-crossing sampling (LCS) and the conventional schemes that transmit zkt , φkt ,ukt via uniform sampling (US) and quantization.

7 that the scheme that transmits zkt using uniform sampling,on the contrary, attains an acceptable performance. Since therange of zkt values is much smaller than that of φkt (see Fig.8), the approximate φ

k

t that is obtained from the recoveredzkt yields a better result than the quantized and recoveredφk

t . Note that the number of dimensions in φkt and zkt aren = 57 and m = 80, respectively. In other words, in theschemes US-φ and US-z, at each time 57 and 80 bits aretransmitted systemwide, respectively, whereas only p = 7bits are transmitted at each time in the scheme US-u andthe proposed scheme LCS-u (on average). We allow thisunfairness in Fig. 7 because LCS-u still perform better thanUS-φ and US-z.

Among the schemes that transmit the local control vectorukt , which is the final information entity in the system, weobserve in the bottom figure in Fig. 7 that the proposedscheme based on level-crossing sampling outperforms the“half” conventional scheme based on uniform sampling. It is“half” conventional because it unconventionally transmits ukt ,as we propose in (52). Quantization threshold of each uk,it isset to zero, and the two quantization levels are set as the meansof the positive (uk,it > 0) and negative (uk,it < 0) values,that are estimated offline. The fluctuations in the conventionalschemes are due to the coarse (one-bit) quantization. It is seenin the bottom figure in Fig. 7 that the performance improves

12

0 20 40 60 80 100 120t

-2

0

2#104

zk;it

?k;it

xk;it

uk;it

0 20 40 60 80 100 120t

-500

0

500

zk;it

xk;it

uk;it

0 20 40 60 80 100 120t

-5

0

5

xk;it

uk;it

Fig. 8. Sample paths of uk,it , xk,it , φk,it , and zk,it . It is seen that uk,it ismuch smoother than φk,it , zk,it (top and middle figures), and xk,it (bottomfigure).

0 10 20 30 40 50 60 70 80

K

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MSE

=E# P 1

00 t=1ku

t!

~utk

2 2

$

US-z systemUS-u nodeLCS systemLCS node

Fig. 9. Mean squared error vs. number of nodes for the resource constraintson the system (i.e., 80 bits per unit time systemwide) and on the nodes (i.e.,1 bit per dimension per unit time per node, which corresponds to 7K bits perunit time systemwide). With the increasing node diversity, the proposed LCS-based scheme, under both system and node resource constraints, significantlyoutperforms its counterparts.

by the transmission of a more finalized information entity, asanticipated in Section V.

B. MSE vs. Number of Nodes

We next discuss the effect of node diversity on the cumula-tive MSE performance

∑100t=1 E[‖ut − ut‖22] for the proposed

scheme and its counterparts based on uniform sampling. Asshown in Fig. 9, the performance of the conventional schemethat transmits zkt via uniform sampling is not affected by theincreasing number of nodes since the number of transmittedmeasurements is constant (m = 80 in this case) regardless ofthe number of nodes. For the controller, it does not matterwho sends the measurements as long as the measurements arethe same regardless of the number and identity of the senders.Hence, in this case, the number of bits transmitted systemwideper unit time, which is 80, does not change with the numberof nodes. Considering this case as a resource (communicationand energy) constraint on the system we use the name “US-z

system” to differentiate between the resource usages of thecontrol schemes.

In the scheme that transmits ukt using uniform sampling,more resources are used as the number of nodes increases.Specifically, each node k transmits 1 bit per dimension i ofuk,it : i = 1, . . . , p per unit time, resulting in Kp bits perunit time systemwide. Since this corresponds to a resourceconstraint on the nodes, we use the name “US-u node” toexpress its resource usage. Despite the increase in resourceconsumption, its MSE performance stays nearly constant withthe increasing node diversity, as shown in Fig. 9. At firstglance this may look counterintuitive as K represents thenumber of information sources. However, in this case, larger Kdoes not mean more information, but means more distributed(or decentralized) information. Specifically, larger K meansmore local signals ukt to report to the controller, hence morequantization losses unless the range of ukt shrinks as fastas 1/K. Apparently, for smaller K values, the range of uktshrinks slower than 1/K.

We next compare the proposed scheme that transmits uktusing level-crossing sampling with the conventional schemes“US-z system” and “US-u node” under the same systemand node resource constraints, respectively. Under the systemconstraints, in which 80 bits are transmitted systemwide perunit time, the performance of “LCS-u system” deterioratesas the number of nodes increases due to the decentralizationof the systemwide optimum control vector ut. However, asseen in Fig. 9, it still significantly outperforms “US-z system”even in the most decentralized case with 80 nodes (i.e., onemeasurement per node). Under the resource constraints oneach node, in which each node on average transmits 1 bitper dimension per unit time, the performance of the proposedscheme “LCS-u node” improves with the increasing nodediversity, as opposed its counterpart “US-u node” (see Fig. 9).This is because as K increases each local control input uk,itgets smaller values, and as a result we decrease the samplingthreshold ∆ to ensure 1 bit is transmitted on average for eachuk,it per unit time. Note that the performances of “LCS-usystem” and “LCS-u node” coincide at around K = 11 asexpected since the resource consumption becomes the same(i.e., 80 bits systemwide per unit time) for both.

C. Discussions

According to (56), under ideal communication channels, itis certain (not probabilistic) that the optimum control inputuit lies in the confidence interval (uit − K∆, uit + K∆),which shrinks with small ∆. Accordingly, in selecting the ∆value, there is a trade-off between performance and resource-efficiency. Specifically, small ∆ improves the performance, asstated in (56), but at the same time results in more frequenttransmissions [see e.g., (25) and (26) in the case of stateestimation], consuming more energy and bandwidth. Hence,∆ should be selected to strike a desired balance between theperformance and resource efficiency.

Considering the LQG cost, given in (47), as the performancemeasure for the control system, the expected tradeoff betweenthe control performance and the resource efficiency, based on

13

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

10

20

30

Bits

/dim

ensi

on/u

nit t

ime

Fig. 10. Tradeoff between the control performance and resource usage.Control performance of the proposed scheme is represented by the additionalcost JLCS100 − Jopt100 incurred due to decentralization at time 100 [see (47)].Resource usage is denoted by the average number of bits transmitted by eachnode per dimension per unit time.

the selection of the sampling threshold ∆, is illustrated inFig. 10. In the figure, the average number of transmitted bitsis given per unit time per node per dimension. For different∆ levels, J100, given by (47), is calculated for the proposeddecentralized scheme, and denoted by JLCS100 . The additionalcost in JLCS100 with respect to the optimum centralized scheme,denoted by Jopt100, in which all information is fully available tothe controller (i.e., ∆→ 0), is shown by JLCS100 − J

opt100 in the

figure.

Performance • minimizes the processing of lossy informationat the controller (see (56) and Fig. 7)

Resource-efficiency

(Energy &Bandwidth)

• smaller number of parallel samplers, and thustransmissions per node (e.g., p = 7 n = 57and m = 80 in IEEE 57-bus)

• smoother signal with smaller jumps, hencesmaller number of transmissions per sample(see Fig. 8 and (26))

TABLE IADVANTAGES OF REPORTING ukt INSTEAD OF zkt , φkt , xkt , GIVEN BY (1),

(12), (22), RESPECTIVELY.

In addition to ∆, the smoothness of uk,it determines thetransmission frequency. In particular, big jumps in uk,it maylead to frequent sampling and/or large number of transmissionsper sample, hence more resource consumption. Fortunately,uk,it , being a control input applied to the system, is muchsmoother than the observation zk,it , the information entity φk,it ,and the state estimate xk,it , as shown in Fig. 8. The advantagesof reporting ukt in the distributed LQG control instead of someearlier products in local processing, such as zkt , φkt , and xkt ,are summarized in Table I.

Finally, the effect of nonzero error probability, p, in termsof the additional LQG cost JLCS100 − JLCS100 , where JLCS100 andJLCS100 are the costs at time 100 [see (47)] for the proposeddecentralized scheme under reliable and unreliable channels,

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

p

0

2

4

6

8

10

12

14

16

BSCBEC

Fig. 11. Additional LQG cost vs. probability of error for BEC and BSC. Thesampling threshold ∆ is taken as 0.01.

respectively, is illustrated in Fig. 11. As expected, as the errorprobability p increases, the additional cost increases as well,i.e., the control performance degrades. Moreover, under BSC,the control performance degrades more compared to the BECcase since an incorrect update on the information entities ismore destructive on the system performance compared to theno-update case (i.e., ±2∆ error vs. ±∆ error).

VII. CONCLUSIONS

In this paper, the distributed dynamic state estimation andLQG control problems have been analyzed for a networkedcontrol system using an event-triggered sampling schemecalled level-crossing sampling. The advantages of processingthe information locally and sampling it with level-crossingsampling have been shown through theoretical analysis andsimulations. Particularly, it has been shown that the informa-tion loss in the proposed decentralized schemes has a deter-ministic upper bound, and thus yields an order-2 asymptoticoptimality under reliable communication channels. Further-more, in transmitting information from sensors to the con-troller, noisy channels have been considered and correspondingoptimal modulation techniques have been proposed to improvenon-asymptotic performance of the proposed schemes.

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Yasin Yılmaz (S’11-M’14) received the B.Sc.,M.Sc., and Ph.D. degrees in Electrical Engineeringfrom Middle East Technical University, Ankara,Turkey in 2008, Koc University, Istanbul, Turkeyin 2010, and Columbia University, New York, NY,in 2014, respectively. He is currently an AssistantProfessor of Electrical Engineering at the Universityof South Florida, Tampa. He received the Collabo-rative Research Award from Columbia University in2015. His research interests include statistical signalprocessing, machine learning, and their applications

to cybersecurity, IoT networks, social networks, communication systems, andcyber-physical systems.

Mehmet Necip Kurt received the B.S. and the M.S.degrees in 2014 and 2016, respectively from the De-partment of Electrical and Electronics Engineeringat Bilkent University, Ankara, Turkey. Currently, heis working towards the Ph.D. degree in electricalengineering at Columbia University, New York. Hiscurrent research interests include statistical signalprocessing with applications in cyber-physical sys-tems.

Xiaodong Wang (S’98-M’98-SM’04-F’08) receivedthe Ph.D degree in Electrical Engineering fromPrinceton University. He is a Professor of ElectricalEngineering at Columbia University in New York.Dr. Wang’s research interests fall in the general areasof computing, signal processing and communica-tions, and has published extensively in these areas.Among his publications is a book entitled “WirelessCommunication Systems: Advanced Techniques forSignal Reception”, published by Prentice Hall in2003. His current research interests include wireless

communications, statistical signal processing, and genomic signal processing.Dr. Wang received the 1999 NSF CAREER Award, the 2001 IEEE Commu-nications Society and Information Theory Society Joint Paper Award, andthe 2011 IEEE Communication Society Award for Outstanding Paper onNew Communication Topics. He has served as an Associate Editor for theIEEE Transactions on Communications, the IEEE Transactions on WirelessCommunications, the IEEE Transactions on Signal Processing, and the IEEETransactions on Information Theory. He is a Fellow of the IEEE and listedas an ISI Highly-cited Author.