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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 1577
Adaptive Kalman Filtering in Networked SystemsWith Random Sensor Delays, Multiple PacketDropouts and Missing Measurements
Maryam Moayedi, Yung K. Foo
, Member, IEEE
, and Yeng C. Soh
Abstract—
In this paper, adaptive ﬁltering schemes are proposedfor state estimation in sensor networks and/or networked controlsystems with mixed uncertainties of random measurement delays,packet dropouts and missing measurements. That is, all three un-certainties in the measurement have certain probability of occur-rence in the network. The ﬁlter gains can be derived by solving aset of recursive discrete-time Riccati equations. Examples are pre-sented to demonstrate the applicability and performances of theproposed schemes.
Index Terms—
Kalman ﬁltering, minimum mean-square errorestimation, missing measurements, networked control systems(NCSs), packet dropouts, sensor delays, sensor networks (SNs).
I. I
NTRODUCTION
I
N a networked control system (NCS), the communicationanddata networksform an integralpartofthesystemwherethecontrolloopisclosedviaacommunicationnetworkchannel.And in a sensor network (SN), which is a network of indepen-dent sensors, the measured data are sent to the estimator, mon-itoring station, or the control station via a communication net-work, usually wireless.While using a communication network in NCS or SN offersmany advantages such as simpler installation, easier mainte-nance, and lowercost [1], it also leads to other problems such asintermittent packet losses and/or delays of the communicatedinformation, [3]. There is another uncertainty that may bepresent in the data received from the network: that is wherethe data packet contains noise only (i.e., the measurementhas missing observations) and the estimator is not capableof directly distinguishing between such packets and packetscontaining valid measurements, [2], [3]. For example, this may
occur in tracking systems [2]. Therefore, it is not surprising thatthe robust state estimation problem involving communicationnetworks has recently attracted considerable attention frommany control researchers.
Manuscript received March 12, 2009; accepted November 05, 2009. Firstpublished December 04, 2009; current version published February 10, 2010.The associate editor coordinating the review of this manuscript and approvingit for publication was Prof. James Lam.M. Moayedi and Y. C. Soh are with the School of Electrical and ElectronicEngineering, Nanyang Technological University, Singapore 639798 (e-mail:mary0008@ntu.edu.sg; eycsoh@ntu.edu.sg).Y.K. Foo is with LW Electrical and Mechanical Engineering Private, Ltd.,Singapore 608608 (e-mail: fooyk@leunwah.com.sg).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TSP.2009.2037853
In general, two main approaches are popular for modelingthe uncertainties. The ﬁrst one is to model the uncertainty by astochasticBernoullibinaryswitchingsequencetakingonvaluesof 0 and 1, [4]. We shall refer to this approach as the stochastic-parametermethod.Thesecondapproachistouseadiscrete-timelinear system with Markovian jumping parameter to representthe random uncertainties, [5]. We shall refer to this approach asthe Markov chain approach. There is another approach wherethe missing data are replaced by zeros and an incompletenessmatrixisthenconstructedinthemeasurement[6].However,thisapproach does not appear to be very popular.There are already many available results on control and stateestimation in NCS and/or SN context. The interested readersmay refer to [1]–[22] and the references therein for further in-formation. We shall review only those works that are closelyrelated to the current work here.The state estimation problem for networked systems withonly one of the aforementioned uncertainties has been studiedextensivelyin the past (see, e.g., [3], [7] and references therein).
For example, Nahi in 1969 [2] ﬁrst developed an optimal re-cursive ﬁlter for systems with missing measurements. In thatpaper, systems with random missing measurement were mod-eled by a binary Bernoulli stochastic parameter and the ﬁlteris derived via solving two Riccati equations. In [9] and [5]
the ﬁltering problem with missing measurements was alsoinvestigated. However, instead of using a stochastic parameterto model the uncertainty, a two-state Markov chain was usedto probabilistically characterize the missing measurements.Thus the measurement loss process was modeled as a Mar-kovian jump linear system [9] and a suboptimal “jump linearestimator” was presented where at each time step, a correctorgain is selected from a ﬁnite set of precomputed ﬁlter gainsand hence the ﬁlter design consists of choosing the switching logic, determining the size of this ﬁniteset and assigning the ﬁlter gains . In [5], the authors employa Riccati equation approach (to compute the ﬁlter) assumingthat the transitional (conditional) probabilities for transitionsfrom one Markov-state to another are known. From a NCS orSN point of view, requiring the knowledge of the transitionalprobabilities may not be too satisfactory because they (forexample, the probability of the next packet arriving will be apacket containing a current measurement given that the currentreceived packet contains a delayed measurement versus theprobability of the next packet arriving will be a packet con-taining a current measurement given that no packet has beenreceived at the current sampling time) may be difﬁcult to be
1053-587X/$26.00 © 2010 IEEE
1578 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010
determined or estimated. In [8], the least mean square ﬁlteringproblem for systems with one random sampling delay has beenstudied using the stochastic parameter approach and an LMIapproach is used for the ﬁlter derivation. Results have also beenreported regarding ﬁltering in networked systems with packetdropout; see [3], [10], and [11]. In [11], an optimal ﬁlter, in
the Kalman sense, for systems with multiple packet dropoutswhere the number of consecutive packet dropouts is limitedby a known upper bound has been proposed. The uncertaintymodel is, again, based on the stochastic parameter approachand the ﬁlter design is based on the Riccati equation. In [10],optimal estimators, which include ﬁlter, predictor and smootherare developed based on an innovation analysis approach andusing stochastic parameter uncertainty model. The estimatorsare computed recursively in terms of the solution of a Riccatidifference equation. In [3], by introducing a new notion of stochastic -norms, the ﬁltering problem involving sensordelay, multiple packet dropouts, and uncertain observations aremodeled by using a stochastic parameter and all treated in auniﬁed framework. A steady-state ﬁlter is then designed via anLMI approach. While the result reported in [3] can handle thecases in which there is a possibility of sensor delay, or packetdropout, or missing measurement in data transmission throughthe network, it is assumed that the packet may be subjectto only one type of the aforementioned uncertainties duringtransmission in the network channel. In other words, the caseof mixed uncertainties is
not
admissible.There are also some recent works that have considered thestate estimation problem in NCS/SN with two random uncer-tainties. In [12], the robust estimation for uncertain sys-tems with signal transmission delay and packet dropout wasconsidered. However, in their approach, the ﬁlter designed isessentially a continuous-time ﬁlter fed-on by an
event-driven
zero-order hold (ZOH). In [1], the ﬁlter design problem isstudied for a class of networked systems where measurementswith random delay and stochastic missing phenomenon (whichis essentially equivalent to the missing measurement or uncer-tain observation phenomena considered in [3] and [10]) are si-
multaneously considered. In [13], the optimal estimation in net-worked controlsystemssubject torandomdelay andpacket lossas well as the stability analysis of the estimator designed hasbeen investigated.To the best of our knowledge, the only work which con-siders the ﬁltering problem for NCS/SN with mixed uncertain-ties, where all of the three aforementioned uncertainties, i.e.,random sensor delay, packet dropout and uncertain observation(missing measurement) are admissible in the data received fromthe network, was presented in [14]. The result of [14] is, how-
ever, based on the LMI approach. This renders it quite unsuit-able forapplications thatcallfor onlinecomputationoftheﬁltergains. For online computation of ﬁlter gains, a Riccati equationapproach is more desirable. This motivates our present work.This paper deals with discrete-time partially observed linearplants where the observations are communicated to the esti-mator via an unreliable channel with possibilities of randomdelays and/or packet dropouts. We also admit the possibilitythat the packet received by the estimator consists of the noiseonly [2]. Therefore, we consider the problem of robust min-imum-varianceﬁlteringinthepresenceofmixedrandomsensordelays, packet dropouts and missing measurement uncertaintieswhere all three types of uncertain observations (sensor delay,packet dropout and missing measurement) can occur in thesystem. A Riccati-like equation approach is adopted here. Thismakes the ﬁlter designed suitable for online applications. Twoadaptive ﬁltering schemes are proposed. In the ﬁrst scheme,we make a distinction between the packet dropout case andthe other two uncertainties. The second scheme is a simpliﬁedscheme that aims to maximize the online computational speed.The organization of the paper is as follows. In the nextsection we present the various state equations used to modelthe uncertain system with measurement delay, packet dropoutand missing measurement. We then show how all these“sub-models” may be combined via Markov chain to modelthe whole uncertain system. In Section III, the general formulawhich is applicable for one-step predictions with sensor delayand missing measurement is derived. In Section IV, we con-sider the problem of one-step prediction with multiple packetdropouts. An adaptive ﬁltering (“Adaptive ﬁlter”) schemewhere we distinguish the packet dropout case from the othertwo uncertainties is developed in Section V. Section VI pro-
poses a simpliﬁed version of the ﬁlter of Section V; the aimis to develop a ﬁlter (the “Simpliﬁed ﬁlter”) that can be easilyand efﬁciently implemented. In Section VII, we discuss thecase of the state estimation with multiple (and possibly non-identical) sensors. Section VIII contains some examples andsimulation results, and we ﬁnally give our concluding remarksin Section IX.The main contribution of the present paper is the develop-ment of a Riccati-like equation approach to the ﬁlter design in anetworked system with mixed uncertainties involving randomsensor delay, missing measurement and packet dropout. Thismakes the ﬁlter more suitable for online
adaptive
applications,since for online computation of the ﬁlter gains a Riccati equa-tion approach is more desirable than an LMI approach.II. S
YSTEM
M
ODELING AND
P
ROBLEM
F
ORMULATION
Consider the following discrete time
linear time-varying
state-space system:(1)(2)where is the state vector, is the measured output, andand are stationary, zero-mean discrete-time whitenoise processes with covariance matrices:(3)and the initial condition satisfying the mean and covarianceconditions:(4)Weassumethattheplantisasymptoticallystableandobservablefrom the measured output .
MOAYEDI
et al.
: ADAPTIVE KALMAN FILTERING IN NETWORKED SYSTEMS 1579
Systems with mixed uncertainties of packet delay, missingmeasurement, and packet dropout may be represented by amodel of the form:(5)of compatible dimensionwith (6)where we have deﬁned(7)and .Let , , de-notethefourmodelscorrespondingrespectivelytosystemswithno uncertainty, sensor delay, missing measurement and packetdropout. These are deﬁned by the following system matrices:
Current measurement
(i.e.,
no uncertainty
):(8)(9)of compatible dimensions with(10)
One-step sensor delay
:(11)(12)of compatible dimensions with(13)
Missing measurement
:(14)(15)of compatible dimensions with(16)and
Packet dropout
:(17)(18)(19)We may then represent asshown in (20) at the bottom of the page.
Remark 1:
There may be some confusion between packetdropout and missing measurement in the literature. In our con-texthere,wedeﬁnemissingmeasurementasonewherethemea-surement is missing before encapsulation into packets. In otherwords,themeasurementitselfisnotavalidone,containingonlynoise (and the estimator is not able to distinguish such error by,for example, examining the “error detection” bits). On the otherhand, a packet dropout is one that occurs at the ﬁlter end.Assumethatbycarefullyanalyzingthesystem, andempiricalexperimentations and observations, we are able to adequatelymodel the real system by assigning to each of the four modelsitsprobability of occurringat time . Let theprobability thatthesystem at time is be givenby .Obviously, .Let . We wish to constructan estimator of the form(21)to generate that minimizeswhere . Note that in (21) maybe considered an estimate vector for .Therefore, an obvious choice for isIn what follows, the arguments of a function may sometimesbe omitted for notational simplicity when there is no dangerof confusion. The reader should note that the system and theprobabilities may be time varying.III. O
NE
-S
TEP
P
REDICTIONS
W
ITH
S
ENSOR
D
ELAYS AND
M
ISSING
M
EASUREMENTS
In this section, we consider only and (i.e., we as-sume cannot happen). The problem of multiple packetdropouts is an exceptional case which warrants separate consid-eration.Let such that .Observing that the right “block column” of is 0, we obtainor (20)
1580 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010
. Similarly, since the left-most block-columnof is zero, we obtain . Left-multiply (5) by , subtract (21) from it, and add
to obtain(22)(23)where .Given and , and since , the propa-gation of the covariance matrices for may be described bythe equation(24)where denotes the covariance matrix of (where ex-pectation operation is taken over both and ) and we havechosen(25)
Remark2:
Intheabsenceofuncertain asinthecaseofstan-dard Kalman ﬁltering, covariance matrices may be obtained bytaking expectation over the uncertain noise. In our case, wherethere are both uncertainties in the noise and , covariance ma-trices have to be deﬁned by taking expectation over both thenoise and in order to reﬂect the true probability distributionof . Hence, we can writeTheproblemof minimizing maybe posedassubject to (24) (26)Let . Left and right multiply (24) by andrespectively, we obtain(27)Differentiate with respect to and set thederivative to zero, we obtain the optimal(28)And the optimal may be chosen as(29)Since is the covariance matrix of , it can be computed from(30)Hence, (27) and (30) give a set of recursive discrete-time Ric-
cati-like equationswhich maybe appliedto computethecovari-ancematricesoftheestimationerrorforany(i.e.,notnecessarilyoptimal) . To obtain the relevant formula for measurementupdate, we let and in (27) to obtain(31) and (32), shown at the bottom of the page.
(31)(32)

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