SICE-ICASE International Joint Conference 2006Oct. 18-2 1, 2006 in Bexco, Busan, Korea

Networked Kalman Filter with Sensor Transmission Interval Optimization

Le Minh Khoa Do, Young Soo Suh, and Vinh Hao NguyenDepartment of Electrical Engineering, University ofUlsan

Namgu, Ulsan, 680-749, KoreaPhone: +82-52-259-2196, Fax: +82-52-259-1686

E-mail: suhgieee.org

Abstract: Sensor transmission interval optimization problem is considered for network monitoring systems. Underlimited network bandwidth, a method to choose an optimal transmission period for each sensor is proposed based on theperiodic solution of Kalman filters. Through simulation and experiments using CAN network, the proposed method isverified.

Keywords: Kalman filter, Multirate estimation, Networked control systems, Rate monotonic scheduling.

1. INTRODUCTIONIn this paper, we consider networked monitoring

systems, where function of sensor nodes is limited toperiodic transmission of sensor data without computa-tional power. In this situation, design parameters aretransmission period of each sensor and schedulingmethods. The rate monotonic algorithm is used as thescheduling method. The performance of estimation isdefined using the periodic solution of Kalman filters.Given the bandwidth allowed for sensor datatransmission, transmission period of each sensor isoptimized.

The paper starts with Section 2, where continuousand discrete time system models are derived. Section 2also proposes a "Sensor Transmission IntervalOptimization" based on a simple "Multirate OptimalState Estimation" method in [5] and an iterativelinearization technique in [6]. In this section, we suggestan estimation performance index relative to errorcovariance matrix in order to find an optimal set ofsensor transmission periods. In Section 3, we mentionabout network schedu- ling constraint. A way to limitregion for searching an optimal set of sensortransmission periods is discussed in Section 4. Animplementation with CAN network is achieved inSection 5 to prove our proposed method. Finally, aconclusion is presented in Section 6.

2. SYSTEM MODELINGConsider the following continuous time system

x(t) = ACx+ Bc (t), (1)

y(t) = Cx(t) + v(t).where x E R' is the state and y E RP is the measure-

ment output. Process noise 0o(t) and measurement

noise v(t) are uncorrelated, zero mean white Gaussianrandom processes satisfyingE{co(t)co(s)'} = Qd(t - s),E{v(t)v(s)'} = Rd(t - s), (2)

E{co1(t)vj(s)}=0, 1<i<n,1< j< p.

where cti and Vj are i-th element of a and j-thelement of v, respectively.

Output y(t) is transmitted through a serial network tothe estimator board, where state x(t) is estimated, as inFig. 1.

Serial network

Sensor 1 Sensork2 Sensor p Estimator

I ~~~~~~~~~board

PLANT

Fig. 1 Networked monitoring system.

The following assumptions are made on the networkdata transmission:

1. Output yi (1 < i < p ) is transmitted periodically tothe estimator board with the period T,.

2. Non-preemptive, priority-based transmission isused.

3. Sensor data packet sizes for all output are identical.Physical packet transmission time without delay isalso identical and denoted by Tp.

The assumption 1-3 are not restrictive and can besatisfied if, for example, CAN is used as a serialnetwork: rigorously speaking, in the case of CAN, Tp] isnot constant even if the data packet sizes are identicaldue to the bit stuffing. The difference in transmissiontime due to bit stuffing, however, small and can beignored.

The transmission period of i-th output (T) is chosenso that it is an integer multiple of a base sampling timeT; that is, Ti = NiT, where Ni is an integer. In theestimator board, the system state is estimated using thediscrete Kalman filter.

The discretized system of (1) with the samplingperiod Tis given as follows:Xk+l =AXk + Ok (3)

Yk =CXk +Vk.

where:A = eAT (4)

89-950038-5-5 98560/06/$10 © 2006 ICASE1047

k=

T

:1~~~~ 3 4 5 6 7

y1output v zp

transmission VfY2output KC

transmission

C C 0 C , 3Cq C l 0 Ck~~~~~Rk R

Fig. 2 Kalman update timing.

Qk = E[wkwj]= f e BcQB'e dr, (5)

Rk = E[vkv *. (6)Since Ti = NiT, the measurement equation is

time-varying unless Ni=1 for all i. For example, ifp=2,N1=2, N2=3 and

C=[Q]R=0 0]

where Ca is the first row of C and C. is the secondrow of C.

As can be seen in Fig. 2, Ck and Rk are periodicallytime-varying with the period NT, where N=6 is the leastcommon multiple ofN1=2, N2=3.

Since the networked monitoring system is period-ically time-varying, there is no steady-state errorcovariance P Instead, error covariance becomesperiodic with the period N, where N is the least commonmultiple of N , Let (P ,P2 PN) bethe periodic error covariance matrices, then they shouldsatisfyPl = APNA'-APN C (CNPNC+RN) CNPNA+Qk

P2 = AP1 A'- APF C[(C1Pl C+ RI) lClPl- A'+ Qk ,

PN APNA' APN-1C-1(CN-1PN-1CN1 + RN-1)1, (7)X CN_1PN-1A'+ Qk .

A way to solve set of equations in (7) is to use aniterative linearization technique in [6].

The linearization of (7) givesP = APA'+ SRS'+Qk,S = APC'(CPC + R)-1, (8)A =A-SC.Using the linearized equation, we can apply the

following iterative linearization techniquePN (0) = I'

SN (0) so that (A - SN (O)CN) is stable,AN (O) = A SN (O)CN,PI (0) = AN (O)PN (O)AN (0) + SN (O)RNSN (0) + Qk' (9)

S1 = APN (O)C1(C1PN(O)C + R )

Al = A - SI (0)CI

Now, we define the estimation performance using thesteady-state periodic error covariance (P1-, P2 ..., PN)

f(NI, N2, ... Np) =max(trace(P, )) (10)1< i <N

3. NETWORK SCHEDULING

Considering our problem, for each output,transmission period and priority should be assigned.Transmission period design issue will be discussed inSection 4 and in this section, priority assignment isdiscussed. Once transmission periods Ti (1 < i < p ) aredetermined, we use rate monotonic algorithm, in whicha packet with shorter transmission period is given ahigher priority than a packet with a longer transmissionperiod. This algorithm is known to be optimal amongfixed scheduling algorithm [3]. By schedulablecondition, it means that transmission delay of eachoutput packet does not exceed its transmission period.

Theorem 1 [4] Suppose that outputs are arranged sothat T.< T2 < ... < TP . The i periodic packet transmis-sions satisfying assumption 1-3 are schedulable if T1satisfy:

Tpt + Tpt + + pt + Tpt < i(21Xi-1), I < i < p .(II)T, T2 Ti Ti

For example, the highest priority output 1 shouldhave a period larger than 2Tp, from Theorem 1. This 2Tp,is for packet transmission time TpI and transmissiondelay by other packet, which could be as long as Tp.Note that a higher priority packet can be delayed bylower priority packets since a non-preemptivescheduling is used.

4. SENSOR TRANSMISSION INTERVALOPTIMIZATION DESIGN

4.1. Compute estimation performance (cost) of set(N1, N2, ..., Np) and limit the searching region

Now we consider a situation when N, changes. Basesampling time T changes lead to measurement errorcovariance varies. Function f (N1, N2 ... NP) in (10) isused to calculate estimation performance index. In ournetwork model, which Ni the best value is observed.

Let NN be a set of k integer numbers, andN, E NN, I < i < p. Set (N1, N2 ,--,NP) is an arbitrary

1048

I OC Ir2 I OC

combination of N1. Note that there are kP combina-tional sets. In addition, let fj(NI, N2,... Np) be theestimation performance of the j-th combinational set of

(NJ, N2 ,.., Np), 1 < j < kV. The optimal performanceindex is defined as follows

foptimal(NI, N2,. N,) = min fj (NI, N2...Np). (12)1<j<k n

We also define a limited utilization as follows

U(NN2 ..IN,)= PT (N + +... + ) (13)T N1 N2 N

We expect to find a good performance index toevaluate the result. Fig. 3 below shows an exampleresult of all combinational sets of (N1, N2) with NN[1,2, 3, 4,6, 8, 12,24].

Estimation performance index: f(N1 N2) = max(trace(P1))

-r r -

0o0360.036

0025.

0.026

0.015.

0.016

o.06630

6 g

Fig. 3 Cost (performance index) of all sets (N1, N2).

Our problem is how to find the optimal set of(N1, N2 ,.., Np) that satisfies a limited network utiliza-tion and has a minimum cost. There are so many sets

(kP sets, as mentioned above). It takes a long time tosearch all sets to find an optimal cost (a minimum one)that satisfies utilization condition. Therefore, the regionof searching should be limited.

Intuitively, from Fig. 3 we can see that cost is ininverse proportion to utilization: cost increases whenutilization of set decreases (i.e. N1, N2 increase).Therefore, we believe that the optimal set will hasutilization equal or near limited utilization. Thus, weonly need to search sets that have utilization in adjacentregion with limited utilization. Let U1 and

U2 < p(2 "-1) be the lower bound and the upperbound (limited) utilization, respectively. We see thatU(NI, N2,.., N'p) in (13) must satisfyU1 < U(N1, N2,-,Np) < U2 The limited region forsearching is (Ul, U2).

4.2. Method of finding the optimal setThe transmission periods of sensor nodes are

important information because they determine both

network utilization and estimation performance. Theseperiods are now given implicitly by the method "SensorTransmission Interval Optimization". The method looksfor a set (N1,N2,..,Np) that satisfies utilization condi-tion and has minimum cost. The problem of this methodis the range for searching over to find this set is wide,especially when number of sensors increases. If allcombinational sets are considered, it will take a longtime. We suggest a method in order to limit searchingregion.

Here is the process:- Find all sets (N1, N2,.., Np) satisfying rate mono-tonic scheduling condition in (11):

( 1+ +. 1 + )<i(2 -1). (14)T Ns1 Ns2 Nsi Nsi

with 1<i<pandNsisaset (N1,N2,.., Np) that isarranged in order so that Ns,< Ns2< ... <Nsp.- Calculate utilization:

U(NI,N2,-Xp)= Pt ( 1+ 1 +~~1'2''~'T~ N1 N2 N

- Get all potential sets that satisfyUl < U(N1,N2 ..,Np) < U2 (15)

- Calculate f(NI, N2 ...Np) (estimation perfor-mance ) of all set belonging to (15).- Find optimal set satisfy (12).

Example: Two sensors case, NN [1, 2, 3, 4, 6, 8, 12,24], T=500,us, Tpt=160,us, U1=0.2, U2=0.45 (in orderto visualize easily, we choose a wide range ofutilization). Here are all sets that satisfy condition (14)&(15).

Table 1 All sets (N1, N2) having utilization satisfies (14)&(15).

Order N1 N2 U Order N1 N2 U1 2 8 0.2000 13 1 18 0.33782 8 2 0.2000 14 18 1 0.33783 2 6 0.2133 15 1 12 0.3467

4 3 3 0.2133 16 12 1 0.34675 6 2 0.2133 17 1 8 0.36006 2 4 0.2400 18 8 1 0.36007 4 2 0.2400 19 1 6 0.37338 2 3 0.2667 20 6 1 0.37339 3 2 0.2667 21 1 4 0.4000

10 2 2 0.3200 22 4 1 0.400011 1 24 0.3333 23 1 3 0.4267

12 24 1 0.3333 24 3 1 0.4267

In the Fig. 4:+ Number of sets havingcondition 0 < U < 0.45: 78+ Number of sets havingcondition 0.2<U<0.45: 24

Each point in Fig. 4 presents

utilization satisfies(i)

utilization satisfies(ii)

for a set (N1, N2)satisfying (i). The region which is displayed in large red(*) is the region containing potential sets (ii). We seethat (ii) takes about 1/3 time compare to (i) to find a setthat has an optimal cost.

1049

PC

Searching region

X *+ + + * + + +

WX+ + +*

5. EXPERIMENT SETUPAND RESULT

5.1 Experiment modeling:

5.1.1 Hardware:

To verify the proposed method, an experiment was

done using CAN (Controller Area Network). The

experiment setup is given in Fig. 5. The plant is

emulated in PC using Matlab real-time workshop. Three

AT90CAN128 microcontroller boards are used as sensor

boards. We do not use the real sensors in the sensors

boards but use A/D converters as sensor and sensor data

are generated by 0 card (NI-6025E and PCI- 1711) in

PC.

Because the base-rate sampling time is very small

(500,us), the model could not estimate data immedi-

ately after receiving sensor data from network. Instead,

sensor data is stored at collector (TMS320F2812 with

256KB RAM). When all sensor nodes on network finish

transmitting sensor data from PC, collector will send

these data and accessories to PC through COM port,

where Kahman filter estimation is achieved. Error covar-

iance and err or with original data are calculated in order

to verify the result.

Fig. 5 Experiment hardware setup with three sensor nodes.

5.1.2 Flow of processFig. 6 show the flow of the process in the

experiment.PC generates analog signals for three sensor nodes

and synchronizing signal as base-rate sampling time. PCalso stores data that it generated in order to calculateerror of estimation data by Kalman filter later.

Sensor nodes receive synchronizing signal fromcomputer, check whether waiting time equals to sensor

transmission period, if equals, convert analog to digital,and send to collector through CAN network.

Collector receives synchronizing signal fromcomputer, saves last sensor data, which it received fromnetwork, and accessories information into memory.

After finishing all collected data (5 seconds),collector will transmit these data to PC through COMport. Then we use Kalman filter to estimate analog datathat generated by PC and check error and error

covariance. The optimal set of sensor periods, whichsatisfies network utilization, will have smallest one.

5.1.3 State-space equations:We use this simple state-space equation for

experiment:

F0 1 0 Olx(t)= 0 0 1 x(t)+ 0 0c(t),

00 0 L1i (16)

-1 0 O-y(t) = 0 1 0 x(t) + v(t).

O 1State-space is defined as follows: x=[position,

velocity, acceleration]'.

Example: N1=1, N2=2,N3=3

Save SaveData Data

SaveI, m2~ Data] ml~

2T

Save2 ave

Data ~ [Data]3T 4T

Save ml m2 m3

Data

5T

At rising or falling edge of synchronizing signal from P/O card, nodes on network start theirs jobs. Collector will save last

received data, sensor nodes will start A/D conversion and send this digital data to network. E is the process at collector to save

network sensor data that it received. 23 is the time for transmitting message mi on CAN network. As we see, E" has N1 equals toone, T1 equals to T, this message takes the highest priority network ID (rate monotonic scheduling), so it is transmitted before others

and appears in every cycle. E has N2 equals to 2, T2 equals to 2T, this message takes the second highest priority network ID and is

transmitted after LE] finished.

Fig. 6 Synchronizing signal from PC and flow of process on network and at collector.

1050

0L---i L---i j;-.L---i L---i L---i

10

We use the following parameters for the experiment:-0.3 0 0-

Q=0.5,;R= 0 0.3 0 T=0.0005s. (17)O O 0.1

Measurement output is a three dimensional vector [yl,Y2, y3] representing for position, velocity, and acceleration sensor output, respectively. Measurement outputshave different sampling rate periods.

5.1.4 Network setup:There are three sensor nodes on the network. Here

are some properties of them:± T= 500,cus (Kalman time-base)± CAN speed: 500kbps. Each message has 80 bits,

include stuffing-bits. Thus, the time transmission of onemessage Tpt equals to 160,us .

+ Utilization: suppose that network utilization mustbe lower than U2=0.54. Choose Ul = 0.8xU2 = 0.432.So potential sets (N1, N2, N3) will satisfy condition:0.432<U<0.54+ T =N1T;TT2=N2T; T3=N3T; (period of sensors

transmission)+ Network priority: comply with rate monotonic

scheduling, shorter transmit period takes higher networkpriority.+NN [1,2,3,4,6,8, 12,24].Here are first 34 sets of (N1, N2, N3) as we used

"Sensor Transmission Interval Optimization" method tosearch for the optimal cost (by Matlab programming).

Table 2 Ranking of sets (N1, N2, N3) by cost (ascending).

Order OrdernubrN,N2N3 U Cost nubrN,N2N3 U Costnumber 23number 231 2 6 1 0.5333 0.005474 18 6 4 1 0.4533 0.0057392 2 8 1 0.5200 0.005482 19 8 3 1 0.4667 0.0058013 2 12 1 0.5067 0.005490 20 8 4 1 0.4400 0.0058474 2 24 1 0.4933 0.005499 21 122 1 0.5067 0.0058705 3 3 1 0.5333 0.005524 22 12 3 1 0.4533 0.0059686 3 4 1 0.5067 0.005543 23 24 2 1 0.4933 0.0061867 3 6 1 0.4800 0.005564 24 24 3 1 0.4400 0.0063498 3 8 1 0.4667 0.005576 25 1 6 2 0.5333 0.0076429 3 121 0.4533 0.005588 26 1 8 2 0.5200 0.00765010 4 3 1 0.5067 0.005590 27 1 12 2 0.5067 0.00765711 3 24 1 0.4400 0.005602 28 1 24 2 0.4933 0.00766512 4 4 1 0.4800 0.005615 29 2 2 2 0.4800 0.00769113 4 6 1 0.4533 0.005643 30 6 1 2 0.5333 0.00778714 6 2 1 0.5333 0.005647 31 8 1 2 0.5200 0.00784715 4 8 1 0.4400 0.005659 32 12 1 2 0.5067 0.00794516 6 3 1 0.4800 0.005704 33 24 1 2 0.4933 0.00816617 8 2 1 0.5200 0.005730 34 1 3 3 0.5333 0.009387

takes about 15.4% of time compares to (i) to find anoptimal set.

5.2. Experiment result:To verify the result clearly, we choose three sets for

the experiment. Sensor data are generated and stored inPC (we used the same data for all sets to guarantee theresult). The results are verified by error and errorcovariance of estimation at Kalman filter framework.

x 10 Error cova riance with NN=[2 6 11, trace(Pii)= .0067641

2

P1 1=00002791

0 0.6 1.6 2 2.5

x lo-,

k 4

3.6 4 4.5 5

P22=0. 0001 0037OI0 0.5 1.5 2 2.5

0.01 _

m 0.005 _-

3.5 4 4.5 5

P33=00063766

0 0.5 1.5 3.5 4 4.5

x10-4 Error covariance with NN=[2 2 2], trace(Pii)=0.008536841

2

P1 0000274, , I,

0 0.6 1.6 2 2.5 3 3.5 4 4.6 6

X 104

41P22=0.0001 5553

0 1I 'I0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0005m0.00=0008I

0 0.6 1.6 2 2.5 3 3.5 4 4.6 6

X 10-4 Error covariance with NN=[1 3 3], trace(Pii)=0.0110652

Pll 008

41L0 0.5 1.5 2 2.5 3 3.5 4 4.5 5

X l1o 44

2xi

- ~~~~~~~~~~P22=0.0001 92]0 0.6 1.5 2 2.6 3.5 4 4.6 5

0.02

l.001

00 0.6 1.5 2 2.6 3.5 4 4.6 5

Fig. 7 Error covariance of sensor data.

- Number of sets havingcondition 0 < U < 0.54: 475- Number of sets havingcondition 0.432 < U < 0.54: 73

utilization satisfies

utilization satisfies(ii)

Look at the Table 2, the set (N1, N2, N3) that hasoptimal cost is (N1, N2, N3) = (2, 6, 1) with U= 0.5333and minimum cost equals to 0.005474013.

There are the first 34 sets ranked by cost. We see thatthe number one has the lowest cost. After doing thesimulation, result shows that there are 475 setssatisfying condition 0 < U < 0.54 and 73 setssatisfying condition 0.432 < U < 0.54 . Therefore, (ii)

Look at three figures above. The title of each figureshows trace(max (P)) value of each set (N1, N2, N3).Checking the first figure, trace (max (P)) equals to0.005756, a little larger than simulation result (on theTable 2, cost of this set equals to 0.005474). Althoughset (2, 6, 1) has the same network utilization with set (1,3, 3), equals to 0.5333, set (2, 6, 1) takes the optimalcost. All costs of other sets are in desired order. It showsthat the experiment result verifies our simulation.

Another thing we can see that set (2, 2, 2) hasutilization smaller than set (1, 3, 3) but its estimationperformance is higher.

1051

n _--

2

I~~~~~~~~~~~~~~~~~~~~~~~~~

n n r

Errorwith NN=[2 6 1 trace(cov)=0.00514310.02

0

0 02 -- -- -------I _-.-=O~~~~~~~~~~~~[E 'lcov 91=7.429e-006

-0.040 0512 5 2 2.5 3 3.5 4 46 5

0.02

0-00266005 165 2.5 3 3.5 4 46 5

O0.5

O | ~~~~~~~~~~~~cov e3=0.0050207° -0.5 __]ll

0 0.5 1.6 2.5 3 3.5 4 4.6 6t

Error vvith NN=[2 2 21. trace(cov)=0.0070444

-0.05

0 16 2 6 4 4T6 6

0.02

-00

0 0.5 1.5 2 2.5 3 3.6 4 4.6 6

20.0 O l co~~~~~~~~~~~~~~v e3=0.57e0068521-0 .0

0 0.5 1.5 2 2.5 3 3.5 4 4.5 5

0. 6-0.02 I0 0.6 1.6 2 2.6 3 3.5 4 4.5 6

0.04

0.02 I--

-0.02 ll0 0.5 1.5 2 2.5 3 3.5 4 4.5 5

80.04

0.0

-0 .00 0.5 1.5 2 2.5 3 3.5 4 4.5 5

[2] G. C. Walsh and H. Ye, "Scheduling of networkedcontrol systems", IEEE Control System Magazine,vol. 21, no. 1, pp. 57 - 65, 2001.

[3] C. L. Liu and J. W. Layland, "Scheduling algorithmsfor multiprogramming in a hard-real-timeenvironment", Journal of the Association forComputing Machinery, vol. 20, no. 1, pp. 46 - 61,1973.

[4] L. Sha, R. Rajkumar, and J. P. Lehoczky, "Priorityinheritance protocols: An approach to real-timesynchronization", IEEE Trans. on Computers, vol.39, no. 9, pp. 1175 - 1185, 1990.

[5] D. J. Lee and M. Tomizuka, "Multirate optimal stateestimation with sensor fusion", Proceeding ofAmerican Control Conference, pp. 2887 - 2892,2003.

[6] S. Bittanti, P. Colaneri, and G. D. Nicolao, "Thedifference periodic Riccati equation for the periodicprediction problem", IEEE Transactions onAutomatic Control, vol. 33, no. 8, pp. 706 - 712,1988.

[7] R. G. Brown and P. Y. C. Hwang, Introduction toRandom Signal and Applied Kalman Filtering. NewYork: John Wiley & Son, 1997.

[8] S. Boyd and L. Vandenberghe, Convex Optimization.New York: Cambridge University Press, 2004.

Fig. 8 Error of sensor data.

6. CONCLUSION

In networked monitoring system, under limitedbandwidth, sensor transmission periods are the majorconcern. The paper proposes a new method to selecttransmission period of each sensor. The methodsuggests an estimation performance based on steady-state periodic error covariance matrices although thesystem is periodically time-varying. Using thisperformance index and a rate monotonic networkscheduling, the optimal set of transmission periods,which has minimum performance index, is chosen. Inaddition, we discuss about the way to reducecomputation time occurring in the algorithm. It is shownthat the proposed method can suggest different sets oftransmission period of sensors and selects an optimal setamong them. Through an experiment with CAN, we seethat the optimal set takes the best estimation errorcovariance and the smallest error.

REFERENCES

[1] M. Y Chow and Y Tipsuwan, "Network-basedcontrol system: A tutorial" in The 27th AnnualConference of the IEEE Industrial ElectronicsSociety, pp. 1593 - 1602, 2001.

1052