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State Estimation with Delayed Measurements Considering Uncertainty

of Time Delay

Minyong Choi, Jinwoo Choi, Jonghoon Park and Wan Kyun Chung

Abstract—State estimation problem with time delayed mea-

surements is addressed. In dynamic system with noise, after

taking measurements, it often requires some time until that is

available in a filter. A filter not considering this time delay

cannot be used since a current measurement is related with a

past state. These delayed measurements problem is solved with

augmented state Kalman filter, and uncertainty of the delayed

time is also resolved based on the probability distribution of the

delay. The proposed method is analyzed by a simple example,

and its consistency is verified.

I. INTRODUCTION

State estimation with noisy measurements is required

in many applications. Researchers have been developed a

number of filtering methods. In order to reduce the effect of

that noise, Extended Kalman Filter (EKF) or particle filter

is widely used.

In general, it is assumed that a measurement is transmitted

to a filter without any time delay. There is, however, a chance

to have the delay between an instant when a measurement

is taken by a sensor and another instant when the measure-

ment is available in the filter. For example, communication

through a network demands a fundamental processing time.

Sometimes, data processing time is required to obtain higher

level information from raw sensor data. This problem that

measurements are available in the filter with delay is referred

as Out Of Sequence Measurement (OOSM) problem [1]. In

addition, uncertainty of the delay should also be resolved.

Even though an average delay is known, it is not an exact

value, and cannot be used in the filter naively.

When there is no time delay, several prediction steps are

carried out using a process model before a measurement

arrives in the filter. Unfortunately, in the OOSM problem,

the predicted state is not meaningful because the delayed

measurement contains information about the past state. In

order to handle it, backward prediction or retrodiction can

be used. The past state can be obtained by applying the

inverse model of the process model to the current state.

The optimal approach of one step delay using the backward

prediction was presented in [1] and multi step delay was

also solved in [2]. For those cases, linear process model was

supposed to get the inverse model and the optimal solution.

However, if the process model is nonlinear and it is hard to

Minyong Choi and Jinwoo Choi are with the Department of Mechanical

Engineering, Pohang University of Science and Technology, San 31, Pohang,

Korea minyong/shalomi@postech.ac.kr

Jonghoon Park is with SimLab Co., Ltd., Seocho3-Dong, Seocho-Gu,

Seoul, Korea coolcat@simlab.co.kr

Wan Kyun Chung is with Faculty of Mechanical Engineering, Po-

hang University of Science and Technology, San 31, Pohang, Korea

wkchung@postech.ac.kr

obtain the inverse model, these methods may require some

modification.

Larsen et al. [3] proposed a method extrapolating the

delayed measurements. As extrapolating the delayed mea-

surements, the current measurement could be calculated and

this value was directly applied to the filtering algorithm. This

approach has low computational cost and does not need any

modification in the filter algorithm. But it might have a weak

point in a dynamic system having fast movements.

A state augmentation for delayed state was also used when

there were several steps of time delay in measurement. After

this augmentation, not only Kalman filter, EKF could be used

directly. Challa et al. [4] solved the OOSM problem using

this augmented state Kalman filter (ASKF). The uncertainty

of the delay was also resolved by means of Probabilistic

Data Association Filter (PDAF). Merwe et al. [5] solved

the problem with almost same formulation but sigma-point

Kalman filter was used instead of EKF. In these methods, as

the time delay increases, the number of augmented states also

increases, and the augmentation has a weak point requiring

more memory and computational power. If, however, the

measurement delay shouldn’t be too large, this method is

easy to implement, and be expended the nonlinear models.

Julier and Uhlmann [6] suggested a method in which

an unknown time delay could be treated by means of the

covariance union algorithm. In this method, the covariance

was estimated conservatively since the worst case, unknown

delay, was considered. But, if the time delay is able to be

modeled as a form of a distribution, it is helpful in estimation

using that quantity considering the uncertainty of the delayed

time.

In vehicle localization, fusion of delayed observations is

also a problem. A practical fusion method for the localization

of an outdoor vehicle is presented in [7].

In this paper, the state estimation with the delayed mea-

surements, OOSM problem, is considered. Several steps of

the delayed measurement will be treated to reinforce the

current state. The uncertainty of the delayed time will also be

considered using its own distribution because the measured

time delay may have noise in practice. First of all, it is

assumed that the delayed time is small enough to accept

for using augmentation, and augmented state Kalman filter

is adopted for dealing with delays. Then, it is supposed

that a Probability Density Function (PDF) representing the

uncertainty of delay can be modeled. A bundle of updates

are carried out for the delay uncertainty based on the PDF.

Although the proposed method has similar form with Prob-

abilistic Data Association Filter (PDAF), both are different

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in detail. Concisely, PDAF calculates weights needed in a

bundle of updates based on likelihood. On the other hand,

the proposed filter calculates the weights based on the PDF

of the delay.

This paper is organized as follows. In section II, the

problem definition is presented about delayed measurements.

Section III suggests the method to handle the delayed mea-

surements with uncertain delays. In section IV, simulation

result and analysis are given to verify the proposed method.

Finally, section V concludes the paper.

II. PROBLEM DEFINITION

A. Delayed Measurements

In state estimation, system dynamics is represented by a

discrete-time equation referred to as the process model.

x(k + 1) = f(x(k),u(k)) + v(k)

(1)

where x(k) is the state at time k, u(k) is the control input,

and v(k) is a zero-mean, white, Gaussian process noise with

covariance matrix Q(k). f(·) represents a relation between

variables in k-th step and x(k + 1). The measurement

equation is

z(k) = h(x(k)) + w(k)

(2)

where z(k) is a measured value by means of sensors,

w(k) is a zero-mean, white, Gaussian measurement noise

with covariance matrix R(k), and h(·) represents a relation

between the state and the measured value.

If an instant when a measurement is taken in a sensor

coincides with another instant when the measurement value

is available in a filter, a general filtering algorithm such

as EKF or particle filter can get a consistent and correct

estimation result. A measurement sequence without delay

is represented in Fig. 1(a). When, however, both instants

does not coincide, there is a time difference, called delay,

between them. Fig. 1(b) shows a sequence of this situation.

The measurement equation is redefined as

z(k) = h(x(k − τ)) + w(k − τ)

(3)

(a) Normal Measurements Sequence

(b) Delayed Measurements Sequence

Fig. 1.Measurements Sequence

where τ is the number of delayed steps. Although z(k)

is available in the filtering method at a time step k, x(k)

cannot be corrected directly because the measurement value

is related with the past state x(k − τ). The filtering method

must consider these delays for the correct estimation.

B. Uncertainty of Time Delay

Suppose that the number of the delayed steps are given,

the filtering method considering the delay estimates the state

correctly. The amount of delay, however, is also a measured

quantity by means of device such as the CPU clock, or an

additional timer. So, this quantity if often corrupted by a

measurement noise. When τ is given as 2 in (3) using a

timer, the real value of τ is able to be 1 or 3. In Fig. 2,

this situation is interpreted. The measurement value arrives

at time tk+2in general, but it sometimes arrives at a different

time step. The delayed steps can be regarded as a random

variable having a corresponding PDF. This uncertainty or

noise of the time delay also should be addressed to obtain

the consistent estimation result.

III. FILTERING METHOD FOR UNCERTAIN TIME

DELAYED MEASUREMENTS

A. Augmented State for Delayed Measurements

As augmenting several steps of past states, the current

measurement having information about a past state can

correct the augmented state directly. This means that the

current measurement value can update the current state which

is the first part of the augmented state. For example, when

there is one step delay, the process model is augmented like

?x(k + 1)

where

multiple step delays case, the process model can be extended

as

xa(k) =?xT(k)

⎢⎢⎢

where n is the maximum number of the delayed steps, I

is the identity matrix, and xa(k) is the augmented state. A

x(k)

?

xT(k)?Tis the augmented state. For

=

?f(x(k),u(k)) + v(k)

x(k)

?

(4)

?xT(k + 1)

···

xT(k − n)?T

0

...

0

(5)

xa(k + 1) =

⎡

⎣

f(x(k),u(k))

00

...

0

0I

⎡

⎣

⎢

I

0

0

⎤

⎥

⎦X(k)

⎤

⎦+

⎥⎥⎥

⎡

⎣

⎢⎢⎢

v(k)

0

...

0

⎤

⎦

⎥⎥⎥

(6)

Fig. 2.Uncertainty of Time Delay

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revised measurement equation is also described as

z(k) = h??0

where I is placed in corresponding with k − τ time step.

After constructing the revised process and measurement

equations like (6) and (7), EKF can be applied to estimating

the state based on these equations. Detailed descriptions

about EKF are appeared in [8].

···

I

···

0?xa(k)?+ w(k − τ) (7)

B. EKF considering Uncertainty of Time Delay

The time delay corrupted by its own noise has a prob-

abilistic property, and can be modeled as a PDF like in

Fig. 3. Although time is a continuous value, that can be

discretized practically. When a measurement arrives in the

filter, a probability that the value is in a time step can be

calculated as integrating the PDF. The probability1that the

measured value is in the k-th time step is

αk= P(tk− Δt/2 ≤ t < tk+ Δt/2)

=

tk−Δt/2

where P(·) means a probability, and p(·) is the PDF. Suppose

that the measurement value z(k) is given, and a correspon-

dence cimeans that the value z(k) is in the i-th time step, a

probability which represents the measurement is in the i-th

step is written as

?tk+Δt/2

p(t)dt

(8)

P(ci|z(k)) = αi

(9)

As considering the correspondence ci, a state estimator

can be derived.

ˆ x(k|k) = E{x(k)|z(k)}

=

x(k)p(x(k)|z(k))dx

?

i

=

P (ci|z(k))E{x(k)|z(k),ci}

?

?

=

x(k)

?

P (ci|z(k))p(x(k)|z(k),ci)dx

?

i

=

i

αiE{x(k)|z(k),ci}

(10)

1This probability has a similar role with weights in PDAF [4]. In the

PDAF, likelihood is used for calculating the weights. On the other hand,

this probability, the area under the PDF, is directly used as weights in the

proposed method.

Fig. 3.Probability Density Function of Time Uncertainty

where E{·} means expectation, x(k), same with xa(k) in

(6), is the real state, and ˆ x(k|k) is the estimated state when

z(k) is taken up. When ci is conditioned, each estimate

ˆ xi(k|k) can be obtained using the state update equation of

Kalman filter.

ˆ xi(k|k) = E{x(k)|z(k),ci}

= ˆ x(k|k − 1) + Wi(k)νi(k)

where νi(k) = z(k) − ˆ zi(k|k − 1), ˆ x(k|k − 1) is predicted

state just before z(k) is taken up, and Wi(k) is Kalman

gain. zi(k|k − 1) is predicted measurement considering the

delayed state. Detailed equations for Kalman gain are also

in [8]. Consequently, the estimated state

(11)

ˆ x(k|k) =

?

i

αiˆ xi(k|k)

= ˆ x(k|k − 1) +

?

i

αiWi(k)νi(k)

(12)

A flow of state estimation is shown in Fig. 4.

The covariance matrix can also be derived in the same

manner like (13). P(k|k) is the estimated covariance matrix,

and each covariance matrix Pi(k|k) is calculated by means

of the covariance update equations in EKF [8] when ci is

conditioned.

Note that the derived update equations (12), (13) can be

used whether the process model or the measurement equation

is linear or not.

IV. EXAMPLE

In order to verify the proposed estimation method, the

position and velocity of a 1D particle which is moving with

a constant velocity are estimated.

Process

Model

Observation

Model

?

? ??

Kalman

Gain

?

Fig. 4.Flow of State Estimation

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P(k|k) = E

?

(x(k) − ˆ x(k|k))(x(k) − ˆ x(k|k))T|z(k)

αiE?x(k)xT(k)|ci,z(k)?− ˆ x(k|k)ˆ xT(k|k)

αi

E

(x(k) − ˆ xi(k|k))(x(k) − ˆ xi(k|k))T|ci,z(k)

?Pi(k|k) + ˆ xi(k|k)ˆ xT

?

=

?

?

?

i

=

i

??

?

+ ˆ xi(k|k)ˆ xT

i(k|k)

?

− ˆ x(k|k)ˆ xT(k|k)

=

i

αi

i(k|k)?− ˆ x(k|k)ˆ xT(k|k)

(13)

TABLE I

PARAMETERS FOR SIMULATION

Parameter

Initial Value

Time Duration Δt

Measurement Frequency

Integration Steps

Acceleration Noise q

Measurement Noise w

Time Delay τ

Value

x1(0) = 0, x2(0) = 10

1

1/10

2000

N(0,1)

N(0,1)

N(5,1)

A. System Description

The position and the velocity of the particle are defined

as the state variables,

?x1(k)

where x is the position of the particle, and ˙ x is the velocity of

the particle. The particle is moving with a constant velocity

but a noise is assumed in the acceleration. If the noise is q,

the process model can be obtained like

?1

where Δt is a time duration in discrete time system, and is

regarded as a constant value. When a sensor measures the

position of the particle, the measurement equation is

z(k) =?1

where w(k) means a measurement noise. The measurement

equation, however, should be transformed like (3) since a

measurement delay is assumed in the following simulation.

x(k) =

x2(k)

?

=

?x(k)

˙ x(k)

?

(14)

x(k + 1) =

Δt

10

?

x(k) +

?(Δt)2/2

Δt

?

q(k)

(15)

0?x(k) + w(k)

(16)

B. Simulation

The process noise, and the measurement noise is assumed

to be Gaussian distribution. The uncertainty of the delay also

is assumed to be Gaussian distribution. The real particle is

moving with a initial velocity according to the process model

(15), and the number of integration steps is 2000. Total 200

updates are carried out in the filter since the sensor measured

a position value with frequency 0.1. Parameters used in

the simulation are described in Table I. In the simulation,

dimension of the augmented state is 22 because the number

of the state is 2 and twice of the mean delay is considered.

Following three cases are tested for a comparison.

050 100 150 200

−250

−200

−150

−100

−50

0

50

100

150

200

Time Step

Position Error

KnownDelay

MeanDelay

ModeledDelay

(a) Sequence of Position Error

050100150 200

−25

−20

−15

−10

−5

0

5

10

15

20

Time Step

Velocity Error

KnownDelay

MeanDelay

ModeledDelay

(b) Sequence of Velocity Error

Fig. 5.Error sequences of the Estimated State

• KnownDelay : The time delay is known. That is, al-

though there is the uncertainty in the delay, the filer

knows the delay perfectly.

• MeanDelay : For compensating the delay, the mean

value of the delay is considered. Simply, the uncertainty

of the delay is not considered in this case.

• ModeledDelay : In order to handle the delay, the pro-

posed filtering method is applied. The uncertainty of the

delay is considered explicitly as PDF.

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0 50100150200

10

−2

10

−1

10

0

10

1

10

2

10

3

Time Step

Log[Normalized Estimated Error]

KnownDelay

MeanDelay

ModeledDelay

Fig. 6.Sequence of Normalized Estimated Error Squared

In Fig. 5, sequences of the estimated position and velocity

error of one trial are plotted. KnownDelay naturally has min-

imum error compared with MeanDelay, and ModeledDelay.

Especially, ModeledDelay in which the proposed method

is applied has better performance than MeanDelay. This

is because the model of the uncertain delay is addressed

explicitly.

For more refined analysis, Normalized Estimated Error

Squared (NEES) is selected for a measure which shows the

performance of the proposed method since the performance

cannot be confirmed just using the sequences of error. NEES,

defined in (17), is a general measure for checking consistency

of the filter [4], [8], [9].

?(k) = {x(k) − ˆ x(k|k)}P−1(k|k){x(k) − ˆ x(k|k)}T(17)

Fig. 6 shows sequences of NEES. MeanDelay has relatively

larger values than the others. It means that just using the

mean value of the delay can cause a crucial effect on the

estimation.

Through Monte Carlo simulation tests, the consistency of

the proposed method is analyzed. N-run average NEES is

calculated as

1

N

¯ ?(k) =

N

?

i=1

?i(k)

(18)

When 50-run Monte Carlo tests are achieved, two sided 95%

probability acceptance region for ¯ ? is [1.5,2.6]. That is, 95%

the of average NEES sequence should be in the acceptance

region. The average NEES sequence of the proposed method

is plotted in Fig. 7, in which red solid line represents

the average NEES values and black dashed line means

acceptance region. We can find out that the filter slightly

overestimate the covariance, but this can also be regarded

as consistent result because the estimated covariance still

contains the estimated error.

Moreover, the quantity ?(k) has a χ2distribution with 2

degrees of freedom in this simulation. So, the optimal value

of E{¯ ?(k)} is 2. This value of ModeledDelay is 1.7932, and

050100150200

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Time Step

Average Normalized Estimated Error Squared

Fig. 7.Average Normalized Estimated Error Squared of Proposed Filter

it can be said that the estimation result is consistent because

of same reason described in the former paragraph.

V. CONCLUSIONS

In this paper, state estimation with delayed measurements

has been addressed when the delay itself is not exact.

When the uncertainty of the delayed time is modeled as

probability density function, a filtering method combined

with augmented state Kalman filter has been derived. A

bundle of updates based on the probability that the measure-

ment is in each time step refine the estimation result. The

simulation result of a simple example has been analyzed,

and consistency of the proposed method has been verified

based on it. Although there is a tendency the proposed

method overestimates the estimated covariance, it is still in

the acceptable range, and seems to be consistent because the

estimated error is inside the estimated covariance.

VI. ACKNOWLEDGMENTS

This work was supported in part by the Acceleration

Research Program of the Ministry of Education, Science

and Technology of the Republic of Korea and the Korea

Science and Engineering Foundation [R17-2008-021-01000-

0], in part by the Korea Health 21 R&D Project, Ministry of

Health & Welfare, Republic of Korea under Grant A020603,

in part by the Agency for Defence Development and by

Unmanned Technology Research Center(UTRC), Korea Ad-

vanced Institute of Science and Technology, in part by the

Korea Science and Engineering Foundation(KOSEF) under

Korea Government Grant MOST R0A-2003-000-10308-0, in

part by the IT R&D program of MKE/IITA [2008-F-038-

1, Development of Context Adaptive Cognition Technol-

ogy], and in part by the Dual-Use Technology Program of

DAPA/DUTC and MIC/IITA. [06-DU-LC-01, Development

of Multi-Purpose Dog-Horse Robot based on the Network].

REFERENCES

[1] Y. Bar-Shalom, “Update with Out-of-Sequence Measurements in

Tracking: Exact Solution,” IEEE Transactions on Aerospace and

Electronic Systems, Vol. 38, No. 3, pp. 769-778, 2002.