VEHICLE PARAMETER IDENTIFICATION
2.4 Application of Bootstrap Particle Filtering Method for Vehicle Parameter Identification
The main idea of this resampling technique is to first construct the cumulative distribution function (CDF) from original random particles{xlik,wli}, as it is given by
=
−
≈
≤ Np
i
li i l
i l l
l x w x x
X
1
) ( )
Pr( µ (2.38)
where (.) is the unit step function.
Uniform random samples vk are drawn from the interval [0,1] and projected onto the inverse cumulative distribution function corresponding to the associated probability and identifying the particular new samples index, ik , and corresponding replacement particle.
The particles with highest weights will be selected more frequently, thereby, replacing particles with lower weights and therefore the new random measure is created. Selection of new particles xˆ according to the multinomial distribution can be expressed as ik
−
= =
− ≤ <
= 1
1 1
1( ))
ˆ ( i
s
i
s s
k s k
ik x F v for w v w
v (2.39)
where, i=1,2,3…N, ws is a normalized weight of particle at time step l. F-1 denotes the generalized inverse of cumulative distribution of normalized particle weights.
2.4 Application of Bootstrap Particle Filtering Method for Vehicle
include those additional parameters. The system states rl are assumed to propagate according to system equation
) ,
;
1 l(l l l
l g r
r+ = Φ η ; l=0,1,2,3…. Nt (2.40) in which, l represents discretized time dimension and Nt is the number of time instants considered. l Rd is a d-dimensional vehicle parameters vector which is considered as constant for small interval of time, rl Rn is a n-dimensional vector denoting the state of the system, a model noise l Rm is the discretized m-dimensional vector of a sequence of independent and identically distributed random variables which are independent of past and current state and whose probability density function are assumed to be known. gl(.) is a system transition function.
When the system measurements become available, the system states are related to these measurements via the observation equation.
; ) ,
; (l l l
l
l f r
Z = Φ ζ l=0, 1, 2, 3…. Nt (2.41) where Zl Rp is a p-dimensional bridge response measurement vector, a measurement noise l Rs is a s-dimensional vector of a sequence of independent and identically distributed random variables and fl(.) is a non linear function that relates the measurements to the system state
Since, state of the system is dependent on system parameters { }, observation equation can be rewritten as
; ) , ( l l
l
l f
Z = Φ ζ l=0, 1, 2, 3…. Nt (2.42)
Vehicle parameters identification problem can now be considered as being equivalent to the determination of the posterior probability density function p( l Zl). According to Bayesian theorem p( l Zl) can be written as (Ristic et. al. 2004)
) ( )
| ( )
|
( l Zl p Zl l p l
p Φ =λ Φ Φ (2.43) where, p( l |Zl) is the posterior PDF, p(Zl | l) is the likelihood of individual parameters, p(Φ) is the prior probability density function and λ is the normalizing parameter evaluated as
]1
) ( )
| (
[ Φ Φ −
= p l Zl p l dφl
λ (2.44) Thus knowing posterior PDFp(Φl|Zl), first few moments of the vehicle parameters l, conditioned on bridge response measurement Zl, at each time step can be determined.
The main steps of the particle filtering algorithm for identifying vehicle parameters has been stated in a sequential manner for implementation in computer program in MATLAB environment as follows:
(i) Initial stage: Draw Np random samples of vehicle parameters {Φ0j}Npj=1 from the assumed PDF{p(Φ0j)}Npj=1
(ii) Prediction: Determine bridge response using available vehicle parameters and employing forward solution at time step l. If Np is sufficiently large, these estimates are approximately distributed as {p(flm[Φlj]|Zlm)}Npj=1 .
where, m=1,2,3…Nm is number of response measurement locations, Zlmdenotes measured bridge response at location m and flm[Φlj] represents simulated bridge response at location m using generated particles.
It may be noted here that present study proposes a semi-analytical scheme to generate response samples in prediction stage with an aim to enhance the performance of the particle filter method. The proposed method of response sample generations has been given in Chapter-3 and Chapter-4.
(iii) Update: Once the measurements are available for different measurement location m (m=1,2,3….Nm) at time l, evaluate the likelihood corresponding to all the samples which depends on the nature of measurement noise probability distribution. Normalized weight for each particle can be obtained as
∏
∏
= =
=
Φ Φ
= Np
j
lj m l m l Nm m
lj m l m l Nm
j m
f Z L
f Z L w
1 1
1
]) [
| (
]) [
| (
(2.45)
in which, ( | [ ])
1 m lj
l m l Nm m
f Z
L Φ
∏
=represents likelihood of measured bridge response given measured bridge response for each particle j at time l for different measurement locations m. The discrete mass probability function for the next iteration is defined as
j lj m l
lj f w
P(Φ = [Φ ])= (2.46) (iv) Resample: From the discrete mass distribution function, a new set of Np samples of lj
are generated. This constitutes the posterior estimates of lj. The mean of estimates are obtained by averaging across the ensemble, and is expressed as
=
Φ
= p
N
j lj
p l
l| N 1
µ 1 (2.47)
The corresponding standard deviation of the estimate is calculated as
=
−
− Φ
= Np
j lj ll
p l
l N 1
2
|
| ( )
1
1 µ
σ
(2.48)
Set l=l+1 and if l < Nt then go to step (ii) and repeat other steps, otherwise stop. In this way, the filtering is carried out for the entire available time history of measurements.
A flow diagram of the Bootstrap Particle Filtering algorithm is shown in Fig 2.3 where the procedures of vehicle parameters identification based on measured bridge response are again illustrated.
Fig.2.3 Flow chart for Bootstrap Particle Filtering method for identification of vehicle parameters
(i) Initial stage
(ii) Prediction
(iii) Update
(iv) Resample
Start
Generate NP particles for each unknown parameter { 0} from assumed PDF p( 0)
Determine bridge response using forward solution for each generated parameter
Take the measurement of bridge response and evaluate likelihood for each parameter
Normalize the likelihood to obtain approximate discrete probability mass function for each parameter
Generate NP new samples from the discrete probability mass function for each parameter using suitable resampling method
Obtain the mean ( ) of each parameter and the corresponding standard deviation ( )
Stop
No
Yes time step (l) If
≤ total time step (Nt)