A Parameter Identification Scheme for Second-Order Highway Traffic Model Based on Differential Algebraic Methodology - Intelligent Computing for Sustainable Energy and Environment - page 90

ρ α

α ( 1 ρ cr ) α ]

ln V =ln v f +[ − (9)

Let ln V = V , ρ α = ρ , θ 1 =ln v f , θ 2 =

α ( ρ cr ) α , then the static speed
equation would be the linear equation of the form:

1

−

V ( t )= θ 1 + θ 2 ρ ( t )

(10)

Step 2: Transform function (10) from time-domain to complex frequency domain
by means of Laplace transformation. The operational calculus of (10) yields:

V ( s )= θ 1

s + θ 2 ρ ( s )

(11)

Step 3: In most parameter identification problem, the initial conditions are
usually dicult to get and regarded as unknown disturbances, so the unknown
initial conditions and the constant disturbance can be treated as structured
terms and be annihilated by differentiating both sides with respect to s .Thatis,
select the derivation operator Δ 1 =

d

ds to multiply both sides of equation (11),

we obtain

V ( s )+ s d

ds V ( s )= θ 2 ρ ( s )+ θ 2 s d ds ρ ( s )

(12)

Step 4: It is well known that the differential of noisy signals would amplify the
noise. Moreover, a (strict) proper estimator can ensure the causality property
of the system. Therefore, we select the derivation operator Δ 2 = s −ν ( ν 1)

to multiply both sides of equation (12), in order to avoid the derivatives of the
input and output signal in the constructed estimator. We obtain the algebraic
expressions that only contain the integral of noisy signal.
V ( s )
s 2

≥

ds V ( s )= θ 2 [ ρ ( s )

+ 1
s

d

+ 1
s

d
ds ρ ( s )]

(13)

s 2

Furthermore, according to the identifiable definition aforementioned, there are
two parameters to be identified, so another equation is required to construct a
square matrix system. According to (11), it is readily obtained as follows:

V ( s )
s s 2 + θ 2 ρ ( s )

= θ 1 (14)

s

Step 5: Transfer equation (13) and (14) simultaneously to time-domain from
frequency domain by inverse-Laplace transformation. Then make use of Cauchy
formulation ( ··· y ( τ 1 ) dτ 1 ··· τ 1 = t

0

( t−τ ) l− 1 y ( τ )
( l− 1)! dτ ) to simplify the results.
Finally, the parameter estimator in time-domain would be:

ρ ( τ ) dτ ] ,θ 2 e = T

T [ T
0

θ 2 e T
0

0 ( T

− 2 τ ) v ( τ ) dτ

θ 1 e = 1 v ( τ ) dτ −

T
0 ( T

− 2 τ ) ρ ( τ ) dτ

where θ 1 e and θ 2 e are the identification results of θ 1 and θ 2 respectively.