Page 304 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Estimation and Control of Magneto-Rheological Damper for Semi-Active Suspensions 307

Then the parameters η and P can be appropriately selected, such that the
performance requirement of the road holding (19.21) can be guaranteed.
Finally, we can obtain the upper bound of suspension spaces as

|x 1 − x 3 | ≤ 2V 1 + (V(0) + β α) λ min (P) ≤ z max (19.42)
Hence, the suspension movement limitation (19.21) can be fulfilled if
the parameters ,k s , 1 ,σ 1 ,η,P are designed appropriately.

19.4 SIMULATIONS

In this section, numerical simulations are provided to illustrate the effec-
tiveness of the proposed estimation and control algorithms. The parameters
of the MR damper and quarter-car model are given as: m s = 600 kg,
m us = 60 kg, k s = 18000 N/m, k sn = 1000 N/m, k t = 200000 N/m, b f =
1000 Ns/m, b e = 2500 Ns/m, b c = 2200 Ns/m, c 0 = 810.78 Ns/m, c 1 =
13.76 s/m, θ I = 457.04 N/A, k 1 = 10.54 1/m, k 0 = 620.79 N/m, z max =
0.15 m.
The following two cases are simulated:

Case 1 (Adaptive parameter estimation of MR damper):Inthissim-
ulation, only the MR damper dynamics (19.11)isconsidered to show the
online modeling method (19.14). Thus, we set the velocity of piston as
˙ z = 0.6cos(6t) and the input current as I = 2. The estimation performance
of the gradient method and the proposed method are compared. For fair
T
comparison, the initial simulation conditions are set as θ(0) =[0,0,0.001] .
The simulation parameters are set as = 30diag[0.065,0.53,6.4] and k =
0.001, = 1as [21]. One may find from Fig. 19.4 that the velocity-force
curves with the estimated parameters are very close to its nominal coun-
terparts. It is clearly shown that the estimated model based on (19.14)can
capture the essential dynamics of the realistic MR damper. This implies that
the estimated parameters converge to their true values.

Case 2 (Adaptive control with external road disturbance): The pro-
posed control and estimation are also simulated under the external road
disturbance, which is given as follows

h 1 − cos 2πV s t , 0 ≤ t ≤ l

2 l
z r = l V s (19.43)
0 t ≥
V s