Page 300 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics
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Adaptive Estimation and Control of Magneto-Rheological Damper for Semi-Active Suspensions 303
1) Ride comfort: This needs to design the input current I of MR damper
to create appropriate force F to regulate the vertical displacement of
vehicle body under the road shocks, i.e., x 1 = z s → 0.
2) Road holding: The firm uninterrupted contact of wheels to road should
be ensured for the safety of passengers, that is
|F t | <(m s + m us )g (19.20)
3) Suspension movement limitation: The suspension space should not exceed
the allowable maximum, i.e., the difference z s −z us should be bounded
by the maximum suspension space z max as
(19.21)
|z s − z us | ≤ z max
The suspension performance 1), 2) and 3) will be studied by introducing
an adaptive control, where the unknown parameters will be also online
estimated.
19.3.2 Adaptive Control Design With Parameter Estimation
To address the regulation of vertical displacement x 1 of system (19.19), we
first define the filtered error variable as
T
s 1 =[ ,1][x 1 ,x 2 ] (19.22)
where > 0 is a positive constant. Thus, s 1 is bounded as long as the
filtered error s 1 is bounded. In particular, |x 1 | ≤ |s 1 | and |x 2 | ≤ 2|s 1 | are
true for zero initial condition.
Furthermore, we can obtain the time derivative of s 1 as
s ˙ 1 = x 2 + 1 [(c 0 − b e )(x 2 − x 4 ) + (k 0 − k s )(x 1 − x 3 )
m s (19.23)
3
−k sn (x 1 − x 3 ) + θ I I tanh c 1 (x 2 − x 4 ) + k 1 (x 1 − x 3 ) ]
In this section, the coefficients of springs, the mass of car body and
the parameters of hyperbolic model are all unknown. We will present an
online estimation algorithm to obtain these unknown parameters. Hence,
we denote the system dynamics as a more compact form as
1 3
T(Z) = c 0 − b e (x 2 − x 4 ) + k 0 − k s (x 1 − x 3 ) − k sn (x 1 − x 3 )
m s
T
= W φ 1 (Z 1 ) (19.24)
1