Page 298 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Estimation and Control of Magneto-Rheological Damper for Semi-Active Suspensions 301

Proof. It hasbeen shownin[18,19]thatif is PE, then the matrix M in
(19.12) is positive definite, i.e., its minimum eigenvalue λ min (M)>σ > 0.
1
We select a Lyapunov function as V = θ θ. Then one can calculate
−1 ˜
˜
2
˙ V along (19.12)as
−1 ˙
T
T
˜
˜
˜
˙ V = θ θ =−θ Mθ ≤−μV (19.15)
˜
2σ
where μ = is a positive constant for all t > 0. Then according to
λ max ( −1 )
Lyapunov theorem and (19.15), the estimation error θ will converge to zero
˜
exponentially.
As shown in the above proof, the variable H used to drive the adap-
˜
tive law (19.14) contains the information of estimation error θ,sothat
it can drive the estimated parameter θ to converge to its true value in
ˆ
an exponential manner. Moreover, the observer or predictor used in the
traditional parameter estimation methods (e.g., gradient method and RLS
approaches [23]) are not needed, which leads to reduced computational
costs. For more details of this new adaptive law and the performance anal-
ysis, we refer to [19,18,22].
19.3 ADAPTIVE ESTIMATION AND CONTROL FOR VEHICLE
SUSPENSION WITH MR DAMPER

19.3.1 Quarter Car Model and Control Objectives
In this section, a non-linear quarter-car model with MR damper will be
used to achieve suspension. The diagram of the studied quarter-car system
is showninFig. 19.3,where m s is the sprung mass, and m us represents the
mass of wheel, respectively. F d and F s are the force produced by the dampers
and springs with the damping coefficient b e, the stiffening coefficients of
linear and non-linear terms with k s ,k sn. F t and F b denote the elasticity and
damping forces of tire with the stiffness and damping coefficients k t ,b f .
z s and z us are the displacements of sprung and unsprung masses. z r is the
input of road displacement. F is the control force of the semi-active sus-
pension system, which is generated by hyperbolic model (19.11).
According to Newton’s second law, the dynamics of the studied suspen-
sion system shown in Fig. 19.3 are obtained as [14]

m s ¨z s + F s + F d = F
(19.16)
m us ¨z us − F d − F s + F t + F b =−F