Page 302 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Estimation and Control of Magneto-Rheological Damper for Semi-Active Suspensions 305

The adaptive law for updating ˆ is given by

˙ ˆ

= 1s 1 − 1 κH 1 (19.31)

where 1 > 0 is a constant diagonal matrix and κ> 0 is a constant scalar.
Now, we have the following results:

Theorem 19.2. For vehicle suspension system (19.19) with control (19.26)and
(19.31), if the regressor vector in (19.25) is PE, then the control error s 1 and
estimation error
=
−
exponentially to zero.
ˆ
˜
Proof. As proved in [22], if in (19.25)isPE, theminimumeigenvalue
of the matrix M 1 fulfills λ min (M 1 )>σ 1 > 0. By substituting (19.26)into
(19.25), the closed-loop error dynamics ˙s 1 can be written as

T
s ˙ 1 =−k ss 1 +
(19.32)
˜
On the other hand, according to (19.28)–(19.30), the vector H 1 defined
in (19.30) equals to H 1 =−M 1
as shown in [22]. Therefore, we select a
˜
Lyapunov function as

1 2 1 T −1
V 1 = s +

(19.33)
˜
˜
1
1
2 2
Thenthetimederivativeof V can be obtained as
T −1 ˙ 2 T
˜ ˜ ˜ ˜
˙ V 1 = s 1 ˙s 1 +

=− k ss − κ
M 1
≤−μ 1V 1 (19.34)
1 1
−1
where μ 1 = min 2k s ,2κσ 1 /λ max is a positive constant. According to
1
Lyapunov theorem, the control error s 1 and estimation error ˜ all converge

to zero exponentially, where the convergence rate depends on the control
gain k s, the excitation level σ 1 and the learning gain 1.
The use of the leakage term 1 κH 1 in adaptive law (19.31)isinspiredby
our work [18,19,22]. As shown in the above proof, the inclusion of variable
˜ T
˜
H 1 leads to a quadratic term (i.e.,
M 1
) of the estimation error
in the
˜
Lyapunov analysis. Thus the estimated parameter can converge to its true
values in an exponential manner. This can help improve the suspension
performance.