Page 37 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

28 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

2.4.2 Finite-Time Parameter Estimation
Now, the problem is to develop an adaptive law to online update the un-
ˆ
known parameters ˆσ 0, ˆσ 1, ˆσ 2, ˆ J,and T d in the control (2.29). To improve
the estimation accuracy and the transient control response, a finite-time
adaptive law [21] is developed based on the initial values of these parame-
ters derived by the above presented GSO.
Let ˆx be the state of the predictor for (2.6), which is given by

˙ ˆ x = F(x) + G(x,T) 0 + k μ (x − x) (2.31)

where 0 is the initial estimation of obtained using GSO and k μ > 0is a
constant matrix.
Define the auxiliary variable as

η = x − x − μ( − 0 ) (2.32)

where μ is the output of the following filter

˙ μ = G(x,T) − k μ μ, μ(t 0 ) = 0 (2.33)

such that we can verify that η fulfills

˙ η =−k μ η, η(t 0 ) = e p (t 0 ) (2.34)

where e p (t 0 ) is the prediction error at the time t 0.

Lemma 2.1. [21]Define Q ∈ R and C ∈ R as intermediate variables, which
are calculated based on the following equations:

T
˙
Q = μ μ, Q(t 0 ) = 0 (2.35)
T
C = μ (μ 0 + x − x − η), C(t 0 ) = 0
˙
and let t c be the time such that Q(t c )> 0, and then the following adaptive law is
given

˙
) =− Q ˜
ˆ
ˆ
= (C − Q ˆ , (t 0 ) = 0 (2.36)
T
with = > 0 being the learning gain. Hence, this adaptive law can guarantee
˜
that the estimation error = − is non-increasing for t 0 ≤ t ≤ t c and exponen-
ˆ
tially converges to zero after t > t c .