Page 52 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Control of Two-Inertia Systems With LuGre Friction Model 43

designs
δ ie − δ e −z i
z i
i
S i (z i ) = . (3.16)
e i + e −z i
z
Then, from (3.16), the transformed error z i is derived as

e i (t) e i (t) 1 e i (t) 1 e i (t)
−1
z i = S = R i = ln + δ − ln δ i − ,
i i
μ i (t) μ i (t) 2 μ i (t) 2 μ i (t)
(3.17)
where R i (·) is the inverse function of S i (·). The transformed error will be
utilized to ensure the prescribed output tracking error performance.

3.2.4 High-Gain Tracking Differentiator

In this chapter, we will design a control based on DSC. Hence, a filter
should be used to avoid the repeated calculation of derivatives of the virtual
control signals. To enhance the convergence speed of this operation, high-
gain tracking differentiator (HGTD) reported in [30] will be used, which
is given by

ϑ 1i (t) = ϑ 2i (t) (3.18)
˙
α
˙
ϑ 2i (t) = H 2 −ρ 1i [ϑ 1 (t) −¯χ i ] − ρ 2i [ϑ 2i (t)/H] β
where ρ ji, α, β,and H are positive design parameters, ¯χ i represents the
input signal of HGTD, which is the virtual control signals in the DSC
design procedure.

(j)
Lemma 3.2. [30]: If the signal ¯χ i satisfies sup|¯χ | < ∞ for j = 1,2, then the
i
HGTD (3.18) is convergent for any initial condition within finite-time T > 0,
i.e., there exists H > H 0 > 0 and t > T, such that

a/b
˙
ϑ 1i (t) −¯χ i è L 1 (1/H) , |ϑ 2i (t) − ¯χ i |≤ L 2 (3.19)
where L 1,L 2, a, and b are constants.
The fast convergence property of HGTD as shown in Lemma 3.2 makes
it superior over linear filters in the other DSC control designs to solve the
problem of “explosion of complexity” encountered in the backstepping
methods.