Page 54 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Control of Two-Inertia Systems With LuGre Friction Model 45

where ˜z = z − z. We can choose appropriate matrix L such that A − LC
is stable to ensure the stability of error dynamics (3.21). A well-known
stability condition of the matrix A − LC is presented in [31]. Then for
the ease of simple analysis, the observer states ˆx 1 , ˆx 2 will be used in the
following error constraint dynamic surface control (ECDSC) design with
friction compensation.

3.3.2 Error Constraint Dynamic Surface Control Design
The control for non-linear two-inertia system is designed based on the idea
of DSC and PPF, which is given in the following steps.
Step 1: Define the output tracking error as

s 1 =ˆx 1 − x d . (3.22)

Then from (3.17), we can obtain

s 1
z 1 = R 1 (3.23)
μ 1
Thetimederivativeof z 1 is

˙ μ 1 ˙ μ 1
˙ z 1 = r 1 ˙ s 1 − s 1 = r 1 ˆ x 2 −¨x d − s 1 (3.24)
μ 1 μ 1
¯
where r 1 = (1/2μ 1 )[1/(ρ 1 + δ ) − 1/(ρ 1 − δ 1 )],and ρ 1 = s 1 /μ 1.
1
To avoid the problem of “explosion of complexity” in the traditional
backstepping design [32], we let x d go through a HGTD as

˙
ϑ 1,1 = ϑ 2,1
(3.25)

α
˙
ϑ 2,1 = H 2 − ρ 1,1 [ϑ 1,1 − x d ] − ρ 2,1 [ϑ 2,1 /H] β
where H,ρ 1,1 ,ρ 2,1 ,α,and β are positive constants, ϑ 1,1 is the filter signal of
the desired trajectory x d . Then the time derivative of z 1 with (3.25)is
˙ μ 1

˙ z 1 = r 1 ˆ x 2 − ϑ 2,1 − s 1 (3.26)
μ 1
By defining s 2 =ˆx 2 −¯χ 1 as the intermediate error, one obtains

ˆ x 2 = s 2 +¯χ 1 . (3.27)