Page 208 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ESO Based Adaptive Sliding Mode Control of Servo Systems With Input Saturation 207

where α 1 ,α 2 ,α 3 and τ are positive numbers between [0,1],and e o1 = z 1 −x 1
is the observer output error.
Now, we will present the design of the observer gain parameters β 1, β 2,
β 3 by using the pole placement method [12]. Define 1 = e o1 = z 1 − x 1,
2 = z 2 −x 2, 3 = z 3 −d, and then from (13.10)and (13.8), we can obtain

⎧
⎪ ˙ = 2 − β 1g( 1 )
⎨ 1
2 = 3 − β 2g( 1 ) (13.11)
˙
⎪
3 =−β 3g( 1 ) − h
⎩ ˙
Since the function h is bounded in practice and g(e o1 ) is smooth and
g(0) = 0, g (e o1 ) = 0, according to Taylor expansion, Eq. (13.11)can be

rewritten as
⎧
⎪ ˙
⎨ 1 = 2 − β 1g ( 1 ) 1

2 (13.12)
˙ = 3 − β 2g ( 1 ) 1

˙
⎪
⎩ 3 =−β 3g ( 1 ) 1 − h

Denote l i = β ig ( 1 )(i = 1,2,3), and we have
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤
˙ 1 0 0
1 −l 1 1
˙
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣ 2 ⎦ = ⎣ −l 2 0 1 ⎦⎣ 2 ⎦ + ⎣ 0 ⎦ h (13.13)
˙ 0 0 −1
3 −l 3 3
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
−l 1 10 0 1
Define the matrices A = ⎣ −l 2 01 ⎦, E = ⎣ 0 ⎦, = ⎣ 2 ⎦,
⎥
⎢
⎢
⎢
⎥
⎥
−l 3 00 −1 3
and then (13.13) can be rewritten as
˙
= A + Eh. (13.14)
Therefore, the determination of the parameters β i is achieved by choos-
ing constants l i. The necessary condition for the asymptotic stability of
(13.13) that is subject to non-linear disturbances is that all eigenvalues of
the matrix A are all in the left half part of the complex plane. Namely, the
poles of (13.13) are all negative. Moreover, the position of eigenvalues of A
also determines the convergence rate of error i . Therefore, according to
the pole placement method, the expected pole p i (i = 1,2,3) can be selected
in a priori so that the parameters l i can be calculated based on the following