Page 210 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ESO Based Adaptive Sliding Mode Control of Servo Systems With Input Saturation 209

where λ 1 > 0 is a positive constant. Hence, the tracking error e c1 is bounded
as long as s is bounded.
Then, from (13.9)and (13.17), the first derivative of s is calculated as

s ˙ =˙e c2 + λ 1 ˙e c1 = x 3 + a 0 + b 0u −¨x 2d + λ 1 (x 2 −¨x 1d ) (13.18)

According to (13.18), a sliding mode control based on ESO (13.16)can be
designed as

1
∗ ∗
u = [−z 3 − a 0 +˙x 2d − λ 1 (z 2 −¨x 1d ) − k sgn(s)] (13.19)
b 0
where k > 0 is the ideal control gain satisfying k ≥| 3 + λ 1 2 | for the
∗
∗
upper bound of observer error | 3 + λ 1 2 |,and z 2 ,z 3 are the states of
ESO given in (13.16).
In practical control implementation, the upper bounds of the estimation
errors 2 and 3 are usually unknown, and it is difficult to determine
the control gain k for (13.19). Hence, we will present an adaptive law to
∗
update the parameter k . Then based on the idea of parametric adaptive
∗
laws in [13], an adaptive sliding mode controller is designed as
1
u = [−z 3 − a 0 +˙x 2d − λ 1 (z 2 −¨x 1d ) − k(t)sg(s)] (13.20)
b 0
with
⎧
⎨ sgn(s) |s|≥ μ
sg(s) = 2|s|
sgn(s) |s| <μ
⎩
|s|+ μ
where μ> 0 is the boundary parameter and k(t) is the adaptive feedback
gain, which is updated based on the following adaptive law

˙
k(t) = k ms · sg(s) (13.21)
where k m > 0 is the adaptive learning gain.

13.3.3 Stability Analysis
This subsection will present the stability of the closed-loop system with the
proposed control (13.20) and adaptive law (13.21). This can be summarized
as follows: