Page 207 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

13.3 ADAPTIVE SLIDING MODE CONTROL DESIGN AND
STABILITY ANALYSIS

13.3.1 Non-linear ESO Design
To design ESO, we first represent system (13.7) into a more compact form.
For this purpose, define x 1 = θ m, x 2 = ω m,and then(13.7) can be rewritten
as:

˙ x 1 = x 2
(13.8)
˙ x 2 = a(x) + bu
where x 1, x 2 are the system states, and u is the output of the controller.
D 1 K t d 1 (u)
T
a(x) =− ω m − T f + with x =[x 1 ,x 2 ] is the lumped unknown
J J J
system dynamics including friction and saturation approximation error d 1,
K t g u ξ
and b = is the input gain.
J
In practice, parts of the system dynamics a(x),b may be known. Hence,
to design the ESO we set a(x) = a 0 + a, b = b 0 + b, d(x,u) = a +
bu,where a 0 and b 0 are the nominal values of a(x) and b, which can be
obtained based on the prior knowledge; and d(x,u) represents the system
uncertainties.
Define the extended state x 3 = d,andthen(13.8) is augmented as:
⎧
⎪ ˙ x 1 = x 2
⎨
˙ x 2 = x 3 + a 0 + b 0u (13.9)
⎪
⎩ ˙ x 3 = h
˙
where h = d is the derivative of uncertainties.
Hence, we can use an observer to estimate x 3.Let z i , i = 1,2,3be
the observation of the state variable x i in the system (13.9). Then, the
non-linear ESO is designed by
⎧
⎪ ˙ z 1 = z 2 − β 1g(e o1 )
⎨
˙ z 2 = z 3 − β 2g(e o2 ) + a 0 + b 0u (13.10)
⎪
˙ z 3 =−β 3g(e o3 )
⎩
where β 1, β 2,and β 3 are the observer gain parameters, g(e oj ) is a function
given by

α j
|e o1 | sgn(e o1 ), |e o1 | >τ
g(e oj ) = e o1 j = 1,2,3
, |e o1 |≤ τ
τ 1−α j