with x = [x1 x˙ 1]
T
and w = g is considered as a disturbance.
The objective of this paper is to design an anti-windup
adaptive PID control of the magnetic levitation system which
has the output y(t) regulates a set point r.
B. Error System with Internal Model Filter
In order to achieve the control objective, we first introduce
an internal model filter.
u(t) = GIM(s)[ua(t)] =
s + α
s
[ua(t)] , (4)
where ua(t) is a designed control input.
Defining an output error by e(t) = y(t)−ym, the obtained
error system form the input ua(t) to the error e(t) can be
expressed by the following form:
e˙(t) = Aee(t) + beua(t) + bηη(t)
η˙(t) = −αη(t) + c
T
η e(t)
e(t) = [1 0]e(t)
(5)
Where e(t) = [e(t), e˙(t)]T
, and Ae, be, bη and cη are
appropriate matrix and vectors.
It is noted that the error system (5) with the internal
model filter can be expressed as 3rd order system with a
stable first order zero dynamics in which the effects from
the disturbance has been eliminated and has the following
transfer function.
G
∗
p2
(s) = G
∗
p1
(s)GIM (s) = b
∗
(s + α)
s
2(s + β
∗)
(6)
In the following section, we first consider to design an
ASPR based adaptive PID control system.
III. ASPR AUGMENTED SYSTEM WITH A PFC
In order to design an ASPR based adaptive PID control
system, the controlled system must be ASPR. Since the error
system (5) is not ASPR, we consider introducing a PFC in
parallel with the error system (5). The PFC is designed so
as to render the augmented controlled system with a PFC in
parallel ASPR.
A. Model-based PFC Design
The model-based PFC design scheme has been provided
as one of the simple PFC design schemes [6], [7]. In this
scheme, the PFC GPFC (s) can be designed using the system
approximated model as follows:
GPFC (s) = GASP R(s) − G
∗
p2
(s) (7)
where GASP R(s) is a given desired ASPR model and G∗
p2
(s)
is an approximated model of the controlled system.
In the considering magnetic levitation system, the approximated
model is given in (6) . Taking this in consideration,
we design an ASPR model as a form of
GASP R(s) = k(s + h0)
s
2
(8