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Sliding Mode Control
for Active Magnetic Bearings
Zdzislaw GOSIEWSKI Mariusz ZOKOWSKI
Dept. of Automatics and Robotics Dept. of Mechanical
Technical University of Bialystok Technical University of Koszalin
Wiejska 45c St., 15-351 Bialystok, POLAND Raclawicka 15-17 St., 75-620 Koszalin, POLAND
gosiewski@pb.bialystok.pl mariusz_zokowski@interia.pl
1 Abstract
The paper evaluates the sliding mode control
methodology as a control law for an active magnetic
bearing.
2 Introduction
Developments in power electronics has made it
possible to implement a modern control theory to
various kinds of electrical and mechanical systems.
This paper describes sliding mode control of a single-
axis magnetic bearing’s actuator.
Sliding mode control (SMC) offers several
advantages over the other control methods as follows:
• robustness;
• good dynamic response;
• simple implementation.
Sliding mode control is a non-linear, model-based
control method supported by the Lyapunov stability
theory. It has been proved to be an efficient technique
to provide good tracking performance, applied to non-
linear systems with external disturbances. It also
allows the plant model to be imprecise, for example a
model that has been simplified due to difficulties in
representing system dynamics. Among applications
where sliding mode control successfully has been
applied there are: submarines, ground vehicles and
aeroplanes.
3 Theory of SMC
Systems such as active magnetic bearings are
second order system particularly. The basic idea is to
replace an n
th
order system (n > 1) by a first order
system to facilitate the control. Instead of getting an n-
dimensional state vector to track an n-dimensional
desired time-varying state vector, the method reduces
the problem to the tracking of a scalar.
The mathematics approach of sliding mode control
can be described as follows. Consider the single input
non-linear system such as active magnetic bearing:
u)x(b)x(fy
)n(
⋅+=
where:
u – the control input;
T)1n(
]y...,,y,y[x
−
=
o
– the state vector.
The function f(x) and control gain b(x) are normally
non-linear and not exactly known, but the extent of
uncertainty is upper bounded by a continuous function
of x. The control gain b(x) is of known sign. The
desired state vector:
T
)1n(
d
d
dd
y...,,y,yx
=
−
o
is to be tracked. A tracking error vector is:
d
T
)1n(
~~~~
xxy...,,y,yx −=
=
−
o
and gives the error in each state. These state errors are
then weighted individually and together define a
function s(x, t), which is the weighted sum of errors:
~
1n
y
dt
d
)t,x(s ⋅
+=
−
λ
where:
λ
– a design parameter. We can find error s(x, t) for
any n
th
order system, for instance:
( )
d
d
~~
d
~
yyyyyys2n
yyys1n
−+
−=+=⇒=
−==⇒=
λλ
oo
o
We see, for example, that the second-order
mechanical system is replaced by a first order system.
A time-varying surface S(t), called sliding surface,
is defined by the scalar-equation s(x, t) = 0. The
sliding surface of a second order system becomes a
line and is presented visually in a phase-plane (fig. 1).
Fig. 1. The sliding surface is described by s = 0.
In control design – the main problem is to get
x = x
d
, which is equivalent to keep system described by
function s(t) on the sliding surface.
4 Plant of active magnetic
bearing
Dynamics of active magnetic bearing can be
describeb by following equations [3]:
• mechanical part:
zxi
Fxk2ik2xm
+
+
=
&&
• electrical part:
dt
dx
LL k
LL Ri
LL u
dt
di
So
i
So
1
So
11
+
−
+
−
+
=
dt
dx
LL
k
LL
Ri
LL
u
dt
di
So
i
So
2
So
22
+
+
+
−
+
=
where:
m – mass of rotor;
k
i
– current stiffness;
k
x
– displacement stiffness;
x – displacement of rotor;
i – current of electromagnetic coil;
F
z
– external force;
u
1,2
– voltage;
R – resistance of coil;
L – inductance.
The model represented by above equations is the
one used in the simualtions.
Fig. 2. A magnetic bearing system.
Figure 2 shows the schematic of a typical magnetic
bearing system. The system consists of two
electromagnets and an object to be levitated. In this
figure:
i
1
and
i
2
respectively represent the coil currents
input to the electromagnets;
x
0
denotes the nominal air
gap, and
x
is the displacement. It is also assumed that
both electromagnets have the same pole area
A
and
identical number of turns
N
in the coil.
5 Design of controller
The sliding mode controller consists of two
components, one that keeps the error at zero and one
that guarantees the error reaches zero. The sliding
mode control law is next:
)s(sign)x(kuu
1
⋅
+
=
where:
u
1
– is equivalent control, a level-keeping component.
It is obtained from equation
0
s
=
o
, which keeps
the
s(x, t)
on the sliding surface, once it is
reached;
)
s
(
sign
)
x
(
k
⋅
-
the striking component that makes sure
that the weighted sum of error
s
, reaches zero. It
also forces the error back to zero if it becomes
non-zero due to external disturbances or model
uncertainties.
Once the system has reached the sliding surface,
the switching component is zero.
k(x)
- the striking control, is chosen to fit the
Lyapunov condition:
|s|s
dt
d
2
1
2
⋅−≤
η
where:
2
s
dt
d
2
1
contains
k(x)
, which makes sure that the
function reaches the sliding surface in finite
time;
η
– is a design parameter that decides about
response time of the system. Greater η means
that the sliding surface is reached quicker, but
it also brings higher control activity (energy).
The sign function sets striking direction:
>
=
<
−
=0sfor1
0sfor0
0sfor1
)s(sign
The striking and the level-keeping component
cooperates to reach the desired state
x
d
(Fig. 3).
Fig. 3. Realization of SMC for second order system
using sliding mode controller.
Classical and modern control theory is based
especially on linear systems and provides very good
design solutions. To obtain the advantage of linear
control theory it is linearize the system at one
operation point (working point). For systems like the
active magnetic bearings it is possible to find a
feedback control law for a controller so that the closed
loop system satisfies predefined specifications.[5] The
mathematical model that has been developed in section
4 has been implemented using Matlab-Simulink in
order to show characteristics and step response of the
system.
Since AMB is a nonlinear system, the deployment
of sliding mode control is considered. By using sliding
mode control, the nonlinearity of the AMB system is
expressed using sliding principles in description.
The sliding mode control is based on the model
structure presented in section 4.
Introducing as a new state-space variable the
following displacement error:
x)t(xxe
01
−
=
=
where
x
0
(t)
is a reference input. On the base dynamic
equations and displacement error we get state-space
model in form:
++=
=
m
F
x
m
k2
u
m
k2
x
xx
z
1
xi
2
21
&
&
where:
•
3
0
2
0
x
i
2
K
k
x
=
;
•
2
0
0
i
x
i
2
K
k=
.
For discontinuous control:[5]
)s(signuu
0
⋅
=
with following switching function in form:
21
xxcs
+
⋅
=
where
c, u
0
are const, the error
x
1
decays exponentially
after reaching of the switching line
s = 0
since its
equation:
0xxc
11
=
+
⋅
&
Taking above equations into account and introduce
to control law we can obtain the following equation:
m
F
x
m
k2
u
m
k2
xcs
z
1
xi
2
+++⋅=
&
m
F
x
m
k2
)s(signu
m
k2
xcs
z
1
x
0
i
2
++⋅+⋅=
&
Can be noticed that the values of switching function
s
and their time derivative ś have opposite signs when
control signal fulfil the following condition:
m
F
x
m
k2
xcu
m
k2
z
1
x
20
i
−−⋅−>
As a result of above the state of plant reaches the
sliding surface ś
= 0
in finite time. It means, the above
equation indicates minimal value of control voltage
which causes the error slides to zero.
c
2
deltae
1
e
-K-
-K-
1du/dt
Derivative
1
e0
e(k) e(k)
e(k)-e(k-1)
Fig. 4. Subsystem to getting switching function in the
form: s = cx
1
+ x
2
.
6 Computer simulation
Control systems with Sliding Mode Controllers
were investigated in the computer simulation.
The dynamic properties of close-loop system with
SMC controllers based on the following control laws:
)s(signuu
0
⋅
=
)e(signuu
0
⋅
=
which shows following time responses. They were
obtained during start phase of the system.
The result of responses is that the order-reduction
does not cause incoreect work of control system. But
time response of the model with SMC controller given
by control law:
)s(signuu
0
⋅
=
has less amplitude of oscilations.
0 0.5 1 1.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
xzad
s=ce+de/dt
Fig. 5. Time responses of systems with control rules:
s = cx
1
+ x
2
.
This control is characterized by slide of state-space
modes on switching line. There is terminal cycle,
which causes non-finite frequency depends of constant
value
c
. The value of this parameter was found
experimentally on the base of computer simulations.
The systems with SMC controllers are
chcracterized by the changes of control signal
according to:
x)t(xxe
01
−
=
=
and
m
F
x
m
k2
xcu
m
k2
z
1
x
20
i
−−⋅−>
and load voltage gets only two values (Fig. 6).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
Time [s]
Supply voltage U [V]
Fig. 6. Power supply.
7 Conclusion
The paper shows possibilities of using sliding mode
control to regulate of displacement of active magnetic
bearing system.
In the future, will be done comparison between
conventional controller such as PID and SMC in
connection with consumption electric energy. It is
required less consumption electric energy to control
system.
8 References
[1] L. Li, P. E. Allaire, Sensorless Sliding Mode
Control of Magnetic Bearing Actuators using
Implicit Switching Surfaces, Seventh
International Symp. on Magnetic Bearings,
August 2000, pp. 311-316;
[2] Z. Gosiewski, D. Hanc, Sliding Mode Control
of an Aircraft Electric Drive, Recent Research
and Design Progress in Aeronautical
Engineering and its Influence on Education,
Warsaw 2000, Poland;
[3] Z. Gosiewski, K. Falkowski, Wielofunkcyjne
ŁoŜyska Magnetyczne (Multifunctional
Magnetic Bearings), Biblioteka Naukowa
Instytutu Lotnictwa, Warszawa 2003;
[4] Z. Gosiewski, M. Zokowski, Sliding Mode
Control of Active Magnetic Bearing System,
AGH Krakow, Mechanics, Vol. 24, No. 2,
2005;
[5] V. Utkin, Sliding Mode Control Design
Principles and Applications to Electric
Drives, IEEE Transactions on Industrial
Electronics, Vol. 40, No. 1, 1993.
