GAIN SCHEDULING CONTROL OF MACHINE TOOLS WITH VARYING STRUCTURAL ...
GAIN-SCHEDULING CONTROL OF MACHINE TOOLS WITH VARYING STRUCTURAL
FLEXIBILITIES
Wim Symens, Hendrik Van Brussel, Jan Swevers
PMA, Mechanical Engineering Department, Katholieke Universiteit Leuven, Celestijnenlaan 300B,
3001 Heverlee, Belgium
Summary In this paper an experimental set-up, consisting of one axis of an industrial pick-and-place machine, driven by a linear motor,
is controlled based on the gain-scheduling approach. The set-up contains a flexible arm of which the stiffness depends on its length. Next
to an ad-hoc scheduled H∞ controller, analytically scheduled controllers are designed using a global linear time varying (LTV) model of
the set-up. Experiments show that scheduling is necessary if high-performance controllers are demanded.
INTRODUCTION
The growing competition on the international markets pushes manufacturers towards faster machine tools while
preserving or improving final accuracy. High accelerations of the fast machines excite the machine structure up to high
frequencies thereby exciting the structure's modes of vibration. These structural vibrations need to be damped if accurate
positioning or trajectory tracking is required.
An additional problem is that the dynamical behaviour of the machine tool depends on the position of the work tool as a
consequence of the varying machine configuration during machining. Such time-varying behaviour cannot be controlled
by classical linear control methods as these methods require a linear time invariant (LTI) model of the system. One
solution to this problem is to ensure that the designed LTI-controller makes the system behaviour robustly stable against
the varying dynamics of the machine tool. The performance of the closed loop behaviour however could be improved if
the variations could be included in the controller by making the controller also dependent on the instantaneous
configuration of the machine tool by gain-scheduling. In classical gain-scheduling, LTI controllers are designed for
several fixed positions of the machine, and these controllers are scheduled ad-hoc. More recent design methods for gain-
scheduling controllers start from a linear parameter varying (LPV) description of the model [1]. Using this kind of
models, criteria for robustness can be derived analytically. The most up-to-date and generalised framework for LPV-
based control design is given in [3].
DESCRIPTION OF THE SET-UP
Rotative motors
Linear Motors
In this paper, the control of the X-Z axis of a Philips 4-axis
pick-and-place machine (FlexCell) is considered (Fig. 1).
The position of the X-motor and the length of the vertical
arm are measured with an optical encoder. An accelerometer
measures the acceleration of the end point of the arm. The
Encoder
Accelerometer
objective is to move the end point of the arm accurate and
L
fast along a prescribed trajectory in the X-Z plane. Fast
movements of the
linear
motor
will
excite
the
eigenfrequencies of the flexible arm.
Fig. 1: Set-up: pick-and-place machine.
EXPERIMENTAL IDENTIFICATION OF THE SET-UP
Models from motor force to end-tip acceleration
]
Fig. 2 shows frequency response functions (FRFs) from the linear
/
N
0
2
10
motor force excitation to end point acceleration, for different
s
[
m
lengths of the arm. This figure clearly shows the dependency of
e
d
the first two eigenfrequencies of the arm on the length of the arm.
i
t
u
n
-2
The FRFs from the motor force to the motor displacement and
g 10
a
from the rotative motor input torque to the Z-position of the tool
M
1
2
10
10
tip do not depend on the length of the beam and are omitted for
Frequency[Hz]
reasons of brevity.
0
]
g
e -200
CONTROL DESIGN
[
d
e
L = 34cm
s
L = 37.75cm
-400
a
h
L = 41.5cm
The objective of the control design is to move the end point of
P
L = 45.25cm
-600
L = 49cm
the arm as accurate and fast as possible using the actuators that
1
2
10
10
are present on the machine tool. For the set-up the motion
Frequency[Hz]
controller is designed with a vibration controller in a HAC-LAC
Fig. 2:FRFs from linear motor force excitation (N) to
structure [2] (Fig. 3). The position of the motor is used for the
end point acceleration (m/s2).
position control of the end point of the arm and the acceleration of the arm tip is used to damp the vibrations.
VIBRATION CONTROLLER
C2(s)
In this section, vibration controllers aimed at damping the
Vibration controller (LAC)
first eigenfrequency of the flexible arm are presented.
acc
F => acc
Ad-hoc scheduled H∞ controller
encdesired
enc
H∞ controllers [4] are designed for constant arm lengths to
C1(s)
F => enc
e
F
suppress both disturbances entering at the motor input and at
Motor controller
System
the end point of the arm. These controllers are linearly
(Hac)
scheduled based on the arm length and result in a controller
Fig. 3: Control scheme.
with high performance as will be shown in the next section.
Analytically scheduled LPV controller
The robustness of the ad-hoc scheduled H∞ controller is not guaranteed theoretically. To guarantee this stability a model
is needed that depends on the length of the arm. Such a model is called a linear parameter varying (LPV) model.
The most general framework for LPV-control design is presented in [3] and writes the design problem as a linear matrix
inequality (LMI) that needs to be solved to obtain the controller. An LPV model of the set-up is identified and used to
design an LPV controller. The performance of this controller is very low compared to the scheduled H∞ controller. The
inherent conservatism of the used LPV-design method and bad numerical conditioning of the resulting LMI design
problem explain this low performance.
As the synthesis of LPV controllers proved to be conservative and
Vibration suppression
the analysis tools for LPV systems are less conservative, an LPV
15
model is identified for the ad-hoc scheduled H∞ controller and the
10
resulting closed-loop system is analysed for stability and
]2
performance. Even the state-of-the-art analysis tools are however
/
s
5
not able to predict the correct stability region of this closed loop
[
m
n
LPV system as they are still conservative to an extended degree.
t
i
o
0
r
a
l
a
e -5
EXPERIMENTAL RESULTS
c
c
A
-10
Without control
To validate the different controllers a reference trajectory is applied
X-trajectory (not on scale)
Z-trajectory (not on scale)
to the set-up. Fig. 5 shows the acceleration signal for this trajectory
-15
With control
with and without vibration control for the scheduled H∞ controller.
0
1
2
3
4
5
6
7
8
The figure clearly shows the damping effect of the controller. For
Time [s]
reasons of brevity the (poor) results of the analytically scheduled
Fig. 5: Response with and without the scheduled H∞
controller.
controller is not shown in the paper.
CONCLUSIONS
In this paper ad-hoc scheduled H∞ controllers and analytically scheduled LPV vibration controllers are designed for a
machine tool with varying dynamic stiffness. The performance of the analytically scheduled controller is much lower
than this of the ad-hoc scheduled controller. This can be explained by the extended degree of conservatism that is
introduced in the LPV design methods and numerical limitations of present day LMI solvers.
This study shows that further development of the LPV-tools is necessary to be able to design high-performance
analytically scheduled LPV controllers for motion and vibration control systems for machine tools.
Acknowledgments
This research is sponsored by the Belgian program of Interuniversity Poles of Attraction by the Belgian State, Prime
Minister's Office, Science Policy Programming (IUAP). W. Symens was a Research Assistant of the Fund for Scientific
Research - Flanders (Belgium) (F.W.O.) in the period that this research was performed (1999-2003).
The authors would like to thank Carsten Scherer for providing a draft version of the LPV-synthesis toolbox.
References
[1] Apkarian P., Gahinet P.: A convex characterization of parameter dependent H∞ controllers. IEEE Transactions on
Automatic Control 40(5):853-864, 1995.
[2] Preumont, A.: Vibration control of active structures. Kluwer Academic Publishers, Dordrecht 1997.
[3] Scherer C.: LPV control and full block multipliers. Automatic 37:361-375, 2001.
[4] Skogestad S., Postlethwaite I.: Multivariable feedback control: Analysis and Design. John Wiley & Sons, 1996.
