29-03-2012, 03:08 PM
Variable Structure Control of a Distributed-Parameter Flexible Beam
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INTRODUCTION
Modelling and control of flexible link manipulators
have been studied intensively in the literature. Since
the vibration of a flexible link is governed by partial
differential equations ŽPDEs., the system is a distributed-
parameter system and possesses infinite
number of dimensions, which makes it difficult to
control. Further, the inherent nonminimum phase
characteristic from the base actuator to the free tip
of a flexible robot makes it very difficult to achieve
high level performance and robustness simultaneously.
2. DYNAMICS OF THE SYSTEM
Let us consider a flexible beam which is clamped at
its base on the rotor of a motor as shown in Figure
1. The flexible beam is rotating in the horizontal
plane and the effect of gravity is neglected. Frame
XOY is the fixed inertia frame and frame xOy is the
local reference frame with axis Ox being tangent to
the beam at the base. System parameters and variables
are defined as follows:
L: the length of the beam;
EI the uniform flexural rigidity of the beam;
: the uniform mass per unit length of the beam;
Ih: the hub inertia;
: the control torque provided by the motor;
: the joint angle;
yŽ x, t.: the elastic deflection measured from the
undeformed beam;
pŽ x, t.: the approximated length-of-arc position
of the beam.
VARIABLE STRUCTURE CONTROLLER
3.1. Controller Design
In this section, the variable structure controller is
presented for the regulation of the flexible beam
system described in Section 2. First, we need to
choose a suitable sliding variable S. In ref. 15, the
sliding variable was chosen based on the truncated
model of the flexible robot. Consequently, the flexible
modes and their first- and second-order time
derivatives are needed for feedback. These signals
are, in general, difficult to obtain. Bearing this in
mind, we construct a sliding variable which leads to
a controller that is relatively easy to implement. The
implementation problems are discussed later in this
section.
CONCLUSION
Regulation of a distributed-parameter flexible beam
was investigated using the VSC technique in this
article. A novel controller was developed based on
the PDEs of the system, and thus the problems
associated with the truncated-model-based controllers
are overcome. The controller can be easily
implemented since all the required signals can be
measured or estimated by some currently available
sensor facilities.