4.5. EXERCISES AND PROJECTS 159
radius of the ball, andgthe gravitational constant, the Lagrange equa-
tion of motion can be written as
∑
J
r^2
+M
∏
R®+Mgsinα−mR(α ̇)^2 =0
The beam angleαmay be approximated by a linear relationshipα=DLθ.
These equations form the set of coupled equations for the system.
Use the following system parameters
Mmass of the ball 0 .2kg
rradius of the ball 0 .02 m
Dlever arm offset 0 .03 m
ggravitational acceleration 9 .8m/s^2
Llength of the beam 1 .5m
Jthe moment of inertia of the ball 2 e−^6 kg m^2
(a) Obtain a linearized model of the system for small variations inα.
(b) For the linearized model, obtain a set of PID control parameters to
satisfy a chosen criteria. Use Simulink inMatlabfor developing
such a model. Test the system performance for various disturbance
conditions.
(c) Using the nonlinear model, obtain a suitable fuzzy controller.
(d) Compare and contrast the performance of the PID and fuzzy con-
trollers.
- Magnetic levitation (MagLev) is a technology presently used in high-speed
transportation systems. The idea is to levitate an object with a magnet
and to propel the object at high speed. This technology has found prac-
tical use in high-speed trains in Japan and is being explored for a similar
purpose in the U.S. NASA is exploring this technology for rocket propul-
sion as a means to reduce the fuel payload during launch.
In this exercise, the idea is to levitate a steel ball using the configuration
shown in Figure 4.18. The objective is to control the vertical positionz
of the steel ball of massMby controlling the current inputito the elec-
tromagnet through a variable voltageV.TheforceF produced by the
electromagnet must overcome the forceMgcreated due to gravity, where
gis the gravitational constant.
