Neural Control of Pendulum Arm
A block diagram
of the system (plant) is shown in Fig. 1. The DC motor and gear train drive
a aluminum rod of 45.9 cm in length with a mass of 118 grams. Additional
loads can be attached to the rod as shown in Fig. 2. See control
bench for a photo and arm
assembly for a closeup photo of the external gear train and arm. The
control platform was obtained from Quanser
Consulting.
A linear model for the plant for small angles (< + 20 deg) is
G(s) = 386 e-0.02s/(s + 3.6)2
with units in degrees/volt. The arm was selected (length and mass) such
that the plant would exhibit a high degree of static and kinetic friction.
The dead-zone due to static friction was modeled as time delay in the G(s)
expression above.
The plant nonlinearity
is shown in Fig. 3. Figure 3 is the ramp response with a proportional controller
(Kp =0.05).
Figure 4 is a
better illustration of the dead-zone as well as the hysteresis in this
system. In Fig. 4, the output was plotted versus the input command.
An adaptive controller
was designed for this plant based on the CMAC (cerebellar model articulation
controller) network by Albus. The step response is shown in Fig. 5. The
command swing is + 90 degrees. The step rise and fall times were
limited to 40 degrees/sec to avoid controller (D/A converter) saturation.
After 20 cycles of training the steady state error is 2 degrees. A PI or
PID controller cannot be implemented because of the development of a limit
cycle due to the large plant dead-zone. A PD controller design with a phase
margin specification of 90 degrees results in a steady state error of 36
degrees for the test conditions given above. The linear and neural controllers
were implemented in a 200 MHz Pentium-based computer (C language) as shown
in Fig. 1. "Using Conventional Controllers with the CMAC Neural Network"
is being developed for submittal to the ANNIE '99 Conference in St. Louis,
Mo.