The objective of this lab is to implement a PD controller for a 2DOF system with an oscillatory mode. Students will gain a better understanding of the limitations of PD/PID control for higher order systems. Students will design, simulate, and implement a non-collocated controller with multiple feedback loops to acquire an acceptable response for the system. The controller will be designed and implemented in LabVIEW using the Simulation Module and Control Design Toolkit.
Collocated / noncollocated control of 2dof rectilinear
system
Objectives
Implement a PD controller for a 2DOF system with a
oscillatory mode.
Understand the limitations of PD/PID control for higher order
systems.
Design, simulate, and implement a noncollocated controller
with multiple feedback loops to acquire an acceptable response forthe system.
Pre-lab
Consider the system shown below. Both mass carriages are
loaded with four
$0.5kg$ brass weights and the medium stiffness
spring is connecting them.
Derive the equations of motion for this system and rewrite them so that
the control effort is
$u\left(t\right)$ (DAC counts) and the respective
positions, velocities, and accelerations are:
${x}_{1e}$ ,
${\stackrel{.}{x}}_{1e}$ ,
${\stackrel{..}{x}}_{1e}$ ,
${x}_{2e}$ ,
${\stackrel{.}{x}}_{2e}$ ,
${\stackrel{..}{x}}_{2e}$
From your EOM derive the appropriate transfer function
numerator and denominator polynomials
${N}_{1}\left(s\right)$ ,
${N}_{2}\left(s\right)$ , and
$D\left(s\right)$ in the
block diagram below:
Using root locus techniques, find the rate feedback gain
${k}_{v}$ that provides satisfactory damping of the complex roots of the
inner loop
${x}_{1}\left(s\right)/{R}^{*}\left(s\right)$ .
With
${k}_{v}$ determined, you can now design for the system given
by
${G}^{*}\left(s\right)$ in the block diagram. Design a notch filter,
${G}_{n}\left(s\right)={N}_{n}\left(s\right)/{D}_{n}\left(s\right)$ with two poles at
$5.0Hz$ and two additional higher
frequency poles at
$8.0Hz$ , using
$\zeta =\sqrt{2}/2$ for both poles. Place the
zeros of
${G}_{n}\left(s\right)$ such that they cancel the oscillatory poles of
${G}^{*}\left(s\right)$ . Finally, normalize the notch filter transfer function to
have unity DC gain.
Write a LabVIEW VI that simulates this plant configuration
with two differentcontrollers. Write your VI so it displays theresponse of both mass carriages.
Collocated: Simulate your critically damped PD controller
from Lab #3 where you are feeding back the position of the first
carriage. Use the PD controller with the differentiator in theinner feedback loop. With these gains, what do you notice about the
behavior of the second mass carriage? Remember to record thesegains so you can implement them in the lab. Now iteratively reduce
the controller gains until you are able to achieve minimalovershoot for both carriages (try for less than
$10\%$ ) with as fast a
response as possible. Again, don't forget to record thegains.
Noncollocated: Simulate the controller you designed in steps
3 and 4 above. Find
${k}_{p}$ and
${k}_{d}$ to meet rise time and overshoot less
than
$0.5\mathrm{sec}$ and
$10\%$ , respectively.
Lab procedure
Configure the Model 210 plant for this experiment. Be sure to
check that you are using the medium stiffness spring between thefirst and second carriages.
Code the two controller structures (collocated PD and
noncollocated PD + notch filter) into the LabVIEW control loop.Again, you can use a case selector to easily switch between the two
algorithms.
Implement the high-gain controller from step 5.1 of the
pre-lab and perform a 3000 count step and save the plot. Notice thebehavior of the second mass carriage. Gently displace the carriages
and note the relative stiffness of the servo system at the firstmass.
Now implement the low-gain controller from step 5.1 and
perform a 3000 count step and save the plot. Manually displace thefirst and second masses and note their relative stiffness. Are they
generally more or less stiff than for the controller from the stepabove? How does the speed compare to the high-gain controller? How
about the steady-state error?
Now implement your noncollocated PD + notch filter controller
from step 5b of the pre-lab and perform a 3000 count step; save theplot. From the response plot, determine the rise time and overshoot
of the second mass carriage.
7.4 post-lab
What was the predominant behavior of the second mass carriage
with the highgain collocated PD controller? Can you give anexplanation for the difference in the responses of the two masses
in terms of their closed-loop transfer functions?
What differences did you observe in the responses between the
low-gain and high-gain collocated PD controllers?
What was the rise time and overshoot for your noncollocated
PD + notch filter controller. Was this better or worse than youwere able to achieve with the collocated controllers? How did the
steady-state error of the system for this controller compare tothat of the low-gain collocated PD controller?
Questions & Answers
anyone know any internet site where one can find nanotechnology papers?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?