The objective of this lab is to review of the behavior of second-order systems. Students will gain a better understanding of the importance system identification. Students will also develop a hands-on understanding of the concept of hardware gain and why it will play a crucial role in controller design. System Identification will be implemented in LabVIEW using the System Identification Toolkit.
System identification for the torsional plant
Objectives
Understand the dynamic equivalence between rotational and
translational systems.
Perform system identification using two different methods and
compare the results.
Pre-lab
Assume that the least squares estimate has already been found
for the unloaded and loaded sine sweep tests, so
${\stackrel{\u02c6}{x}}_{\mathrm{d1}}$ ,
${\stackrel{\u02c6}{x}}_{\mathrm{d2}}$ ,
${\stackrel{\u02c6}{x}}_{\mathrm{d3}}$ ,
${\stackrel{\u02c6}{\stackrel{\u02c9}{x}}}_{\mathrm{d1}}$ ,
${\stackrel{\u02c6}{\stackrel{\u02c9}{x}}}_{\mathrm{d2}}$ ,
${\stackrel{\u02c6}{\stackrel{\u02c9}{x}}}_{\mathrm{d3}}$ are known values. Formulate the linear least
squares equation to estimate the 9 individual plant parameters. In other words, find the
$y$ vector and
$H$ matrix that would go into the equation
$$y=Hx+\epsilon \phantom{\rule{0ex}{0ex}}$$ where the vector of parameters to be estimated,
$x$ , is defined as
$$x=\left[\begin{array}{c}{J}_{\mathrm{d1}}\\ {J}_{\mathrm{d2}}\\ {J}_{\mathrm{d3}}\\ {c}_{1}\\ {c}_{2}\\ {c}_{3}\\ {k}_{1}\\ {k}_{2}\\ {k}_{hw}\end{array}\right]\phantom{\rule{0ex}{0ex}}$$
Outline the experimental steps you will take to identify the torsional plant using the second-order model method similar to Lab #2. Your procedure should allow you to find
${J}_{\mathrm{d1}},{J}_{\mathrm{d3}},{c}_{1},{c}_{3},{k}_{1},{k}_{2},$ and
${k}_{hw}$ . You may exclude the procedure for identifying the inertia and damping for disk 2. When formulating your procedures, remember that disks 2 and 3 can be clamped, disk 1 cannot.
Lab procedure
System identification using least squares
Configure the plant with all three disks rotating freely and
no brass weights attached.
Perform a 1638 count (0.5V) linear sine sweep from
$0$ to
$8Hz$ with a sweep time of
$20$ seconds. When the execution is complete,
enter a file name such as
$\mathit{3DiskSweepUnloaded}$ and save the raw data from the front panel.
Now load two
$0.5kg$ brass weights onto each of the three disks
so their centers are
$9.0cm$ from the axis of rotation.
Perform the sine sweep again. Enter a file name such as
$\mathit{3DiskSweepLoaded}$ and save the raw data.
You are now ready to identify the system parameters using least squares estimation.
System identification using second-order model
Follow the steps you outlined in the pre-lab to identify the
system parameters using the second-order model method.
Post-lab
Complete the table below; remember to include units.
How close are your least-squares values compared to your
second-order model values. Can you explain any discrepanciesbetween them. Which method do you think is more accurate?
Questions & Answers
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