# 0.7 System identification for the torsional plant

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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.

## 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 ${\stackrel{ˆ}{x}}_{\mathrm{d1}}$ , ${\stackrel{ˆ}{x}}_{\mathrm{d2}}$ , ${\stackrel{ˆ}{x}}_{\mathrm{d3}}$ , ${\stackrel{ˆ}{\stackrel{ˉ}{x}}}_{\mathrm{d1}}$ , ${\stackrel{ˆ}{\stackrel{ˉ}{x}}}_{\mathrm{d2}}$ , ${\stackrel{ˆ}{\stackrel{ˉ}{x}}}_{\mathrm{d3}}$ 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.

## 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?

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