0.7 System identification for the torsional plant

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 ${\hat{x}}_{\mathrm{d1}}$ , ${\hat{x}}_{\mathrm{d2}}$ , ${\hat{x}}_{\mathrm{d3}}$ , ${\widehat{\stackrel{ˉ}{x}}}_{\mathrm{d1}}$ , ${\widehat{\stackrel{ˉ}{x}}}_{\mathrm{d2}}$ , ${\widehat{\stackrel{ˉ}{x}}}_{\mathrm{d3}}$ 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 $$y=Hx+\epsilon$$ where the vector of parameters to be estimated, $x$ , is defined as $$x=\left[\begin{array}{c}{J}_{\mathrm{d1}}\\ J_{\mathrm{d2}}\\ J_{\mathrm{d3}}\\ c_1\\ c_2\\ c_3\\ k_1\\ k_2\\ k_{hw}\end{array}\right]$$
• 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?