0.6 Collocated / noncollocated control of 2dof rectilinear system

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 $0.5\sec$ 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?