Question

Problem 3 Nonlinear System Control (25pts) Given the following inverted pendulum. Modeled as a point mass of negligible size connected to a massless rod. With a motor connected at the base, providing a moment torque. ? m M 1 With m = 3kg, l = 500cm a. (10%) Derive the equations of motion of the system. Then linearize and derive the trans- fer function of the system, relating torque to angular position: $\frac{\Theta(s)}{M(s)}$ b. (30%) Using the transfer function, design a PID controller to achieve the following specifica- tions: - $T_s$ = 1s - $\omega_s$ = 2rad/s c. (30%) Simulate the system with your controller, on the nonlinear model (numerical integra- tion), and test it by using it as a regulator (r = 0) for each of the following intial conditions $\theta_0$ = 30°, 60°, 90°, 145°, 180° Plot all the responses on the same subplot. Plot the torque applied for each case, in a different subplot. d. (30%) In practice, you are bounded by the actuation effort. Repeat the simulation in part. c, but limit the magnitude of u (which is the moment torque) to 200 Nm. This is called a saturation limit. Discuss how you would change your PID gains after you've added this saturation limit.

          Problem 3
Nonlinear System Control (25pts)
Given the following inverted pendulum. Modeled as a point mass of negligible size connected to
a massless rod. With a motor connected at the base, providing a moment torque.
?
m
M
1
With m = 3kg, l = 500cm
a. (10%) Derive the equations of motion of the system. Then linearize and derive the trans-
fer function of the system, relating torque to angular position: $\frac{\Theta(s)}{M(s)}$
b. (30%) Using the transfer function, design a PID controller to achieve the following specifica-
tions:
- $T_s$ = 1s
- $\omega_s$ = 2rad/s
c. (30%) Simulate the system with your controller, on the nonlinear model (numerical integra-
tion), and test it by using it as a regulator (r = 0) for each of the following intial conditions
$\theta_0$ = 30°, 60°, 90°, 145°, 180°
Plot all the responses on the same subplot.
Plot the torque applied for each case, in a different subplot.
d. (30%) In practice, you are bounded by the actuation effort. Repeat the simulation in part. c,
but limit the magnitude of u (which is the moment torque) to 200 Nm. This is called a saturation
limit. Discuss how you would change your PID gains after you've added this saturation limit.

Problem 3
Nonlinear System Control (25pts)
Given the following inverted pendulum. Modeled as a point mass of negligible size connected to
a massless rod. With a motor connected at the base, providing a moment torque.
?
m
M
1
With m = 3kg, l = 500cm
a. (10%) Derive the equations of motion of the system. Then linearize and derive the trans-
fer function of the system, relating torque to angular position: (Θ(s))/(M(s))
b. (30%) Using the transfer function, design a PID controller to achieve the following specifica-
tions:
- Ts = 1s
- = 2rad/s
c. (30%) Simulate the system with your controller, on the nonlinear model (numerical integra-
tion), and test it by using it as a regulator (r = 0) for each of the following intial conditions
θ0 = 30°, 60°, 90°, 145°, 180°
Plot all the responses on the same subplot.
Plot the torque applied for each case, in a different subplot.
d. (30%) In practice, you are bounded by the actuation effort. Repeat the simulation in part. c,
but limit the magnitude of u (which is the moment torque) to 200 Nm. This is called a saturation
limit. Discuss how you would change your PID gains after you've added this saturation limit.

Added by Joshua R.

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