EE 4343 Lab#4
PID Control Design of Rigid Bodies
Prepared by: Stacy Cason
E-mail: [email protected]
Updated: July 13, 2017
This lab demonstrates some key concepts associated with proportional plus derivative
(PD) control and subsequently the effects of adding integral action (PID). The block
diagram for the forward path PID control of a rigid body plant is shown in Figure 1. Note
that friction is neglected here. This control scheme, acting on plants modeled as rigid
bodies, finds broader application in industry than any other. It is employed in such
diverse areas as machine tools, automobiles (cruise control), and spacecraft control
(attitude and gimbal control).
r s 

Reference
Input

k
k p  i  kd s
s
PID
Controller
xs 
k hw
PLANT
Output
Hardware
Gain
Figure 1. Rigid Body PID Control – Control Block Diagram
Before proceeding, check with lab assistant as to which system you are using for this
lab. The three different systems are Model 210 (Rectilinear Spring/Mass Control
System), Model 205 (Torsional Control System) and Model 505 (Inverted Pendulum
Control System). Next, follow the appropriate design procedures below for your
particular model.
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1. Rigid Body PID Control for Model 210 (Rectilinear Spring/Mass Control System)
The closed loop transfer function is given as
cs  


k hw m k d s 2  k p s  k i
xs 

r s  s 3  k hw m  k d s 2  k p s  k i


(1.1)
For the first portion of your lab we shall consider PD control only ( k i  0 ). With this
assumption in mind, one may express (1.1) above as:
k hw m k d s  k p
xs 
cs  

r s  s 2  k hw m k d s  k p




(1.2)
Notice that one may equate the expression in (1.2) to
cs  
by defining the following:
2 n s  2n
xs 

r s  s 2  2 n s  2n
n 
(1.3)
k p k hw
m
(1.4)
k k
k d k hw
  d hw 
2m n 2 mk p k hw
The effects of k p and k d on the roots of the denominator (damped second-order
oscillator) of Eq. (1.2) is studied in the work that follows.
PD Control Design:
a. From Eq’s (1.2, 1.3 and 1.4) design a PD controller (i.e., find k p and k d ) for a system
with natural frequency  n  15 rad/s and   0.707 .
(Do not exceed a value of
k p  0.08 and k d  0.04 .) Assume k hw  12800 N/m and m  2.77 kg .
b. Implement your controller by performing a step response. Set up a trajectory for a
2500 count closed-loop STEP with 2000ms duration (1 rep). Ask your lab assistant to
check your setup before proceeding further.
c. Now, execute this trajectory and plot the commanded position and encoder position #1.
Plot them both on the same vertical axis so that there is no graphical bias. Save your
plots for later comparison.
Adding Integral Action:
a. Implement a PID controller with a value of k i  0.01 along with the previous values of
k p and k d found in (a.) above. Be certain that the following error seen in the
background window is within 20 counts prior to implementing this controller (if not
choose Zero Position from the Utility Menu).
b. Now, execute a trajectory for a 2500 count closed-loop step of 2000ms duration (1
rep). Plot the commanded position and encoder position. Plot them both on the same
vertical axis so that there is no graphical bias. Save your plots (export raw data from
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the data menu) for later comparison.
Questions:
a. Show all calculations for k p and k d .
b. Derive the natural frequency  n and damping ratio  in Eq. 1.4.
c. What is the effect of the system hardware gain k hw , mass m , and control gains k p
and k d on the natural frequency  n and damping ratio  ?
d. Describe the effects of natural frequency  n and damping ratio  on the
characteristic roots of Eq. 1.1. Use a root-locus diagram in your answer to show the
effect of changing  from 0 to  for any given  n .
e. What is the effect of adding integral action?
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2. Rigid Body PID Control for Model 205 (Torsional Control System)
The closed loop transfer function is given as
cs  


k hw J  k d s 2  k p s  k i
xs 

r s  s 3  k hw J  k d s 2  k p s  k i


(2.1)
For the first portion of your lab we shall consider PD control only ( k i  0 ). With this
assumption in mind, one may express (2.1) above as:
k hw J  k d s  k p
xs 
cs  

r s  s 2  k hw J  k d s  k p




(2.2)
Notice that one may equate the expression in (2.2) to
cs  
by defining the following:
2 n s  2n
xs 

r s  s 2  2 n s  2n
n 
(2.3)
k p k hw
J
k k
k d k hw
  d hw 
2 J n
2 Jk p k hw
(2.4)
The effects of k p and k d on the roots of the denominator (damped second-order
oscillator) of Eq. (2.2) is studied in the work that follows.
PD Control Design:
a. From Eq’s (2.2, 2.3 and 2.4) design a PD controller (i.e., find k p and k d ) for a system
with natural frequency  n  10 rad/s and   0.707 .
(Do not exceed a value of
k p  0.08 and k d  0.1 .) Assume k hw  17.4 N - m/rad and J  0.0108 kg - m 2 .
b. Implement your controller by performing a step response. Set up a trajectory for a
2500 count closed-loop STEP with 2000ms duration (1 rep). Ask your lab assistant to
check your setup before proceeding further.
c. Now, execute this trajectory and plot the commanded position and encoder position.
Plot them both on the same vertical axis so that there is no graphical bias. Save your
plots for later comparison.
Adding Integral Action:
a. Implement a PID controller with a value of k i  0.05 along with the previous values of
k p and k d found in (a.) above. Be certain that the following error seen in the
background window is within 20 counts prior to implementing this controller (if not
choose Zero Position from the Utility Menu).
b. Now, execute a trajectory for a 2500 count closed-loop step of 2000ms duration (1
rep). Plot the commanded position and encoder position #1. Plot them both on the
same vertical axis so that there is no graphical bias. Save your plots (export raw data
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from the data menu) for later comparison.
Questions:
a. Show all calculations for k p and k d .
b. Derive the natural frequency  n and damping ratio  in Eq. 2.4.
c. What is the effect of the system hardware gain k hw , mass m , and control gains k p
and k d on the natural frequency  n and damping ratio  ?
d. Describe the effects of natural frequency  n and damping ratio  on the
characteristic roots of Eq. 2.1. Use a root-locus diagram in your answer to show the
effect of changing  from 0 to  for any given  n .
e. What is the effect of adding integral action?
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3. Rigid Body PID Control for Model 505 (Inverted Pendulum Control System)
The closed loop transfer function is given as
cs  


k hw m k d s 2  k p s  k i
xs 

r s  s 3  k hw m  k d s 2  k p s  k i


(3.1)
For the first portion of your lab we shall consider PD control only ( k i  0 ). With this
assumption in mind, one may express (3.1) above as:
k hw m k d s  k p
xs 
cs  

r s  s 2  k hw m k d s  k p




(3.2)
Notice that one may equate the expression in (3.2) to
cs  
by defining the following:
2 n s  2n
xs 

r s  s 2  2 n s  2n
n 
(3.3)
k p k hw
m
(3.4)
k k
k d k hw
  d hw 
2m n 2 mk p k hw
The effects of k p and k d on the roots of the denominator (damped second-order
oscillator) of Eq. (3.2) is studied in the work that follows.
PD Control Design:
a. From Eq’s (3.2, 3.3 and 3.4) design a PD controller (i.e., find k p and k d ) for a system
with natural frequency  n  10 rad/s and   0.707 .
(Do not exceed a value of
k p  0.35 and k d  0.012 .) Assume k hw  2088 N/m and m  0.139 kg .
b. Adjust the position of the balance masses to lt  10 cm being sure to secure them on
the threaded rode by counter rotating them. Verify that the donut weights are in
place and secure on the sliding rod.
c. Implement your controller by performing a step response. Set up a trajectory for a
1000 count closed-loop STEP with 1000ms duration (1 rep). Ask your lab assistant to
check your setup before proceeding further.
d. Now, execute this trajectory and plot the commanded position and encoder position
#2. Plot them both on the same vertical axis so that there is no graphical bias. Save
your plots (export raw data from the data menu) for later comparison.
Adding Integral Action:
a. Implement a PID controller with a value of k i  0.3 along with the previous values of k p
and k d found in (a.) above. Be certain that the following error seen in the background
window is within 20 counts prior to implementing this controller (if not choose Zero
Position from the Utility Menu). Now, execute a trajectory for a 1000 count closed6
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loop step of 1000ms duration (1 rep). Plot the commanded position and encoder
position. Plot them both on the same vertical axis so that there is no graphical bias.
Save your plots for later comparison.
Questions:
a. Show all calculations for k p and k d .
b. Derive the natural frequency  n and damping ratio  in Eq. 3.4.
c. What is the effect of the system hardware gain k hw , mass m , and control gains k p
and k d on the natural frequency  n and damping ratio  ?
d. Describe the effects of natural frequency  n and damping ratio 
on the
characteristic roots of Eq. 3.1. Use a root-locus diagram in your answer to show the
effect of changing  from 0 to  for any given  n .
e. What is the effect of adding integral action?
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