EEE 304 Transfer Functions Lab Report

EEE 304, LAB 3The objective of this sequence of laboratory exercises is to explore a few fundamental
problems arising in Linear Feedback Systems.
1. Lab 3 Assignment
#0. Write an overview of your results.
#1. For the control system shown in Fig. 1,
d
r
e
+
C(s)
u
+
P(s)
y
–
Figure 1. A closed loop control system with disturbance
where
•
•
•
•
•
•
•
P(s) is the transfer function of the plant
C(s) is the transfer function of the controller
r is the reference signal
e is the error signal
u is the output signal from the controller
d is the disturbance signal
y is the output signal
1.
Find the transfer function Tdy from the disturbance d to the output y (assume r = 0). Find the transfer
function Try from the reference r to the output y (assume d = 0).
If the transfer functions of the plant and controller are given as
2.
1
Plant: 𝑃(𝑠) = , (e.g., a normalized description of car velocity with force as the input; this can represent a
𝑠
cruise control system.)
Controller: 𝐶(𝑠) = 𝐾
i)
ii)
𝑠+𝑧
𝑠
, where K and z are constants.
Verify that this is a PI controller. Show your work and state the expression for the P and I gains in
terms of K and z.
Show that if the input d is a step disturbance and 𝐶(0) = ∞, the effect of d on y approaches zero as
𝑡 → ∞ (Hint: Assume K,z > 0. Use the final value theorem.) Demonstrate your result by choosing the
following different sets of values for K and z (varied by orders of magnitude) and plotting their closedloop step response.
• K = 0.1, z = 0.1
• K = 1, z = 1
• K = 10, z = 10
Include MATLAB code and plots. Discuss your observations.
iii) Design a controller (select values of K and z) such that the following specifications are met:
• Target crossover frequency = 1 rad/s (approximately equal to the closed-loop bandwidth).
• Target phase margin is 60⁰.
(1) Show all your work and demonstrate your design meets these specifications by the use of
“allmargin” command on the open-loop transfer function L(s) = P(s)C(s). (hint: “help allmargin” in
Matlab). Include MATLAB code and results.
(2) Plot the closed-loop frequency and step responses from the reference input r to the output y.
Show the achieved bandwidth by marking on the Bode plot with a “data cursor”. Comment on
how different, if so, it is from the specified bandwidth. Include MATLAB code and plots. (Read
http://www.mathworks.com/help/matlab/creating_plots/data-cursor-displaying-data-valuesinteractively.html on how to use the data cursor.)
(3) Plot the closed-loop frequency and step responses from the disturbance d to the output y. Does
the steady state error e become 0 when a step disturbance d is applied to this closed loop
transfer function? Include MATLAB code and plots.
NOTE: in order to form a closed loop transfer function you need to utilize the “feedback” command. Assuming you
have created transfer function objects for both the plant and controller, i.e., P and C, the closed-loop transfer
function from r to y, Try, will be computed as
>> T_ry = feedback(P*C,1)
Subsequently the bode plot and step response for Try can be found with
>> bode(T_ry)
>> step(T_ry)
In order to compute the transfer function from the disturbance d to the output y, use
>> T_dy = feedback(P, C)
#2 This problem explores the effect of sensor noise on the input to the plant. Plant inputs are typically provided by
actuators, e.g., valves, motors, and other mechanical components. High-frequency inputs to these actuators cause
more stress, leading to shorter actuator lifetime and higher probability of actuator failure. The sensors that
measure the output and feed it back to the controller are prone to high-frequency measurement noise. This
problem will show the trade-off between bandwidth and high-frequency noise attenuation. Fig. 2 shows how
sensor noise enters into the closed-loop system as the signal n(t):
r
+
e
C(s)
u
P(s)
y
–
+
n
Figure 2. A closed loop control system with noise
1
𝑠+𝑧
𝑠
𝑠
1) Given the plant 𝑃(𝑠) = , design a PI controller 𝐶(𝑠) = 𝐾
as in #1.2.iii, same phase margin but for a target
crossover frequency of 10 rad/s. Repeat #1.2.iii.
2) Compare the closed-loop frequency responses from the sensor noise n to the plant input u (Tnu = feedback(C,
P)) for this controller and the controller you designed in #1.2.iii with crossover frequency 1. Comment on the
difference and what you think is happening. Include MATLAB code and plots.
3) In order to visualize the effects, you will now simulate the two closed loop responses in Simulink. At this point,
we assume that you have transfer functions for both controllers. Create a Simulink model as shown below:
In the top loop, use the controller with crossover frequency 1 and the bottom loop use the controller with
crossover frequency 10. Use the following Simulink blocks:
•
•
•
•
•
Sum (under Simulink->Commonly Used Blocks): “List of Signs”, “|+-“ for +- block and “|++“ for ++ block
Step (under Simulink->Sources)
Transfer function (under Simulink->Continuous): specify numerator and denominator coefficients
Band-limited white noise (under Simulink->Sources): noise power = 0.0001
Scope (under Simulink->Sinks): double click open scope, click parameters (next to the printer icon),
change “number of axes” to 2. Now it can display two graphs.
Run the simulation and provide plots of the output. Justify your answer to problem #2.2 using these step
responses.
#3. This problem studies the effect of unmodeled lag on closed-loop system behavior. The following transfer
function represents a perturbation to the system in Fig. 1.
•
Plant: 𝑃(𝑠) =
1
𝑠(𝑎𝑠+1)
The term 1/as+1 can be viewed as a 1st order lag introduced by unmodeled actuator dynamics (engine; e.g.,
throttle to torque/force response) or sensor dynamics. The objective of this problem is to demonstrate that when
such lags alter the system significantly around the crossover frequency, the closed-loop may become unstable and
the controller must be redesigned. Notice that the achievable performance may no longer be the same.
1) Consider the cases where a = 0, 0.1, 1, 10 and plot the closed-loop frequency and step responses from r to y
for the same controller designed in #1.2.c. (Note: The case of a = 0 is the nominal plant with no lag.) Include
MATLAB code and plots.
2) For the case a = 10, redesign the controller to achieve 60o phase margin and crossover frequency of 0.036
rad/s. Demonstrate your design meets these specifications by the use of “allmargin” command on L(s) =
P(s)C(s). Plot the closed-loop frequency and step responses from r to y. Include MATLAB code and plots.
As you see, you were able to stabilize the controller but at the cost of bandwidth. The system is stable but sluggish
now. You do not need to perform any further experiment but know that the sacrifice in speed would not have
been necessary if a PID controller is used in this case rather than a PI.