CHEE **03: Process Dynamics and Control Fall 2024
Lab 3: Root Locus Diagram and Controller Tuning
Process Background
Consider a blending process with two inlet streams and a single (overflow) outlet stream. The
schematic diagram of the process is shown in Figure 1, where x1, x2 and x represent the mass
fraction of component A and w1, w2 and w represents the overall mass flow rate. One of the inlet
streams, stream 1, is made up of compound A (and the balance compound B). The mass fraction
of A is a disturbance variable and has a steady state value of 20% (with a total steady state flow
rate of 1 kg/min). The other inlet stream, stream 2, is made up of pure B, where the flow rate is a
manipulated variable. The outlet mass fraction of A is a controlled variable with a target of 10%.
Assume that there is 10 kg of liquid in the tank (constant volume with a density similar to water).
Figure 1. A blending process in a CSTR.
Process Parameters
The process operating conditions are as follows:
Constants Input Steady State Conditions
V 10 L **;**;**;1**; 1 kg/s
ρ 1 kg/L **;**;1**; 0.2
x2 0 **;**;**;2**; 1 kg/s
Output Steady State Condition
**; 0.1
CHEE **03: Process Dynamics and Control Fall 2024
Kathy Isaac, Stanislav Sokolenko Page 2 of 7
From Example 10 in Topic 2, the process, Gp, and disturbance, Gd, transfer functions are:
The process is controlled by a PI controller, Gc. Model the actuator, Gv, with a variable delay, a,
and assume all other transfer functions are unity (Gm = Gs = 1).
(1/1 𝑃𝑎𝑑é 𝑎w**1;w**1;w**3;w**0;w**9;𝑖𝑚𝑎w**5;𝑖w**0;𝑛)
Figure 2. General representation of a closed loop process.
Objectives
1. Determine the critical controller parameters using a root locus plot.
2. Evaluate the effect of delay on stability and critical controller parameter using root locus plots.
3. Apply the direct synthesis controller tuning method to the process and evaluate the response
to disturbance rejection.
CHEE **03: Process Dynamics and Control Fall 2024
Kathy Isaac, Stanislav Sokolenko Page 3 of 7
Controller Setup
1. Implement a closed loop PI controller in Simulink to control the outlet mass fraction of
component A by controlling the flow rate of stream 2 as illustrated in Figure 2.
• Use Transfer Fcn blocks to implement the process, Gp, disturbance, Gd, and the actuator,
Gv.
• Refer to Prelab 2 to set up the appropriate controller in a closed loop process.
• Use Setpoint blocks for the setpoint and disturbance inputs and set the setpoint equal to a
constant value of 0 and the disturbance input to a constant value of 0.1.
Root Locus Diagram
2. Add a Pole-Zero Plot block and set the Disturbance input signal as the Input Perturbation and
the Process output signal as the Output Measurement. See the example provided at the end of
this document for an example on setting up the root locus plot.
Questions
1. For the closed loop process with a PI controller with a delay of 1 s, set Kc = 1 and find the
critical τI. Make plots of the poles and zeros showing the transition from stable to unstable at
the critical τI. Include 3 plots: stable, critical and unstable. Repeat for delays of 3 and 5 s. Note:
Set the Setpoint block constant at 0 and the Disturbance Step block constant at 0.1
a. How does the critical τI change with increase in delay?
2. For the closed loop process with a PI controller with a delay of 1 s, set τI = 10 and find the
critical Kc. Make a plot of the poles and zeros showing the transition from stable to unstable at
the critical Kc. Include 3 plots: stable, critical and unstable. Repeat for delays of 3 and 5 s.
Note: Set the Setpoint block constant at 0 and the Disturbance Step block constant at 0.1
a. How does the critical Kc change with increase in delay? CHEE **03: Process Dynamics and Control Fall 2024
Kathy Isaac, Stanislav Sokolenko Page 4 of 7
3. When there is no delay in the actuator, tune the PI controller using the direct synthesis method
and evaluate the response to a step change of 0.1 in the disturbance variable to different values
of τc between 10 and 100. Note: set the initial value of the disturbance input to 0 and the final
value to 0.1.
a. How does τc affect the process response?
b. What τc should be chosen if the process must reject a step disturbance of 0.1 in under
60 seconds with no large oscillations.
Report Guidelines
1. Use the lab report template provided.
2. The report must seek to concisely answer the questions in the previous section.
3. The text of the report body must be within 1 page. It is recommended to use a 12-point font,
1.5 spaced but please use 1**point font, single spaced at a minimum.
4. Do not break up the text. Add all the text to page 1 and refer to figures and tables on subsequent
pages to aid your discussion.
5. Include a screenshot of your complete Simulink model for the PI controller set up in Question1.
Pole-Zero Plot Example for the Heating Tank Process
1. Set up the closed loop for the given process and controller. Set the Setpoint block constant at
0 and the Disturbance Step block constant at 1.
CHEE **03: Process Dynamics and Control Fall 2024
Kathy Isaac, Stanislav Sokolenko Page 5 of 7
2. Add a Pole-Zero Plot block to the workspace
3. Double click on the Pole-Zero Plot block and click on the + symbol to add inputs and outputs
4. Click on the disturbance signal (highlighted in blue) and press the << symbol to add the
selected signal. Repeat for the Output signal. CHEE **03: Process Dynamics and Control Fall 2024
Kathy Isaac, Stanislav Sokolenko Page 6 of 7
5. Once added, change the Configuration of the input signal to Input Perturbation and the
output to Output Measurement and click Apply. Change the snapshot time to 1.
6. Click on Show Plot
CHEE **03: Process Dynamics and Control Fall 2024
Kathy Isaac, Stanislav Sokolenko Page 7 of 7
7. Click on the green Run button to display the poles and zeros. Poles are represented by x and
zeros by o. Click on them to see their exact values
8. Change controller parameters and assess how the poles and zeros change.
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