CHEE 4703: Process Dynamics and Control

CHEE 4703: 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 ̅̅̅1̅ 1 kg/s 

ρ 1 kg/L ̅̅1̅ 0.2 

x2 0 ̅̅̅2̅ 1 kg/s 

Output Steady State Condition 

 ̅ 0.1 

 CHEE 4703: 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 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛) 

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 4703: 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 4703: 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 11-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 4703: Process Dynamics and Control Fall 2024 

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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 4703: Process Dynamics and Control Fall 2024 

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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 

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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.