2024 Summer Process Control 
Homework #6 (100) 
 
Problem 1 (25) 
 
A heat transfer process has the following transfer function between a temperature T and an inlet flow rate q 
where the time constants have units of minutes: 
 
The inlet flow rate varies as sinusoidal function with amplitude of 2.5 L/min and a period of 0.5 min. 
What is the amplitude of the temperature after a long time? 
 
Problem 2 (25) 
 
Consider a closed loop system with negative feedback, with the process 
The time constant of the process is in seconds. A sensor has unit steady state gain and zero time constant. 
 
Obtain the steady-state output of the system when it is subjected to each of the following inputs in set point: 
 
 a) x(  )=sin(  +30°); 
b) x(  )=3cos(2  −45°); 
c) x(  )=sin(  +30°)+3cos(2  −45°). 
 
For all these cases, angular velocities are in radians per second. 
 
Problem 3 (50) 
 
For control system on the Figure, 
 
a) determine the offset for a unit-step change in set-point; 
b) determine the characteristic equation of the closed-loop response; 
c) determine the range of     
 for stability (use the Routh stability criterion and Direct substitution 
Method); 
d) using MATLAB, plot Bode diagrams for the closed-loop system and determine the values of the 
ultimate gain and critical frequency; 
e) evaluate the stability of the closed-loop control system using the analytical Bode stability criterion; 
f) determine the tuning parameters of P, PI and PID controllers (    
) using Ziegler-Nichols method; 
g) draw the response curves ( Y versus t ) for a unit-step change in set-point for cases when P, PI and PID 
controllers were used (plot all three curves on one plot using Matlab). 
  
 
Bonus Problem (5) 
Draw the Bode plots for following system. Format the plots and label the axes. Determine the values of the 
ultimate gain and critical frequency.



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