Process Control: Understanding Dynamic Behavior
 Jun 27, 2003
 Background
 Linear State Space Models
 Introduction to Laplace Transforms
 Transfer Functions
 FirstOrder Behavior
 Integrating System
 SecondOrder Behavior
 LeadLag Behavior
 Poles and Zeros
 Processes with Dead Time
 Padé Approximation for Dead Time
 Converting State Space Models to Transfer Functions
 Matlab and Simulink
 Summary
 References
 Student Exercises
Student Exercises
1:  Solve for the timedomain output of a firstorder transfer function to a step input change. 
2:  A secondorder process with one pole at the origin has the transfer function Find the output as a function of time, for a unit step input change. Sketch the expected behavior. 
3:  Use the initial value theorem to find the immediate response of a leadlag transfer function to a step input change at t = 0. Also, use the final value theorem to find the longterm response of a leadlag transfer function to a step input change. 
4:  For the following secondorder process with numerator dynamics, solve for the timedomain output response to a step input change of magnitude Du at t =0. For k_{p} = 1°C/Lpm, t_{1} = 3 min, t_{2} = 15 min, t_{n} = 20 min find the peak temperature and the time that it occurs. 
5:  Consider an inputoutput transfer function that mimics two firstorder processes in parallel where If the gain of g_{1} is positive and the gain of g_{2} is negative, find the conditions (relationship between gains and time constants for the two transfer functions) that cause a righthalfplane zero (resulting in inverse response to a step input change) in g_{p}(s). 
6:  Consider the state space model Find the secondorder differential equation in y. Hint: first solve for x_{1} from the second equation, then take the derivative and substitute into the first equation. 
7:  Consider the following statespace model Which has the following inputoutput transfer function relationship For a unit step change in the input, u(s) = 1/s:

8:  As a process engineer with the Complex Pole Corporation, you are assigned a unit with an exothermic chemical reactor. In order to learn more about the dynamics of the process, you decide to make a step change in the input variable, the coolant temperature, from 10°C to 15°C. Assume that the reactor was initially at a steady state. You obtain the following plot for the output variable, which is reactor temperature (notice that the reactor temperature is in °F). Use Figure 39 to help answer the following questions.

9:  Match the transfer functions with the responses to a unit step input, shown in the figure. 
10:  Consider the following state space model:

11:  As a process engineer, you decide to develop a firstorder + time delay model of a process using a step test. The process is initially at steady state, with an input flow rate of 5 gpm and an output of 0.75 mol/L. You make a step increase of 0.5 gpm at 3:00 p.m. and do not observe any changes until 3:07 p.m. At 3:20 p.m., the value of the output is 0.8 mol/L. You become distracted and do not have a chance to look at the output variable again, until you leave for happy hour at a local watering hole at 6:30 p.m. You note that the output has ceased to change and has achieved a new steadystate value of 0.85 mol/L. What are the values of the process parameters, with units? Show your work. 
12:  Use the initial and final value theorems of Laplace transforms to determine the initial and final values of the process output for a unit step input change to the following transfer functions. 
13:  Consider the following state space model for a biochemical reactor. Since there are two states (the A matrix is 2 x 2) we expect that the process transfer function will be secondorder. Show that polezero cancellation occurs, resulting in a firstorder transfer function. Find the values of the gain and time constant. 
14:  Match the transfer functions with the responses to a unit step input, shown in the figure. 
15:  Consider Example 3.4. For an impulse input of 30 kJ, find the value of the vessel temperature immediately after the impulse input is applied. 
16:  Consider the following secondorder transfer function For a unit impulse input, find the output response as a function of time. What is the peak change and when does it occur? 