2. Solution to Homework 2 Chapter 2
Examples of Dynamic Mathematical Models
Solution to Homework 2

3. Chapter 2
General Process Models
State Equations
A suitable model for a large class of continuous theoretical processes is a set of ordinary differential equations of the form:
t : Time variable
x1,...,xn : State variables
u1,...,um : Manipulated variables
r1,...,rs : Disturbance, nonmanipulable variables
f1,...,fn : Functions

4. Chapter 2
General Process Models
Output Equations
A model of process measurement can be written as a set of algebraic equations:
t : Time variable
x1,...,xn : State variables
u1,...,um : Manipulated variables
rm1,...,rmt : Disturbance, nonmanipulable variables at output
y1,...,yr : Measurable output variables
g1,...,gr : Functions

5. State Equations in Vector Form
Chapter 2
General Process Models
State Equations in Vector Form
If the vectors of state variables x, manipulated variables u, disturbance variables r, and vectors of functions f are defined as:
Then the set of state equations can be written compactly as:

6. Output Equations in Vector Form
Chapter 2
General Process Models
Output Equations in Vector Form
If the vectors of output variables y, disturbance variables rm, and vectors of functions g are defined as:
Then the set of algebraic output equations can be written compactly as:

7. Heat Exchanger in State Space Form
Chapter 2
General Process Models
Heat Exchanger in State Space Form Tl q V ρ T cp T q Tj
If ,
then
State Space Equations

8. Double-Pipe Heat Exchanger in State Space Form
Chapter 2
General Process Models
Double-Pipe Heat Exchanger in State Space Form
Processes with distributed parameters are usually approximated by a series of well-mixed lumped parameter processes.
This is also the case for the heat exchanger, as shown in the next figure, which is divided into n well-mixed heat exchangers.
The space variable is divided into n equal lengths within the interval [0, L].
After rearrangement, the mathematical model of the heat exchanger is of the form:
After rearrangement, the mathematical model of the heat exchanger is of the form:
where
and

9. Double-Pipe Heat Exchanger in State Space Form
Chapter 2
General Process Models
Double-Pipe Heat Exchanger in State Space Form
We introduce the state parameters

10. Chapter 2
General Process Models
Difference Quotient
The derivation with respect to space, δT/δτ, will now be approximated by using a difference quotient.
The difference quotient itself is the equation that can be used to approximately calculate the slope of a function at a certain point.
There are three formations of difference quotient:
Forward Difference
Backward Difference
Central Difference

11. Double-Pipe Heat Exchanger in State Space Form
Chapter 2
General Process Models
Double-Pipe Heat Exchanger in State Space Form
Replacing δT/δτ with its corresponding difference will result a model that consists of a set of ordinary differential equations only.
The equation set can then be re-written in state space form.
State Space Equations

12. Taylor series expansion
Chapter 2
Linearization
Linearization
Linearization is a procedure to replace a nonlinear original model with its linear approximation.
Linearization is done around a constant operating point.
It is assumed that the process variables change only very little and their deviations from steady state remain small.
Operating point
Linearization
Taylor series expansion
Nonlinear Model
Linear Model

13. Chapter 2
Linearization
Linearization
The approximation model will be in the form of state space equations
An operating point x0(t) is chosen, and the input u0(t) is required to maintain this operating point.
In steady state, there will be no state change at the operating point, or x0(t) = 0

14. Taylor Expansion Series
Chapter 2
Linearization
Taylor Expansion Series
Scalar Case
A point near x0
Only the linear terms are used for the linearization

15. Taylor Expansion Series
Chapter 2
Linearization
Taylor Expansion Series
Vector Case
where

16. Taylor Expansion Series
Chapter 2
Linearization
Taylor Expansion Series
n : Number of states
m : Number of inputs

17. Taylor Expansion Series
Chapter 2
Linearization
Taylor Expansion Series
Performing the same procedure for the output equations,

18. Taylor Expansion Series
Chapter 2
Linearization
Taylor Expansion Series
r : Number of outputs

19. Taylor Expansion Series
Chapter 2
Linearization
Taylor Expansion Series
Nonlinear Model
Linear Model

20. Single Tank System qi The model of the system is already derived as: V
Chapter 2
Linearization
Single Tank System qi
The model of the system is already derived as: V h qo v1
The relationship between h and h in the above equation is nonlinear.
An operating point for the linearization is chosen, (h0,qi,0).

21. Chapter 2
Linearization
Single Tank System
The linearization around (h0,qi,0) for the state equation can be calculated as:

22. Single Tank System The linearization for the ouput equation is:
Chapter 2
Linearization
Single Tank System
The linearization for the ouput equation is:
Note that the input of the linearized model is now Δqi.
To obtain the actual value of state and output, the following equation must be enacted:

23. Chapter 2
Linearization
Single Tank System
The Matlab-Simulink model of the linearized system is shown below. All parameters take the previous values.

24. Single Tank System The simulation results : Original model
Chapter 2
Linearization
Single Tank System
The simulation results
: Original model
: Linearized model

25. Chapter 2
Linearization
Single Tank System
If the input qi deviates from the operating point, the linearized model will deliver inaccurate output.
: Original model
: Linearized model

26. Chapter 2
Linearization
Single Tank System
If the input qi deviates from the operating point, the linearized model will deliver inaccurate output.
: Original model
: Linearized model

27. Chapter 2
Linearization
Homework 3
Linearize the the interacting tank-in-series system for the operating point resulted by the parameter values as given in Homework 2.
For qi, use the last two digits of your Student ID. For example: 08  qi= 8 liters/s.
Submit the complete calculation of the linearization.
Submit also the mdl-file and the screenshots of the Matlab-Simulink file + scope. qi h1 h2 q1 a1 a2 v1 v2

28. Homework 3A qi2 qi1 hmax h a qo v
Chapter 2
Linearization
Homework 3A
Linearize the the triangular-prism-shaped tank as given in Homework 2A.
For the operating point, use qi1 = 5 liter/s and qi2 equals the last two digits of your Student ID divided by For example: 03  qi2= 0.3 liter/s, 17  qi2= 1.7 liter/s.
Submit the complete calculation of the linearization.
Submit also the mdl-file (softcopy) and the screenshots of the Matlab-Simulink file + scope (hardcopy). v
qi1 qo a
qi2
hmax h