1. Mathematical Models of Control Systems
Dr. Mohammed Abdulrazzaq
October

2. YA ALLAH ! ADVANCE ME IN KNOWLEDGE AND TRUE UNDERSTANDING
بسم الله الرحمن الرحيم
(قَالَ رَبِّ اشْرَحْ لِي صَدْرِي وَيَسِّرْ لِي أَمْرِي وَاحْلُلْ عُقْدَةً مِّن لِّسَانِي يَفْقَهُوا قَوْلِي) 
صدق الله العلي العظيم
سورة طه 25-28
YA ALLAH ! ADVANCE ME IN KNOWLEDGE AND TRUE UNDERSTANDING

3. Introduction
1- Why？
1) Easy to discuss the full possible types of the control systems —only in terms of the system’s “mathematical characteristics”.
2) The basis of analyzing or designing the control systems.
2- What is ？
Mathematical models of systems — the mathematical relation- ships between the system’s variables.
3- How get？
1) theoretical approaches
2) experimental approaches
3) discrimination learning

4. Non-Linear Models Tractable non-linearity Intractable non-linearity
Equation may be transformed to a linear model.
Intractable non-linearity
No linear transform exists

5. 1) Differential equations 2) Transfer function
4- types
1) Differential equations
2) Transfer function
3) Block diagram、signal flow graph
4) State variables
The input-output description of the physical systems — differential equations
The input-output description—description of the mathematical relationship between the output variable and the input variable of physical systems.

6. Contents Introduction Differential Equations of Physical Systems
The Laplace Transform
Transfer Function of Linear Systems
Block Diagram

7. Tractable Non-Linear Models
Several general Types
Polynomial
Power Functions
Exponential Functions
Logarithmic Functions
Trigonometric Functions

8. Polynomial Models Linear Parabolic Cubic & higher order polynomials
All may be estimated with OLS – simply square, cube, etc. the independent variable.

9. Power Functions
Simple exponents of the Independent Variable
Estimated with

10. Exponential and Logarithmic Functions
Common Growth Curve Formula
Estimated with
Note that the error terms are now no longer normally distributed!

11. Trigonometric Functions
Sine/Cosine functions
Fourier series

12. Introductions
A mathematical model is a set of equations (usually differential equations) that represents the dynamics of systems.
In practice, the complexity of the system requires some assumptions in the determination model.
The equations of the mathematical model may be solved using mathematical tools such as the Laplace Transform.
Before solving the equations, we usually need to linearize them.

13. Differential Equations
Examples:
How do we obtain the equations?
Physical law of the process  Differential Equation
Mechanical system (Newton’s laws)
Electrical system (Kirchhoff’s laws)

14. Differential Equations
Example 1: Springer-mass-damper system
Assumption: Wall friction is a viscous force.
The time function of r(t) sometimes called forcing function
Linearly proportional to the velocity

15. Differential Equations
Example 2: Springer-mass-damper system
Newton’s 2nd Law:

16. Example 3 : A passive circuit
define: input → ur output → uc。
we have：

17. Example 4 : A mechanism Define: input → F ，output → y. We have:
Compare with example 4: uc→y, ur→F---analogous systems

18. Differential Equations
Example 5: RLC Circuit

19. Differential Equations
Example 6: An operational amplifier (Op-amp) circuit
Input →ur output →uc

20. Differential Equations
Example 7 : A DC motor
Input → ua， output → ω1

21. The Laplace Transform
The differential equations are transformed into algebraic equations, which are easier to solve.
The Laplace transformation for a function of time, f(t) is:
If, , then,
Similarly,
Thus,

22. The Laplace Transform Example 8: Spring-mass-damper dynamic equation
Laplace Transform for the equation above:
When r(t)=0, y(0)= y0 and (0)=0:

23. The Laplace Transform Example 9: Spring-mass-damper dynamic equation
Some Definitions
q(s) = 0 is called characteristic equation (C.E.) because the roots of this equation determine the character of the time response.
The roots of C.E are also called the poles of the system.
The roots of numerator polynomial p(s) are called the zeros of the system.

24. The Laplace Transform Transform table: Impulse function Step function
f(t)
F(s) 1.
δ(t) 1 2.
u(t) 3.
t u(t) 4.
tn u(t) 5.
e-at u(t) 6.
sin t u(t) 7.
cos t u(t)
Impulse function
Step function
Ramp function

25. The Laplace Transform
Transform
Properties

26. The Laplace Transform
Example: Find the Laplace Transform for the following.
Unit function:
Ramp function:
Step function:

27. The Laplace Transform Transform Theorem Differentiation Theorem
Integration Theorem:
Initial Value Theorem:
Final Value Theorem:

28. The Laplace Transform
The inverse Laplace Transform can be obtained using:
Partial fraction method can be used to find the inverse Laplace Transform of a complicated function.
We can convert the function to a sum of simpler terms for which we know the inverse Laplace Transform.

29. The Laplace Transform
We will consider three cases and show that F(s) can be expanded into partial fraction:
Case 1:
Roots of denominator A(s) are real and distinct.
Case 2:
Roots of denominator A(s) are real and repeated.
Case 3:
Roots of denominator A(s) are complex conjugate.

30. The Laplace Transform
Case 1: Roots of denominator A(s) are real and distinct.
Example:
Solution:
It is found that:
A = 2 and B = -2

31. The Laplace Transform
Case 1: Roots of denominator A(s) are real and distinct.
Problem: Find the Inverse Laplace Transform for the following.

32. The Laplace Transform
Case 2: Roots of denominator A(s) are real and repeated.
Example:
Solution:
It is found that:
A = 2, B = -2 and C = -2

33. The Laplace Transform
Case 3: Roots of denominator A(s) are complex conjugate.
Example:
Solution:
It is found that:
A = 3/5, B = -3/5
and C = -6/5

34. The Laplace Transform
Case 3: Roots of denominator A(s) are complex conjugate.
Example:
Solution:

35. The Laplace Transform
Problem: Find the solution x(t) for the following differential equations.

36. The Transfer Function Transfer Functions and Linear Systems
in which a, b, c are given constants and f(t)is a given function. In this arena f(t)is often called the input signal or forcing function and the solution y(t)is often called the output signal. We shall assume that the initial conditions are zero (in this case y(0)= 0, y’ (0)= 0). Now, taking the Laplace transform of the differential equation, gives:

37. in which we have used y(0)= y'(0)= 0 and where we have designated L{y(t)} = Y (s)and L{f(t)} = F(s).
We define the transfer function of a system to be the ratio of the Laplace transform of the output signal to the input signal with the initial conditions as zero. The transfer function (a function of s), is denoted by H(s). In this case
Now, in the special case in which the input signal is the delta function f(t) = δ(t)we have F(s)= 1 and so,

38. Modelling of Linear Systems by Transfer Functions:::
- To begin, we can imagine a differential equation:
It is the system that changes the input signal into the output signal. This is easy to describe pictorially.
in the t−domain
After the Laplace transform of the differential equation is taken the differential equation is transformed into :
in the s−domain

39. - Consider a spring-mass-damper dynamic equation with initial zero condition.
The transfer function is given by the following.
Y(s)
R(s)

40. The Transfer Function Electrical Network Transfer Function Component
V-I
I-V
V-Q
Impedance
Admittance

41. The Transfer Function
Problem: Obtain the transfer function for the following RLC network.
Answer:

42. The Transfer Function
Problem: Obtain the transfer function for the following RC network.

43. The Transfer Function Mechanical System Transfer Function
Problem: Find the transfer function for the mechanical system below.
The external force u(t) is the input to the system, and the displacement y(t) of the mass is the output.
The displacement y(t) is measured from the equilibrium position.
The transfer function of the system.

44. Block Diagram Feedback Control System Therefore,
The negative feedback of the control system is given by:
Ea(s) = R(s) – H(s)Y(s)
Y(s) = G(s)Ea(s)

45. Transfer Function G(s)
Block Diagram
A block diagram of a system is a practical representation of the functions performed by each component and of the flow of signals.
Cascaded sub-systems:
Transfer Function G(s)
Output
Input

47. Block Diagram
Feedback Control System

48. Block Diagram
Reduction Rules

49. Block Diagram
Reduction Rules

50. Block Diagram
Problem:

51. Block Diagram
Problem:

52. “The whole of science is nothing more than a refinement of everyday thinking…”
The End…