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8/11/2019 Bab 3 slide 1.ppt
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1st Revision 2007 Taufik
Cal Poly State University, San Luis Obispo 25
Mathematical Modeling
RECALL:
The Design of Control System requires Formulation of Mathematical
Model of the System
Why Mathematical Model?So that we will able to Design and Analyze the control system
Examples:
• How would we know the relationship between Input and Output?
• How would we predict or describe the dynamic behavior of the
control system?
Two Methods to develop the Mathematical Model of Control System
1. Transfer Function in Frequency Domain (Using Laplace Transform)
2. State Space Equations in Time Domain
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1st Revision 2007 Taufik
Cal Poly State University, San Luis Obispo 26
Review of Laplace Transform and Linearity
V(t)
L
R
Ci(t)
RECALL:
In EE301, We learned that Ordinary Differential Equation may describe
mathematically dynamic behavior, input vs. output relationship ofphysical system such as Mechanical System using Newton’ Law and
Electrical System using Kirchoff’s Law
Example: RLC Network
Let’s assume that V(t) is Input; i(t) is Output
Using KVL:
0
( ) 1
( ) ( ) ( )
t
R
di t v t v t L i d
dt C
( ) ( ) ( ) ( ) R L C
v t v t v t v t
Using the differential equation (derived from KVL):
• Is it a simple algebraic equation?
• Is it easy to describe relationship between Input and Output and System?
i.e. Can you make them as a Separate Entities?
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Cal Poly State University, San Luis Obispo 27
Laplace Transform
• If you answered NO to all previous questions, then YOU NEED
LAPLACE TRANSFORMS
• Laplace Transform provides:
– Representation of Input, output, and system as separate entities
– Simple Algebraic interrelationship between Input, Output, and
System
• Limitation of Laplace Transfrom:
– Works in Frequency Domain
– Valid when the system is LINEAR
• A System is called LINEAR when it possesses:
– Superposition
– Homogeneity
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Homogeneity
• Describes the response of the system to a Multiplication of the input
by a scalar as compared to a Multiplication of output to the same input
by the same scalar
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• Will any LINEAR EQUATION/FUNCTION represent a LINEAR
SYSTEM?
Example: Try y = mx + b (where m and b are constants) is
a LINEAR function, but check to see
if it yields a LINEAR System
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Linearization
BAD NEWS is Real World Physical Systems are NON-LINEAR
GOOD NEWS:
1. A great majority of Physical Systems are LINEAR within some range ofoperating point
2. We have a Mathematical Tool that could perform Linear Approximation
of Non-Linear Systems, a process called LINEARIZATION
NOTE:
Linearization does NOT necessarily
yield a LINEAR system!
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