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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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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 Kirchoffs Law

Example: RLC Network

Lets assume that V(t) is Input; i(t) is Output

Using KVL:

0

( ) 1

( ) ( ) ( )

t

R

di tv 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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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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Superposition

The output response of a system to the sum of inputs is the sum of

the responses to individual inputs

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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 NEWSis 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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