csl2 16 j15

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Lecture 2


Today’s class

• System• Transfer Function • Laplace transform• Differential equations• Modelling mechanical systems


System

SystemInputx(t)

Output (response)y(t)

Rules of operations

By writing the rules of operation, we get a differential equation as a combination of inputs and outputs

dn y𝑑𝑡𝑛

+𝑎𝑛−1dn− 1 y𝑑𝑡𝑛− 1 +…+𝑎0 y=𝑏𝑚

dmx𝑑𝑡𝑚

+…+𝑏0 x


CLASSIFICATION OF SYSTEMS

• Linear and Non-linear – Linear - Having properties of Additivity and

scalability• Time invariant and time varying

– Time invariant – system parameters do not change with time.

• Networks with RLC components

– Time varying – system parameters vary with time• Space shuttle losing mass due to fuel


CLASSIFICATION OF SYSTEMS• Controls are classified with respect to

– technique involved to perform control (i.e. human/machines): manual/automatic control

– Time dependence of output variable (i.e. constant/changing): regulator/servo,

(also known as regulating/tracking control)– fundamental structure of the control (i.e. the information used

for computing the control): Open-loop/feedback control,

(also known as open-loop/closed-loop control)


Mathematical Modelling of physical systems


Transfer function

System g(t)Inputx(t)

Output (response)y(t)

𝑌 (𝑠)𝑋 (𝑠)

=𝐺(𝑠)Transfer function

How to write the transfer function?


Fundamentals to transfer function

• Laplace TransformA technique to solve differential equation

Transforming time domain function to frequency domain function

Laplace Transform Definition

0

)()( dtetfsF stSolving differential equation is easy that is through algebra. No need to carry out differentiation or integration.


Example of Laplace Transform technique

atatadtty

adt

dy

tt

0

0)(

condition initial zero

Integral Approach


Example of Laplace Transform technique

2)(

)(

][

Transform Laplace Taking

condition initial zerowith

s

asY

s

assY

aLdt

dyL

adt

dy

Laplace Transform Approach

atty

s

aLsYL

)(

)]([

Transform Laplace Inverse Taking

211


Partial Fraction Expansion

A mathematical technique to help taking Inverse Laplace Transform



tt eKeKtfsFL

s

K

s

KsF

sssF

221

1

21

)()]([

21)(

Expansion,Fraction PartialBy

)2)(1(

2)(

Solve



Three cases:1.Roots of the denominator of F(s) are

real number and distinct2.Roots of the denominator of F(s) are

real number and repeated 3.Roots of the denominator of F(s) are

complex or imaginary


Partial Fraction Expansion – Case 1: Real and distinct roots

5)5(

)2()(

s

B

s

A

ss

ssY

5

2

)5(

)2(

0

ss

sA

5

3

)(

)2(

5

ss

sB

55

35

2)(

ss

sY


Partial Fraction Expansion – Case 1: Real and distinct roots

55

35

2)(

ss

sY

ttt eeety 550

5

3

5

2

5

3

5

2)(

Taking Inverse Laplace Transform


Partial Fraction Expansion - MATLAB

2)2)(1(

2)(

sssY

)52(

3)(

2

ssssY

Real and repeated roots

Imaginary and complex roots


What is Transfer Function?

Mathematical model that separates input from output

)()(2)(

trtcdt

tdc

2

1

)(

)(

ssR

sC


What is Transfer Function?

2

1

)(

)(

)()(2)(

)()(2)(

ssR

sC

sRsCssC

trtcdt

tdc


Mathematical modelling of physical systems

• Systems to be modelled– Mechanical– Electrical – Electro mechanical– Pneumatic– Thermal– Hydraulic


Mechanical system modelling

• Translational• RotationalExample

Automobile suspension systemAlong the road1. The vertical displacements at the tires act as the motion

excitation to the automobile suspension system2. Motion consists of a translational motion of the center of

mass3. Rotational motion about the center of mass


Translational system

Example 1


Element Law Expression Laplace

Spring Spring Force α change in length

F(t) = K x(t)K – Stiffness constant in N/m

F(s) = K X(s)

Viscous Damper or Dashpot

Force α velocity F(t) = fv v(t)F(t) =fv dx/dt fv - Friction or damping coefficient ,Ns/m

F(s) = fvsX(s)

Mass Newton’s second lawForce α acceleration

F(t) = M d2x/dt2

F(s) = Ms2X(s)

Translational system


Step 1: Decide input and output

Input variable:

)(tf

Output variable:

)(txMass position

)(txMass velocity

)(txMass acceleration

Applied force


Translational systems

Newton’s second law


Step 2: The Time and frequency response representation

dt

tdxftkxtf

dt

txdM v

)()()(

)(2

2

Taking Laplace Transform


Example 2

F(t)

x1(t) x2(t)

x3(t)

csl2 16 j15

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