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Control Systems 0108400 Lect 03 Math Models, LT, Electric Element, Transfer Function Dr. M. N. Najem Summer 2010-2011 May 18, 2011
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Lect 03 TF- Modeling in Freq Domain

Nov 27, 2014

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Page 1: Lect 03 TF- Modeling in Freq Domain

Control Systems 0108400

Lect 03 Math Models, LT, Electric

Element, Transfer Function

Dr. M. N. Najem

Summer 2010-2011

May 18, 2011

Page 2: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 2

Mathematical models

• In the previous lecture after obtaining a

schematic, the control engineer makes

simplifying assumptions in order to keep

the ensuing model manageable and still

approximate physical reality.

• Next step is to develop mathematical

models from schematics of physical

systems

Page 3: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 3

Mathematical models …

• Transfer functions in frequency domain (classical approach).

• State-space in time domain (modern approach).

• In every case the first step in developing a mathematical model is to apply the fundamental physical laws of science an engineering:– In modeling electric networks: Ohm, KVL, KCL, Nodal,

Equivalent circuit, ..

– in modeling mechnical systems: Newton laws (forces, torques)

• From these equations we obtaing the relationship between system’s output and input

Page 4: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 4

Mathematical models …

• Differential equations can describe the relationship between the input and output of a system.

• The form of the D.E and its coefficients are a formulation of the system.

• Although the D.E relates the system to its input and output, it is not a satisfying representation from a system perspective. Look at a LTI nth order D.E:

Page 5: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 5

Mathematical models …

• The system parameters (the coefficients), the output, as well as the input appear through out the D.E.

)()(

...)()(

)()(

...)()(

011

1

1011

1

1 trbdt

tdrb

dt

trdb

dt

trdbtca

dt

tdca

dt

tcda

dt

tcdm

m

mm

m

mn

n

nn

n

• Prefer a mathematical representation such that the input, output, and system are distinct and separate parts; ( conveniently represent interconnection of several subsystems). Mathematical function that appears inside the block called the transfer function, denoted, H(s), or G(s).

SystemInput

r(t)

Output

c(t)

Page 6: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 6

Review of Laplace transform

Page 7: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 7

Laplace transform

• Addressed in MATH IV course

00

01)(

)()()(2

1)()]([

)()()]([

1

0

t

ttu

tutfdsesFj

tfsFL

dtetfsFtfL

j

j

st

st

• Multiplication f(t) by u(t) yields a time function that is zero

for t < 0.

Page 8: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 8

Laplace transform

• Refer to Laplace transform in Math IV to

review the various operations and to relate

to Laplace transform tables.

• Inverse of Laplace transform

– Partial-fraction expansion

– Use of various theroems

• Also, refer to solving D.E.s using Laplace

transform.

Page 9: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 9

LT examples

• Presented and discussed on board

Page 10: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 10

Review of resistive electric

networks

Page 11: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 11

Some elements

Independent sources: Resistor:

Page 12: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 12

Superposition example

2kW1kW

2kW12V

I0

2mA

4mA

– +

• Given the following circuit

• Calculate I0 using superposition

Page 13: Lect 03 TF- Modeling in Freq Domain

Superposition example …

Dr. Najem 13

• Solve the problem by leaving one independent source at a time and

opening the other Independent current sources and short circuiting

the other independent voltage sources

2kW1kW

2kW

I’0

2mA

I’0 = -4/3 mA

Open current source

Short circuit

voltage source

Page 14: Lect 03 TF- Modeling in Freq Domain

Superposition example …

Dr. Najem 14

2kW1kW

2kW

I’’0

4mA

I’’0 = 0

Short circuit

Open current source

Page 15: Lect 03 TF- Modeling in Freq Domain

Superposition example …

Dr. Najem 15

2kW1kW

2kW12V

I’’’0

– +

I’’’0 = -4 mA

Final result:I’0 = -4/3 mA

I’’0 = 0

I’’’0 = -4 mA

I0 = I’0+ I’’0+ I’’’0 = -16/3 mA

Open current source

Open

current

source

Page 16: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 16

Use of KVL and KCL

Loop1:

Node1:

Node1

Loop1

Page 17: Lect 03 TF- Modeling in Freq Domain

17

Equivalent circuits USE THEVENIN TO COMPUTE Vo

W kRTh 3/1024//2

0)(246

2

211

2

IIkkIV

mAI

circuitbuttonleftonAnalysisLoop

mAmAI

I3

5

6

26 21

][3/3243/20*2*4 21 VVIkIkVOC

VocCalculate V0

using Voltage

division

Page 18: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 18

Nodal analysis and Mesh analysis

Mesh analysis

224111

213212111 )()(

SS

S

VIRIRV

IIRIIRIRVSolve for I1 and I2.

Page 19: Lect 03 TF- Modeling in Freq Domain

Dynamic networks

Dr. Najem 19

Page 20: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 20

Characteristics of dynamic network

• Dynamic Elements Ohm’s Law: ineffective

• Inductor:

Page 21: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 21

Characteristics of dynamic network ...

• Dynamic Elements Ohm’s Law: ineffective• Capacitor:

Page 22: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 22

Characteristics of dynamic network ...

• Example:

t

tRidtiCdt

tdiLtv

0)()(

1)()(

then

If

v(t)

Page 23: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 23

Dynamic relationships into

Algebraic operation LT

• Using Laplace Transform on previous circuit

)()(1

)()(

)()0()(1

))0()(()]([)()1(

sRIsICs

sILs

sRIs

i

s

sI

CissILtLsVs

• Then generalize resistor R into impedance Z

LssZ )(1Cs

sZ1

)(2

Page 24: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 24

RsZsZ

sVsI

sRIsIsZsIsZsV

s

s

)()(

)()(

)()()()()()(

21

21

Dynamic relationships into

Algebraic operation LT …

Page 25: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 25

Dynamic elements Laplace

transform models

• Capacitor LT model

Page 26: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 26

• Inductor LT model

Dynamic elements Laplace

transform models

Page 27: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 27

Resistor LT model

• V(s) = RI(s)

Sources as LT models

Page 28: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 28

Laplace and analysis methods

• Laplace application to analysis techniques

analysismesh and analysis Nodal

circuit Equivalent

KCL and KVL

ionsuperposit

Law Ohms dGeneralize

These will be

applied as in

Resistive Networks

Key: Laplace transform models of (dynamic) elements.

Page 29: Lect 03 TF- Modeling in Freq Domain

Transfer Functions

Dr. Najem 29

Page 30: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 30

System representations

• Mathematical form of representing (modeling) the system.

• In this course we are concerned with the following system representation methods:

– Differential equations

– Transfer functions

– State-space

Page 31: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 31

Transfer Functions• System analysis: emphasize is on relation between

input and output (using blocks)

• Circuit analysis: detailed analysis (voltages, branch currents)

• System analysis: how the system processes the input to form the output, or how the system transforms the input into output.

• Output: variable to be controlled

• Input: variable used and to be adjusted to change or influence the output

Dynamic systemInput

r(t)

Output

c(t)

Page 32: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 32

Transfer Function H(s) or G(s)

• Gives Quantitative Description of ‘ how

the system processes the input to form the

output’.

Page 33: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 33

input to H(s) or G(s)

• Input is (t) impulse fucntion

• The resultant output y(t) is called: the unit impulse response.

• In this case: X(s) = L [ (t)] = 1 and

Y(s) = Laplace Transform of the unit impulse response

Therefore, H(s) = Y(s)/X(s) = Y(s)

Therefore, the transfer function of a system is the Laplace transform of the unit impulse response of the system

Page 34: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 34

Facts on Transfer Functions• Independent of input, a property of the system structure

and parameters.

• Obtained with zero initial conditions.

• Generates a mathematical model of the system and algebraically relates the representation of the output to the representation of the input,.

• Applies only on LTI systems.

• Rational Function of s (Linear, lumped, fixed)

• H(s): Transfer function can be easily converted into frequency response function of the system H(jw) or H( j2f ). Simply replace s = jw.

• |H( j2f )| or |H( j )|: amplitude response function and H(j2f) or H( j ): Phase response function

Page 35: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 35

Properties of Transfer Function

• Properties of Transfer Function for Linear, Lumped stable systems.

)(

)(

...

...)(

01

1

01

1

sD

sN

asasa

bsbsbsH

nn

nn

mm

mm

• Corresponding differential equations:

)(...)()(

)(...)()(

01

)1(

1

)(

01

)1(

1

)(

txbdt

txdb

dt

txdb

tyadt

tyda

dt

tyda

m

m

mm

m

m

n

n

nn

n

n

coef. ai, bj: all real! Why? Results from real system components.

Page 36: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 36

• Differential equations do not separate the representation of input, output, and system; transfer functions do.

• Replacing D.Es. with algebraic equations simplifies the representation of individual subsystems and simplifies modeling interconnected subsystems.

• Roots of N(s), D(s): are real, or complex conjugate pairs.

• Zeros of the transfer function:= roots of N(s)

• Poles of the transfer function:= roots of D(s)

Properties of Transfer Function …

1,2:

)1)(2()()13()( 2

poles

sssDsssD

Page 37: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 37

H(s) of BIBO stable system

H(s) = G(s) = N(s) / D(s)

• Degree of N(s) must be Degree of D(s).

– If degree N(s) > Degree D(s) then divide:

)(

)('...)( 01

sD

sNcscscsH k

k

• but, under a bounded-input x(t) = u(t) (unit step function) => L[u(t)] = X(s) = 1/s, then:

Page 38: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 38

H(s) of BIBO stable system …

...)(...)(

)(

)('...)(

1

01

1

tcty

ssD

sN

s

ccscsY k

k

( )(t not bounded!

so y(t) unbounded)

Page 39: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 39

H(s) of BIBO stable system …

• Poles: must lie in the left half of the s-

plan.

0)Re(

))...()(()( 21

j

nssssD

then:

• Question: Any restriction on zeros?

• Answer: No (for BIBO stable system)

WHY? …..

Page 40: Lect 03 TF- Modeling in Freq Domain

Dr. Najem 40

– Electric Networks Transfer Functions

– Translational Mechanical Systems Transfer Functions

– Rotational Mechanical Systems Transfer Functions

– Transfer Functions for Systems with Gears

– Electromechanical Systems Transfer Functions

– Electric/Mechanical Circuit Analogs

– You do linearization on your own

Next 2 lectures modeling of: