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EE 1265 Control Engineering

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EE1265 Control Engg -R Essakiraj

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S ystem

A system is a collection of components

which are co-ordinate together to perform

a function.

Dynamic system: A system with a memory,

i.e., the input value at time t will influence

the output at future instants.

Systems interact with their environmentacross a separating boundary.

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A control system is a system that is

used to realize a desired output or objective

Control is an essential element of almost

all engineering systems.

Control System Concepts

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T ypes of Control S ystem

Open loop control system

Closed loop control system

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Any physical system which does not

automatically correct for variation in its

output is called open loop system.

The output may be changed to any

desired value by changing the input

signal manually.

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EE1265 Control Engg -R Essakiraj 7


Example

1. Fan with regulator

2. Traffic light control

Controller Process

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Closed loop control system

A system that uses a measurement of

the output and compares it with the

desired output.I/P O /P

CONROLLER PLANT

FEEDBACK

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T ypes of feedback controlsystems

1.Negative feedback

2.Positive feedback

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Examples of feedback

system1. Room temperature control system.

2. Speed control of switched reluctance

motor.

3. Automatic aircraft landing system.

4. Automatic car driven system.

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T ransfer function

For linear time invariant system

transfer function is defied as the ratio of

Laplace transform of output to the Laplace

transform of input with initial zero condition.

Transfer function T(S) = C(s) / R(s)

G(S)

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Mechanical systems

T ypes of mechanical system

1.Translation system.

2.Rotational system.

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Mechanical systemsTranslation system

Input Force

Output Displacement

Physical mass, spring,Element dash pot

Physical acceleration

variable velocity.

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Mechanical systems

Rotational system

Input Torque

Output Angular Displacement

Physical Moment of inertia,Element spring, dash pot

Physical Angular velocity,variable acceleration

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Force-Current Analogy

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Force-Voltage Analogy

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UNIT-2

OPEN AND CLOSED LOOP SYST EMS

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FEEDBACK CONTROLSYST

EM

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BLOCK DIAGRAM

A block diagram for a system is not

unique, meaning that it may manipulated

into new forms.

Typical a block diagram will be developed

for a system. The diagram will then be

simplified through a process that is both

graphical and algebraic.

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RULES FOR BLOCK

DIAGRAM REDUCT ION Negative feedback reduction

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Positive feedback reduction

Moving branches before block

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Moving branches after block

Moving summation function before block

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Moving summation function after block

Combining sequential function block

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EE1265 Control Engg -

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Example for block diagram

reduction (car speed control)

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Example for block diagramreduction (car speed control)

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Example for block diagramreduction (car speed control)

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Signal flow graph(SFG)

A signal-flow graph is a diagram consisting of

nodes that are connected by several directed

branches and is a graph representation of a set of linear relation.

Only valid for linear system

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Basic elements of SFG

Branch: A unidirectional path segment.

Nodes: The input and output points or junctions.

Path: A branch or a continuous sequence of

branches that can be traversed from one node to

anther node.

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Basic elements of SFG

Loop: A closed path that originates and

terminates on the same node, and along the path

no node is met twice.

Non-touching loops: If two loops do not have a

common node.

Touching loops: Two touching loops share one

or more common nodes.

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Mason·s gain formula

The linear dependence (Tij) between the

independent variable xi (input) and the dependent

variable (output) x j is given by Mason¶s SF gain

formula

ijk ijk

ijk

k

ijk ijk

ij

P

k P

P

T

paththeof cofactor graphtheof tdeterninan

xtoxfrom path ji

th

!(!(

!

(

(

!§

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Mason·s gain formula

(=1 ±(sum of all different loop gains) +(sum

of the gain products of all combinations of

2 nontouching loops)-(sum of the gain

products of all combinations of 3nontouching loops)«

The cofactor is the determinant with

loops touching the kth path removed

ijk (

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Example

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Solution for Example

Two paths :P1, P2

Four loops

P1 = G1G2G3G4, P2= G5G6G7G8

L1=G2H2 L2=G3H3 L3=G6H6 L4=G7H7

Det (!- (L1+L2+L3+L4)+(L1L3+L1L4+L2L3+L2L4)

Cofactor for path 1:(

1= 1- (L3+L4) Cofactor for path 2: (2= 1-(L1+L2)

T(s) = (P1(1 + P2(2)/(

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Unit ² III(CHARACT ERIST ICEQUAT ION AND FUNCT IONS)

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Laplace transform of a

derivative term The Laplace transform of the derivative of a

function

where y(0-) is the initial condition associated with

y(t).

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Laplace transformation

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Different T ypes of inputs

Step input,

Impulse input,

Ramp input,

Parabolic and

Sinusoidal inputs,

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Step input

t ime

)(t x s

Figure 3.1. Step input.

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Ramp input

Figure 3.2. Ramp input.

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Sinusoidal inputs

Figure 3.3 Sinusoidal inputs

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Impulse input

Figure 3.4 Sinusoidal inputs

Here,

It represents a short, transient disturbance

. I U t t H !

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Response of First-OrderS

ystemsThe standard orm or a irst-order TF is:

Where

Find y(s) and y(t) for someparticular input x(s)

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ystems1. Step response

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ystems.

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ystems2 . Ramp response.

R f Fi t O d

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Response of First-OrderS ystems

3. Sinusoidal response.

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Response of second-Order

S ystems Standard form of the second-order transfer

function.

Where

is the process gain.

is the time constant which determines the

speed of response of the system

is the damping factor

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S ystems The characteristic polynomial is the denominator

of the transfer function:

=0

Roots:

2 2

2 1 s s

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S ystems The type o behavior that occurs depends on the

numerical value o damping coe icient,

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S ystems

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S ystems

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S ystems

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T ime domain specification

1. Rise time( ) is the time the process output

takes to first reach the new steady-state value.

2. peak time( ) is the time required for the outputto reach its first maximum value.

3. Settling time( ) is defined as the time required

for the process output reach and remain inside a

band whose width is equal to 5% of the total

change in y .

r t

pt

st

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4. Overshoot: OS = a/b (% overshoot is 100a/b).

5. Decay Ratio: DR = c /a (where c is the height of

the second peak).

6. Period of Oscillation: P is the time between

two successive peaks or two successive valleys

of the response.

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1. Rise time

2. peak time.

3. Overshoot:

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4. Decay Ratio:

5. Period of Oscillation:

d

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Steady state errors(Step Reference)

d

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Steady state errors(RAMP Reference)

S d

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Steady state errors(Parabolic Reference)

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Summary of steady stateerrors for different systems

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Unit ² IV(CONCEPT OF ST ABILITY)

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Basic concept of stability

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Stability

A system is stable if any bounded

input produces a bounded output for

all bounded initial conditions.

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Stability of the system androots of characteristic equations

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Routh-Hurwitz Criterion

It is a technique that one can use to check thestability of the system from the characteristicequation without solving it.

Let¶s consider

Determine the closed-loop stability of this system

R th H it C it i

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Routh-Hurwitz Criterionexample

The characteristic equation of the system is:

1 + G(s) = 0

s3 + s2 + 2s +24 = 0

The Routh-Hurwitz table can be formulated as

follows:

R th H it C it i

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Routh-Hurwitz Criterionexample

R h H C

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Routh-Hurwitz Criterion

exampleSince there are two changes in the sign of the first

column of the Routh-Hurwitz table, there are two

unstable poles in the closed-loop system.In fact, the roots are:

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Bode plot

Plots of the magnitude and phase characteristics

are used to fully describe the frequency response.

A Bode plot is a (semilog) plot of the transfer

function magnitude and phase angle as a function

of frequency

The gain magnitude is many times expressed in

terms of decibels (dB)dB = 20 log10 A

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Bode plot procedure

There is 4 basic forms in an open-loop transfer

function G( j)H( j).

1.Gain factor, K

2.( j) p factor: pole and zero at origin

3.(1+ jT ) q factor

4.Quadratic factor r

nn

j

s

¼¼½

»

¬¬-

« 2

2

21[

[

[

[ ^

Th i d d h

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T he magnitude and phaseplots for some typical factorsFactor Magnitude: Phase:

Gain, K

j

dB jG )( [ )( [ j

0dB20logK

0dB

120 dB/decade

Factor Magnitude: Phase:

Gain, K

j

90º

45º

0º

0dB

Th i d d h

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T he magnitude and phaseplots for some typical factorsFactor Magnitude: Phase:

Gain, K

j

dB jG )( [ )( [ j

0dB20logK

0dB

120 dB/decade


Gain, K

j

90º

45º

0º

0dB

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T he magnitude and phase

plots for some typical factorsFactor Magnitude: Phase:

j2

1 /j

dB jG )( [ )( [ j

0dB

-20 dB/decade

0dB

1

40log

40 dB/decade

180º

90º

0º

0º

-45º

-90º

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T he magnitude and phaseplots for some typical factors


1 /j2

1+jT

90º

45º

0º

0dB

-40log

-40 dB/decade

0.1/T 1/T 10/T

dB jG )( [ )( [ j

Th it d d h l t

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T he magnitude and phase plotsfor some typical factors


1 / (1+jT

dB jG )( [

0.1/T

)( [ j

Th it d d h l t

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T he magnitude and phase plotsfor some typical factors


n[

0dB

0º

-90º

-180º

0dB

40 dB/decade

180º

90º

0º

22

2

2

21log20 ¹¹ º

¸©©ª

¨

¹¹

º

¸

©©

ª

¨

nn[

[ ^

[

[

n[

22

2

2

21log20 ¹¹ º

¸©©ª

¨

¹¹

º

¸

©©

ª

¨

nn[

[^

[

[

-40 dB/decade

dB jG )( [ )( [ j

2

2

21

1

nn

j[

[

[

[ ^

G i i d Ph

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Gain margin and Phase

marginGain margin:

The gain margin is the number of dB that is

below 0 dB at the phase crossover frequency (ø=-

180º). It can also be increased before the closed-

loop system becomes unstable.

Phase margin:

The phase margin is the number of degrees

the phase of that is above -180º at the gain

crossover frequency..

G i i d Ph

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Gain margin and Phase

margin0dB

20dB

-20dB

-40dB

-60dB

-90º

0º

-180º

-270º

Gain margin

Phase marginPhase Crossover

with -180º line

Gain Crossover with 0dB line

Magnitudecurve

Phasecurve

Log Phase

(Degree)

Magnitude(dB)

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Root locus

The root locus is the path of the roots of the

characteristic equation traced out in the s-plane

as a system parameter is varied.

Root locus and system performanc e

1. Stability

2. Dynamic performance

3. Steady-state error

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T he Root Locus Procedure

S tep 1: Write the characteristic equation as

S tep 2 : Rewrite preceding equation into the form of

poles and zeros as follows:

0)(1 ! s F

0)(

)(

1

1

1!

!

!

n

i

i

j

j

p s

z s

K

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S tep 3:

Locate the poles and zeros with specific

symbols, the root locus begins at the open-loop

poles and ends at the open-loop zeros asK

increases from 0 to infinity.

If open-loop system has n-m zeros at infinity,

there will be n-m branches of the root locus

approaching the n-m zeros at infinity.

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S tep 4:The root locus on the real axis lies in a section

of the real axis to the left of an odd number of

real poles and zeros.

S tep 5:The number of separate loci is equal to the

number of open-loop poles.

S tep 6:

The root loci must be continuous andsymmetrical with respect to the horizontal real

axis.

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S tep 7:The loci proceed to zeros at infinity along

asymptotes centered at and with angles

mn

z p

n

i

m

j

ji

a

!

§ §! !1 1

W

)1,2,1,0()12( !

! mnk mn

k a .T J

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T he Root Locus ProcedureS

tep 8:The actual point at which the root locus crosses

the imaginary axis is readily evaluated by using

Routh criterion.

S tep 9:Determine the breakaway point d (usually on

the real axis):

§ §! ! !

m

j

n

i i j pd z d 1 1

11

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T he Root Locus ProcedureS tep 10 :

Determine the angle of departure of locus from

a pole and the angle of arrival of the locus at

a zero by using phase angle criterion

i p

i z

)(180,11

0 §§{!!

!n

i j j

p p

m

j

p z p i ji jiUN U

)(180 1,1

0

§§ !{!

!

n

j z p

m

i j j z z z i ji ji UN N

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T he Root Locus ProcedureStep 11:

Plot the root locus that satisfy the phase criterion.

Step 12:Determine the parameter value K 1 at a specific

root using the magnitude criterion.

.,2,1)12()( !! k k s P T

1 s

11

11

)(

)(

s s

m

j

j

n

i

i

z s

p s

K

!!

!

!

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ExampleSketch the root locus of the following system

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UNIT-V

SAMPLED DAT A S YST EMS

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Digital control

1. More useful for computer systems

2. Time is discrete

denoted k instead of t

3. Main tool is z-transform

f (k ) ® F (z ) , where z is complex

Analogous to Laplace transform for s-domain

§g

!

!!

0

)()()]([k

k z k f z F k f Z

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Digital Controller Adv antage

1. Modern digital technology(DSP, FPGA,ADC)

2. Cost effective and precise power supply

3. High flexibility and modularity

4. High functionality5. Built in all functions for PS

Current regulator(PI, feed forwarder)

Direct PWM generation

Fast control port

Analog, Digital I/O

Block diagram of digital

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Block-diagram of digitalsystem

Two views of uniform rate

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T wo views of uniform-ratesampling:

Spectrum of a Sampled

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Spectrum of a SampledSignal

Spectrum

± Consider a cont. signal r(t)

± with sampled signal r*(t)

± Laplace transform R*(s) can be calculated

§

§

g

g!

g

g!

!

!

k

s

k

jn s RT

s R

k T t t r t r

)(1

)(

)()()(

[

H r(t) r*(t)

T


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Signal

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Z transform

Consider a sequence of values: {xk : k = 0,1,2,... }

These may be samples of a function x(t),

sampled at instants t = k T ; thus x k = x( k T).

The Z transform is simply a polynomial in z

having the x k as coefficients:

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Uses of Z transform

Analysis of Discrete & Sampled-data Systems

Transfer Function & Block Diagram

Representations

Exploit the z-plane

± Dynamic Analysis

± Root Locus Design

Z-transform of a sampled

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Z-transform of a sampledsignal

? A

f t f kT t kT

F s L f t f kT e

z e

F z f kT z

k

k sT

k

sT

k

k

*

* *

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

!

! !

!

!

!

g

!

g

!

g

§

§

§

H 0

0

1

0

Partial table of z- and s-

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Partial table of z- and s-transforms

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z -transform theorems

Sampled-data systems and

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Sampled data systems andtheir z -transforms

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