ist and 2nd order system analysis

7/27/2019 ist and 2nd order system analysis

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First & Second Order

System Analysis


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CONTENTS

Introduction

Influence of Poles on Time Response

Transient Response of First-Order System Transient Response of Second-Order

System


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INTRODUCTION

The concept of poles andzeros, fundamental to the analysisof and design of control system, simplifies the evaluation of

system response.

The polesof a transfer function are:

i. Values of the Laplace Transform variables s, that cause thetransfer function to become infinite.

The zerosof a transfer function are:

i. The values of the Laplace Transform variable s, that cause

the transfer function to become zero.


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INFLUENCE OF POLES ON TIME RESPONSE

The output response of a system is a sum ofi. Forcedresponse

ii. Naturalresponse

a) System showing an input and an output

b) Pole-zero plot of the system


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c) Evolution of a system response. Follow the

blue arrows to see the evolution of system

component generated by the pole or zero


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System transfer function :

Step response :

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90

100Step Response

Time (sec)

Amplitude



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Re(s)

Im(s)


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Unstable

Re(s)

Im(s)


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Unstable

Re(s)

Im(s)

-1


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Unstable

Re(s)

Im(s)

-2


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Unstable

Re(s)

Im(s)

faster response slower response

constant


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FIRST-ORDER SYSTEM

General form:

Block diagram representation:

By definition itself, the input to the system should be a step

function which is given by the following:

C(s)R(s)1s

K

ssR 1)(

Where,K : Gain

: Time constant

1)()()(

sK

sRsCsG


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FIRST-ORDER SYSTEM

General form:

Output response:

1)(

)()(

s

K

sR

sCsG

1

1

1)(

s

B

s

A

s

K

ssC

te

BAtc

)(

)()()( sRsGsC


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c(t) = 1 -

At point when t =

c() = 1 -

= 1 0.37

= 0.63

Time constant is the time it takes for thestep-response to increase to 63% of its

final value. In short time constant

measures how fast system can respond.

FIRST-ORDER SYSTEM


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FIRST-ORDER SYSTEM

First-order system response to a unit step


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FIRST-ORDER SYSTEM

Problem: Find the forced and natural responses for thefollowing systems


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TRANSIENT RESPONSE SPECIFICATIONS

Time constant, The time for e-atto decay 37% of its

initial value.

Rise time, tr The time for the waveform to go

from 0.1 to 0.9 of its final value.

Settling time, ts The time for the response to reach,

and stay within 2% of its final value.

a

1

atr

2.2

ats

4


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Problem: For a system with the transfer function shownbelow, find the relevant response specifications

i. Time constant,

ii. Settling time, ts

iii. Rise time, tr

50

50)(

s

sG


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FIRST ORDER SYSTEMSIMPLE BEHAVIOR.

No overshoot

No oscillations


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SECOND ORDER SYSTEM

(MASS-SPRING-DAMPER SYSTEM)

k b

y(t)

F(t)=ku(t)

m

ODE

Transfer Function


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SECOND-ORDER SYSTEM

Natural frequency,

n

Frequency of oscillation of the system without damping.

Damping ratio,

It is measure of the degree of resistance to change in thesystem output.


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SECOND-ORDER SYSTEM

General form:

Roots of denominator:

22

2

2 nn

n

ss

KsG

Where,

K : Gain : Damping ratio

n : Undamped natural frequency

02 22 nnss

122,1 nns Generically

2

2,1 1 nn js


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POLAR VS. CARTESIAN REPRESENTATIONS.Cartesian representation :

Imaginary part (frequency)

Real part (rate of decay)


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Polar representation :

damping ratio

natural frequency


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Polar representation :

damping ratio

natural frequency


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SECOND-ORDER SYSTEM


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SECOND-ORDER SYSTEM


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SECOND-ORDER SYSTEM

Step responses for second-order system damping cases


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SECOND-ORDER SYSTEM

Second-order response as a function of damping ratio


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SECOND-ORDER SYSTEM

Second-order response as a function of damping ratio


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SECOND ORDER SYSTEM RESPONSE.

Unstable

Re(s)

Im(s)

Undamp

ed

Overdamped or Critically damped

Underdamped

Underdamped


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SECOND-ORDER SYSTEM

Second-order underdamped responses for damping ratiovalue


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Second-order underdamped response specifications


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Rise time, Tr The time for the waveform to go from 0.1 to 0.9 of its final

value.

Peak time, Tp The time required to reach the first

or maximum peak.

Settling time, Ts The time required for the transients

damped oscillation to reach and stay

within 2% of the steady-state value.

2

1

n

pT

n

sT

4


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Percent overshoot, %OS The amount that the waveform overshoots the steady-state, or

final value at peak time, expressed as a percentage of the

steady-state value.

%100% )1/(2

eOS

)100/(%ln

)100/ln(%

22OS

OS


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SYSTEM PERFORMANCE

Percent overshoot versus damping ratio


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SYSTEM PERFORMANCE

Lines of constant peak time Tp, settling time Tsand percentovershoot %OS

Ts2< Ts1Tp2< Tp1

%OS1< %OS2


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SYSTEM PERFORMANCE

Step responses of second-order underdamped systems aspoles move

a) With constant

real part

b) With constant

imaginary part


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SYSTEM PERFORMANCE

Step responses of second-order underdamped systems aspoles move

c) With constant dampingratio

Th ti f t l t i t


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40

The time response of a control system consists

of two parts:

1. Transient response

- from initial state to the final

statepurpose of control

systems is to provide a desired

response.

2. Steady-state response

- the manner in which the

system output behaves as t

approaches infinitythe error

after the transient response has

decayed, leaving only the

continuous response.


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THEOREMS

Initial Value Theorem

Final Value Theorem (Steady state) If all poles of sX(s) are in the left half plane

(LHP), then


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FURTHER READING

Chapter 4

i. Nise N.S. (2004). Control System Engineering (4th Ed), John

Wiley & Sons.

Chapter 5

i. Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9thEd), Prentice Hall.


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THE END

ist and 2nd order system analysis

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