Chapter 2 System IInd Order Responses

8/22/2019 Chapter 2 System IInd Order Responses

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Chapter 2 (page#119)

Continuous-TimeSystem Responses

2.3 Response of Second-Order Systems

2.5 Stability Testing

2.4 Higher- Order System Response


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System Response

First-order system time response

transient

dc steady-state

Second-order system time response

transientdc steady-state


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Ist & IInd order systems if stable:

the forced response is the steady-state

responseand the natural response is the transient

response

For unstable responses, "steady-state" and"transient" are meaningless


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2.3 (p 126) RESPONSE OF SECOND-ORDER SYSTEMS

)......(asas

b

)s(R

)s(C)s(G 174

012

0

The Standard form of the second-order transfer function is given by:

)......(ss)s(R

)s(C)s(G

nn

n 1842 22

2

, the damping ratio, will determine how much the systemoscillates as the response decays toward steady state.

n, the undamped natural frequency (natural frequency) , willdetermine how fast the system oscillates during any transientresponse.

Note: All system chracteristics of the 2nd-order system are

functions of only and n.


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22

2

2 nn

n

ss)s(R

)s(C

)s(G

)unityisdcgain()(R

)(C)(G 1

0

00

G(0): DC gain of the system, will determine the size of steady

state response when the input settles out to a constant value.

We will also consider 2nd-order systems with other than unity

gain and with numerator other than a constant.

2.3 (p 126) RESPONSE OF SECOND-ORDER SYSTEMS..cont


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Unit-step response for standard Second-0rder system

)s(R)s(G)s(C R(s) C(s)22

2

2 nn

n

ss

sss)s(C

nn

n 1

222

2

s)s(G)s(C1

)tsin(e)t(c ntn

11

Assume poles of G(s) are complex, taking inverse LLT

21

1tan

n)(constatntTime

1

ns )T(meSettlingTi

44

usoidesindampedoffrequencyn

Step response R(s) = A/s = 1/s unit step response

G(s)

2.3 (p 126) RESPONSE OF SECOND-ORDER SYSTEMS..cont


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Three cases:

Overdamped case: >1, two real

distinct polesUnderdamped case:


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4 cases to be considered

Case 1: Distinct real roots

tsts ecectx

ss

2

2

1

1

21

)(

Case 2: Equal roots & real

stst tecectx

sss

21

21

)(

Case 3: Imaginary roots

tktktx

ecectx

js

tjtj

cossin)(

)(

21

21

Case 4: Complex conjugate roots

tktketx

js

t

cossin)( 21


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>1Distinct

real roots

Timeconstant () ForcingFunction and n

Forcing

Function

=1


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>1

=1


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3


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4


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Step Response

t

)1(

)2(

)(c

rt

pt

st

rt

dt

pMovershoot


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Exponential decay generated by real

part of the complex pole pair

Exponential decay

generated by real

part of the complex

pole pair (=n)

Sinusoidal oscillation

generated by imaginary

part of the complex pole

pair

C(t)

t

)tsin(e)t(c ntn

11

-=-n n

jd

-jd

S-Plan


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t

)1(

)2(

)(c

rt

pt

st

rt

dt

pMovershoot


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P bl 2


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40012

4002

ss

)s(G)a(

Problem

90090

9002

ss

)s(G)b(

22530

2252

ss

)s(G)c(

625

6252

s

)s(G)d(

t.t.

CeBeA)t(c46115478

21 /t/t teCBeA)t(c

)tcos(eBA)t(c ntn

)tcos(BA)t(c 25

)t.cos(eBA)t(ct

08196

21 /t/t CeBeA)t(c

tt teCBeA)t(c 1515

21

22

2

2 nn

n

ss)s(G

N t


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1. The impulse response ofany system does give an indication of the nature of

the initial-condition (IC) response, and thus the transient response, of the system.

3. An Initial condition excitation of a Higher-order system cannot be modeled as

simply as that of the first-order system.

2. An Initial condition on a first-order system can be modeled as an impulse

function input

4. The unit-impulse response of the II-order system is given by the equation:

).(responseimpulse.....tsine)t(c ntn n 254

2

2

22 n

n

ns)t(c -1

).(responsestepUnit).....tsin(e)t(c ntn 20411

Equation 4.25 is the derivative of equation 4.20

5. The impulse response of the second order system can also be considered to

be the response to certain initial conditions, with r(t)=0

Note:-


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Third-order systems (Higher Order System)

Consider one real pole plus a pair ofcomplex conjugate poles

Complex poles dominant (close toorigin), real pole non-dominant

or

Real pole dominant, complex poles

non-dominant


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Complex poles dominant:

approximate as secon-dordersystem

Real pole dominant:

approximate as first-order system


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All three poles dominant:

approximation difficult

Complex conjugate poles

Dominant:zero tends to minimize

effect of nearby real pole


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Real pole dominant,

zeros tend to minimize

effects of nearby poles

Unstable due to rhp pole,

nearby zero is not usefulin canceling pole


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Lhp zero will amplify overshoot

Stable system, rhp zero can

give response that starts inopposite direction from the

steady-state resp


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X

S = -a0

Pole =1/=-a0

Characteristics

Equation

X

S = -a0

Pole =1/=-a0X

S = -a0

Pole =1/=-a0

Chapter 2 System IInd Order Responses

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