chapter5 system analysis

Signals & Systems

Chapter 5Time-domain and frequency-domain

analysis of LTI systems using TF

INC212 Signals and Systems : 2 / 2554

Overview Stability and the impulse response Routh-Hurwitz Stability Test Analysis of the Step Response Fourier analysis of CT systems Response of LTI systems to sinusoidal inputs Response of LTI systems to periodic inputs Response of LTI systems to nonperiodic

(aperiodic) inputs

INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF

Chapter 5 The analysis of LTI systems using TF

Stability and the Impulse Response

011

1

011

1)(asasasa

bsbsbsbsH

NN

NN

MM

MM

Numerator

Denominator

Assumption : N ≥ M

: H (s) dose not have any common poles and zeros

INC212 Signals and Systems : 2 / 2554


H(s) has a real pole, p

H(s) has a complex pair poles, σ±jω

H(s) has a repeated poles

h(t) contains cept

h(t) contains ceσtcos(ωt+θ)

h(t) contains ctiept or ctieσtcos(ωt+θ)

Chapter 5 The analysis of LTI systems using TFINC212 Signals and Systems : 2 / 2554


cept

ceσtcos(ωt+θ)

ctiept or ctieσtcos(ωt+θ)

h(t) 0 as t ∞

p < 0

σ < 0

p < 0 or σ < 0

Stability



h(t) 0 as t ∞ Stable

|h(t)| ≤ c for all t Marginally Stable

|h(t)| ∞ as t ∞ Unstable

bounded

unbounded



Stable

Marginally Stable

Unstable

OLHP ORHP


Stability and the Impulse Response Example 8.1 Series RLC Circuit

L

Rb

L

Rb

bL

R

2

2

02

2

02

),Re( 21 L

Rppb < 0

LCL

Rb

bL

Rpp

LCsLRs

LCsH

1

2

2,

1)(

1)(

2

21

2

02

bL

R

b ≥ 0


Routh-Hurwitz Stability Test

1,,2,1,0for0

0,)( 011

1

Nia

aasasasasA

i

NN

NN

N

2

615

2

61525

2

413

2

41323

N

NNN

N

NNNNN

N

NNN

N

NNNNN

b

baa

b

baabc

b

baa

b

baabc

1

54

1

5414

1

32

1

3212

N

NNN

N

NNNNN

N

NNN

N

NNNNN

a

aaa

a

aaaab

a

aaa

a

aaaab

Table 8.1 Routh Array

sN aN aN-2 aN-4 …

sN-1 aN-1 aN-3 aN-5 …

sN-2 bN-2 bN-4 bN-6 …

sN-3 cN-3 cN-5 cN-7 …

s2 d2 d0 0 …

s1 e1 0 0 …

s0 f0 0 0 …

… …… …


Routh-Hurwitz Stability TestTable 8.1 Routh Array

sN aN aN-2 aN-4 …

sN-1 aN-1 aN-3 aN-5 …

sN-2 bN-2 bN-4 bN-6 …

sN-3 cN-3 cN-5 cN-7 …

s2 d2 d0 0 …

s1 e1 0 0 …

s0 f0 0 0 …

… …… …

Stable

Marginally Stable

Unstable

If all elements > 0

If sign changes

If 1 or more elements = 0& no sign changes


Routh-Hurwitz Stability Test Example 8.2 Second-Order Case

012)( asassA

01

012

)0)(1(a

a

aabN

Routh Array in the N=2 Case

s2 1 a0

s1 a1 0

s0 a0 0

Stable

1 pole in ORHP

Unstable

If a1 & a0 > 0

If a1 > 0 & a0 < 0 or a1 < 0 & a0 < 0

If a1 < 0 & a0 > 0


Routh-Hurwitz Stability Test Example 8.3 Third-Order Case

012

23)( asasassA

Routh Array in the N=3 Case

s3 1 a1

s2 a2 a0

s1 a1-(a0/a2) 0

s0 a0 0

0,,0

0

02

012

2

01

2

01

aa

aaa

a

aa

a

aa


Routh-Hurwitz Stability Test Example 8.4 Higher-Order Case

Example 8.4

s5 6 4 2

s4 5 3 1

s3 0.4 0.8 0

s2 -7 1 0

s1 6/7 0 0

s0 1 0 0

123456)( 2345 ssssssA


Routh-Hurwitz Stability Test Example 8.5 Fourth-Order Case

Example 8.5

s4 1 3 2

s3 1 2 0

s2 1 2 0

s1 0 ≈ε 0 0

s0 2 0 0

223)( 234 sssssA


Analysis of the Step Response

ssA

sBsY

ssX

sXsA

sBsY

)(

)()(

1)(

)()(

)()(

0),0()()(

)(

)()(

)0()]([

)(

)()(

1

11

0

tHtyty

sA

sELty

HsYsc

s

c

sA

sEsY

s

Transient part Steady-state value (if stable)


Analysis of the Step Response First-Order Systems

ps

pk

s

pksY

ssX

ps

ksH

)(

1)(

)(

p

kH

tep

kty

tep

kty

pt

pt

)0(

0,)(

0),1()(

1


Analysis of the Step Response First-Order Systems :

Unstable

Without boundp = 3, 2, 1

0),1()( tep

kty pt



StableBoundp = -5, -2, -1

k = -p H(0) = 1

Steady-state value = 1

0),1()( tep

kty pt



(time constant, )

p = -5, -2, -1≈ 63% of H(0)

=0.2 sec

= 1 sec

= 0.5 sec


Analysis of the Step Response Determining the pole location from the step

response

y(t) ≈ 1.73 ; t ≈ 0.1 s

t = 0.1 sec

20

]1[273.1)1.0( )1.0(

p

ey p


Analysis of the Step Response Second-Order Systems

22 2)(

nnss

ksH

1

1

22

21

nn

nn

p

p

is called the damping ratio

n is called the natural frequency


Analysis of the Step Response Second-Order Systems :

Case when both poles are real

spsps

ksY

psps

ksH

))(()(

))(()(

21

21

212

2121

2121

)0(

0),()(

0),1()(

21

21

pp

kkH

tekekpp

kty

tekekpp

kty

n

tptptr

tptp



Case when both poles are real

0,2)(

0,12)(

1

2

1

1

2)(

)2)(1(

2)(

)2)(1(

2)(

2,1,2

2

2

21

teety

teety

ssssY

ssssY

sssH

ppk

tttr

tt



Case when poles are real and repeated

0,)1()(

0,)1(1)(

)()(

)()(

2

2

2

2

tetk

ty

tetk

ty

ss

ksY

s

ksH

t

n

n

tr

t

n

n

n

n

n

n



Case when poles are real and repeated

0,)21(1)(

)2(

4)(

)2(

4)(

2,,2,4

2

2

2

21

tetty

sssY

ssH

ppk

t

n



Location of poles in the complex plane

dn

dn

jp

jp

2

1

222

21

)2()(

))(()(

))(()(

dnn

dndn

s

ksH

jsjs

ksH

psps

ksH



Case when poles are a complex pair

0,sincos)(

)(

)(

)(

))(()(

2)()(

)()(

)()(

22

2

2222

2

2

22

2

2222

tk

tek

tek

ty

s

k

s

k

s

sks

k

s

ksksY

ss

ksY

s

ksH

nd

t

dnd

t

n

n

dn

n

dn

nn

n

dn

nn

dndn

nn




0,)sin(1)(

0),(tan

0),(tan)sin(sincos

0,sincos)(

2

1

1

22

22

ttek

ty

CDC

CDCwhere

DCDC

tk

tek

tek

ty

dt

d

n

n

nd

t

dnd

t

n

n

nn




0,)326.14sin(4

171)(

41

4,17,242.0,17

172

17)(

2

ttety

jp

k

sssH

t

dn



Effect of Damping Ratio on the Step Response

7.0,25.0,1.0for

1,1

)()(

22

k

s

ksH

n

dn



Effect of ωn on the Step Response

srad

k

s

ksH

n

n

dn

/2,1,5.0for

,4.0

)()(

2

22



Comparison of cases

22 2)(

nnss

ksH

0 < < 1 underdamped

overdamped

Critically damped

> 1

= 1



Comparison of cases

5.1,1,5.0for

2,4

2)(

22

n

nn

k

ss

ksH


Analysis of the Step Response Higher-Order Systems

011

1

011

1)(asasas

bsbsbsbsH

NN

N

MM

MM


Fourier Analysis of CT Systems LTI systems

h(t)x(t) y(t) H()X() Y()

h(t) is Impulse Response H() is Frequency Response function

dtxhtxthty )()()()()( )()()( XHY

)()()( XHY

)()()( XHY

dtth )(

Assume that the system is stable:

Amplitude :

Phase :

Time domain Frequency domain

F


Response of LTI System to Sinusoidal Inputs H()X() Y()

)cos()( 0 tAtx )]()([)( 00 jj eeAX

)()()( XHY

)]()([)(

)]()()()([

)]()([)()(

0))((

0))((

0

0000

00

00

HjHj

jj

jj

eeHA

HeHeA

eeAHY

))(cos()()( 000 HtHAty

F

F -

1 Response to Sinusoidal Input


Response of LTI System to Sinusoidal Inputs h(t)x(t) y(t)

)cos()( 0 tAtx ))(cos()()( 000 HtHAty h(t)

)3000cos()100cos()( tttx

)cos()cos()( 222111 tAtAtx

))(cos()())(cos()()( 2222211111 HtHAHtHAty

h(t)

))3000(3000cos()3000())100(100cos()100()( HtHHtHty

h(t) where 1 = 100and 2 = 3000


Response of LTI System to Sinusoidal Inputs

)()()(

)()()(

inoutout

inoutout

VVRCVj

tvtvdt

tdvRC

)()()(

)()()(

inout VHV

XHY

)(

)()(

in

out

V

VH

)()()1( inout VVRCj )1(

1

)(

)(

RCjV

V

in

out

RCHRC

HRCj

H

1

2tan)(;

1)(

1)(;

)1(

1)(

H()X() Y()



H(

)

H(

)

1)(lim0

H

0)(lim

H

Low frequency

High frequency


Response of LTI System to Sinusoidal Inputs h(t)x(t) y(t)


))3000(3000cos()3000())100(100cos()100()( HtHHtHty

h(t)


))3000(3000cos()3000())100(100cos()100()( HtHHtHty

RCH

RCH

1

2

tan)(

1)(

1)(

= 1 = 100 = 2 = 3000




))3000(3000cos()3000())100(100cos()100()( HtHHtHty

RCH

RCH

1

2

tan)(

1)(

1)(

RC = 0.001

H(

)

H(

)

x(t)

y(t)




))3000(3000cos()3000())100(100cos()100()( HtHHtHty

RCH

RCH

1

2

tan)(

1)(

1)(

RC = 0.01

H(

)

H(

)

x(t)

y(t)


Response of LTI System to Periodic Inputs h(t)x(t) y(t)

)( 0kHAA xk

yk

1

00 ),cos()(k

kk ttkAatx h(t)

1

0000 )),(cos()()0()(k

kk tkHtkkHAHaty

)( 0 kHxk

yk

)(2

10kHAc x

kyk

)( 0 kHc xk

yk

y(t)A

x(t)Ay

kyk

xk

xk

for FS tric trigonome theof tscoefficien theis,

for FS tric trigonome theof tscoefficien theis,


Response of LTI System to Periodic Inputs Response to a rectangular pulse train

1

00 ),cos()(k

kk ttkAatx

0.5-0.5 2.02.0 t

x(t)

……

1

0 ),cos()(k

k ttkaatx

,5,3,1,1

,6,4,2,0

0,5.0

kk

k

k

c xk

k

k

k

kkA

xk

xk

other all,0

,11,7,3,

,6,4,2,0

,5,3,1,2



0.5-0.5 2.02.0 t

x(t)

……

k

TT

k

2,2

0

0

kRCk

kRCk

k

kRCkkA

aHa

yk

yk

xy

other all,tan

,11,7,3,tan

,6,4,2,0

,5,3,1,1)(

12

5.0)0(

1

1

2

00



• RC = 1



• RC = 0.01


Response of LTI System to Nonperiodic Inputs

dteXHty

XHty

XHY

tj

)()(2

1)(

)()()(

)()()(1 -F

H()X() Y()

Response to a rectangular pulse


Response of LTI System to Nonperiodic Inputs

H()

x(t)

t

1

-1/2 1/2

2

sinc)( X

F

1

1)(

RCjH

)()()( XHY

dteXHty

XH

Yty

tj

)()(2

1)(

)()(

)()(1 -

1 -

FF

F-1

Response to a rectangular pulse


Response of LTI System to Nonperiodic Inputs Response to a rectangular pulse

• RC = 1

F

H()

F-1


Response of LTI System to Nonperiodic Inputs Response to a rectangular pulse

• RC = 0.1

F

H()

F-1


chapter5 system analysis

Education

s y s s

s s y t

s p1 s p2 kh s

p1 p2 s p1 s p2 s

n jd s

order systems kkh s

step responsebse s cy

analysis of lti systems