chapter 4 transfer function and block diagram operations § 4.1 linear time-invariant systems § 4.2...

Chapter 4 Transfer Function and Block Diagram Operations

§ 4.1 Linear Time-Invariant Systems

§ 4.2 Transfer Function and Dynamic Systems

§ 4.3 Transfer Function and System Response

§ 4.4 Block Diagram Operations for Complex Systems

1

§ 4.1 Linear Time-Invariant Systems (1)

• LTI Systems:

y(t)r(t)LTI

I.C.

Differential Equation Formulation

input) system, .,C.I(y)t(y

)0(r),......0(r r(0),

)0(y),......0( y y(0),I.C.

system causal for mn

rp......rprpyq...... yqy

)2m((1)

)1n((1)

0)2m(

2m)1m(

1m0)1n(

1n)n(

2


• Solution Decomposition:

y(t)=y(I.C., system)+y(system, input)

y(I.C., system)=yh(t)

I.C.-dependent solution

Homogeneous solution

Natural response

Zero-input response

y(system, input)=yp(t)

Forcing term dependent solution

Particular solution

Forced response

Zero-state response

3


• Solution Modes: Characteristic equation

Eigen value

Solution modes

0q...... λ qλ 01n

1nn

iβ±α=λ roots Complex

......=λ=λ roots repeated Real

1,2,......=j ,λ roots distinct Real

jjj

21

j

ii

i

tλk

λ eigenvalue ofty Multiplici :n

1n ......, 2, 1, ,0k

et i

0λ roots alRe j

4


exponentialdecay

t

Amplitude

exponentialgrowth

t

Amplitude

sinusoidal

t

Amplitude

modulated decaysinusoidal

t

Amplitude

modulated growthsinusoidal

t

Amplitude

0λ roots Real j 0λ j

constant

t

Amplitude

0λ j

iβλ rootsmaginary I jj

iβαλ rootsomplex C jjj

0α j 0α j

5


• Output Response: (1) y(t)=yh(t)+yp(t)

yh(t): Linear combination of solution modes

yp(t): Same pattern and characteristics as the forcing function

The RH side of LTI model affects only the coefficients of solution modes.

The LH side of LTI model dominates the solution modes of the transient

response.

(2) y(t)=ys(t)+yt(t)

yt(t): Transient solution

ys(t): Steady state solution

Transient solution is contributed by initial condition and forcing function.

0)t(ylim tt

6

§ 4.2 Transfer Function and Dynamic Systems (1)

Input is transfered through system G to output.

• Definition:

Key points: Linear, Time-Invariant, Zero initial condition

y(t)r(t)

InputSystem

Output

G

system. a of relaxation Initial

ignored. is condition initial of nInformatio :0I.C.

Transform Laplace L

input of Transform

response forced of Transform

trL

tyL

inputL

outputLsG

0CI0CI

t - Domain s - DomainL

L-1

7

§ 4.2 Transfer Function and Dynamic Systems (2)Pierre-Simon Laplace (1749 ~ 1827)

Monumental work “ Traite de mécanique céleste ”

8

§ 4.2 Transfer Function and Dynamic Systems (3) Laplace Transform• Definition:

Time function

• Existence Condition

• Inverse Laplace Transform

Signals that are physically realizable (causal) always has a Laplace transform.

0t f(t),

f(t)

0t ,0

st

0L[f(t)] F(s) f(t)e dt

s j

0σ for ,dte)t(f 1

0

tσ1

j1 st

j

1L [F(s)] F(s)e ds

2 j

0+0- t

f(t)

0

9


• Important Properties: t – Domain s – Domain Linearity Time shift

Scaling

Final value theorem

Initial value theorem

Convolution

Differentiation

Integration

1 2 1 2 af (t) bf (t) aF (s) bF (s)

a)F(s tfe -at t

f( ) aF(as)a

s f(0 ) lim sF(s)

t t

0 0f(t- )g( )d or f( )g(t )d F(s)G(s)

df(t) sF(s)-f(0)

dt

t t

0 0 t 0

F(s) 1 f( )d f( )d

s s

t s 0 lim f(t) lim sF(s)

10

§ 4.2 Transfer Function and Dynamic Systems (5)• Signal:

Unit impulse Unit step

Ramp

Exponential decay

Sine wave

Cosine wave

1 t

s

1 )t(U s

2s

1 t

as

1 0a ,e at-

20

20

0ωs

ω tωins

20

20ωs

s tωcos

F(s) 0t f(t), ≥

t0

t0

1

t0

1

t0

t0

t0

1

0ω

π2

0ω

π2

11


• Fundamental Transfer Function of Mechanical System:

Elements Function Block Diagram T.F. Example Static element(Proportional element)

Integral element

Differential element

Transportation lag

12


Flow(r) Quantity Effort(E) Flux Edt

E Electrical i Q e

M Mechanical v x F Impulse H Fluid q Q P T Thermal qt Qt T

Effort (E)

=f(r)=Lr

Flow (r)

Quantity (Q)

Q=f(E)C=Q/E

R=E/rE=f(r)

RC L

Flux( )s

1

s

1

• States and Constitutive Law of Physical Systems:

13


• Analog Physical Systems:

C

K M

y=x

f(t)

frictionless

0 ( )tdv

M cv k vdt f tdt

01

( )tdi

L Ri idt e tdt c

R L C

e(t)

i

14


• Inverse Laplace Transform and Partial Fraction Expansion:

Rb ,a n,m ,asa......sas

bsb......sbsb

)s(D

)s(N)s(F

ii01

1n1n

n01

1m1m

mm

Roots of D(s)=0:

(1) Real and distinct roots

From Laplace transform pairs

)]s(F)rs[(limc

rs

c......

rs

c

rs

c

)rs)......(rs)(rs(

)s(N)s(F

r...... ,r ,rs

irs

i

n

n

2

2

1

1

n21

n21

i

n

1i

tri

1 iec)]s(F[L)t(f

15

§ 4.2 Transfer Function and Dynamic Systems (10)(2) Real repeated roots

From Laplace transform pairs

(1) followingby obtained are ......d ,d

......k 2, 1,i )]},s(F)rs[(ds

d

)!ik(

1{limc

)]s(F)rs[(limc

)rs(

d ......

)rs(

d

rs

c......

)rs(

c

)rs(

c

)rs)......(rs()rs(

)s(N)s(F

kty multiplici ithw ,rs

1-n1

k1)ik(

ik

rsi

k1

rsk

n

1-n

2

1

1

11k

1

1kk

1

k

n2k

1

1

1

1

n

2i

tr1i

k

1i

tr1ii i1 edet)!1i(

c)t(f

16

§ 4.2 Transfer Function and Dynamic Systems (11)(3) Complex conjugate pairs with real distinct roots

From Laplace transform properties and pairs

20

20

221

i21

n

n

1

12

21

n12

22

ω)cs(

ωB)cs(A

bass

csc with

equation balancingby obtained are d ,c ,c

rs

d......

rs

d

bass

csc

)rs)......(rs)(bass(

)s(N)s(F

0b4a ,0bass

n

1i

tri0

ct0

ct iedtωsinBetωcosAe)t(f

17


• Dynamic System Equation and Transfer Function: Differential Equation and Transfer Function

Differential Equation: Transfer Function:

Problems associated with differentiation of noncontinuous functions, ex. step function, impulse function.

y(t)r(t)DifferentialEquation

I.C.

Y(s)R(s)G(s)

0r0r r(0),

0y0 y y(0),

: I.C.

rprprp

yq yqy

2m(1)

1n(1)

02m

2m1m

1m

01n

1nn

01n

1nn

02m

2m1m

1m

0CI

qsqs

pspsp

trL

tyL(s)G

18


Integral Equation and Transfer Function

The transfer function of a system is the Laplace transform of its impulse response

Y(s)R(s)G(s)

r(t)g(t)

y(t)

‧

* d)(r)t(g)t(y

t

0

1R(s) response, Impulse

expansion fraction partial usingby obtained is g(t) ),t(g)]s(G[L 1

(t)r(t) response, Impulse

g(t) y(t)

)s(GL[g(t)]0.C.I

G(s)Y(s)

19

§ 4.3 Transfer Function and System Response (1)

• Transfer Function G(s):

Rational T.F.

Irrational T.F.

Proper T.F.

Y(s)R(s)G(s)

s of spolynomial are N(s) D(s), , )s(D

)s(N)s(G

01n

1nn

02m

2m1m

1m

q......sqs

p......spsp)s(G

constant a is T ,eG(s) :Ex , )s(D

)s(N)s(G Ts-

n)(m

D(s) of Order N(s) of Order i.e. ,G(s) lims

20


• Response by T.F.:

Partial fraction expansion is employed to find y(t).

R(s)q(s)

p(s)

q(s)

m(s) Y(s)

q(s)

p(s)G(s) ,)s(R)s(G)s(Y

(t)y(t)y(t)y[Y(s)]L y(t)

d(s)

k(s) (s) Y,

q(s)

l(s) (s) Y,

q(s)

m(s)(s) Y

d(s)

k(s)

q(s)

l(s)

q(s)

m(s)

d(s)

n(s)

q(s)

p(s)

q(s)

m(s) Y(s)

d(s)

n(s))s(R If

3211

321

response Natural response Forced

response Transient response stateStedy

21


Ex:

3 2 ( ), r(t) is a unit step function

y(0) 1, (0) 0

By Laplace transform

y y y r t

y

2

2 2

System : [s Y(s) sy(0) y(0)] 3[sY(s) y(0)] 2Y(s) R(s)

1 Input : R(s)

ss 3 1

Y(s)s 3s 2 s(s 3s 2)

5 12 1 2 2 2 By partial fraction expansion :Y(s) ( ) ( ) s 1 s 2 s 1 s 2 s

]s2

1[L)]s(Y[L)t( y],

2s2

5

1s

2[L)]s(Y[L)t(y

]2s

1

1s

2[L)]s(Y[L)t(y:transform Laplace Inverse By

13

13

12

12

11

11

1 t 2t t 2t5 1 y(t) L [Y(s)] 2e e 2e e

2 21

Steady state response: 2

22

§ 4.3 Transfer Function and System Response (4)• Poles, Zeros, and Pole-zero Diagram:

For an irreducible proper rational transfer function G(s), a number (real or complex) is said to be

Pole-zero diagram Representation of poles and zeros distribution by using “x” and “o”, respectively in complex plane along with static gain.

Ex:

Ex:

Characteristic Equation i.e. characteristic roots: The roots of characteristic equation i.e. The poles of G(s).

o"" notation ,0)λG( if zero a

x"" notation ,)λ(G if pole a

-2s :pole-2s if ,)2(G2s

1

2s3s

1sG(s)

2

2

1 s 5 1 s 5G(s)

5 s s 1 5 1 3i 1 3i(s )(s )

2 2

1 3ipoles: s , zeros : s 52

0qsq......sqs ,0)s(D 011n

1nn

ωj

-2

Static gain = 1/2

ωj

-5

1

diagram zero-Pole

diagram zero-Pole

λ

23


• Impulse Response of Poles Distribution

ωj

σ

24


• Effects of Poles and Zeros

A pole of the input function generates the form of the forced response.

A pole of the transfer function generates the form of the natural response.

The zeros and poles of transfer function generate the amplitude for both the forced and natural responses.

The growth, decay, oscillation, and their modulations determined by the impulse response of the poles distribution.

25

y(t)r(t) 2s

1

1

1 -2t

Sol: Y(s) G(s) R(s)

1 1

s 2 s1 12 2 2 s

1 1y(t) [Y(s)] e

2 2

s

L

Ex: Find y(t)

pattern response Find :Ex

y(t))5s)(4s)(2s(

1s

1

Y(s)s

1

y (t)

t

1/21/21/2

tt

- =2t-e2

1

-2t -4t -5t1 2 3

-2t -4t -5t1 2 3

By inspection, solution modes are

k e , k e , k e

1 y(t) k e k e k e

40

ωj

-2

1/2

ωj

-5 -4 -2 -1

1/40

diagram zero-Pole

diagram zero-Pole

Response


response state-steady

26

§ 4.4 Block Diagram Operations for Complex Systems (1)

• Fundamental Operations: Signal operation

Summer Y(s)=X1(s)+X2(s)

Comparator Y(s)=X1(s)-X2(s)

Take-off point Y(s)=X1(s)

Component combinations

Serial

Parallel

Feedback

X Y

X Y

X Y

21 GG

21 GG

GH1

G

X1 Y

Y

X1 Y+

-

X2

X1 Y+

+

X2

X Y1G 2G

YX

1G

2G

X Y

GG

H27


Moving junction / sequence

Ahead of a block

Past a block

Exchange sequence

GZ1

Z2

X

GX

+

+

XG1 G2

XG

X

GX

G

+

1/G

+

G1G2

Z2

Z1

Z1

Z2

Z2

Z1

Z3 Z2Z2Z1

28


• Negative Feedback System:

GY(s)R(s) +

H

aE (s)

Y(s) G

R(s) 1 GH

H1 Y(s)R(s) +

GHH

1 E(s)

Y(s) 1 GH( )

R(s) H 1 GH

G :Gain Forward GH :Gain Forward

GH+1

G function transfer system loop-Closed

GH function transfer loop -Open :Note

True Error Signal: E(s)

H :Gain Feedback

GH :Gain Loop

aActuating Error Signal: E (s)

H

1 :Gain Cascade

GH :Gain Loop

29


• Loading Effect:

Cascade

C1

R1

V1(s) C2

R2

V4(s)

KK=1

Isolated Amp

C1

R1

V1(s) V2(s) C2

R2

V3(s) V4(s)

G1 G2

21

1 1 1

V (s) 1G (s)

V (s) R C s 1

4

23 2 2

V (s) 1G (s)

V (s) R C s 1

1s)CRCR(s)CRCR(

1)s(G)s(G

22112

221121

+-

K=1

Realization Isolated Amp by 741OP

30


• History of Operational Amplifier:

1965

Fairchild develops the first OpAmp (operational amplifier) generally used throughout the industry--a milestone in the linear integrated circuit field.

OP was first built with vacuum tubes. Originally designed by C. A. Lovell of Bell Lab. and was used to control the movement of artillery during World War .Ⅱ

1968

Fairchild introduces an OpAmp (operational amplifier) that is one of the earliest linear integrated circuits to include temperature compensation and MOS capacitors.

31

§ 4.4 Block Diagram Operations for Complex Systems (6)• Operational Amplifier:

+

-

(+) Supply voltage,Vcc (usually DC 15V)

(-) Supply voltage, Vee(usually DC -15V)

Output

Invertinginput

inputNoninverting

741 A

Offsetnull adjust

2

3

4

7

6

1 5

iZ

oZv

A

vA

iZ

oZ

ideal

0

1

2

3

4 5

6

7

8

ln-

ln+ Out

Vcc-

Vcc+

Offset null adjust

Offset null adjust NC

32


• Network 1:

• Network 2:

C1

R1

V1(s) V4(s)

R2

C2I(s)

V(s)

V2(s)C1

R1

V3(s)

R2

C2I(s)

V(s)

V1(s) V4(s)V(s)

I(s)

V3(s) V2(s)V(s)

I(s)

1s)CRCRCR(s)CRCR(

1)s(G

2122112

2211

1

⇒

Loading effect

)s(G)s(G)s(G)s(G 221

1

1+s)CR+CR+CR(+s)CRCR(

1=)s(G

1222112

2211

2⇒

Loading effect

33


)s(G

)s(R1

)s(R2

1Y (s)

2Y (s)

Note: For MIMO System

Y(s) G(s)R(s)

Output

Vector

Transfer

Matrix

Input

Vector

1 11 12 1

2 21 22 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Y s G s G s R s

Y s G s G s R s

G11

G22

G 12G

21

+

+

+

+

R1(s)

R2(s) Y2(s)

Y1(s)

34

Ex: Armature control DC servomotor

Static characteristics (Ideal)

La

ia(t)

if+

eb(t)

Ra

va(t) M

-

=const.

,Ttm(t) , Jm, Bm,m(t) m(t)

or Permanent magnet

Ttd(t)

tT

max

maxv

tmaxT (Stall)

tT BT K v K

t

tmax

T maxmax 0

max Bb

max T maxT 0

tmax

i a T maxmax

T K (Stall)

v

v K K (No load)

K

T K R K

i


35


I/O Block Diagram Reduction

t v T

m i

a a a m m b i

m m m

(s) K

V (s) s[(R L s)(J s B ) K K ]

(s) (s) (s)

Dynamic characteristics

Total

Response

Command

Response

Disturbance

Response

BmRa

Kb

Ki

+

-

+

-a

1

L s m

1

J s1

s

m(s)ai m(s)

Electronics Mechatronics Mechanics

tdT (s) Load disturbance

aV (s) -

36

S1

1

m S

1 m(s)m(s)aV (s)Km


iK

bK

S

1

SJ

1

maR

1 m(s) m(s)+

-

aV (s)

0B ,0L )2( ma

T

J=

KK

J= ,

KK

JR=

Κ

1=K

max

mmax

maxbT

m

maxmbi

mam

bm

im

a m i b

a mm

a m i b

m m

a m

K Define K , Motor gain constant

R B K K

R J , Motor time constant

R B K K

(s) K

V (s) s(1 s)

0L (1) a Model Reduction

37


0 ,0B ,0L If )3( mma

S1 m(s)m(s)

KmaV (s)

Servomotor Ideal

S1 m(s)m(s)

Integrator Pure

Static gain is dominated by feedback gain Kb=1/Km in system dynamics.

Key points: Linear time-invariant motor

No load

No delay

No damping

No inertia

No resistance

No inductance38

Gaintatic S

chapter 4 transfer function and block diagram operations § 4.1 linear time-invariant systems § 4.2...

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