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Transfer FunctionsTransfer FunctionsTime Domain Laplace DomainTime Domain Laplace Domain

U(s)G(s)

Y(s)u(t)System

y(t)

System model = linear ODE System model = transfer function

G(s)y

Sys e ode e O Sys e ode s e u c o

G(s) Y(s)U(s)

=

Assuming all initial conditions are zero

Transfer Function of a Standard 1st Order System

Assumptions

y( )0 0=T dydt

y Ku+ = y( )0 0=dt

( ) ( )( ) ( ) ( )T sY s y Y s KU s− + =0Solution ( ) ( )( ) ( ) ( )

( ) ( ) ( )

T sY s y Y s KU s

T Y KU

− + =0

1( ) ( ) ( )Ts Y s KU s+ =1

KsT+1

U(s) Y(s)

Transfer Function of a Gear Box

θ θo in=

Solution

( ) ( ) ( )( )

Θ ΘΘΘo i

os n s Y sU

s n= ⇒ = =( )( )

Solution

( )ΘiU s s( )

A gear box is an example of a system that can be modelled by a simple GAIN term

θ i θon

i o

Transfer Function of an Integratorg

y udt= ∫∫u y

Solution

( ) ( )Y ss

U s G s Y sU s s

= ⇒ = =1 1( ) ( )

( )

1 Y(s)U(s) 1s

Y(s)U(s)

Lecture 5 - Block DiagramLecture 5 Block Diagram Manipulation

• Introduction of block manipulation.S i R l– Series Rule

– Closed Loop Rule– Blocks around summing junctions

Block Diagram Manipulation

• The idea is to use a set of rules to gradually reduce a complex system consisting of a p y gnumber of subsystems down to a single block that contains the transfer function for the whole system.

Blocks in Series

GA(s) GB(s)A( ) B( )

GA(s)GB(s)

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA(s) GB(s)U(s) Y(s)X(s)

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA(s) GB(s)U(s) Y(s)X(s)

Y(s)= GB(s)X(s)

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA(s) GB(s)U(s) Y(s)X(s)

X(s)= G (s)U(s)

Y(s)= GB(s)X(s)

X(s)= GA(s)U(s)

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA(s) GB(s)U(s) Y(s)X(s)

X(s)= G (s)U(s)

Y(s)= GB(s)X(s)Y(s)= GB(s) GA(s)U(s)

X(s)= GA(s)U(s)

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA(s) GB(s)U(s) Y(s)X(s)

X(s)= G (s)U(s)

Y(s)= GB(s)X(s)Y(s)= GB(s) GA(s)U(s)

X(s)= GA(s)U(s)

Y s( )Y sU s

G s G sB A( )( )

( ) ( )=

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA(s) GB(s)U(s) Y(s)X(s)

X(s)= G (s)U(s)

Y(s)= GB(s)X(s)Y(s)= GB(s) GA(s)U(s)

X(s)= GA(s)U(s)

U( ) Y(s)U(s) Y(s)GB(s) GA(s)

Proof for Blocks in Series

U(s) Y(s)X(s)

Proof for Blocks in Series

GA (s) GB (s)U(s) Y(s)X(s)

X(s)= G (s)U(s)

Y(s)= GB (s)X(s)Y(s)= GB (s) GA (s)U(s)

X(s)= GA (s)U(s)

U( ) Y(s)U(s) Y(s)GA (s) GB (s)

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1

q1

Tank 1 q12

Tank 2 q2

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1 q12q12

12dqq1 12 1 1

12 = q + A Rdqdt

NB Standard 1st order TF,K=1 and T =A1R1 (sec)

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1 q12q12

12dq

K q12q1

q1 12 1 112 = q + A R

dqdt

K1+Ts

q121NB Standard 1st order TF,K=1 and T =A1R1 (sec)

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1 q12q12

12dq

1 q12q1

q1 12 1 112 = q + A R

dqdt

11+T1s

q121NB Standard 1st order TF,K=1 and T =A1R1 (sec)

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1 q12q12

Tank 2 q

1 1q2q12q1

q2

11+T1s

11+T2 s

q12q1

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1 q12q12

Tank 2 q

1 q2q1

q2

111+T1s

q1 11+T2s

×

Two Water Tanks in Series (Problem 1 Q3a)

q1

( Q )

Tank 1 q12q12

Tank 2 q

1 q2q1

q2

1(1+T1s)(1+T2s)

q1

Closed Loop

G(s)+

-H(s)

G(s) 1 + G(s)H(s)

G( )+U(s) Y(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

H(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

H(s)X(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

Y(s) = G(s) Z(s)

H(s)X(s)

Y(s) = G(s) Z(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

Y(s) = G(s) Z(s)

H(s)X(s)

Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

Y(s) = G(s) Z(s)

H(s)X(s)

Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

Y(s) = G(s) Z(s)

H(s)X(s)

Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s) {U(s) - X(s)}

Hence;

(s) G(s) {U(s) (s)}

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

Y(s) = G(s) Z(s)

H(s)X(s)

Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s) {U(s) - X(s)}

Hence;

(s) G(s) {U(s) (s)}

X(s) = H(s)Y(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

Y(s) = G(s) Z(s)

H(s)X(s)

Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s) {U(s) - X(s)}

Hence;

(s) G(s) {U(s) (s)}

X(s) = H(s)Y(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

H(s)X(s)

Y(s) = G(s) Z(s)Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s)U(s) - G(s)H(s)Y(s)

Y(s) = G(s) {U(s) - X(s)}

Hence;

(s) G(s) {U(s) (s)}

X(s) = H(s)Y(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

H(s)X(s)

Y(s) = G(s) Z(s)Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s)U(s) - G(s)H(s)Y(s)

{1 + G( )H( )}Y( ) G( )U( )

Y(s) = G(s) {U(s) - X(s)}

{1 + G(s)H(s)}Y(s) = G(s)U(s)Hence;

(s) G(s) {U(s) (s)}

X(s) = H(s)Y(s)

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

H(s)X(s)

Y(s) = G(s) Z(s)Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s)U(s) - G(s)H(s)Y(s)

{1 + G( )H( )}Y( ) G( )U( )

Y(s) = G(s) {U(s) - X(s)}

{1 + G(s)H(s)}Y(s) = G(s)U(s)Hence;

G( )Y s( )(s) G(s) {U(s) (s)}

X(s) = H(s)Y(s)G(s)

1 + G(s)H(s)Y sU s

( )( )

=

G( )+U(s) Y(s)Z(s)

Proof for G(s)-

H(s)

Proof forClosed Loop Rule

H(s)X(s)

Y(s) = G(s) Z(s)Y(s) = G(s) Z(s)

Z(s) = U(s) - X(s)

Y(s) = G(s)U(s) - G(s)H(s)Y(s)

{1 + G( )H( )}Y( ) G( )U( )

Y(s) = G(s) {U(s) - X(s)}

{1 + G(s)H(s)}Y(s) = G(s)U(s)Hence;

G( )(s) G(s) {U(s) (s)}

X(s) = H(s)Y(s)G(s)

1 + G(s)H(s)

Flow rate Controlqd

Gear box MotorC llController

Tank

qi

Flow rate sensorTapTank qo

Flow rate Control

+ 1qi qoqd

-

+10

11+100s

qi qoqd

Controller + motor + gearbox + tap

Flow rate Control

qi+ 1 qoqdqi

-

+10 1

1+100s

qoqd

+?

qd qo

-

Flow rate Control

+ 1qi qoqd

-

+10

11+100s

qi qoqd

+ 101+100s

qd qo

-

Flow rate Control

+ 1qi qoqd

-

+10

11+100s

qi qoqd

+ 101+100s

qd qo

-

Closed Loop

G(s)+

-H(s)

G(s) 1 + G(s)H(s)

Flow rate ControlG(s)

+qd q

G(s)

-

+ 101+100s

qd qo

G(s)1 G( ) ( )

10

1 + G(s)H(s)

101+100sqd qo

H(s) = 1

1 + 101+100s

Flow rate Control

101+100s

1 + 10

qd qo

1 + 101+100s

10(1+100 )10(1+100s)1+100sqd qo

1(1+100s) + 10(1+100s)1+100s

F l o w r a t e C o n t r o l

+ 1qd qo

-

+10

11+100s

qd qo

1011 + 100s

qdqo

11 + 100s

Moving a block round a summerMoving a block round a summer++

G(s)-

G(s)- =

G(s)

G(s)+

= G(s)+

- -1/G(s)

Satellite Tracking AntennaSatellite Tracking Antenna

τ α ω= +J Bτ α ωω

ω

= +

= +

J B

J ddt

B

Antenna

Kτ ωK1 + sT

τ

Torque Angularvelocity

where T = J/B, and K = 1/B

K 1KK+ θθd

Motor + Gears Antenna Integratorresolver

1 + sT sKmgKr

-

Kr

resolver

desiredangle

resolver

K 1KK+ θθd

Motor + Gears Antenna Integratorresolver

1 + sT sKmgKr

-

Kr

resolverresolver

K 1KK+ θθd

Motor + Gears Antenna Integratorresolver

1 + sT sKmgKr

-

Kr

resolver

K 1K+ θθd K

resolver

1 + sT sKmg

-

1

Kr

KrKr

1

K 1K+ θθd K

1 + sT sKmg

-

Kr

K 1K+ θθd K

1 + sT sKmg

-

Kr

K K K θθd + Kr Kmg K

(1 + sT)s

d +

-

K K K θθd + Kr Kmg K

(1 + sT)s

+

-

θθ Kr Kmg K

(1 + sT)s + Kr Kmg K

θθd

K K K θθd Kr Kmg K

Ts2 + s + Kr Kmg K

Summary of Lecture 5Summary of Lecture 5• Block manipulation involves reducing a

d b hsystem made up from many subsystems each with their own transfer function down to a single transfer function for the whole systemsingle transfer function for the whole system.

• Use the rules on the formula sheet working from the middle outwardsfrom the middle outwards.

• The open loop transfer function is what you get if there were a switch in the feedbackget if there were a switch in the feedback path that is OPEN.