8/6/2019 FOC Lect5 Feedback

1/16

FUNDAMENTALS OF

CONTROL SYSTEMS

LECTURE 5:FEEDBACK CONTROL SYSTEM

CHARACTERISTICS

8/6/2019 FOC Lect5 Feedback

2/16

OPEN-AND CLOSED-LOOPCONTROL SYSTEMS

An open-loop system operates withoutfeedback and directly generates the output inresponse to an input signal.

c ose - oop sys em uses a measuremen othe output signal and a comparison with thedesired output to generate an error signal that

is applied to the actuator.

8/6/2019 FOC Lect5 Feedback

3/16

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

Open-loop controlsystem

R(s) + Ea(s)

G(s)

H(s)

C(s)

-

systems = s s

)()()(1

1)(

)()()(1

)()(

)]()()()[()()()(

sRsHsG

sE

sRsHsG

sGsC

sCsHsRsGsEsGsC

a

a

+=

+

=

==

8/6/2019 FOC Lect5 Feedback

4/16

Sensitivity toParameter Variations

A process, represented by G(s), is subject to a changing

environment, aging, ignorance of the exact values of the processparameters. In the open-loop system, all these errors and changesresult in a changing and inaccurate output.

However, a closed-loop system senses the changes in the outputue o process c anges an a emp s o correc e ou pu .

Consider a change in the process as: )()( sGsG +

In the open-loop case, the change in theoutput is: )()()( sRsGsC =

In the closed-loopsystem:

)())()(1))(()()()(1(

)()(

)()())()((1

)()()()(

sRsHsGsHsGsHsG

sGsC

sRsHsGsG

sGsGsCsC

+++

=

++

+=+

8/6/2019 FOC Lect5 Feedback

5/16

)())()(1(

)()(

),()()()(If

2sR

sHsG

sGsC

sHsGssG

+

=

The change in the output of theclosed-loop system is reduced by afactor [1+G(s)H(s)]2, which isgreater than one over the range offre uencies of interest.

System Sensitivity: Ratio of the percentage change inthe system transfer function to the percentage change ofthe process transfer function.

)()(

)()(

)(

)(

)(sG

sG

sTsT

SsR

sC

sT

==

In the limit, for

small incrementalchanges

T

G

G

T

GG

TT

S

=

=

Clearly, sensitivity of the open-loop system is equalto one.

8/6/2019 FOC Lect5 Feedback

6/16

GHGH

G

G

GHT

G

G

TS

GH

GsT T

G+

=

+

+

=

=

+=

1

1

)1()1(

1

1)(

2

Result: Sensitivity of the system may be reduced belowthat of the open-loop system by increasing G(s)H(s) over

the frequency range of interest.The sensitivity of the feedback system to changes inthe feedback element H(s) is

GH

GH

GHG

H

GH

G

T

H

H

TSTH

+

=

+

+

=

=

1)1(1

2

When GH is large, the sensitivity approaches unity and thechanges in H(s) directly affect the output. Therefore it isimportant to use feedback components that will not vary withenvironmental changes.

8/6/2019 FOC Lect5 Feedback

7/16

DISTURBANCE SIGNALS IN A

FEEDBACK CONTROL SYSTEM

A disturbance signal is an unwanted input signal that

affects the systems output signal. As a steel barapproaches the rolls, the rolls turn unloaded. However,when the bar engages with the rolls, the load on the rollsincreases immediately to a large value. This loading effectcan e approx ma e y a s ep c ange o s ur ancetorque.

Steel bar

Rolls

Conveyor

1Ra

Ia(s)Km

Tm(s)

+

Td(s)

TL(s) 1Js+B

Kb

Va(s)

+ Speed

W(s)

Motor back emf

Open-loop speed control system

8/6/2019 FOC Lect5 Feedback

8/16

The error for the system is E(s)= R(s)-w(s) and

R(s)=wd(s). For simplicity in calculation, let R(s)=0, andexamine E(s)=-w(s).The change in speed due to the loaddisturbance is

( ))(

1)()( sT

RKKBJsssE d

abm

++==

( ) )()(lim)(lim)( 0 =+=== oabmstd

RKKB

DssEtEs

DsT

8/6/2019 FOC Lect5 Feedback

9/16

KmRa

Tm(s)+

Td(s)

TL(s) 1Js+B

Kb

Va(s)+ SpeedW(s)

+Ka

Amplifie

r

Closed-loop speed controlsystem

Kt

Tachometer

( )abta

ma

d

KKKsHBJs

sGR

KKsG

sTsHsGsG

sGssE

+=+

==

+==

)(,1

)(,)(

)()()()(1

)()()(

21

21

2

8/6/2019 FOC Lect5 Feedback

10/16

( )( ) sD

KKKRKBJss

sRs

DsT

batam

d

+++

=

==

1)(

0)(and)(If

( )( )D

KKKRKBsst

batamst

++==

)(lim)(lim0

When the amplifier gain Ka is sufficientlyhigh,

tma

bmac

c

tma

a

KKK

KKBR

DKKK

R

+=

=

)(

)(

)()(

0

8/6/2019 FOC Lect5 Feedback

11/16

The Cost of Feedback

Increased number of components and complexity.

Loss of gain: Open-loop gain is G(s) and is reduced toG(s)/(1+G(s)) in a unity negative feedback system. We may needan amplifier to increase the gain of the feedforward path.

. - ,

the closed-loop system may not be always stable.

8/6/2019 FOC Lect5 Feedback

12/16

EXAMPLE

R(s) + E(s) K

D(s)

+

+ )12(

1

+ssC(s)

Plot the unit-step input

response and response tounit-step disturbance ofthe system

)(12

1)(

12

)()(1

)()(

)(1

)(

)()()()()(

22sD

KsssR

Kss

K

sDsKG

sGsR

sKG

sKG

sDsTsRsTsC d

+++

++=

++

+=

+=

ii)K=100.

Compare the results.

8/6/2019 FOC Lect5 Feedback

13/16

Disturbance response for both K values is acceptable, butstep-input response has less overshoot for K=50 and thesettling time is one second. For K=100, overshoot is

greater and settling time is about 0.65 seconds.

8/6/2019 FOC Lect5 Feedback

14/16

54

)3)(1(2 ++

++

ss

ssK)3)(1(

1

++ ss

D(s)

+

+ Compare the open-loop and closed-loopcontrol systems fori) sensitivity relative

C(s)

- o n -s epdisturbance

Rover

)3)(1(

1

++ ss+

++ C(s)

Closed-loop control of MarsRover

D(s)

R(s)K

T sK

s s

T s K s s K

o

c

( )

( )

=+ +

=+ + +

2

2

4 5

4 3

8/6/2019 FOC Lect5 Feedback

15/16

For K=2, T s T s T s

s so c( ) ( ) ( )= = =

+ +

2

4 52

Hence we can compare the sensitivity of the open-loop and closed-loop systems for the same transferfunction.

S dTdK

KT

SdT

dK

K

T

s s

s s K

KT o

o

K

T c

c

c

0 1

4 3

4 3

2

2

= =

= =+ +

+ + +

To examine the effect of thesensitivity at low frequencies,let s=jw

Sj

K jK

Tc = +

+ +

( )

( )

3 4

3 4

2

2

For K=2, the sensitivity at low frequencies, w

8/6/2019 FOC Lect5 Feedback

16/16

The effect of disturbance can be determined by setting

R(s)=0 and letting D(s)=1/s. For the open-loopsystem, the steady-state value is

y s s s ss

( ) lim( )( )

=+ +

=0

1

1 3

1 1

3

y s s s K s K s

( ) lim( )

=+ + +

=+0 2

1

4 3

1 1

3

The output of the closed-loop system with a unit stepdisturbance is

ForK=2,

y( ) = 15