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Chapter 33.6 (Page 215) System Sensitivity

3.6.1 (page 215) Calculating the Effects of

Change in Parameter

3.6.2 (page 216-220) Sensitivity Function

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Control systems System should be

stable

insensitive to system parameter values

able to reject disturbances

able to track reference inputs

Now

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Introduction - System Sensitivity

One of the main advantages of feedback system is that it can be

used to make the response of a system relatively independent ofcertain types of changes or inaccuracies in the plant model

Characteristics of a physical plant possibly change with time and

with environmental conditions such as temperature, humidity, andaltitude. We would like that the closed-loop physical systems

characteristics to be insensitive to these changes. Hence in this

section we are interested in the sensitivity of the closed-loop

system to parameter changes in the system.

Robustness: The closed-loop system characteristics are to be

insensitive to small inaccuracies in the system model.

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Amplifier

100+_

R(s) Y(s)

Output

Sensor 1

2

4

s

Plant

(a)

100+_R(s) Y(s)

1

kis

Plant

2

4

(b)

Figure 3.14 (page 215) Determine the effects of changes in the parameters of afeedback system. (a) The nominal system (b) Perturbation of plant pole

402

400

24001

2400

s)s/(

)s/()s(T

3.6.1 (page 215) Calculating the Effects of Change in Parameter

ii

i

ks)ks/(

)ks/()s(T

2400

400

24001

2400

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R(s)100+_

Y(s)

k3

2

4

s

Plant

(d)

100+_R(s) Y(s)

1

2

42

s

k

Plant

(c)

Figure 3.14 Determine the effects of changes in the parameters of a feedback system.(c) Perturbation of plant gain (d) Perturbation of the sensor gain

2400

400

24001

2400

2

2

2

2

ks

k

)s/(k

)s/(k)s(T

2400

400

24001

2400

33

ks)s/(k

)s/()s(T

Cont Calculating the Effects of Change in Parameter

K3 is quite sensitive as

compare to the above

variations

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3.6.2 (page 216) Sensitivity Function

Sensitivity:

Characteristics changes with system parameter variation is calledSensitivity

a/a

T/TSFunctionySensitivit Ta

a

T

T

alimSFunctionySensitivita

Ta

0

aT

TaSFunctionySensitivit Ta

polebecouldorS,Sbecoulda T )s(HTGP

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Sensitivity: of Transfer function to change in GP

HGG

GG

)s(R

)s(C)s(T

pC

pC

1

T

G

G

TSFunctionySensitivit P

P

TGP

2211

1)HGG(

G)HGG(

)HG(GGG)HGG(GT

PC

C

PC

CPCCPC

P

C

PC

PCPC

PP

G

)HGG(

HGG

GG

G

T

G

1

1

HGGPC

1

1

T

G

G

TS P

P

TGP

21 )HGG(

G

PC

C

C

PC

G

)HGG( 1

2D

NDDN //

100+_R(s) Y(s)

1

2

42

s

k

Plant

(c)

100+_R(s) Y(s)

1

2

42

s

k

Plant

100+_R(s) Y(s)

1

2

42

s

k

Plant

(c)

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10

30

3101

10

1

/GH

G)s(T

T

G

G

TSFunctionySensitivit TG

GH1

1

Example from Book: Sensitivity: of TF to change in GP

GHST

G 1

1

13

3

3

101

1

G=10 and H=1/3

Transfer Function (T(S)) with feedback changes only

3/13 with small changes in G as it would be without

feedback. H>0 means SG

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Sensitivity: of Transfer function to change in H

HGG

GG

)s(R

)s(C)s(T

pC

pC

1

T

H

H

TSFunctionySensitivit TH

21 )HGG(

)GG(GGHT

PC

PCPC

PC

PC

PC

PC GG

)HGG(H

HGG

GG

H

T

H

1

1

HGG

HGG

PC

PC

1

T

H

H

TSTH

21 )HGG(

)GG(GG

PC

PCPC

PC

PC

GG

)HGG(H 1

R(s)100+_

Y(s)

k3

2

4

s

Plant

(d)

R(s)100+_

Y(s)

k3

2

4

s

Plant

R(s)100+_

Y(s)

k3

2

4

s

Plant

(d)

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13

30

3101

10

1

/GH

G)s(T

T

H

H

T

SFunctionySensitivit

T

H

GH

GH

1

Example from Book: Sensitivity: of TF to change in H

GH

GHSTH

13

10

3

101

310

/

G=10 and H=1/3

As H increases TF decreases due to ive sign. TF is

effected by changes in H much more than by change in

G. Loop gain must be small i,.e choose high quality sensors

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Sensitivity: of Transfer function to change in Pole (a)

)as/(K

)as/(

)s(R

)s(C)s(T

1

1

T

a

a

TSFunctionySensitivit Ta

2

1

)Kas(aT

)Kas(a

)Kas(

a

T

a

1

Kas

a

T

a

a

TSTa

2

1

)Kas(

)Kas(s

100+_R(s) Y(s)

1

kis

Plant

2

4

(b)

100+_R(s) Y(s)

1

kis

Plant

2

4

100+_R(s) Y(s)

1

kis

Plant

2

4

(b)

Kas

1

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)s/(K

)s/(

GH

G)s(T

21

21

1

T

a

a

T

SFunctionySensitivit

T

a

)Kas(

a

Example from Book: Sensitivity: of TF to change in a

)Kas(

aSTa

Ks

2

2

)as/(K

)as/(

)s(R

)s(C)s(T

1

1

a=2

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