Preparation for Final_Exam Solutions SS2011

8/22/2019 Preparation for Final_Exam Solutions SS2011

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Hochschule Ulm Prof. Dr. A. Beckmann

University of Applied Sciences Control Technology

Page 1

Preparation for Final Exam - Solutions

Describe using your own words:

1. How can we assess whether the closed loop control system is stable or not?(2)

If the plant is stable we can use the simple Nyquist criterion:

We draw the frequency response locus of the open loop. If the critical point is always on the

left side of the frequency response locus of the open loop then the closed loop is stable.

2. What is the yquist plot? (4)

In a Nyquist plot we plot the imaginary part of the frequency response as a function of its real

part. If the critical point is always on the left side of the plot, the control system is stable.

3. Describe the behaviour of the different parts of a PID-controller. (6)

a) P-controller:

The P-controller with gain KP,c is amplifying the error signal by a factor ofKP,c. This

amplified signal is the input of the process. If the chosen gain is too high, the controlsystem may become unstable.

b) I-controller:

The I-controller with time constant Ti,c integrates the error signal of a time interval of

length Ti,c. In this way the controller will respond to small deviations of the control

systems output to the reference signal.

c) D-controller:

The D-controller with time constant Ti,c will compute the derivative of the error signal

with respect of time. In this way the controller will respond to fast changes of the error

signal, e.g. if a step input is applied or in case of disturbancies.


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4. The transfer functions of 2 dynamical systems are:

93

2)(1+

=s

ssG and55

3)(2+

=s

sG .

a) What type of element corresponds to system 1 and to system 2? (2)

system 1 : DT1-element; system 2: PT1-element

b) What is the transfer function, if the 2 elements are connected in series? Is this systemstable? (4)

55

3

93

2)()()( 21

+

+==

ss

ssGsGsGs

1

3

3

2

15

1

+

+=

ss

s

2 poles: p1 = -3 ; p2 = -1 both have negative real part => stable

c) What is the transfer function, if the 2 elements are connected in parallel? Is the

system stable?

55

3

93

2)()()( 21

++

+=+=

ss

ssGsGsGs

1

3

5

1

3

2

3

1

++

+=

ss

s

( ) ( )( )( )31

39110151

++

+++=

sssss

( )( )312791010

15

1 2

++

+++=

ss

sss

( )( )31271910

15

1 2

++

++=

ss

ss

2 poles: p1 = -3 ; p2 = -1 both have negative real part => stable


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5. A PT2-plant with( )( ) ( )( )1715

1

11

1)(

21 ++=

++=

sssTsTsGP and an I-controller

ssTsG

c

C12

11)(

,1

=

= build up a unity negative feedback system.

a) Calculate the spectral magnitude characteristics and the spectral phasecharacteristics for the open loop system.

Transfer function:( )( ) sss

sGsGsG CPOL12

1

1715

1)()()(

++==

( ) ssssGOL

12

1

11235

1)(

2

++

=

ssssGOL

12144420

1)(

23++

=

Frequency Response:

( ) ( )3223 420121441

12144420

1)(

+=

+=

iiiiGOL

Trick 3: idcG +=

1

;

1

22 dcG +=

c

d

=tan

( ) ( )2322 42012144

1)(

+

=iGOL

36

1053

144

42012)(tan

2

2

3

+=

=

b) Write down the transfer function of the closed loop control system and its systemparameters. (4)

112144420

1

12144420

12144420

12144420

11

12144420

1

)(2323

23

23

23

+++=

++

++

+++

++=

ssssss

sss

sss

ssssGcl

System parameters: ;1;0;0;0;1;1;144;420 01230123 ======== bbbbaaaa


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Page 4


6. A plant with the transfer function( )36

3)(

+=

ssGP shall be controlled. The plant and

the controller build up a unity negative feedback system. Check, if the transfer

function of the closed loop control system will fulfill the requirement 1)0( =sGcl ,

a) if we use a P-controller: transfer function of a P-controller: CPCP KsG ,, )( =

( ) ( )121

36

3)()()( ,,

+=

+==

sK

sKsGsGsG CPCPPCo

( )

( )

( )( )12

12

12

1

1

12

1

)(1

)()(

,

,

+

+

++

+

=+

=s

s

sK

sK

sG

sGsG

CP

CP

o

ocl

CP

CP

clKs

KsG

,

,

12)(

++=

CP

CP

CP

CP

clK

K

K

KsG

,

,

,

,

110)0(

+=

++==

The requirement cannot be fulfilled with a finite gain of the controller. We will

have a constant control error in steady state. Therefore a P-controller is not a

good choice for this plant.

b) if we use an I-controller: transfer function I-controller:sT

sGCI

CI

=

,

,

1)(

( ) ( )1211

36

31)()()(

,, +

=

+

==

ssTssTsGsGsG

CICI

PCo

( )

( )

( )

( )12

12

12

11

1

12

11

)(1

)()(

,

,

,

,

+

+

+

+

+

=

+=

ssT

ssT

ssT

ssT

sG

sGsG

CI

CI

CI

CI

o

ocl

( ) 112

1)(

, ++=

ssTsG

CI

cl

12

1)(

,

2

, ++=

sTsTsG

CICI

cl

11002

1)0(

,,

=++

==

CICI

clTT

sG The requirement is fulfilled.


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7. The transfer function of a plant is:3

22)(

+

+=

s

ssGP . It is controlled by a controller

withs

sGC6

)( = . How can we find out whether the unity feedback system is stable or

not.

First we have to check whether the plant is stable itself:

The transfer function of the plant is:3

22)(

+=

s

ssGP

This transfer function has 1 pole: p1 = s = -3 pole has negative real part

This plant is stable.

Therefore we can apply the Simple Nyquist criterion:

we have to check if the critical point is on the left of the frequency response locus. If this

is true, the closed loop will be stable even if the open loop system is not stable.

8. In the following diagram we see the frequency response locus of an open loop controlsystem. Is the closed loop control system stable?

a) Im b) Im

-2 -1 1 2 Re -1 -0,5 0,5 1 Re

stable x not stable x stable not stable


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Bode Diagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

-40

-20

0

20

40

60

10-1

100

101

-225

-180

-135

-90

Bode Diagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

-30

-20

-10

0

10

10-1

100

101

-90

-45

0

45

90

135

180

225

270

9. Find out from the Bode plots the gain margin and the phase margin of each controlsystem.

a)

Gain margin: - 10 dB Phase margin: - 40

Is the system stable? No

b)

Gain margin: + 25 dB Phase margin: + 90

Is the system stable? Yes

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Preparation for Final_Exam Solutions SS2011