Class11_-_2011

8/12/2019 Class11_-_2011

1/28

8/12/2019 Class11_-_2011

2/28

UNIVERSIDAD POLITECNICA SALESIANA

CHAPTER 4The Frequency Respond Design Method

4.1 Introduction.4.2 Frequency response.4.3 Neutral stability.4.4 The Nyquist stability criterion.4.5 Stability margins.4.6 Bodes Gain-Phase relationship.4.7 Closed-Loop frequency response.4.8 Compensation.4.9 Nichols charts.4.10 Specifications in terms of the sensitivity functions.

8/12/2019 Class11_-_2011

3/28


4.7 Closed-Loop frequency response.

2 2

0.5( ) ( )

K G s G s

s s s s

-100

-50

0

50

100

M a g n

i t u d e

( d B )

10-2

10-1

100

101

102

-180

-90

P h a s e

( d e g

)

Bode DiagramGm = Inf dB (at Inf rad /sec ) , Pm = 65.5 deg (at 0.455 rad/s ec)

Frequency (rad/sec)

10-2

10-1

100

101

102

-180

-90

0

P h a s e

( d e g

)

Bode Diagram

Frequency (rad/sec)

-100

-50

0

System: sys1Frequency (rad/sec): 0.704

Magnitude (dB): -2.98

System: sys1Frequency (rad/sec): 0.01

Magnitude (dB): -5.92e-007

M a g n

i t u d e

( d B )

0.455[ / ] g rad seg

0.704[ / ] BW rad seg

0[ / ] 0[ ]r rad seg Mr db

8/12/2019 Class11_-_2011

4/28



2 2

1( ) ( )

K G s G s

s s s s

0.786[ / ] g rad seg

1.27[ / ] BW rad seg

-100

-50

0

50

M a g n

i t u d e

( d B )

10-2

10-1

100

101

102

-180

-135

-90

P h a s e

( d e g

)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 51.8 deg (at 0.786 rad/sec)

Frequency (rad/sec)

-100

-50

0

50

System: sys2Frequency (rad/sec): 1.27Magnitude (dB): -3.03

System: sys2Frequency (rad/sec): 0.682Magnitude (dB): 1.23

M a g n

i t u d e

( d B )

10-2

10-1

100

101

102

-180

-90

0

P h a s e

( d e g

)

Bode Diagram

Frequency (rad/sec)

0.682[ / ] 1.23[ ]r rad seg Mr db

8/12/2019 Class11_-_2011

5/28



2 2

5( ) ( )

K G s G s

s s s s

2.13[ / ] g rad seg


2.12[ / ] 7.21[ ]r rad seg Mr db

-100

-50

0

50

100

M a g n

i t u d e

( d B )

10-2

10-1

100

101

102

-180

-135

-90

P h a s e

( d e g

)


Frequency (rad/sec)

10-1

100

101

102

-180

-90

0

P h a s e

( d e g

)

Bode Diagram

Frequency (rad/sec)

-100

-50

0

50



M a g n

i t u d e

( d B )

8/12/2019 Class11_-_2011

6/28



2 2

15( ) ( )

K G s G s

s s s s

3.81[ / ] g rad seg


3.81[ / ] 11.58[ ]r rad seg Mr db

-50

0

50

100

M a g n

i t u d e

( d B )

10-2

10-1

100

101

102

-180

-135

-90

P h a s e

( d e g

)


Frequency (rad/sec)

10-1

100

101

102

-180

-90

0

P h a s e

( d e g

)

Bode Diagram

Frequency (rad/sec)

-60

-40

-20

0

20



M a g n

i t u d e

( d B )

8/12/2019 Class11_-_2011

7/28

UNIVERSIDAD POLITECNICA SALESIANA4.8 Compensation.

N

j j

M

ii

c

p s

z s K sG

1

1

)(

)()( 1( ) ; 1

1cTsG sTs

A lead compensator is generally usedwhenever a substantial improvementin damping of the system is required.

1( )

1cTj

G jTj

1 1tan ( ) tan ( )T T

4.8.1 A lead compensator.

8/12/2019 Class11_-_2011

8/28


The frequency where

the phase is maximumis given by

1m

T

The maximum phasecontribution

1sin

1m

1 sin1 sin

m

m

8/12/2019 Class11_-_2011

9/28


1m

T 1/ 1 1 1 1 1

log log logm mT

T T T T T T T

1 1 1log (log log )

2m T T

The maximum frequency occurs midway betweenthe break-point frequencies (sometimes called

corner frequencies) on a logarithmic scale.

( )( )

( )c s z

G s s p

Rewriting G c(s) in the form used for root-locus analysis we obtain:

1log (log | | log | |)

2m z p | || |m z p

8/12/2019 Class11_-_2011

10/28


( 2)( )

( 10)c s

G s s

11

( )11c c c

sTs T G s K K Ts s

T

12 0.5

1 110

5

T T

T

20 4.47[ / ]m rad seg

1sin

1m

1 1 (1/5)sin 41.821 (1/5)

om

8/12/2019 Class11_-_2011

11/28


Phase-Lead Design Procedure

1. Determine open-loop gain K to satisfy error or bandwidth requirements.

2. Evaluate the phase margin of the uncompensated system using the value of K obtained from step1.3. Allow for extra margin (about 10 o), and determine the needed phase lead.4. Determine 5. Pick m to be at the crossover frequency.6. Draw the compensated frequency response and check the PM.7. Iterate on the design. Adjust compensator parameters (poles, zeros and gain) until all

specifications are met.

The block diagram of the sun-seeker control is shown in figure. The system may be mounted on aspace vehicle so that it will track the sun high accuracy. The variable r represents the referenceangle of the solar ray, and o denotes the vehicle axis. The objective of the sun-seeker system is tomaintain the error between r and o near zero.

Example

1/ 1/ /m z T p T

8/12/2019 Class11_-_2011

12/28


)25(2500

)( s s K

sG p

The steady-state error due to a unit-ramp function input should

be 0.01 The phase margin has to be greater than 45 degrees.

)(lim;1

0 s sG K K ess sv

v

K K

s s K

s K sv 100252500

)25(2500

lim 0

K 1001

01.0 1)100)(01.0(

1 K

)25(2500

)( s s

sG p

8/12/2019 Class11_-_2011

13/28


Matlab:

>> G=zpk([],[0 -25],[2500])

Zero/pole/gain:2500

--------s (s+25)

>> margin(G);grid-60

-40

-20

0

20

40

60

M a g n i t u d e ( d B )

100

101

102

103

-180

-135

-90

P h a s e ( d e g )

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 28 deg (at 47 rad/sec)

Frequency (rad/sec)

8/12/2019 Class11_-_2011

14/28


om

m

mm

25

82845

2845

We will design a compensation network with a maximum phase lead

Then, calculating , we obtain:

)sin(11 m

0.4059 The magnitude of the lead network at m is: 10 log(1/ 0.4059) 3.9162 dB

The compensated crossover frequency is then evaluated where the magnitude of G(j ) is-3.9162 dB

1 sin( ) 1 sin(25)0.4059

1 sin( ) 1 sin(25)m

m

8/12/2019 Class11_-_2011

15/28


100

101

102

103

-180

-135

-90

P h a s e ( d e g )


Frequency (rad/sec)

-60

-40

-20

0

20

40

60

System: GFrequency (rad/sec): 60.2Magnitude (dB): -3.91


8/12/2019 Class11_-_2011

16/28


1/ 60.2 0.4059 38.35

1/ / 60.2 / 0.4059 94.49

m

m

z T

p T

)494.94(

)35.38(4639.2)(

s

s sGc

)494.94)(25()35.38(75.6159

)()( s s s s

cG sGc

Matlab:

>> G=zpk([-38.35],[0 -25 -94.494],[6159.75])

Zero/pole/gain:6159.75 (s+38.35)------------------s (s+25) (s+94.49)

>> margin(G);grid

8/12/2019 Class11_-_2011

17/28


-100

-50

0

50


100

101

102

103

104

-180

-135

-90

P h a s e ( d e g )


Frequency (rad/sec)

8/12/2019 Class11_-_2011

18/28


4.8.2 A lag compensator.

N

j j

M

ii

c

p s

z s K sG

1

1

)(

)()( 1( ) ; 1

1cTs

G sTs

8/12/2019 Class11_-_2011

19/28


8/12/2019 Class11_-_2011

20/28


4.8.2 Phase-Lag Design using the Bode Diagram.

We determine the compensation network by completing the following steps:

1. Evaluate the uncompensated system phase margin when the error constants are satisfied.2. Assuming that the phase margin is to be increased, the frequency at which the desired phase

margin is obtained is located on the Bode plot. This frequency is also the new gain crossoverfrequency ng , where the compensated magnitude curve crosses the 0-dB axis.

3. To bring the magnitude curve down to 0 dB at the new gain-crossover frequency ng,

the phase-lag controller must provide the amount of attenuation equal to the value of the magnitude curveat ng. In other words,

4. Draw the compensated frequency response, check the resulting phase margin, and repeat thesteps if necessary. Finally, for an acceptable design, raise the gain of the amplifier in order toaccount or the attenuation.

( )

201/10

1

p ng G

101 ng T

8/12/2019 Class11_-_2011

21/28


The block diagram of the sun-seeker control is shown in figure. The system may be mounted on a

space vehicle so that it will track the sun high accuracy. The variable r represents the referenceangle of the solar ray, and o denotes the vehicle axis. The objective of the sun-seeker system is tomaintain the error between r and o near zero.

Example

)25(2500

)( s s

K sG p

The steady-state error due to a unit-ramp function inputshould be 0.01 The phase margin has to be greater than 45 degrees.

8/12/2019 Class11_-_2011

22/28


)(lim;1

0 s sG K K ess sv

v

K K

s s K

s K sv 100252500

)25(2500

lim 0

K 1001

01.0 1)100)(01.0(

1 K

)25(2500

)( s s

sG p

Matlab:

>> G=zpk([],[0 -25],[2500])

Zero/pole/gain:

2500--------s (s+25)

>> margin(G);grid

8/12/2019 Class11_-_2011

23/28


-60

-40

-20

0

20

40

60


100

101

102

103

-180

-135

-90

P h a s e ( d e g )


Frequency (rad/sec)

8/12/2019 Class11_-_2011

24/28


Bode DiagramGm = Inf dB (at Inf r ad/sec) , Pm = 28 deg (at 47 rad/sec)

Frequency (rad/sec)10

010

110

210

3-180

-150

-120

-90

System: GFrequency (rad/sec): 20.7Phase (deg): -130 P

h a s e ( d e g )

-60

-40

-20

0

20

40

60

System: G

Frequency (rad/sec): 20.7Magnitude (dB): 11.4


8/12/2019 Class11_-_2011

25/28


0.57

20 log( ) 11.4 dB11.4

log( ) 0.5720

10 0.27

Calculating the controllers constants.

(1 0.483 )( )(1 1.789 )c

sG s s

)789.11()483.01(

)25(2500

)()( s s

s s sG sG pc

Matlab:

>> g=tf([1208 2500],[1.789 45.73 25 0])

Transfer function:1208 s + 2500

----------------------------1.789 s^3 + 45.73 s^2 + 25 s

>> margin(g);grid

789.15589.01

)07.2(27.01

07.210

7.20110

1

T T

T ng


8/12/2019 Class11_-_2011

26/28


-100

-50

0

50

100


10-2

10-1

100

101

102

103

-180

-135

-90

P h a s e ( d e g )


Frequency (rad/sec)


8/12/2019 Class11_-_2011

27/28


Effects of Phase-Lead Compensation

The phase-lead controller adds a zero an a pole, with the zero to the right of pole, to the forward-path transfer function. The general effect is to add more damping to the close-loop system. The risetime and settling time are reduced in general. This controller improves the phase margin of the closed-loop system. The bandwidth of the closed-loop systems is increased. This corresponds to faster time response. The steady-state error of the system is not affected.

Effects of Phase-Lead Compensation

For a given forward-path gain K, the magnitude of the forward-path transfer function is attenuatednear the above the gain-crossover frequency, thus improving the relative stability of the system. The gain-crossover frequency is decreased, and thus the bandwidth of the system is reduced. The rise time and settling time of the system are usually longer, because the bandwidth is usually

decreased. The system is more sensitive to parameter variation because the sensitivity function is greater thatunity for all frequencies approximately greater than the bandwidth of the system.


8/12/2019 Class11_-_2011

28/28


Bibliography:

[1] Golnaraghi F, Kuo B , Automatic Control Systems, Wiley, John & Sons, 9 th

Edition, 2009.

[2] Dorf R. & Bishop R., Modern Control Systems, Prentice Hall, 11 st Edition,2007.

Documents

Class11_-_2011