capitulo 4 - control ii

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5/4/12 Teoría de Control II - Ing.

UNIVERSIDAD POLITECNICA SALESIANA

Semestre: Septiembre – Febrero 2010Ing. Walter Orozco. [email protected]

THE DESIGN OFFEEDBACK CONTROL

SYSTEMS

5/4/12 11 Teoría de Control II - Ing.




CHAPTER 4The design of feedback control systems

4.1 Introduction.4.2 Cascade Compensation Networks.4.3 Phase-Lead Design using the Bode Diagram.4.4 Phase-Lag Design using the Bode Diagram.4.5 Design on the Bode Diagram using Analytical Methods.4.6 System Design using control design software.

UNIVERSIDADPOLITECNICA

SALESIANA






SALESIANA4.1

Introduct

ion.In this chapter, we address the central issue of the design of compensators. Usingthe methods of the previous chapters, we develop several design techniques in thefrequency domain that enable us to achieve the desired system performance.

Desired outcomes:

•Be familiar with the design of lead and lag compensator using Bode plot methods.•

Understand the value of prefilters and how to design for deadbeat response.•Have a greater appreciation for the varied approaches available for control systemdesign.

Important:

The performance of a feedback control system is of primary importance. Thus, the

design of a control system is concerned with the arrangement, or the plan, of thesystem structure and the selection of suitable components and parameters.

The alteration or adjustment of a control system in order to provide a suitableperformance is called compensation; that is, compensation is the adjustment of asystem in order to make up for deficiencies or inadequacies.




5/4/12 Teoría de Control II - Ing.5/4/12 Teoría de Control II - Ing. 44


SALESIANAA compensator is an additional component or circuit that is inserted into acontrol system to compensate for a deficient performance.

Types of compensation





SALESIANA4.2 Cascade CompensationNetworks.

∏

∏

=

=

+

+=

N

j

j

M

i

i

c

p s

z s K

sG

1

1

)(

)(

)(

The problem reduces to the judiciousselection of the poles and zeros of thecompensator.

A compensator Gc(s) is used with aprocess G(s) so that the overall loopgain can be set to satisfy the steady-state error requirement, and then Gc(s)is used to adjust the system dynamicsfavorably without affecting the steady-state error.

Consider a first-order compensator with the transfer

function

)(

)()(

p s

z s K sGc +

+= z p >

Phase-leadnetwork

z p > >





SALESIANA

p z < <

If the pole was negligible according to theequation, and the zero occurred at the origin of the s-plane, we would have a differentiator sothat

s p

K sGc ≈)(

090)( j

c e p

K

p

K j jG

== ω ω ω

Similarly, the frequency response of the first order compensator networkis:

T j

T j K

p j

z j p Kz

p j

z j K sGc

ω

ωα

ω

ω

ω

ω

++

=+

+=

++

=1

)1(

1)/(

)1)/()(/(

)(

)()( 1

α α /;;/1 1 K K z p pT ===

The angle of the frequency characteristic is

)(tan)(tan)( 11T T ω αω ω φ −− −=




The frequency response of this phase-lead networkis shown.


SALESIANA

Because the zero occurs first on the frequency axis, we obtain a phase-leadcharacteristic, as shown in figure. The slope of the asymptotic magnitude curve

is +20 dB/decade.

m

mm

sen

sen

T φ

φ α

α ω

++==

1

1;

1





SALESIANA The phase-lead compensation transfer function can be obtained with the networkshow in figure. The transfer function of this network is:

22

11

2

1

1

1

)(

)()(

C R s

C R s

C

C

s E

s E sG

in

oc

+

+==

++

==1

1

)(

)()(

22

11

1

2

sC R

sC R

R

R

s E

s E sG

in

oc

+

+==

Ts

Ts

s E

s E sG

in

oc

1

1

1

)(

)()(

α

α

2

1

R

R=α




4.3 Phase-Lead Design using the BodeDiagram.


SALESIANA

We determine the compensation network by completing the following steps:

1. Evaluate the uncompensated system phase margin when the error constantsare satisfied.

2. Allowing for a small amount of safety, determine de necessary additionalphase lead m.φ

3. Evaluate fromα

1. Evaluate 10 log and determine the frequency where the uncompensatedαmagnitude curve is equal to 10 log dB. Because the compensation networkαprovides a gain of 10 log at m, this frequency is the new 0-db crossoverα ωfrequency and m simultaneously.ω

2. Calculate the pole:

1. Draw the compensated frequency response, check the resulting phasemargin, and repeat the steps if necessary. Finally, for an acceptable design,raise the gain of the amplifier in order to account or the attenuation.

α α ω /; p z p m ==

α /1





SALESIANA

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 variabler represents the reference angle of the solar ray, and o denotes the vehicle axis.θ θ

The objective of the sun-seeker system is to maintain the error between r andα θo near zero.θ

Example4.1

)25(

2500)(

+=

s s

K sG p

• The steady-state error due to a unit-ramp functioninput should be ≤ 0.01• The phase margin has to be greater than 45degrees.





SALESIANA

)(lim;1

0 s sG K

K

ess sv

v

→==

K K

s s

K s K sv

10025

2500

)25(

2500lim 0 ==

+

=→

K 100

101.0 = 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





SALESIANA

-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 )

BodeDiagram

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

Frequency (rad/sec)





SALESIANA

o

m

m

mm

25

82845

2845

=

+−=∆+−=

φ

φ

φ φ

We will design a compensation network with a maximum phase lead

Then, calculating , we obtain:α

)sin(1

1mφ

α

α =

+−

4639.2=α The magnitude of the lead network atm is:ω

dB9162.34639.2log10 =

The compensated crossover frequency is then evaluatedwhere the magnitude of G(j ) is -3.9162 dBω

4639.2)25sin(1

)25sin(1

)sin(1

)sin(1=

−+

=−+

=m

m

φ

φ α





SALESIANA

100

101

102

103

-180

-135

-90

P h a s e

( d e g )

Bode Diagram

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

Frequency (rad/sec)

-60

-40

-20

0

20

40

60

System: G

Frequency (rad/sec): 60.2

Magnitude (dB): -3.91


UNIVERSIDAD





SALESIANA

35.384639.2/494.94/

494.944639.22.60

===

===

α

α ω

p z

p m )494.94(

)35.38(4639.2)(

+

+=

s

s sGc

)494.94)(25(

)35.38(75.6159)()(

+++

= s s s

scG 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

UNIVERSIDAD





SALESIANA

-100

-50

0

50


100

101

102

103

104

-180

-135

-90

P h a s e ( d e g )

BodeDiagram

Gm=Inf dB(at Inf rad/sec) , Pm=47.6deg(at 60.2rad/sec)

Frequency (rad/sec)

UNIVERSIDAD





SALESIANA4.4 Phase-Lag Design using the BodeDiagram.

We determine the compensation network by completing the following steps:

1. Evaluate the uncompensated system phase margin when the error constants aresatisfied.

2. Assuming that the phase margin is to be increased, the frequency at which thedesired phase margin is obtained is located on the Bode plot. This frequency isalso the new gain crossover frequency ng , where the compensated magnitudeω

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 toωthe value of the magnitude curve at ng.ω In other words,

1. Draw the compensated frequency response, check the resulting phase margin,and repeat the steps if necessary. Finally, for an acceptable design, raise thegain of the amplifier in order to account or the attenuation.

1

10 20

)(

<=

−

α

α

ω ng pG

10

1 ng

T

ω

α =

UNIVERSIDAD





SALESIANA

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 variabler represents the reference angle of the solar ray, and o denotes the vehicle axis.θ θ

The objective of the sun-seeker system is to maintain the error between r andα θo near zero.θ

Example4.2

)25(

2500)(

+=

s s

K sG p

• The steady-state error due to a unit-ramp functioninput should be ≤ 0.01• The phase margin has to be greater than 45degrees.

UNIVERSIDAD




)(lim;1

0 s sG K

K

ess sv

v

→==

K K

s s

K s K sv

10025

2500

)25(

2500lim 0 ==

+

=→

K 100

101.0 = 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


SALESIANA

UNIVERSIDAD





SALESIANA

-60

-40

-20

0

20

40

60


100

101

102

103

-180

-135

-90

P h a s

e ( d e g )

BodeDiagram

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

Frequency (rad/sec)

UNIVERSIDAD





SALESIANABode Diagram

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

Frequency (rad/sec)

100

101

102

103

-180

-150

-120

-90

System: GFrequency (rad/sec): 20.7

Phase (deg): -130 P h a s e

( d e g )

-60

-40

-20

0

20

40

60

System: G

Frequency (rad/sec): 20.7

Magnitude (dB): 11.4


UNIVERSIDAD





SALESIANA

27.010

57.020

4.11log

dB4.11log20

57.0 ==

−=−=

=−

−α

α

α

789.15589.0

1

)07.2(27.0

107.2

10

7.201

10

1

===⇒==

=

T T

T

ng

α

ω

α

Calculating the controller´sconstants.

)789.11(

)483.01(

)789.11(

)789.271.01()(

s

s

s

s sGc +

+=

++

=

)789.11(

)483.01(

)25(

2500)()(

s

s

s s sG sG pc +

++

=

Matlab:

>> g=tf([1208 2500],[1.789 45.73 250]) Transfer function:

1208 s + 2500----------------------------1.789 s^3 + 45.73 s^2 + 25 s >> margin(g);grid

UNIVERSIDAD





SALESIANA

-100

-50

0

50

100


10-2

10-1

100

101

102

103

-180

-135

-90

P h a s

e ( d e g )

Bode Diagram

Gm= Inf dB(at Inf rad/sec) , Pm=46.1 deg (at 20.8 rad/sec)

Frequency (rad/sec)

UNIVERSIDAD





SALESIANAEffects of Phase-Lead Compensation

• The phase-lead controller adds a zero an a pole, with the zero to the right of pole, tothe forward-path transfer function. The general effect is to add more damping to theclose-loop system. The rise time 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 fastertime 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 transferfunction is attenuated near the above the gain-crossover frequency, thus improvingthe relative stability of the system.•

The gain-crossover frequency is decreased, and thus the bandwidth of the system isreduced.• The rise time and settling time of the system are usually longer, because thebandwidth is usually decreased.• The system is more sensitive to parameter variation because the sensitivity functionis greater that unity for all frequencies approximately greater than the bandwidth of the system.

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SALESIANA4.5 Design on the Bode Diagram using AnalyticalMethods.

An analytical technique of selecting the parameters of a lead or lag netowrk hasbeen developed for the Bode diagram. For a single-stage compensator

Ts

Ts sGc +

+=

1

)1()(

α

Where < 1 yields a lag compensator and >1 yields a lead compensator. Theα α

phase contribution of the compensator at the desired crossover frequency c isωgiven by

( ) α ω

ω αω φ

21

tanT

T T p

c

cc

+−

==

The magnitude M (in dB) of the compensator at c is given byω( )

( ) 2

2

10/

1

110

T

T c

c

c M

ω

α ω

++==

Eliminating cT from the last equations, weωobtain the nontrivial solution equation for αas ( ) 021 222222 =−++++− ccc pc pc p α α

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SALESIANAFor a single-stage compensator, it is necessary that c > p2+1. If we solve for fromαthe last equation on the preview slide, we can obtain T from

2

11

α ω −−=

c

cT

c

The design step for a lead compensator are:

1. Select the desired cω2. Determine the phase margin desired and therefore the required phase .φ3. Verify that the phase lead is applicable: >0 and M>0.φ4. Determine whether a single stage will be sufficient by testing c>p2+15. Determine α6. Determine T.

If we nee to design a single-lag compensator, then <0 and M<0.. Step 4 willφrequire c<p2+1. Otherwise, the method is the same.

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SALESIANAExample:

Let consider the following system, and design a lead network by the analytical

technique .2

10)(

s sG p = 1)( = s H

We require a 450 phasemargin.

145tan 0 == p

100

101

-181

-180.5

-180

-179.5

-179

P h a s e ( d e g )

Bode Diagram

Gm = 2.01e-015 dB (at 3.16 rad/sec) , Pm = 0 deg (at 3.16 rad/sec)

Frequency (rad/sec)

-20

-10

0

10

20

30

System: G

Frequency (rad/sec): 5.03Magnitude (dB): -8.06 M a g n i t u d e ( d B )

397.610 10/06.8 ==c

( ) 021 222222 =−++++− ccc pc pc p α α

( ) 0397.685.81794.121397.61 2 =−+++− α α

046.75794.12397.4 2 =+⋅+⋅− α α

9359.2

8456.5

−==

α

α

0876.08456.5397.6

397.61

5

11122

=−

−=

−−

=α ω c

cT

c

UNIVERSIDAD





SALESIANA

s

s

s

s

Ts

Ts sGc

0876.01

5123.01

0876.01

)0876.08456.51(

1

)1()(

++

=+

⋅+=

++

=α ( )

( ) s s

s sG sG pc

0876.01

5123.0110)()(

2 ++

=

-100

-50

0

50

100

M a g n i t

u d e ( d B )

10-1

100

101

102

103

-180

-150

-120

P h a s e (

d e g )

Bode Diagram

Gm = -Inf dB (at 0 rad/sec) , Pm = 45 deg (at 5.03 rad/sec)

Frequency (rad/sec)

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4.6 System Design using control design software.


SALESIANA

The SISO Design Tool is made up of the following:

• The SISO Design Task in the Control and Estimation Tools Manager, a userinterface (UI) that facilitates the design of compensators for single-input, single-output feedback loops through a series of interactive pages.

• The Graphical Tuning window, a graphical user interface (GUI) for displaying and

manipulating the Bode, root locus, and Nichols plots for the controller currentlybeing designed. This window is titled SISO Design for Design Name.

• The LTI Viewer associated with the SISO Design Task.

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SALESIANA4.6.1 Design Options in the SISO Tool.

The SISO Design Tool facilitates the design of compensators for single-input, single-output feedback loops, and lets you iterate rapidly on your designs and perform thefollowing tasks:

•Manipulate closed-loop dynamics using root locus techniques.•Shape open-loop Bode responses.•Add compensator poles and zeros.•Add and tune lead/lag networks and notch filters.•Inspect closed-loop responses (using the LTI Viewer).•Adjust phase and gain margins.•Convert models between discrete and continuous time.•Automate compensator design.

4.6.2 Opening the SISO Design Tool

Type: >>sisotool to open the SISO Design Task node in the Control andEstimation Tools Manager and the Graphical Tuning Window.

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SALESIANA

Design a controller for the followingsystem:

2

1.5( )

14 40.02G s

s s=

+ +

For this example, the design criteria area as follow:

•Rise time of less than 0.5 seconds.•Steady-state error of less than 5%.•Overshoot of less than 10%.•Gain margin greater than 20 dB.•Phase margin greater than 40 degrees.

4.6.3 Example:

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5/4/12 Teoría de Control II Ing5/4/12 Teoría de Control II Ing 3232

Bibliograp

hy[1] Kuo B.C and Golnaraghi F. , “Automatic control systems”, Willey and Sons,Eight edition, 2003.[2] Dorf R y Bishop R., “Sistemas de Control moderno”, Prentice Hall, 10 edición,2005.[3] Burns R., “Advanced Control Engineering”, Butterworth Heinemann, Firstedition, 2001.

[4] Nobajas H y Diaz-Cordovez Angel, “Ingenieria de Control – Control de Sistemascontinuos”, Universidad de Navarra.


SALESIANA

capitulo 4 - control ii

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