Modern Control SystemEKT 308

Design of Feedback Control Systems

Control System Design: Concerned with the arrangement , or the plan , of the system structure and the selection of suitable components and parameters.

Compensator: Additional component or circuit that is inserted into a control system to compensate for deficient performance.

Types of Compensation: i) Cascade Compensation ( figure 1.a)ii) Feedback Compensation (figure 1.b)iii) Output Compensation (figure 1.c)iv) Input Compensation (figure 1.d)

Types of Compensation

Figure 1: Types of compensation

Cascade Compensation Networks

).( process

specified with thecascaded is )(function network on Compensati

sG

sGc

n

jj

M

ii

c

c

ps

zsKsG

sG

1

1

)(

)()(

, )( of form General

Consider the following first order compensation transfer function

)(

)()(

ps

zsKsGc

The compensation network is called phase-lead network if |z| < |p|

1 Clearly,

}or , {here, 1

]1[

1

]1[

1)/(

]1)/()[/()()(

by,given isnetwork theof responsefrequency The

11

zpj

jK

j

jK

pj

zjpKz

pj

zjKsGc

)(tan)(tan)()(

by,given is responsefrequency of Angle11 jGc

Because zero occurs first on the frequency axis, we obtain a phase-lead characteristic, as shown in the following figure 3.

Figure 3: Bode diagram of phase lead network.

Example of phase lead compensation network

2

212121

2121

1

21

2

1

12

2

12

2

1

2

and )]/([ define, usLet

1)]/([

1

)/(1)/(

)1

||()(

)()(

network, theoffunction Transfer

R

RRCRRRR

CsRRRR

CsR

RR

R

CsRCsR

R

R

sCRR

R

sV

sVsGc

1 clearly Where,

1

11)(

function,r on transfecompensati lead-phase aobtain weThus,

2

21

R

RR

s

ssGc

Phase – Lag network

Consider the following first order compensation transfer function

)(

)()(

ps

zsKsGc

The compensation network is called phase-lag network if |z| > |p|

Example of phase lag network

1)(

1

)/(1

)/(1

)(

)()(

network, lag-phase theoffunction Transfer

21

2

21

2

CsRR

CsR

CsRR

CsR

sV

sVsG

in

oc

network. lag-phase aget weThus, plane.- in theorigin theto

closer lies pole theand |z| |p| ,1/)( As

)/(1 and /1,

1

1

1)(

obtain, we,/)( and defining,By

221

2212

s

RRR

pzwhere

ps

zs

s

ssG

RRRCR

c

Frequency response of the transfer is given by,

j

jjGc

1

1)(

) and ( of valuesdifferent for two

figure in theshown is )( negative with diagram Bode

21

Figure 5 : Bode diagram of the phase – lag network.

Phase Lag Design using Bode Diagram

Tjw

jwTjwGc

1

1)(

network, lag phase offunction Transfer

Steps to design an appropriate phase lag network.

1. Obtain Bode diagram for the uncompensated system. (Gain adjusted for desired error constant).

2. Determine the phase margin. If insufficient, follow the remaining steps.

3. Determine the frequency where the phase margin would be satisfied, if the magnitude curve crosses 0-db at this freq. (allow 5 degree safety)

4. Place the zero one decade below the new cross over frequency.5. Measure the necessary additional attenuation at the new

crossover freq

!!!

/)/(1 as pole theCalculate 7.

.frequency crossover new at the log20 isnetwork

lag phaseby introducedn attenuatio that notingby Calculate 6.

c

Completed

zp

Phase lag design example

)5.01()2(

function, transfer loopopen following

with theated) (uncompens systemfeedback unity aConsider

jwjw

K

jwjw

K v

o45 :margin Phase

20 :Target vK

Steps:

Uncompensated Bode diagram is shown in figure 6.

Figure 6:

increased. bemust margin phase So,

20 plot) Bode (from system ated uncompens in themargin Phase o

10

log 20 db 20

:follows as find We

6) figure (from db 20

frequency,crossover new be to cause tonecessary n Attenuatio

5.1 frequency,crossover new this6, figureIn

-13050-180)( where,frequency Locate

50for design lag),or (compensatfactor safety for 5 Allowing

c

o

oo

c

66.66

66.615.0/1 Then,

015.010/15.0 and

15.010/5.1 So,

zero. below decade one pole theand

crossover thebelow decade oneat zeroput weTherefore,

p

z

)66.661)(5.01(

)166.6(20)()(

then,is system dcompensate The

jwjwjw

jwjwGjwGc

Modern Control SystemEKT 308

Design of Feedback Control Systems

(contd…)

Lead-Lag Compensation

Figure 1: RC lead-lag network.

Provides the attenuation of phase lag network. Provides the phase angle of the phase lead network.

Transfer function of lead-lag network

)/(1)/()1(

)1(

))/(1||())/(1(

))/(1(

)(

)(

11

11

2

22

2

22

1122

22

1

2

sCRsCR

sCsCR

sCsCR

sCRsCR

sCR

sV

sV

)/(1)/())/(1)(1(

)1(

11

1121122

22

sCRsCRsCsCRsCR

sCR

121122

1122

1

11

1121

1122

22

)1)(1(

)1)(1(

1

)/()1

)(1(

)1(

RsCsCRsCR

sCRsCR

sCsCR

sCRsCsCsCR

sCR

sCR

)(

)(

1

2

sV

sV

1)(

)1)(1(

2221112

2121

1122

sCRCRCRsCCRR

sCRsCR

Denote,

1 and 1 ,

)1)(1(

)1)(1(

)(

)( Now,

1. ,

and

, ,

21

21

1

2

221121

22211121222111

where

ss

ss

sV

sV

Clearly

CRCR

CRCRCRCRCR

network lag-Phase)1(

)1(

network. lead-phase)1(

)1(

2

2

1

1

s

s

s

s

Finding from root - locus

)cos(

axis. real theside negative thefrom

measuredorigin at locusroot on point by the made angle theof

cosine theis locusroot on thepoint any for ,

So

Thank You