modern control system
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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
sGsGc
n
jj
M
ii
c
c
ps
zsKsG
sG
1
1
)(
)()(
, )( of form General
Consider the following first order compensation transfer function
)()()(
pszsKsGc
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
zpjjK
jjK
pjzjpKz
pjzjKsGc
)(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
RRRCRRRR
CsRRRRCsR
RRR
CsRCsRR
R
sCRR
RsVsVsGc
1 clearly Where,
111)(
function,r on transfecompensati lead-phase aobtain weThus,
2
21
RRR
sssGc
Phase – Lag network
Consider the following first order compensation transfer function
)()()(
pszsKsGc
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
CsRRCsR
CsRRCsR
sVsVsG
in
oc
network. lag-phase aget weThus, plane.- in theorigin thetocloser lies pole theand |z| |p| ,1/)( As
)/(1 and /1,
111)(
obtain, we,/)( and defining,By
221
2212
sRRR
pzwherepszs
sssG
RRRCR
c
Frequency response of the transfer is given by,
jjjGc
11)(
) 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
TjwjwTjwGc
11)(
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
Completedzp
Phase lag design example
)5.01()2(
function, transfer loopopen following with theated) (uncompens systemfeedback unity aConsider
jwjwK
jwjwK 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 We6) figure (from db 20
frequency,crossover new be to cause tonecessary n Attenuatio5.1 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 theandcrossover thebelow decade oneat zeroput weTherefore,
p
z
)66.661)(5.01()166.6(20)()(
then,is system dcompensate The
jwjwjwjwjwGjwGc
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
sCRsCRsCR
sVsV
)/(1)/())/(1)(1(
)1(
11
1121122
22
sCRsCRsCsCRsCR
sCR
121122
1122
1
11
1121
1122
22
)1)(1()1)(1(
1
)/()1)(1(
)1(
RsCsCRsCRsCRsCR
sCsCR
sCRsCsCsCRsCR
sCR
)()(
1
2
sVsV
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
wheressss
sVsV
ClearlyCRCR
CRCRCRCRCR
network lag-Phase)1()1(
network. lead-phase)1()1(
2
2
1
1
ssss
Finding from root - locus
)cos(
axis. real theside negative thefrommeasuredorigin at locusroot on point by the made angle theof
cosine theis locusroot on thepoint any for ,
So
Thank You