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Digital Control
The s plane root locus design technique concentrates on two figures of merit: time to peak and percent overshoot. From these two figures of merit we determine values for the
damping ratio and natural frequency.
(See equations 5.14 and 5.16)
In digital electronic we learnt how to transform from s plane to z plane. In this lecture we use
another way for transformation called
Ad Hoc (Pole/Zero) Mapping.
Lead Compensation
• Has the same purpose as PD compensator to improve the transient response.
• It consists of zero and a pole with the zero closer to the origin of the s plane than the pole.
• The zero of the lead compensator like the zero of the PD compensator reshapes a portion of the root locus to achieve the desired transient response.
• The pole influence the overall shape of the root locus but its impact on the portion being reshaped by the zero is minimal.
• A lead compensator can be implemented by opamps.
Example
18
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0)(
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1
0.1 %20 are ionsspecificatdesign The
)1(
10 gain heConsider t
s
sscG
babs
ascKscGd
spTPO
sspG
s
s+1s+as+b
1
23 Re (s)
Im (s)
-b -a -1 0
18
)3(8.7)(
8.733
310
181 :
18
)3()(;18153
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33
o3.113;o90.With o180213:
try. trigonomesimpleby poler compensato thefindcan We
seen! be toremains choice good a is isWhrther th
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rcompensato lead theof zero and pole theoflocation theChose
s
sscG
jss
ssscK
s
scKscGb
Criteria Amplitude
Criteria Angle
Lag Compensator• Identical to PI compensator.
• Improves steady state accuracy.
• The pole of the compensator is close to the origin.
• The pole close to the origin is a near approximation to a perfect integrator.
The Ad Hoc (Pole/Zero) Mapping
• Translate all poles from s plane to the z plane using z = esT
• Translate all finite zeros using z = esT
• If the s plane transfer function has any zeros at infinity, place a zero at z = 0 for each zero at
infinity.• Select a gain G(z) so that
TjezzGjssG )()(
Example of Transformation
)6703.0)(9048.0(
)8187.0(8654.0)(;8654.0
778.5
5
5 ofgain DC a achieve order toIn 5. is )( ofgain DC The
7.5)3297.0)(0952.0(
)8187.0(
1)4)((
)2(1)(
isgain DC theG(z)For ).( of that to(z) ofgain DC match the to use willWe
)6703.0)(9048.0(
)8187.0(
1.0)4)((
)2()(
infinity.at zero one has )( Therefore zeros. finiteover poles
of excess the toequal isinfinity at zeros ofnumber The
1.0 ;)4)(1(
)2(10)(
zz
zzzGK
sG
KK
zT-z-e-Tz-e
T-z-eKzzzG
sGGK
zz
zKz
TT-z-e-Tz-e
T-z-eKzzG
sG
Tss
ssG
Design in the s PlaneIn this example, we may do the design ignoring the effect of the zero-
order hold, and then redo the design to include its effect. We may ignore the effect of the zero-order hold, if the sampling rate is high enough.
22at thesatisyingby determined is
r compensato theof pole The
-2.at placed isr compensato lead theof zero The
20%under overshoot thekeep order toin 2/1 Choose
.25.1
frequency dampedenough large a choosingby
achieved becan peak to timedesired The
input) stepfor 0( system. 1 typefor this 20% than less 1.5s;
satifiesr that compensato adersign wish to We:)1(
1)(
js
s
pTdω
ssePOpT
ssspG
condition angle
-b -2 -1 0
s+b s+2s+1
s
2
8
)2(20
7.17) and 7.16 equations (See 20222
1
;843.18tan
22
43.1821o1803
o180321
s
scG
jss
bssscK
ob
o
2 13
o180)( ;1)(
o18011)(
sGHsGH
sGH
Choosing a Sampling Rate
• We say we require 10 samples per cycle of the output.• Since the damped frequency at which the closed loop system will
oscillate is 2 rad/, this yield a sample rate of 20 rad/s, or 3.18 Hz.• Then the intersample period T is 0.314s. This is slow sampling rate.
We will increase it to 10 Hz.
• We use bilinear mapping in order to map the compensator Gc(s) to the z plane
4286.0
)8182.0(7.1511
208
)2(20)(;
1
1
z
zzz
ss
sscGz
z
T
as
The Bilinear Mapping• We usually map the lines of damping constant and
natural frequency from the s plane to the z plane using the mapping z = esT.
• Compensators can be mapped from the s plane to the z plane in variety of ways.
• The bilinear mapping is a frequent choice for mapping compensators from the s plane to the z plane. The usual choice for a is 2. For this choice of a, the bilinear mapping is the inverse of the trapezoidal rule for numerical integration (Tustin’s method).