1 Modern Control Theory (Digital Control) Lecture 1.
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Transcript of 1 Modern Control Theory (Digital Control) Lecture 1.
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Modern Control Theory
(Digital Control)
Lecture 1
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Course Overview
Analog and digital control systems MM 1 – introduction, discrete systems, sampling. MM 2 – discrete systems, specifications, frequency
response methods. MM 3 – discrete equivalents, design by emulation. MM 4 – root locus design. MM 5 – root locus design.
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Outline Short repetition of analog control methods Introduction to digital control
Digitization Effect of sampling
Sampling Spectrum of a sampled signals Sampling theorem
Discrete Systems Z-transform Transfer function Pulse response Stability
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Digitization Analog Control System
ctrl. filterD(s)
plantG(s)
sensorH(s)
r(t) u(t) y(t)e(t)+
-
continuous controllerFor example, PID control
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System caracteristics Transfer function
Characteristic equation 1+D(s)G(s)H(s) = 0
Poles are the roots of the characteristic equation
)()()(1
)()(
)(
)(
sHsGsD
sGsD
sR
sY
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Time functions associated with poles
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Second-order system Transfer function
is the damping ratio
is the undamped natural frequency
n
22
2
2)(
nn
n
sssH
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Rise time, overshoot and settling time
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Response og second-order system versus
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Bode-plot design Determin the open loop gain end phase as function
of
Evaluate the phase margin and gain margin
Adjust the margins by use of poles, zeros and gain scheduling.
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Bode plot )22.3(
07.13
22.3
07.13)(
223
sssssG
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1054.0
123.3051.0
22.3
07.13)()(
23
s
s
sssDsG
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Digitization Analog Control System
ctrl. filterD(s)
plantG(s)
sensor1
r(t) u(t) y(t)e(t)+
-
continuous controllerFor example, PID control
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Digitization Digital Control System
T is the sample time (s) Sampled signal : x(kT) = x(k)
control:differenceequations
D/A andhold
sensor1
r(t) u(kT) u(t)e(kT)
+-
r(kT) plantG(s)
y(t)
clock
A/DT
T
y(kT)
digital controller
voltage → bit
bit → voltage
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Digitization Continuous control vs. digital control
Basically, we want to simulate the cont. filter D(s) D(s) contains differential equations (time domain) –
must be translated into difference equations.
Derivatives are approximated (Euler’s method)
T
kxkxkx
)()1()(
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DigitizationExample (3.1)Using Euler’s method, find the difference equations.
bs
asK
sE
sUsD
0)(
)()(
Differential equation
)()()()()( 00
1
aeeKbuusEasKsUbs L
Using Euler’s method
)1()()1()()1()1(
)()1()()1(
00
0
keKkeaTKkubTku
aeT
kekeKbu
T
kuku
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Digitization
sssssG
s
ssD
2
1
)1(
1)(,
10
270)(
Compare – investigate using Matlab
1) Closed loop step response with continuous controller.
2) Closed loop step response with discrete controller. Sample rate = 20 Hz
3) Closed loop step response with discrete controller.Sample rate = 40 Hz
Significance of sampling time TExample controller D(s) and plant G(s)
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DigitizationMatlab - continuous controllernumD = 70*[1 2]; denD = [1 10];numG = 1; denG = [1 1 0];sysOL = tf(numD,denD) * tf(numG,denG);sysCL = feedback(sysOL,1);step(sysCL);
Matlab - discrete controllernumD = 70*[1 2]; denD = [1 10];sysDd = c2d(tf(numD,denD),T);numG = 1; denG = [1 1 0];sysOL = sysDd * tf(numG,denG);sysCL = feedback(sysOL,1);step(sysCL);
sssG
s
ssD
2
1)(
10
270)(
Controller D(s)and plant G(s)
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Digitization
Notice, high sample frequency (small sample time T )gives a good approximation to the continuous controller
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Effect of samplingD/A in output from controller
The single most important impact of implementing a control digitally is the delay associated with the hold.
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Effect of sampling Analysis
Approximately 1/2 sample time delay
Can be approx. by Padè
(and cont. analysis as usual)
ctrl. filterD(s)
PadéP(s)
sensor1
r(t) u(t) y(t)e(t)+
-
plantG(s)
Ts
TsP
/2
/2)(
2
TTd
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Effect of sampling
Example of phase lag by sampling
Example from before with sample rate = 10 Hz
Notice PM reduction
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Spectrum of a Sampled Signal Spectrum
Consider a cont. signal r(t) with sampled signal r*(t) Laplace transform R*(s) can
be calculated
ks
k
jnsRT
sR
kTttrtr
)(1
)(
)()()(
r(t) r*(t)
T
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Spectrum of a Sampled Signal
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Spectrum of a Sampled Signal High frequency signal and low frequency signal
– same digital representation.
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Spectrum of a Sampled Signal Removing (unnecessary) high frequencies –
anti-aliasing filter
control:differenceequations
D/A andhold
sensor1
r(t) u(kT) u(t)e(kT)
+-
r(kT) plantG(s)
y(t)
clock
A/DT
T
y(kT)
digital controller
anti-aliasing
filter
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Spectrum of a Sampled Signal
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Sampling Theorem Nyquist sampling theorem
One can recover a signal from its samples if the sampling frequency fs=1/T (s=2/T) is at least twice the highest frequency in the signal, i.e.
s > 2 b (closed loop band-width)
In practice, we need 20 b < s < 40 b
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Discrete Systems
Discrete Systems Z-transform Transfer function Pulse response Stability