introduction to control system theory for engineers
DESCRIPTION
Introduction to Control System Theory For Engineers. This talk assumes: No prior background in control systems Working knowledge of Fourier Transforms and frequency-domain analysis Some familiarity with complex numbers (will review) The willingness to ask questions!. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/1.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Introduction to Control System
Theory
For Engineers
This talk assumes:• No prior background in control systems• Working knowledge of Fourier Transforms and
frequency-domain analysis• Some familiarity with complex numbers (will
review)• The willingness to ask questions!
![Page 2: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/2.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Section 1:
Introduction to Control System Theory:
![Page 3: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/3.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCTerminology: dB & the Complex Plane
R
Euler’s Equation:𝑖𝑅𝑠𝑖𝑛(𝜃)
Re
Im
𝑅𝑐𝑜𝑠(𝜃)
i
R
𝑿=𝟏𝟎𝑳𝒐𝒈𝟏𝟎 (𝑿 )
𝐿𝑜𝑔 ( 𝑋 ∗𝑌 )=𝐿𝑜𝑔 ( 𝑋 )+𝐿𝑜𝑔(𝑌 )
𝐿𝑜𝑔 ( 𝑋𝑁 )=𝑁∗𝐿𝑜𝑔(𝑋 )
𝐵𝑒𝑙𝑙=𝐿𝑜𝑔10( 𝑃𝑃0)
𝑑𝑒𝑐𝑖𝐵𝑒𝑙𝑙(𝑑𝐵)=10×𝐿𝑜𝑔10( 𝑃𝑃0)
𝑑𝐵=10× 𝐿𝑜𝑔10( 𝐴2
𝐴0❑2 )=10×𝐿𝑜𝑔10( 𝐴𝐴0
)2
=20×𝐿𝑜𝑔10( 𝐴𝐴0)
Some handy Amplitude ratios expressed in dB:0dB = 1x (often used instead of ‘unity gain’ in controls)
6dB ~= 2x [ -6dB ~= 1/2x ]10dB ~= 3x [ -10dB ~= 1/3x ]12dB ~= 4x [ -12dB ~= 1/4x ]20dB = 10x [ -20dB = 1/10x ]
Lets Practice:16dB = 6dB + 10dB ~= (2x) x (3x) = 6x22dB = 10dB + 12dB ~= (3x) x (4x) = 12x30dB = 10dB + 20dB ~= (3x) x (10x) = 30x80dB = 4 x 20dB ~= (10x) x (10x) x (10x) x (10x) = 104x16 bits ~= 16*6dB = 96dB of dynamic range“20dB/Decade” = f(+/-)1 & “40dB/Decade” = f(+/-)2, etc.
The Complex Plane:
![Page 4: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/4.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
What is a Control System?
System to control
Means to control it
OutputInput
Example Inputsand Outputs:• Force• Volts• Heat• Light• Pressure
Example Systems:• House • Hard drive head• Clock osc.• Radio PLL• Power supply• Amplifiers
Example Means:(sensors & actuators)• Thermistors• Photodiodes• Strain gauge• Disp. Sensor• Heaters• Voice coils
But what does this box mean?
Input Output
![Page 5: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/5.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Boxes Represent Transfer Functions:
T.F.()B(A
Re
Im
i
B/A
(phasor diagram)
This already means we have made assumptions:• The magnitude of the transfer function (B/A) is constant independent of the
magnitude of A. Linearity and superposition• A single frequency input () produces only a single frequency output without
any harmonic distortion, etc. Linearity again….• The ratio of B()/A() and the phase () are constant in time. No saturation! • The input is unaffected by the output and the output is unaffected by load. ”The signal diode” assumption & ZERO output impedance!
..where B/A is the “Gain” at
![Page 6: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/6.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Basic Configuration of a Feedback System:
G(s)
H(s)
OutIn
Transfer function is:
)()(1
)(
)(
)(
sHsG
sG
sIn
sOut
-Becomes large when:
unitysHsG |)()(| 180))()(( sHsG
)()(1 sHsG-The “Characteristic Equation” is:
)()( sHsG-The “Loop Transfer Function” or “Loop Gain” is:
“G(s)” is the “Plant” transfer function“H(s)” is the “Compensation”
M
LKF
Xout
Fin
0
MXout
0
F H)( LKF
L
Fin
Code for “Laplace
transform”
![Page 7: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/7.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
OutG1
H1
In
H2
G’=OutIn
H2
“Block Math”: Multiple Sensors, Nested Loops, & Noise
211
11
11
11
)1(1
)1(
HHG
HG
HGHG
)1( HG
HGX N
)1( 11
11
HG
HG
F
X_noise
H
++
G
H
OutIn
Xn
G
H
OutXn H
F
H1
FH2
H4
H5
H3-The sensor / actuator with thehighest gain wins (you can’t havetwo loops controlling the same DOFat the same time.)
+
+
![Page 8: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/8.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Before we Look Under the Hood…
2) The Laplace Transform is used only in continuous-time models;Discretely-sampled (digital) systems require using the Z-Transform.
1) Question: What is on the cover of the Hitchhiker’s Guide to the Galaxy?
3) You will get into trouble if you try to use CT techniques in discrete-time(DT) systems. Discrete sampling effects such as aliasing becomevery significant and compensation filters need special techniques to design.
4) Sorry, but I won’t cover DT and Z-Transforms here. Understanding CTtechniques is critical to DT and they will get you “90%” of the way there.
Don’t PanicThe Laplace Transform IS the engine, but you can drive the car without itand you will never need to actually calculate one.
![Page 9: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/9.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCWhy Use the Laplace Transform?
)())((
)())(()(
21
21
N
N
PSPSPS
ZSZSZSKsTF
||||||
|||||||)(|
21
21
N
N
PSPSPS
ZSZSZSKsTF
1) The real frequency response is just theLaplace transform evaluated along the positiveimaginary axis in the S-plane: L(s) =F(ω) = L(iω)
Unstable!!!
)()()()()()())(( 2121 NN PSPSPSZSZSZSsTF
Pole-Zero representation:The factorized transfer function takes the form:
Then the magnitude of the frequency response is:
And the phase angle of the frequency response is:
iAeZ
Think:
S
2) The S-plane provides a simple and absolutestability criterion: if any right-half-plane (RHP) polesexist, the system is unstable!!! X
X
X
X
0
0
0
0
0
X
X
Re
Im
S
X
0
0
.(S-P1)
S = iω
1
Re
Im
![Page 10: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/10.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Figuring Out Transfer FunctionsMag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
M
LKF
Xin
Xout
0
0
Minimal-Phase (θ=90*slope exponent):
(plus damping)
f -2
f 0
θ = -2 x 90
θ = 0 x 90
S
X
X
Re
Im
Non-Minimal-Phase (RHP zeros or poles):
Examples:
All-pass filters Time delays (DT) “Zero-order hold” (DT) FIR filters (DT)
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
Non-minimal phaseis always bad in control systems!
f 0
S
X 0Re
Im
![Page 11: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/11.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
The Nyquist Stability Criterion:Recall:
)()(1
)(
)(
)(
sHsG
sG
sIn
sOut
)()(1 sHsGIf the Characteristic Equation:
Has any RHP zeros, the closed-loop transferfunction will be unstable.
S-plane1+G(s)H(s)
X0 Re
ImPolar plot of Loop Transfer Function GH:
0
Real frequency response
-0 freq.
Conjugate frequency response
Contour enclosingall RHP poles and zeros
+0 freq.
+∞ freq.
-∞ freq.
A B
D
C
0
A
B
CD
UNSTABLE!
Gain=1
-1 point!
)()()()()()())(( 2121 NN PSPSPSZSZSZSsTF
![Page 12: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/12.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
A Simple Example…..
Gain=1
STABLE!
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dBf -1
f -2
-0 freq.+0 freq.
Gain=1
UNSTABLE!
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
f -1
f -3
-0 freq.
+0 freq.
![Page 13: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/13.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Phase is Enemy #1 – Time Delays
Gain=1
UNSTABLE!
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
f -1
phase w/o delay
-0 freq.
+0 freq.Time delays cause rapidly dropping phase with higherfrequency
Alternate Bode-based stability requirement:The phase must be less than ±180 degreesat unity gain.
![Page 14: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/14.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCSources of Non-Minimal Phase:
The Zero-Order Hold (ZOH) and Transport Delays
100
101
102
103
-60
-50
-40
-30
-20
-10
0Phase lag vs oversampling factor for ZOH
Oversampling factor (fsamp/f)
Pha
se la
g, d
egre
es
Based on 4th order Pade approximation
10-3
10-2
10-1
-40
-35
-30
-25
-20
-15
-10
-5
0Phase lag vs time delay
Time delay (1/f)
Pha
se la
g, d
egre
es
Based on 4th order Pade approximation
∆t
ADC In DAC out
t
Cts.
Sampling at 10x unity gaingives ~18 degrees of extraphase!
A time delay of 10ms gives ~36 degrees of extra phase at 10Hz!
Both effects exist in all digital control systems!
![Page 15: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/15.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Summary of Section 1: Theory
• Plants with RHP zeros or poles. These are unusual, but can occur, often deliberately! This requires the ‘full version’ of the Nyquist criteria…. (yes, I have lied to you….)
• “MIMO” systems: Multiple Input, Multiple Output. This talk only covers “SISO”.• Digital systems require use of the “Z-Transform”. We don’t cover, but its important to
learn when you are ready.
Things we didn’t cover:
• Pick your Plant (G) to represent the inputs and outputs you care about.
• System Magnitude and Phase responses are represented by complex numbers. In the S-Plane (Laplace Transform Space) by complex poles and zeros.
• Systems with Right Half Plane (RHP) poles are unstable.
• Introduced a working form of the Nyquist Stability Criterion: || < 180• → Phase is the enemy. Beware time delays and ZOH!
The “God Equation”)()(1
)(
)(
)(
sHsG
sG
sIn
sOut
• Introduced Minimal Phase Networks: = n*90, n=power of slope.
• Non-minimal phase networks are useless as compensation (exercise for reader).
![Page 16: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/16.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Section 2:
Performance
![Page 17: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/17.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Dynamic Response: Phase and Gain Margins
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
Phase Margin
Gain MarginGain=1
(STABLE)
Phase Margin
Gain Margin
• Phase Margin usually dominates the closed-loop response
• All the information required for dynamic response is in the Bode diagram
)()(1
)(
sHsG
sG
![Page 18: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/18.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCA Simple Phase-Margin Calculation:
0 10 20 30 40 50 60 70 80 9010
-1
100
101
102
103
Phase Margin, Deg.
Am
plifi
catio
n at
Uni
ty G
ain
Amplification of Control System Responseat the Unity Gain Frequency vs Phase Margin
)(2222 COScbcba
Phasor diagram for 1+GH:
The Law of Cosines:
1+GH
GH at |GH|=1
The vector (1)
θ
Gives the amplification:
when |GH|=1 (at unity gain)
Phase Margin
))cos(1(2
1
1
1
GH
But the real situation is a bit more complex…..
![Page 19: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/19.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCDynamic Response: The Nichols Chart
G
H
OutIn
The Nichols chart plots:
)()(1
)()(
sHsG
sHsG
which is also called the control
signal
![Page 20: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/20.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
The Designer’s Choice – No Right Answer…
10-1
100
101
-40
-20
0
20
40Loop Transfer Function (GH) of System
Frequency, Hz
Mag
nitu
de,
dB
10-1
100
101
-200
-150
-100
-50
0
Frequency, Hz
Pha
se,
Deg
.
Phase margins of: 10, 30, 50, 70 and 90 Degrees
10-1
100
101
-40
-30
-20
-10
0
10
20System Response vs Phase Margin
Frequency, Hz
Res
pons
e, d
B
Phase margins of: 10, 30, 50, 70 and 90 Degrees
0 0.5 1 1.5 2 2.5 3-6
-5
-4
-3
-2
-1
0
1
2
3
4Impulse Response vs Phase Margin
Time, sec
Impu
lse
Res
pons
e
Phase margins of: 10, 30, 50, 70 and 90 Degrees
![Page 21: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/21.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
1/f^2 : The Optimal Servo?
- saturation
+ saturation
No gain(or negative!)
Nom. gain
Reduced gain
sensor out
sensor in
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
f -3
f -<2
f -3
Sensor (or actuator) Saturation:NEW 0dB
UNSTABLE!
Solution: Keep the slope above unity gain to less than 2 powers of frequency.
This is called a “Conditionally Stable” servo
![Page 22: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/22.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCHigh Bandwidth & Sensor Co-Location:
F …
H
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
Gain=1
+0 freq.
180 degreesfor each resonance
4 New unity-gain points!~10x
• Eliminate resonances between sensing and actuation• Damp resonances if possible• Increase resonant frequencies (hard!)• Can use filters for isolated resonances
OK
![Page 23: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/23.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Sensors Lie #1: Representation
Filt & Amp
Consider a temperature-control servo:
Outer housing
Thermistor
Inner housing
Heater windings
Insulation
Heater ground
~60C set pt.
~30C outside
Conclusion:Thermal performance is limited by the balancing ofheat loss and heat input to ~10% (?!?!). This means servo gains higher than ~20-30dB are a waste!!
![Page 24: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/24.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Sensors Lie #2: Sensor Noise
MXout
0
F H
+
Xn
M1
X2
0
F
M2
+
Xn
H
X1
0
(Sensor Out)X1
Log(f)
f 2f 0
0 dB
Conclusions:1. Mass tracks sensor noise2. Sensor output (noise) suppressed
by the loop gain
Conclusions:1. Sensor noise is amplified by 1/f2 below the
M2 resonance!!!!!2. With ‘1/f’ noise in sensor, mass motion
grows by 1,000x each decade (down) to unity gain.
3. Sensor output (noise) suppressed by gain.
Servo wants to minimizethe signal here...
![Page 25: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/25.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Summary of Section 2: Performance
• Integrators can reduce errors to zero, and the more integrators the better this works. However ‘integrator wind-up’ is a difficult problem.
• The importance of modeling systems. Straight-line approximations only go so far…• PID controllers are horribly non-parametric and only work well for free-mass systems
(i.e.: moving stage & motion control). Otherwise they mostly suck. You can do better!• Issues with actually closing the loop on high gain servos – dynamic range and noise!
Things we didn’t cover:
• Showed Phase and Gain Margins are useful parameters to describe performance. There is ALWAYS a Phase Margin, but not always a gain margin.
• Overshoot (~Q) is ~3x at 20pm, ~2x at 30pm, and critically damped at ~60pm.
• Demonstrated there is no ‘right answer’ when it comes to choosing a phase margin.
• Proved that a transfer function slope of 1/f2 is the best performing system possible without introducing conditional stability.
• Showed mechanical resonances always limit bandwidths and that sensor co-location can mitigate that.
• Used the Nichols Plot to show that feedback systems rarely look like simple second-order systems. Will demonstrate in Section 3.
• Sensors are lying bastards. If you need to verify performance, use an independent sensor.
![Page 26: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/26.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Section 3:
Practical Servo Design
![Page 27: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/27.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCHow I Design a Servo:
Step 1:Model, then measure the planttransfer function (G):
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
f -2
f 0
θ = -2 x 90
θ = 0 x 90
Step 3:Measure loop transfer function(GH) to confirm.
G
Step 4:
Close the loop!
Step 2:Design a compensation (H) which pulls the phase down to -180º with enough phase lead at unity gain to give me the desired stability & impulse response.
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
f -2
f 0
H
phase lead
![Page 28: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/28.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCNow With Some More Detail:
M
Fin
Xout
0
F H
L
Remember from start of talk:
Requirements for our servo:• Have a spring-like restoring force• Provide damping• Always bring the sensor to null • Filter noise
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
Compensation
f -1
f 1
f -2
Null sensor Damping(phase lead)
Filter HF noise
H
Always start with physics: F=ma
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
Plant
f -2Units:X/F !
θ = -2 x 90 = -180
G
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
Loop Transfer Function
GH
f -2
f -1f -2
f -4
f -3
![Page 29: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/29.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLCBut is it stable?
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
0dB
Loop Transfer Function
GH
f -2
f -1f -2
f -4
f -3
Gain=1
UNSTABLE?CCW encirclement?
-0 freq.
+0 freq.
Gain=1
STABLE
-0 freq.
+0 freq.
S
X0 Re
Im
0
Real frequency response
-0 freq.
Conjugate frequency response Contour enclosingall RHP poles and zeros
+0 freq.
+∞ freq.
-∞ freq.
0
We have THREE poles at the origin!
Recall:
![Page 30: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/30.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
The Magic Disappearing Resonance:
Consider a mass on a spring:
MXout
Fin
0
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
f -2
f 0
θ = -2 x 90
θ = 0 x 90
G
ω0
Recall:)()(1
)(
)(
)(
sHsG
sG
sIn
sOut
Plant:
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
f -2
f 0
θ = -2 x 90
θ = 0 x 90
GH
ω0
LoopT.F.:
0dB
Closed-loopresponse:
Mag.(dB)
+180
+90
0
-90
-180
Phase
Bode Diagram:
Log(f)
Log(f)
ω0
Original resonance is GONE!
F G
![Page 31: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/31.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
The “Super Spring” Servo
10-1
100
101
102
103
-400
-200
0
200Plant Transfer Function
Frequency, HzM
agni
tude
, dB
10-1
100
101
102
103
-200
-100
0
100
200
Pha
se,
Deg
Frequency, Hz
M
Xout
Sensor
0
F G
M
Mref
Xin
0
Optical Corner Cube
InjectTestSignal
Meas.Output
200Hz
![Page 32: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/32.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Summary of Section 3: Practical Design
• Did I already mention modeling? Good. I needed to at least twice!• Sensor noise can also limit bandwidth. In the Super Spring, it grows like 1/f3!!• So many other things. I hope this is just a start!
Things we didn’t cover:
• Start design with a simple, linear, minimal-phase approximation to get 1/f2.
• Work backwards from the Plant (G) and the desired LTF (GH) to get H.
• Closed-loop systems often show no sign of open-loop resonances.
• You can add 360 to the phase and get the same plot. Phase works on a circle! High-Freq. resonances can sometimes be tamed by ADDING phase!!!
• Use the alternate version of the Nyquist Stability criteria because polar plots make your head hurt too much.
![Page 33: Introduction to Control System Theory For Engineers](https://reader035.vdocuments.us/reader035/viewer/2022081501/56812fab550346895d952fa1/html5/thumbnails/33.jpg)
Dr. Pete NelsonSierra Scientific Solutions, LLC
Thank You