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Chapter 6 Pole Place Synthesis
© National Instruments Corporation 6-3 Xmath Interactive Control Design Module
where
d
p
(s)=s
n
+ a
1
s
n–1
+ a
2
s
n–2
+...+a
n
n
p
(s)=b
0
s
n
+ b
1
s
n–1
+...+ab
n
Notice that the order of the plant is n, and allow the possibility that the plant
transfer function is not strictly proper; that is, the plant can have as many
zeros as poles.
Normal Mode
In normal mode, the order (number of poles) of the controller is fixed and
equal to n (the order of the plant), so there are a total of 2n closed-loop
poles. In this case, the 2n degrees of freedom in the closed-loop poles
exactly determine the controller transfer function, which also has 2n
degrees of freedom.
In normal mode, the controller transfer function has order n and is strictly
proper:
C(s)=n
c
(s)/d
c
(s)
where
d
c
(s)=s
n
+ x
1
s
n–1
+ x
2
s
n–2
+...+x
n
n
c
(s)=y
1
s
n–1
+ y
2
s
n–2
+...+2y
n
Therefore, the closed-loop characteristic polynomial has degree 2n:
where λ
1
, …, λ
2n
are the closed-loop poles chosen by the user.
χ s() n
c
s()n
p
s() d
c
s()d
p
s()+=
s λ
1
–()s λ
2
–()…s λ
2n
–()=
s
2n
α
1
s
2n 1–
…α
2
n+++()=
- NI MATRIXx 1
- Important Information 3
- Conventions 4
- Chapter 3 6
- Chapter 4 6
- Chapter 11 9
- Chapter 12 9
- LQG/H-Infinity Synthesis 9
- Contents 10
- Introduction 11
- Chapter 1 Introduction 12
- Commonly-Used Nomenclature 13
- Related Publications 13
- ICDM Overview 14
- Introduction to SISO Design 16
- Overview of ICDM 18
- ICDM Main Window 19
- PID Synthesis Window 19
- Root Locus Synthesis Window 19
- Pole Place Synthesis Window 19
- LQG Synthesis Window 20
- H-Infinity Synthesis Window 20
- History Window 20
- Alternate Plant Window 20
- Origin of the Controller 21
- Using ICDM 24
- General Plotting Features 26
- Ranges of Plots and Sliders 26
- Data-Viewing Plots 27
- Interactive Plot Re-ranging 28
- Editing Poles and Zeros 28
- Complex Poles and Zeros 29
- Isolated Real Poles and Zeros 29
- Communicating with Xmath 33
- Most Common Usage 34
- Default Plants 34
- Chapter 3 ICDM Main Window 35
- ICDM Plots 36
- Selecting Plots 36
- Ranges of Plots 37
- Plot Magnify Windows 38
- Edit Menu 40
- PID Synthesis 41
- Chapter 4 PID Synthesis 42
- Ranges of Sliders and Plots 45
- Integral Term Normalization 45
- Derivative Term Normalization 46
- Rolloff Term Normalization 46
- Root Locus Synthesis 47
- Terminology 49
- Plotting Styles 50
- Phase Contours 51
- Magnitude Contours 51
- Manipulating the Parameters 52
- Adding a Pole-Zero Pair 53
- Deleting Pole-Zero Pairs 53
- Pole Place Synthesis 56
- Pole Place Modes 57
- Normal Mode 58
- Integral Action Mode 59
- State-Space Interpretation 60
- Opening the Pole Place Window 60
- Time and Frequency Scaling 60
- Butterworth Configuration 61
- Editing the Closed-Loop Poles 61
- Slider and Plot Ranges 61
- LQG Synthesis 62
- Chapter 7 LQG Synthesis 63
- Synthesis Modes 64
- Output Weight Editing 67
- R in the LQG window 69
- window 69
- H-Infinity Synthesis 70
- Opening the Synthesis Window 72
- Central H-Infinity Controller 73
- Infeasible Parameter Values 75
- Editing the Comments 78
- Deleting History List Entries 79
- Cycling Through Designs 79
- Using the History List 80
- Normalization 84
- Adding Unmodeled Dynamics 85
- Ranges of Sliders and Plot 86
- Introduction to MIMO Design 87
- Transfer Functions 88
- ICDM MIMO Windows 91
- MIMO Plot Window 92
- LQG/H-Infinity Weights Window 96
- Decay Rate Window 99
- H-Infinity Performance Window 99
- Frequency Weights Window 100
- Setup and Terminology 102
- Integral Action 105
- Exponential Time Weighting 106
- Weight Editing 106
- How to Select w, u, y, and z 107
- H-Infinity Solution 108
- Main Window 110
- in the LQG window 111
- Multi-Loop Synthesis 112
- Setup and Synthesis Method 114
- Graphical Editor 118
- Editing and Deleting Loops 119
- Loop Gain Magnitude and Phase 119
- Using an Xmath GUI Tool 120
- GUI Functions 122
- GUI Objects 122
- and the left mouse button 123
- Technical Support and 126
- Professional Services 126
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