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Chapter 8 H-Infinity Synthesis
Xmath Interactive Control Design Module 8-4 ni.com
Figure 8-2. Block Diagram Showing the Basic Setup for H-Infinity Synthesis
Figure 8-2 shows a block diagram with the basic setup for H∞ synthesis
where closed-loop transfer matrix H relates the two exogenous inputs w
1
and w
2
to the two outputs z
1
and z
2
.
The design is based on H, the closed-loop transfer matrix relating the noises
w
1
and w
2
to the signals z
1
and z
2
. H is given by the following equation:
The entries of the closed-loop transfer matrix can be interpreted as the
(normalized) transfer functions from the process and sensor noises to
the actuator and output, respectively.
The singular values of H are shown in the top left plot of the H∞ Synthesis
window.
Central H-Infinity Controller
The controller C is chosen to minimize the γ-entropy of the closed-loop
transfer matrix H, given by:
where σ
1
and σ
2
are the singular values of H(jω).
P(s)
r=0
W(s)
y
C(s)
u
w
1
w
2
z
1
z
2
H
1
1 PC+
-----------------
ρPC– ρυC–
PW νPCW–
=
I
γ
H()
γ
2
π
----
1
1 σ
1
ω()γ⁄()
2
–
------------------------------------
1
1 σ
2
ω()γ⁄()
2
–
------------------------------------log+log
⎝⎠
⎜⎟
⎛⎞
dω
0
∞
∫
=
- 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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