Benutzerhandbuch für National Instruments ni matrixx xmath

Inhaltsverzeichnis

• Xmath Model Reduction Module

  1
• Support

  2
  • Worldwide Technical Support and Product Information

    2
  • National Instruments Corporate Headquarters

    2
  • Worldwide Offices

    2
• Important Information

  3
  • Warranty

    3
  • Copyright

    3
  • Trademarks

    3
  • Patents

    3
  • WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS

    3
• Conventions

  4
• Contents

  5
• Chapter 1 Introduction

  8
  • Using This Manual

    8
    • Document Organization

      8
    • Bibliographic References

      9
    • Commonly Used Nomenclature

      9
    • Conventions

      9
    • Related Publications

      10
    • MATRIXx Help

      10
  • Overview

    10
    • Functions

      11
      • Figure 1-1. MRM Function

        11
      • Table 1-1. MRM Restrictions

        12
    • Nomenclature

      13
  • Commonly Used Concepts

    14
    • Controllability and Observability Grammians

      14
    • Hankel Singular Values

      15
      • Table 1-2. Calculating Hankel Singular Values

        16
    • Internally Balanced Realizations

      17
    • Singular Perturbation

      18
    • Spectral Factorization

      20
  • Low Order Controller Design Through Order Reduction

    22
    • Figure 1-2. Low Order Controller Design for a High Order Plant

      22
• Chapter 2 Additive Error Reduction

  24
  • Introduction

    24
  • Truncation of Balanced Realizations

    25
  • Reduction Through Balanced Realization Truncation

    27
    • Figure 2-1. Different Approaches for Obtaining the Same Reduced Order Model

      28
  • Singular Perturbation of Balanced Realization

    28
  • Hankel Norm Approximation

    29
  • balmoore( )

    31
    • Related Functions

      34
  • truncate( )

    34
    • Related Functions

      34
  • redschur( )

    35
    • Algorithm

      35
    • Related Functions

      37
  • ophank( )

    37
    • Restriction

      37
    • Algorithm

      38
    • Behaviors

      38
      • Table 2-1. Orders of G

        39
    • Onepass Algorithm

      41
    • Multipass Algorithm

      43
    • Discrete-Time Systems

      44
    • Impulse Response Error

      45
    • Unstable System Approximation

      46
    • Related Functions

      46
• Chapter 3 Multiplicative Error Reduction

  47
  • Selecting Multiplicative Error Reduction

    47
    • Figure 3-1. Controller C(s) Designed for Multiplicative Error Reduction

      47
    • Multiplicative Robustness Result

      48
  • bst( )

    49
    • Restrictions

      50
    • Algorithm

      50
    • Algorithm with the Keywords right and left

      51
    • Securing Zero Error at DC

      54
    • Hankel Singular Values of Phase Matrix of Gr

      55
    • Further Error Bounds

      55
    • Reduction of Minimum Phase, Unstable G

      55
    • Imaginary Axis Zeros (Including Zeros at ∞)

      56
      • Figure 3-2. Bilinear Mapping from G(s) to (Gs)(Case 1)

        57
      • Figure 3-3. Bilinear Mapping from G(s) to (Gs)(Case 2)

        57
    • Related Functions

      60
  • mulhank( )

    60
    • Restrictions

      60
    • Algorithm

      60
    • right and left

      61
    • Consequences of Step 5 and Justification of Step 6

      64
    • Error Bounds

      66
    • Imaginary Axis Zeros (Including Zeros at ∞)

      67
    • Related Functions

      70
• Chapter 4 Frequency-Weighted Error Reduction

  71
  • Introduction

    71
    • Controller Reduction

      72
    • Controller Robustness Result

      72
    • Fractional Representations

      75
      • Figure 4-1. Controller Representation Through Stable Fractions

        75
      • Figure 4-2. C(s) Implemented to Display Right MFD Representation

        77
      • Figure 4-3. C(s) Implemented to Display Left MFD Representation

        78
      • Figure 4-4. Redrawn; Individual Signal Paths as Vector Paths

        79
  • wtbalance( )

    80
    • Table 4-1. Types versus Error Measures

      81
    • Table 4-2. Error Measure Interpretation for wtbalance

      82
    • Algorithm

      82
    • Related Functions

      85
  • fracred( )

    85
    • Restrictions

      85
    • Defining and Reducing a Controller

      86
      • Figure 4-5. Internal Structure of Controller

        87
    • Algorithm

      88
    • Additional Background

      90
      • Table 4-3. HSVs for an Eighth Order Controller

        91
    • Related Functions

      91
• Chapter 5 Utilities

  92
  • hankelsv( )

    92
    • Related Functions

      93
  • stable( )

    93
    • Algorithm

      93
    • Related Functions

      95
  • compare( )

    95
• Chapter 6 Tutorial

  96
  • Plant and Full-Order Controller

    96
    • Figure 6-1. Loop Gain Constraints

      97
    • Figure 6-2. Frequency Response for Plant, Compensator, and Compensated Plant

      99
  • Controller Reduction

    100
    • Figure 6-3. Open-Loop Gain Using redschur

      100
    • Figure 6-4. Closed-Loop Gain with redschur

      102
    • Figure 6-5. Step Response with redschur

      103
    • ophank( )

      104
      • Figure 6-6. Open-Loop Gain Using ophank

        104
      • Figure 6-7. Closed-Loop Gain with ophank

        105
      • Figure 6-8. Step Response with ophank

        106
    • wtbalance

      107
      • Figure 6-9. Open-Loop Gain with wtbalance

        108
      • Figure 6-10. Closed-Loop Gain with wtbalance

        109
      • Figure 6-11. Step Response with wtbalance

        110
      • Figure 6-12. Frequency Response of the Weight V(jω)

        111
      • Figure 6-13. Open-Loop Gain from wtbalance with "input spec"

        112
      • Figure 6-14. System Singular Values of wtbalance with "input spec"

        113
      • Figure 6-15. Step Response of wtbalance with "input spec"

        114
    • fracred

      115
      • Figure 6-16. Open-Loop Gain Using fracred

        115
      • Figure 6-17. Closed-Loop Response with fracred

        116
      • Figure 6-18. Step Response with fracred

        117
• Appendix A Bibliography

  119
• Appendix B Technical Support and Professional Services

  124
• Index

  125
  • Symbols

    125
  • A-F

    125
  • G-S

    126
  • T-W

    127