Document Outline

• Xmath Model Reduction Module
• Support
  • Worldwide Technical Support and Product Information
  • National Instruments Corporate Headquarters
  • Worldwide Offices
• Important Information
  • Warranty
  • Copyright
  • Trademarks
  • Patents
  • WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS
• Conventions
• Contents
• Chapter 1 Introduction
  • Using This Manual
    • Document Organization
    • Bibliographic References
    • Commonly Used Nomenclature
    • Conventions
    • Related Publications
    • MATRIXx Help
  • Overview
    • Functions
      • Figure 1-1. MRM Function
      • Table 1-1. MRM Restrictions
    • Nomenclature
  • Commonly Used Concepts
    • Controllability and Observability Grammians
    • Hankel Singular Values
      • Table 1-2. Calculating Hankel Singular Values
    • Internally Balanced Realizations
    • Singular Perturbation
    • Spectral Factorization
  • Low Order Controller Design Through Order Reduction
    • Figure 1-2. Low Order Controller Design for a High Order Plant
• Chapter 2 Additive Error Reduction
  • Introduction
  • Truncation of Balanced Realizations
  • Reduction Through Balanced Realization Truncation
    • Figure 2-1. Different Approaches for Obtaining the Same Reduced Order Model
  • Singular Perturbation of Balanced Realization
  • Hankel Norm Approximation
  • balmoore( )
    • Related Functions
  • truncate( )
    • Related Functions
  • redschur( )
    • Algorithm
    • Related Functions
  • ophank( )
    • Restriction
    • Algorithm
    • Behaviors
      • Table 2-1. Orders of G
    • Onepass Algorithm
    • Multipass Algorithm
    • Discrete-Time Systems
    • Impulse Response Error
    • Unstable System Approximation
    • Related Functions
• Chapter 3 Multiplicative Error Reduction
  • Selecting Multiplicative Error Reduction
    • Figure 3-1. Controller C(s) Designed for Multiplicative Error Reduction
    • Multiplicative Robustness Result
  • bst( )
    • Restrictions
    • Algorithm
    • Algorithm with the Keywords right and left
    • Securing Zero Error at DC
    • Hankel Singular Values of Phase Matrix of Gr
    • Further Error Bounds
    • Reduction of Minimum Phase, Unstable G
    • Imaginary Axis Zeros (Including Zeros at ∞)
      • Figure 3-2. Bilinear Mapping from G(s) to (Gs)(Case 1)
      • Figure 3-3. Bilinear Mapping from G(s) to (Gs)(Case 2)
    • Related Functions
  • mulhank( )
    • Restrictions
    • Algorithm
    • right and left
    • Consequences of Step 5 and Justification of Step 6
    • Error Bounds
    • Imaginary Axis Zeros (Including Zeros at ∞)
    • Related Functions
• Chapter 4 Frequency-Weighted Error Reduction
  • Introduction
    • Controller Reduction
    • Controller Robustness Result
    • Fractional Representations
      • Figure 4-1. Controller Representation Through Stable Fractions
      • Figure 4-2. C(s) Implemented to Display Right MFD Representation
      • Figure 4-3. C(s) Implemented to Display Left MFD Representation
      • Figure 4-4. Redrawn; Individual Signal Paths as Vector Paths
  • wtbalance( )
    • Table 4-1. Types versus Error Measures
    • Table 4-2. Error Measure Interpretation for wtbalance
    • Algorithm
    • Related Functions
  • fracred( )
    • Restrictions
    • Defining and Reducing a Controller
      • Figure 4-5. Internal Structure of Controller
    • Algorithm
    • Additional Background
      • Table 4-3. HSVs for an Eighth Order Controller
    • Related Functions
• Chapter 5 Utilities
  • hankelsv( )
    • Related Functions
  • stable( )
    • Algorithm
    • Related Functions
  • compare( )
• Chapter 6 Tutorial
  • Plant and Full-Order Controller
    • Figure 6-1. Loop Gain Constraints
    • Figure 6-2. Frequency Response for Plant, Compensator, and Compensated Plant
  • Controller Reduction
    • Figure 6-3. Open-Loop Gain Using redschur
    • Figure 6-4. Closed-Loop Gain with redschur
    • Figure 6-5. Step Response with redschur
    • ophank( )
      • Figure 6-6. Open-Loop Gain Using ophank
      • Figure 6-7. Closed-Loop Gain with ophank
      • Figure 6-8. Step Response with ophank
    • wtbalance
      • Figure 6-9. Open-Loop Gain with wtbalance
      • Figure 6-10. Closed-Loop Gain with wtbalance
      • Figure 6-11. Step Response with wtbalance
      • Figure 6-12. Frequency Response of the Weight V(jω)
      • Figure 6-13. Open-Loop Gain from wtbalance with "input spec"
      • Figure 6-14. System Singular Values of wtbalance with "input spec"
      • Figure 6-15. Step Response of wtbalance with "input spec"
    • fracred
      • Figure 6-16. Open-Loop Gain Using fracred
      • Figure 6-17. Closed-Loop Response with fracred
      • Figure 6-18. Step Response with fracred
• Appendix A Bibliography
• Appendix B Technical Support and Professional Services
• Index
  • Symbols
  • A-F
  • G-S
  • T-W