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National Instruments Xmath manual

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NI MATRIXxTM

XmathTM Model Reduction Module

Xmath Model Reduction Module

April 2007

370755C-01

Contents

Main Page Important Information Warranty Copyright Trademarks Patents Conventions Contents Chapter 1 Introduction Chapter 2 Additive Error Reduction Chapter 3 Multiplicative Error Reduction Chapter 4 Frequency-Weighted Error Reduction Page Introduction Using This Manual Document Organization Bibliographic References Commonly Used Nomenclature Conventions Related Publications MATRIXx Help Overview Functions Page Nomenclature Commonly Used Concepts Controllability and Observability Grammians Hankel Singular Values Page Internally Balanced Realizations Singular Perturbation Page Spectral Factorization Page Low Order Controller Design Through Order Reduction Page Additive Error Reduction Introduction Truncation of Balanced Realizations Page Reduction Through Balanced Realization Truncation Singular Perturbation of Balanced Realization Hankel Norm Approximation Page balmoore( ) Page Page truncate( ) redschur( ) Page ophank( ) Restriction Behaviors Page Page Onepass Algorithm Page Multipass Algorithm Discrete-Time Systems Impulse Response Error Unstable System Approximation Multiplicative Error Reduction Selecting Multiplicative Error Reduction CG Multiplicative Robustness Result bst( ) Page Algorithm with the Keywords right and left Page Page 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 ) G G s() Gs() valuesvalues -a a b s() Gs() valuesvalues Page Page mulhank( ) right and left Page Page Consequences of Step 5 and Justification of Step 6 Page Error Bounds Imaginary Axis Zeros (Including Zeros at ) Page Page Page Frequency-Weighted Error Reduction Introduction Controller Reduction Controller Robustness Result Page Page Fractional Representations e d --- n d ---Cs() Page Page KrsI A KEC+() Ps() B K + - K +() +Ps()I s() Cs() P K + wtbalance( ) Page Page Page Page fracred( ) Defining and Reducing a Controller Page Page Page Additional Background Page Utilities hankelsv( ) stable( ) Page compare( ) Tutorial Plant and Full-Order Controller 40 dB/decade Frequency (rad/sec) Loop Gain (dB) 0.3 0.07 40 dB/decade Page Page Controller Reduction Page Page Page ophank( ) Page Page wtbalance Page Page Page Page Generate Figure 6 -13 : Page Page fracred Page Page Page A Bibliography Page Page Page Page B Technical Support and Professional Services Index Symbols A B C G H I K L