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TM
NI MATRIXx
TM
Xmath Model Reduction Module
Xmath Model Reduction Module
April 2007
370755C-01
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Important Information Warranty The media on which you receive National Instruments software are warranted not to fail to execute programming instructions, due to defects in materials and workmanship, for a period of 90 days from date of shipment, as evidenced by receipts or other documentation. National Instruments will, at its option, repair or replace software media that do not execute programming instructions if National Instruments receives notice of such defects during the warranty per
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Conventions The following conventions are used in this manual: [ ] Square brackets enclose optional items—for example, [response]. Square brackets also cite bibliographic references. » The » symbol leads you through nested menu items and dialog box options to a final action. The sequence File»Page Setup»Options directs you to pull down the File menu, select the Page Setup item, and select Options from the last dialog box. This icon denotes a note, which alerts you to important information. b
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Contents Chapter 1 Introduction Using This Manual.........................................................................................................1-1 Document Organization...................................................................................1-1 Bibliographic References ................................................................................1-2 Commonly Used Nomenclature ......................................................................1-2 Conventions.............
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Contents Onepass Algorithm ......................................................................................... 2-18 Multipass Algorithm ...................................................................................... 2-20 Discrete-Time Systems ................................................................................... 2-21 Impulse Response Error .................................................................................. 2-22 Unstable System Approximation ..............
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Contents fracred( ) ........................................................................................................................4-15 Restrictions......................................................................................................4-15 Defining and Reducing a Controller................................................................4-16 Algorithm ........................................................................................................4-18 Additional Bac
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1 Introduction This chapter starts with an outline of the manual and some useful notes. It also provides an overview of the Model Reduction Module, describes the functions in this module, and introduces nomenclature and concepts used throughout this manual. Using This Manual This manual describes the Model Reduction Module (MRM), which provides a collection of tools for reducing the order of systems. Readers who are not familiar with Parameter Dependent Matrices (PDMs) should consult the Xm
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Chapter 1 Introduction � Chapter 5, Utilities, describes three utility functions: hankelsv( ), stable( ), and compare( ). � Chapter 6, Tutorial, illustrates a number of the MRM functions and their underlying ideas. Bibliographic References Throughout this document, bibliographic references are cited with bracketed entries. For example, a reference to [VODM1] corresponds to a paper published by Van Overschee and De Moor. For a table of bibliographic references, refer to Appendix A, Bibliog
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Chapter 1 Introduction Related Publications For a complete list of MATRIXx publications, refer to Chapter 2, MATRIXx Publications, Online Help, and Customer Support, of the MATRIXx Getting Started Guide. The following documents are particularly useful for topics covered in this manual: � MATRIXx Getting Started Guide • Xmath User Guide � Control Design Module � Interactive Control Design Module � Interactive System Identification Module, Part 1 � Interactive System Identification Module, Part
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Chapter 1 Introduction As shown in Figure 1-1, functions are provided to handle four broad tasks: � Model reduction with additive errors � Model reduction with multiplicative errors � Model reduction with frequency weighting of an additive error, including controller reduction � Utility functions Functions Additive Error Multiplicative Frequency Weighted Model Reduction Model Reduction Model Reduction balmoore redschur ophank bst wtbalance truncate mulhank fracred balance mreduce Utility Functi
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Chapter 1 Introduction Certain restrictions regarding minimality and stability are required of the input data, and are summarized in Table 1-1. Table 1-1. MRM Restrictions balance( ) A stable, minimal system balmoore ( ) A state-space system must be stable and minimal, having at least one input, output, and state bst( ) A state-space system must be linear, continuous-time, and stable, with full rank along the jω-axis, including infinity compare( ) Must be a state-space system fracred( ) A s
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Chapter 1 Introduction � L approximation, in which the L norm of impulse response error (or, 2 2 by Parseval’s theorem, the L norm of the transfer-function error along 2 the imaginary axis) serves as the error measure � Markov parameter or impulse response matching, moment matching, covariance matching, and combinations of these, for example, q-COVER approximation � Controller reduction using canonical interactions, balanced Riccati equations, and certain balanced controller reduction algor
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Chapter 1 Introduction � An inequality or bound is tight if it can be met in practice, for example 1 + logxx – ≤ 0 is tight because the inequality becomes an equality for x =1. Again, if F(jω) denotes the Fourier transform of some ft ()∈ L , the 2 Heisenberg inequality states, 2 ft () dt ∫ --- ---- ---- ---- ---- --- ---- ---- ---- --- ---- ---- ---- ---- --- ---- ---- ---- ---- --- ---- ---- ---- --- - ≤ 4π 12 ⁄ 12 ⁄ 2 2 2 2 t ft () dt ω Fj() ω dω ∫ ∫ 2 and the bound is tight since it is attai
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Chapter 1 Introduction � The controllability grammian is also E[x(t)x′(t)] when the system · – x = Ax + Bw has been excited from time ∞ by zero mean white noise withEwt[] ()w′() s = Iδ() ts – . � The observability grammian can be thought of as measuring the information contained in the output concerning an initial state. · If x = Ax, y = Cxwith x(0) = x then: 0 ∞ y′() t yt ()dt = x′ Qx 0 0 ∫ 0 Systems that are easy to observe correspond to Q with large eigenvalues, and thus large output en
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Chapter 1 Introduction � Suppose the transfer-function matrix corresponds to a discrete-time system, with state variable dimension n. Then the infinite Hankel matrix, 2 CB CAB CA B 2 CAB CA B H = 2 CA B has for its singular values the n nonzero Hankel singular values, together with an infinite number of zero singular values. The Hankel singular values of a (stable) all pass system (or all pass matrix) are all 1. Slightly different procedures are used for calculating the Hankel singular valu
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Chapter 1 Introduction Internally Balanced Realizations Suppose that a realization of a transfer-function matrix has the controllability and observability grammian property that P = Q = Σ for some diagonal Σ. Then the realization is termed internally balanced. Notice that the diagonal entries σ of Σ are square roots of the eigenvalues of PQ, i that is, they are the Hankel singular values. Often the entries of Σ are assumed ordered with σ ≥ σ . i i+1 As noted in the discussion of grammians,
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Chapter 1 Introduction This is almost the algorithm set out in Section II of [LHPW87]. The one difference (and it is minor) is that in [LHPW87], lower triangular Cholesky 1/2 1/2 factors of P and Q are used, in place of U S and U S in forming H c c O O in step 2. The grammians themselves need never be formed, as these Cholesky factors can be computed directly from A, B, and C in both continuous and discrete time; this, however, is not done in balmoore. The algorithm has the property that: –
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Chapter 1 Introduction and also: –1 Reλ (A )<0 and Reλ() A – A A A < 0. i 11 12 22 21 i 22 Usually, we expect that, –1 Reλ() A « Reλ() A – A A A i 22 i 11 12 22 21 in the sense that the intuitive argument hinges on this, but it is not necessary. · Then a singular perturbation is obtained by replacing x by zero; this 2 means that: –1 –1 A x++ A x B u = 0 x = –A A x – A B u or 21 1 22 2 2 2 22 21 1 22 2 Accordingly, –1 –1 · x =() A = A A A x +() B – A A B u 1 11 12 22 21 1 1 12 22
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Chapter 1 Introduction Similar considerations govern the discrete-time problem, where, x() k + 1 x () k B A A 1 1 1 11 12 = + uk () x() k + 1 x () k B A A 2 2 2 21 22 x () k 1 yk () = + Du() k C C 1 2 x () k 2 can be approximated by: –1 x() k + 1 =[] A + A() IA – A x () k + 1 11 12 22 21 1 –1 [] B + A() IA – B uk () 1 12 22 2 –1 y =[] C + C() IA – A x () k + k 1 2 22 21 1 –1 [] DC +() IA – B uk () 2 22 2 mreduce( ) can carry out singular perturbation. For further discussion, refer to Chapter 2,
