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182 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2 Subsection 10.6.3 describes an example of a globally optimal H 2 model-order reduction of order 4 to order 2. The co-order k = 1 and k = 2 techniques are applied again and for this example the results turn out to be quite different. The co-order k = 2 technique turns out to exhibit the best performance in terms of the H 2 - approximation criterion in comparison with the co-order k = 1 technique. 10.6.1 Example 1 The transfer function H(s) of a continuous time system studied in this example is of order 6. We shall compare two different ways to arrive at an approximation G(s) of order 4: (i) by applying the co-order two technique once, and (ii) by applying the co-order one technique, described in Chapter 9, twice consecutively. The transfer function H(s) is given in an exact format by: −48 − 184s + 216s 2 − 124s 3 +16s 4 +24s 5 H(s) = 445 2916 + 27383 21870 s + 30001 7290 s2 + 3503 486 s3 + 11773 1620 s4 + 182 45 s5 + s . (10.43) 6 The poles of H(s) are δ 1 = − 8 9 − 5 9 i, δ 2 = − 8 9 + 5 9 i, δ 3 = − 2 3 − 1 2 i, δ 4 = − 2 3 + 1 2 i, δ 5 = − 3 5 , δ 6 = − 1 3 , which are all in the open left-half plane Π− . The zeros of H(s) are located at −3.27927, −0.205398, 0.666667 − 1.24722i, 0.666667 + 1.24722i and 1.48466. The impulse response and the Bode diagrams of this system are shown as plots in blue in Figures 10.5 and 10.6. (i) We first apply the co-order two technique and construct the corresponding system of quadratic equations (10.2). This system has 6 equations that each involve the variables x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 , and the parameter ρ 1 . To use the Stetter-Möller matrix method, a quotient space C(ρ 1 )[x 1 ,...,x 6 ]/I(ρ 1 ) is needed with an appropriate basis. The dimension of this quotient space C(ρ 1 )[x 1 ,...,x 6 ]/I(ρ 1 )is2 6 = 64. The basis B contains 64 elements: (1, x 1 ,x 2 ,x 1 x 2 ,x 3 ,x 1 x 3 ,x 2 x 3 ,x 1 x 2 x 3 ,...,x 1 x 2 x 3 x 4 x 5 x 6 ). The linear constraint ã N−1 (x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ) = 0, is computed using the elements of the last row of the inverted Vandermonde matrix as defined in (8.45). It attains the following exact form: ã N−1 = ( 31709313 12721865 + 88691598 12721865 i)x 1 + ( 31709313 12721865 − 88691598 12721865 i)x 2 + −( 18895680 1467661 + 5190480 1467661 i)x 3 − ( 18895680 1467661 − 5190480 1467661 i)x 4 + 6834375 181826 x 5 − 2187 130 x 6 =0. (10.44) The next step is to construct the matrix AãN−1 (ρ 1 ) T , which represents multiplication by the polynomial ã N−1 (x 1 ,...,x 6 ) in the quotient space C(ρ 1 )[x 1 ,...,x 6 ]/ I(ρ 1 ). Here I(ρ 1 ) denotes the ideal generated by the 6 polynomials of the quadratic system of equations. The rows of this matrix are then multiplied elementwise with suitable scalar expressions given in Theorem 10.1 to arrive at the polynomial matrix ÃãN−1 (ρ 1 ) T . The dimension of the polynomial matrix is 64 × 64 and it is sparse and structured: only 2123 of the 4096 entries are non-zero. See Figure 10.1 for a representation of the sparsity and the structure of the matrix Ãã N−1 (ρ 1 ) T .
10.6. EXAMPLES 183 Figure 10.1: Sparsity and structure of the matrix Ãã N−1 (ρ 1 ) T To find the polynomial eigenvalues ρ 1 of this matrix, we rewrite the polynomial eigenvalue problem as a generalized eigenvalue problem (B + ρ 1 C)z = 0 which is linear in ρ 1 and with matrices B and C obtained from Ãã N−1 (ρ 1 ) T according to the Equations (10.12) – (10.14). The matrices B and C are of dimension (N − 1) 2 N × (N − 1) 2 N = 320 × 320. See Figure 10.2 for a representation of the sparsity structure of the matrices B and C. Figure 10.2: Sparsity structure of the matrices B and C Although the matrix pencil B + ρ 1 C is singular and highly ill-conditioned, we can in this case avoid to compute the Kronecker canonical form using a numerical
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Stellingen behorende bij het proefs
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ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
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ALGEBRAIC POLYNOMIAL SYSTEM SOLVING
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For all those I love...
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ii CONTENTS II Global Optimization
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iv CONTENTS 12.2 Directions for fur
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Chapter 1 Introduction 1.1 Motivati
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1.3. RESEARCH QUESTIONS 5 The goal
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1.4. THESIS OUTLINE 7 Jacobi-Davids
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Part I General introduction and bac
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12 CHAPTER 2. ALGEBRAIC BACKGROUND
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Chapter 3 Solving polynomial system
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3.1. SOLVING POLYNOMIAL EQUATIONS I
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3.2. METHODS FOR SOLVING POLYNOMIAL
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3.3. THE STETTER-MÖLLER MATRIX MET
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3.5. EXAMPLE 37 applying the Stette
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3.5. EXAMPLE 39 be: I = 〈13x 2 2
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Chapter 4 Global optimization of mu
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4.1. THE GLOBAL MINIMUM OF A DOMINA
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4.3. AN EXAMPLE 47 Using the matric
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4.3. AN EXAMPLE 49 the Stetter-Möl
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Chapter 5 An nD-systems approach in
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5.1. THE ND-SYSTEM 53 cerned with t
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5.1. THE ND-SYSTEM 55 Example 5.1.
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5.1. THE ND-SYSTEM 57 Figure 5.2: T
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5.3. EFFICIENCY OF THE ND-SYSTEMS A
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Chapter 6 Iterative eigenvalue solv
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6.2. THE JACOBI-DAVIDSON METHOD 83
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6.2. THE JACOBI-DAVIDSON METHOD 85
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6.4. PROJECTION TO STETTER-STRUCTUR
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6.5. A JACOBI-DAVIDSON METHOD FOR C
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6.5. A JACOBI-DAVIDSON METHOD FOR C
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108 CHAPTER 7. NUMERICAL EXPERIMENT
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Page 145: Part III H 2 Model-order Reduction
Page 148 and 149: 134 CHAPTER 7. NUMERICAL EXPERIMENT
Page 150 and 151: 136 CHAPTER 8. H 2 MODEL-ORDER REDU
Page 175 and 176: Chapter 10 H 2 Model-order reductio
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Page 183 and 184: 10.3. THE KRONECKER CANONICAL FORM
Page 193 and 194: 10.4. COMPUTING THE APPROXIMATION G
Page 195: 10.6. EXAMPLES 181 5. Compute the n
Page 199 and 200: 10.6. EXAMPLES 185 Figure 10.3: The
Page 201 and 202: 10.6. EXAMPLES 187 Figure 10.5: Imp
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12.1. CONCLUSIONS 233 solver as des
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12.2. DIRECTIONS FOR FURTHER RESEAR
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Bibliography [1] W. Adams and P. Lo
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BIBLIOGRAPHY 239 [26] A. Cayley. On
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BIBLIOGRAPHY 241 [58] M.E. Hochsten
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BIBLIOGRAPHY 243 [90] S. Prajna, A.
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Appendix A A linearization techniqu
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A.2. LINEARIZATION WITH RESPECT TO
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A.3. EXAMPLE 251 The first step of
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A.3. EXAMPLE 253 where the eigenvec
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256 APPENDIX B. POLYNOMIAL TEST SET
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258 SUMMARY vector. This routine is
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260 SUMMARY globally optimal approx
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262 SAMENVATTING De matrix-vrije im
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264 SAMENVATTING eigenwaarde proble
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List of Publications [1] I.W.M. Ble
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List of Symbols and Abbreviations A
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LIST OF FIGURES 271 7.7 Residual no
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Index H 2 model-order reduction pro
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