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190 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2 Of the 321 eigenvalues ρ 1 , only 26 are real-valued. The values for x 1 ,...,x 7 are read off from the corresponding eigenvec<strong>to</strong>rs. The 26 tuples (x 1 ,...,x 7 ,ρ 1 ) are substituted in<strong>to</strong> the available third order polynomial V H (x 1 ,...,x 7 ,ρ 1 ), which coincides with the H 2 -criterion at the solutions of the system of equations. It turns out that there is only one solution which yields a real and feasible approximation G(s) of order five: ρ 1 =2.03804, x 1 = −0.0583908, x 2 =0.0960792 − 0.0601039i, x 3 = 0.0960792 + 0.0601039i, x 4 = −0.0207708, x 5 = 0.0179519 − 0.0274311i, x 6 = 0.0179519 + 0.0274311i, x 7 = −0.00383800 with an H 2 -criterion value of 0.228778. This system G(s) is the H 2 globally optimal approximation of order 5 of the original system H(s) of order 7 and is computed as: 6.109 − 12.02s +14.93s 2 − 14.41s 3 +1.257s 4 G(s) = 0.007068 + 0.10286s +0.4956s 2 +1.210s 3 +1.573s 4 + s 5 . (10.54) The system G(s) has its poles at −0.366841± 0.192807i, −0.358033± 0.454934i, and −0.122789 and its zeros at 0.161729± 0.781139i, 0.734038, and 10.4062. Computing the globally optimal approximation of order 5 of the original system H(s) of order 7 by applying the co-order one technique twice, results in virtually the same (numerical) approximation G(s). When comparing the poles of the system H(s) and G(s) one finds that effectively two real poles and two real zeros are removed and the remaining poles and zeros are only adapted slightly. Figure 10.8 shows the poles of the transfer function H(s) and its approximation G(s) obtained by the co-order k = 2 technique. Figure 10.9 shows the poles of the transfer function H(s) and the poles of its approximations G 1 (s) of order 6 and G 2 (s) of order 5 obtained by co-order k = 1 techniques. For completeness the impulse responses and Bode diagrams of the systems H(s) (blue) and G(s) (green) are given in the Figures 10.10 and 10.11, which show highquality approximations. 10.6.3 Example 3 In this example a system H(s) of order 4 is given which has four complex poles. The globally optimal approximations of order 2 are computed by applying the co-order one technique twice, which results in the systems G 1 (s), and G 2 (s), and the co-order two technique once, which results in the system G(s). The results are given in Table 10.1 which presents the locations of the poles and zeros, and the H 2 -criterion values (V H = ||H(s) − G(s)|| 2 H 2 ) of the systems H(s), G 1 (s), G 2 (s) and G(s). Applying the co-order one technique twice, yields a significantly worse performance than applying the co-order two technique once: the H 2 -error of G(s) is9.612, whereas the H 2 -error of G 2 (s) is10.92. Moreover, the system G 2 (s) has a completely different behavior from the system G(s), as shown in Figures 10.13 and 10.14, which show the impulse responses and the Bode diagrams of the systems H(s) (blue), G 2 (s) (green), and G(s) (red).
10.6. EXAMPLES 191 Figure 10.8: Poles of H(s) and its approximation G(s) computed by the co-order two technique Figure 10.9: Poles of H(s) and its approximations G 1 (s) and G 2 (s) computed by repeated application of the co-order one technique
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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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Part III H 2 Model-order Reduction
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136 CHAPTER 8. H 2 MODEL-ORDER REDU
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138 CHAPTER 8. H 2 MODEL-ORDER REDU
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Page 181 and 182: 10.2. LINEARIZING A POLYNOMIAL EIGE
Page 183 and 184: 10.3. THE KRONECKER CANONICAL FORM
Page 193 and 194: 10.4. COMPUTING THE APPROXIMATION G
Page 195 and 196: 10.6. EXAMPLES 181 5. Compute the n
Page 197 and 198: 10.6. EXAMPLES 183 Figure 10.1: Spa
Page 199 and 200: 10.6. EXAMPLES 185 Figure 10.3: The
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Page 239 and 240: 11.5. EXAMPLE 225 of both matrices
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Page 251 and 252: Bibliography [1] W. Adams and P. Lo
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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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