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156 CHAPTER 9. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-1 Following the approach of [50], all the 1024 tuples x 1 ,...,x 10 are determined from the mutual eigenvectors of the matrices A T x 1 ,...,A T x 10 . Each such tuple yields an approximation of order N − 1 which is either feasible or infeasible. By substituting these tuples into the polynomial V H , we compute the H 2 -criterion value for each feasible solution. The solution with the smallest real H 2 -criterion value is used to compute the polynomials b(s) and a(s) which yield the feasible globally optimal stable approximation G(s) = b(s) a(s) of order 9 with an H 2-criterion value of 0.000489593: G(s) = 1.69597 + 260.566 s + 1254.59 s2 + 899.268 s 3 + 1246.84 s 4 + ... 0.0538714 + 17.9585 s + 147.756 s 2 + 138.523 s 3 + 252.714 s 4 + ... ... +181.474 s 5 + 276.659 s 6 − 1.22433 s 7 +15.7699 s 8 ... +99.6209 s 5 +71.4287 s 6 +19.6358 s 7 +5.15532 s 8 + s 9 . (9.8) The solution used to compute this approximation is: x 1 = 78.6020 x 6 = 0.000289407 x 2 = −0.180875 − 0.343180i x 7 = −0.00193041 + 0.00371561i x 3 = −0.180875 + 0.343180i x 8 = −0.00193041 − 0.00371561i x 4 = −0.00155184 − 0.00377204i x 9 = 5.99264 × 10 −6 x 5 = −0.00155184 + 0.00377204i x 10 = 2.91654 × 10 −6 (9.9) The Hankel singular values of the approximation G(s) are now changed to the values: 9.68, 8, 7,...,1. Note that the coefficients of the approximation G(s) are very similar to the coefficients of the original problem H(s). Effectively, one pole/zero pair of the original transfer function at s = 0 is removed by the model-order reduction algorithm and has left the other poles and zeros nearly unchanged. Another computational approach is by working with the matrix A T V H . First, we explicitly construct this matrix and compute all its eigenvalues. Second, we do not use the explicit matrix A T V H but we use the matrix-free approach for the associated operator of the matrix A T V H together with an iterative eigenvalue solver and an nDsystem, to compute only the smallest real eigenvalue of A T V H which immediately will lead to a feasible globally optimal approximation of order 9. To construct the matrix A T V H explicitly and to compute all its eigenvalues and eigenvectors is computationally a hard job: it takes, respectively, 27164 and 82 seconds. Furthermore, it takes 275 MB of internal memory to store all the data. The matrix A T V H has 45 real eigenvalues, including the zero eigenvalue. The smallest positive real eigenvalue λ min is computed as 0.000489593. This smallest real eigenvalue yields the solution x 1 ,...,x 10 since it is read off from the corresponding eigenvector. The solution x 1 ,...,x 10 computed here turns out to be the same solution as found before in (9.9). This leads to the same globally optimal approximation G(s) of order 9 as in (9.8). The H 2 -criterion is again 0.000489593, which exactly equals the eigenvalue λ min =0.000489593. In Figure 9.1 all the eigenvalues of the matrix A T V H , excluding the zero eigenvalue (as it will not lead to a feasible approximation), are plotted in the complex plane.
9.3. EXAMPLE 157 The best stable approximation G(s) corresponds to the real eigenvalue which is as small as possible. Figure 9.2 shows the eigenvalues of the matrix A T V H near the real axis around the origin. Figure 9.1: All eigenvalues of the matrix A T V H Figure 9.2: Zoomed in on the smallest real eigenvalues of the matrix A T V H The matrices involved in these computations are all sparse. All the matrices contain 1024 2 = 1048576 elements of which 98415 are non-zero in the matrices A T x i
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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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Page 150 and 151: 136 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
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11.3. KRONECKER CANONICAL FORM OF A
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11.4. COMPUTING THE APPROXIMATION G
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11.5. EXAMPLE 223 AãN−2 (ρ 1 ,
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11.5. EXAMPLE 225 of both matrices
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11.5. EXAMPLE 227 of solutions (ρ
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Chapter 12 Conclusions & directions
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12.1. CONCLUSIONS 231 used first. T
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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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