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140 CHAPTER 8. H 2 MODEL-ORDER REDUCTION Because of the order preserving isometry between the discrete time and continuous time cases, we can restrict to the continuous time case without loss of generality. In this thesis we shall restrict to real transfer functions in continuous time, both for the given system H(s) and for the candidate approximations G(s). Following the approach and notation of [50], we start from a given function: H(s) = e(s) d(s) (8.12) of McMillan degree N ≥ 1, in which d(s) =s N + d N−1 s N−1 + ... + d 1 s + d 0 is a monic real polynomial of degree N that is Hurwitz (i.e., all its zeros are in Π − ) and e(s) =e N−1 s N−1 + ...+ e 1 s + e 0 is a real polynomial of degree ≤ N − 1 that is co-prime with d(s). I.e., it does not identically vanish and it does not have any zeros in common with d(s). For a given approximation of order n
8.3. A REPARAMETERIZATION OF THE MODEL REDUCTION PROBLEM 141 Remark 8.3. Note that in the MIMO case, the corresponding H 2 -inner product is obtained by applying the trace operator to the expressions given above for the SISO case. The state-space formulation needs to be adapted in a similar fashion, by including the trace operator. 8.3 A reparameterization of the H 2 model-order reduction problem From an algebraic point of view, note that the squared H 2 -norm of H(s) − G(s) in (8.13) can be worked out to constitute a (rather complicated) rational expression in terms of the 2n coefficients of a(s) and b(s), when G(s) = b(s) a(s) . (8.15) In Section 8.4 of this chapter we show how this criterion can be evaluated at some special points of interest by means of a third order homogeneous polynomial. From a geometric point of view it is appropriate to regard the problem of minimizing the criterion function in (8.13) as a projection problem within the Hardy space H 2 (Π + ) of functions that are analytic on the open right-half plane Π + and which satisfy the property that their H 2 -norm attains a well-defined finite value. Proceeding as in [50], it is well known that the space S n constitutes a smooth submanifold of H 2 (Π + ). Therefore, if b(s)/a(s) is an optimal approximation in S n to e(s)/d(s) with respect to the H 2 -norm, then b(s)/a(s) is an orthogonal projection of e(s)/d(s) ontoS n . The difference e(s) d(s) − b(s) a(s) (8.16) is perpendicular to the tangent space to S n at the point G(s). The tangent space to S n at G(s) is easily computed as the set {t(s)/a(s) 2 | t(s) polynomial of degree ≤ 2n−1}. Now, from the theory of Hardy spaces it is known that the orthogonal complement of this tangent space in H 2 (Π + ) is obtained as the set {a(−s) 2 R(s) | R(s) ∈H 2 (Π + )}. Consequently, G(s) is a stationary point if and only if there exists a function R(s) ∈ H 2 (Π + ) for which: e(s) d(s) − b(s) a(s) = a(−s)2 R(s) (8.17) (see [8], [9], and [80]). Clearly R(s) needs to be a real and rational function and its poles are in the open left-half plane Π − . Since the factor a(−s) 2 has its zeros in the open right-half plane Π + none of its factors cancel and it follows upon multiplication by a(s)d(s) that there exists a non-vanishing real polynomial q(s) =a(s)d(s)R(s) such that: e(s)a(s) − b(s)d(s) =a(−s) 2 q(s). (8.18) Note that q(s) 0 since H(s) and G(s) have different degrees. Note also that the degree of q(s) is≤ k − 1. The value k = N − n ≥ 1 is the co-order of the approximation G(s).
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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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Page 122 and 123: 108 CHAPTER 7. NUMERICAL EXPERIMENT
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Page 199 and 200: 10.6. EXAMPLES 185 Figure 10.3: The
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10.6. EXAMPLES 191 Figure 10.8: Pol
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10.6. EXAMPLES 193 Table 10.1: Redu
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Chapter 11 H 2 Model-order reductio
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11.2. LINEARIZING THE TWO-PARAMETER
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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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