Type of Document Master's Thesis Author Flagg, Garret Michael Author's Email Address garretf@vt.edu URN etd-05222009-124513 Title An Interpolation-Based Approach to Optimal H∞ Model Reduction Degree Master of Science Department Mathematics Advisory Committee

Advisor Name Title Gugercin, Serkan Committee Chair Beattie, Christopher A. Committee Member Borggaard, Jeffrey T. Committee Member Keywords

- Model Reduction
- Rational Interpolation
- Optimization
Date of Defense 2009-05-05 Availability unrestricted AbstractA model reduction technique that is optimal in the H∞-norm has long been pursued dueto its theoretical and practical importance. We consider the optimal H∞ model reduction

problem broadly from an interpolation-based approach, and give a method for finding the

approximation to a state-space symmetric dynamical system which is optimal over a family

of interpolants to the full order system. This family of interpolants has a simple parameterization

that simplifies a direct search for the optimal interpolant. Several numerical

examples show that the interpolation points satisfying the Meier-Luenberger conditions for

H_2 -optimal approximations are a good starting point for minimizing the H∞-norm of the

approximation error. Interpolation points satisfying the Meier-Luenberger conditions can

be computed iteratively using the IRKA algorithm [12]. We consider the special case of

state-space symmetric systems and show that simple sufficient conditions can be derived

for minimizing the approximation error when starting from the interpolation points found

by the IRKA algorithm. We then explore the relationship between potential theory in the

complex plane and the optimal H∞-norm interpolation points through several numerical experiments.

The results of these experiments suggest that the optimal H_2 approximation of

order r yields an error system for which significant pole-zero cancellation occurs, effectively

reducing an order n+r error system to an order 2r+1 system. These observations lead to a

heuristic method for choosing interpolation points that involves solving a rational Zolatarev

problem over a discrete set of points in the complex plane.

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