Browsing by Author "Flagg, Garret Michael"

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    Interpolation Methods for the Model Reduction of Bilinear Systems
    Flagg, Garret Michael (Virginia Tech, 2012-04-30)
    Bilinear systems are a class of nonlinear dynamical systems that arise in a variety of applications. In order to obtain a sufficiently accurate representation of the underlying physical phenomenon, these models frequently have state-spaces of very large dimension, resulting in the need for model reduction. In this work, we introduce two new methods for the model reduction of bilinear systems in an interpolation framework. Our first approach is to construct reduced models that satisfy multipoint interpolation constraints defined on the Volterra kernels of the full model. We show that this approach can be used to develop an asymptotically optimal solution to the H_2 model reduction problem for bilinear systems. In our second approach, we construct a solution to a bilinear system realization problem posed in terms of constructing a bilinear realization whose kth-order transfer functions satisfy interpolation conditions in k complex variables. The solution to this realization problem can be used to construct a bilinear system realization directly from sampling data on the kth-order transfer functions, without requiring the formation of the realization matrices for the full bilinear system.
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    An Interpolation-Based Approach to Optimal H∞ Model Reduction
    Flagg, Garret Michael (Virginia Tech, 2009-05-05)
    A model reduction technique that is optimal in the H∞-norm has long been pursued due to 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 H₂-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∞ 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.