Browsing by Author "Zigic, Dragan"

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    Contragredient Transformations Applied to the Optimal Projection Equations
    Zigic, Dragan; Watson, Layne T.; Beattie, Christopher A. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. It is not obvious from their original form how they can be differentiated and how some algorithm for solving nonlinear equations can be applied to them. A contragredient transformation, a transformation which simultaneously diagonalizes two symmetric positive semi-definite matrices, is used to transform the equations into forms suitable for algorithms for solving nonlinear problems. Three different forms of the equations obtained using contragredient transformations are given. An SVD-based algorithm for the contragredient transformation and a homotopy algorithm for the transformed equations are given, together with a numerical example.
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    Homotopy Approaches to the H2 Reduced Order Model Problem
    Zigic, Dragan; Watson, Layne T.; Collins, Emmanuel G.; Bernstein, Dennis S. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1991)
    The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. The algorithms proposed herein utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial system all the systems along the homotopy path will be asymptotically stable, controllable and observable. One method, which solves the matrix equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix. An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given. Several strategies for choosing the homotopy maps and the starting points (initial systems) are discussed and compared, in the context of some reduced order model problems from the literature. Numerical results are included for ten test problems, of sizes 2 through 17.
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    Homotopy Methods for Solving the Optimal Projection Equations for the H2 Reduced Order Model Problem
    Zigic, Dragan; Watson, Layne T.; Collins, Emmanuel G.; Bernstein, Dennis S. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1991)
    The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. Due to the resemblance of these equations to standard matrix Lyapunov equations, they are called modified Lyapunov equations. The algorithms proposed herein utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable, controllable and observable. One method, which solves the equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix. It is shown that the appropriate inverse is a differentiable function. An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given. Another class of methods considers the equations in a modified form, using a decomposition of the pseudogramians based on a contragredient transformation. Some freedom is left in making an exact match between the number of equations and the number of unknowns, thus effectively generating a family of methods.
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    Homotopy methods for solving the optimal projection equations for the reduced order model problem
    Zigic, Dragan (Virginia Tech, 1991)
    The optimal projection approach to solving the reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. Due to the resemblance of these equations to standard matrix Lyapunov equations, they are called modified Lyapunov equations. The proposed algorithms utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable, controllable and observable. One method, which solves the equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix. It is shown that the appropriate inverse is a differentiable function. An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given. Another class of methods considers the equations in a modified form, using a decomposition of the pseudogramians based on a contragredient transformation. Some freedom is left in making an exact match between the number of equations and the number of unknowns, thus effectively generating a family of methods. Three strategies are considered for balancing the number of equations and unknowns. This approach proved to be very successful on a number of examples. The tests have shown that using the ‘best’ method practically always leads to a solution.