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Browsing by Author "Bernstein, Dennis S."

Now showing 1 - 8 of 8
  • Convergence Theory of Probability-one Homotopies for Model Order Reduction
    Wang, Chang Y.; Bernstein, Dennis S.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1996-06-01)
    The optimal H-square model reduction problem is an inherently nonconvex problem and thus provides a nontrivial computational challenge. This paper systematically examines the requirements of probability-one homotopy methods to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems, and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the optimal projection equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computation- al implementation of the homotopy algorithms.
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    Globally Convergent Homotopy Algorithms for the Combined H-squared/ H-to Infinity Model Reduction Problem
    Yuzhen, Ge; Watson, Layne T.; Collins, Emmanuel G.; Bernstein, Dennis S. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1993-06-01)
    The problem of finding a reduced order model, optimal in the H-squared sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H-to infinity constraint to the H-squared optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of probablity-one homotopy methods the combined H-squared/H-to infinity model reduction problem is difficult to solve. Several approaches based on homotopy methods have been proposed. The issues are the number of degrees of freedom, the well posedness of the finite dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. Homotopy algorithms based on several formulations -- input normal, Ly, Bryson, and Cannon's 2 x 2 block parametrization -- are developed and compared here.
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    A Homotopy Algorithm for the Combined H-squared/H-to Infinity Model Reduction Problem
    Yuzhen, Ge; Collins, Emmanuel G.; Watson, Layne T.; Bernstein, Dennis S. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1993-05-01)
    The problem of finding a reduced order model, optimal in the H-squared sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H-to infinity constraint to the H-squared optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of probability-one homotopy methods the combined H-squared/H-to infinity model reduction problem is difficult to solve. Several approaches based on homotoppy methods have been proposed. The issues are the number of degrees of freedom, the well posedness of the finite dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. Homotopy algorithms based on two formulations - input normal form; Ly, Bryson, and Cannon's 2 x 2 block parametrization - are developed and compared here.
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    A Homotopy Algorithm for the Combined H2/H&infin Model Reduction Problem
    Yuzhen, Ge; Collins, Emmanuel G.; Watson, Layne T.; Bernstein, Dennis S. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    The problem of finding a reduced order model, optimal in the H2 sense, to a given system model is a fundamental one in control system analysis and design. The addition of an H∞ constraint to the H2 optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of probability-one homotopy methods the combined H2 /H∞ model reduction problem is difficult to solve. Several approaches based on homotopy methods have been proposed. The issues are the number of degrees of freedom, the well posedness of the finite dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. Homotopy algorithms based on two formulations---input normal form; Ly, Bryson, and Cannon's 2x2 block parametrization are developed and compared.
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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.
  • Probability-One Homotopy Algorithms for Full and Reduced Order H-squared/H-to Infinity Controller Synthesis
    Yuzhen, Ge; Watson, Layne T.; Collins, Emmanuel G.; Bernstein, Dennis S. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1994)
    Homotopy algorithms for both full- and reduced-order LQG controller design problems with an H-to infinity constraint on disturbance attenuation are developed. The H-to infinity constraint is enforced by replacing the covariance Lyapunov equation by a Riccati equation whose solution gives an upper boundary on H-squared performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H-squared performance. The algorithms are based on two minimal parameter formulations: Ly, Bryson, and Cannon's 2X2 block parametrization and the input normal Riccati form parametrization. An over-parametrization formulation is also proposed. Numerical experiments suggest that the combination of a globally convergent homotopy method and a minimal parameter formulation applied to the upper bound minimization gives excellent results for mixed-norm H-squared/H-to infinity synthesis. The nonmonocity of homotopy zero curves is demonstrated, proving that algorithms more sophisticated that standard continuation are necessary.
  • Probability-one Homotopy Algorithms for Solving the Coupled Lyapunov Equations Arising in Reduced-Order H^2/H^(infinity) Modeling, Estimation, and Control
    Wang, Chang Y.; Bernstein, Dennis S.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 2000-04-01)
    Optimal reduced order modeling, estimation, and control with respect to combined H^2/H^(infinity) criteria give rise to coupled Lyapunov and Riccati equations. To develop reliable numerical algorithms for these problems this paper focuses on the coupled Lyapunov equations which appear as a subset of the synthesis equations. In particular, this paper systematically examines the requirements of probability-one homotopy algorithms to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled Lyapunov equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms.