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Browsing by Author "Collins, Emmanuel G."

Now showing 1 - 10 of 10
  • Cost-Effective Parallel Processing for H-squared/H-to infinity Controller Synthesis
    Yuzhen, Ge; Watson, Layne T.; Collins, Emmanuel G. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1995-07-01)
    A distributed version of a homotopy algorithm for solving the H-squared/H-to infinity mixed-norm controller synthesis problem is presented. The main purpose of the study is to explore the possibility of achieving high performance with low cost. Existing UNIX workstations running PVM (Parallel Virtual Machine) are utilized. Only the Jacobian matrix computation is distributed and therefore the modification to the original sequential code is minimal. The same algorithm has also been implemented on an Intel Paragon parallel machine. Our implementation shows that acceptable speedup is achieved and the larger the problem sizes, the higher the speedup. Comparing with the results from the Intel Paragon, the study concludes that utilizing the existing UNIX workstations can be a very cost-effective approach to shorten computation time. Furthermore, this economical way to achieve high performance computation can easily be realized and incorporated in a practical industrial design environment.
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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.
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    An Input Normal Form Homotopy for the L2 Optimal Model Order Reduction Problem
    Yuzhen, Ge; Collins, Emmanuel G.; Watson, Layne T.; Davis, L. D. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1993-06-01)
    In control system analysis and design, finding a reduced order model, optimal in the L-squared sense, to a given system model is a fundamental problem. The problem is very difficult without the global convergence of homotopy methods, and a homotopy based approach has 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. A homotopy algorithm based on the input normal form characterization of the reduced order model is developed here and is compared with the homotopy algorithms based on Hyland and Bernstein's optimal projection equations. The main conclusions are that the input normal form algorithm can be very efficient, but can also be very ill conditioned or even fail.
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    Minimal Parameter Homotopies for the L2 Optimal Model Order Reduction Problem
    Yuzhen, Ge; Collins, Emmanuel G.; Watson, Layne T.; Davis, L. D. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    The problem of finding a reduced order model, optimal in the L2 sense, to a given system model is a fundamental one in control system analysis and design. The problem is very difficult without the global convergence of homotopy methods, and a number of homotopy based approaches 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 are developed and compared here. The main conclusions are that dimensionality is inversely related to numerical well conditioning and algorithmic efficiency is inversely related to robustness of the algorithm.
  • 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 Robust Controller Analysis and Synthesis with Fixed-Structure Multipliers
    Collins, Emmanuel G.; Haddad, Wassim M.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1996)
    To enable the development of M-K (i.e., multiplier-controller) iteration schemes that do not require (suboptimal) curve fitting, mixed structured singular value analysis tests that allow the structure of the multipliers to a priori be specified, have been developed. These tests have recently been formulated as linear matrix inequality (LMI) feasibility problems. The least conservative of these tests always results in unstable multipliers and hence requires a stable coprime factorization of the multiplier before the control synthesis phase of the M-K iteration. This paper first reviews the LMI formulations of robustness analysis. It then develops alternative formulations that directly synthesize the stable factorizations and are based on the existence of positive definite solutions to certain Riccati equations. These problems, unlike the LMI problems, are not convex. The feasibility problem is approached by posing an associated optimization problem that cannot be solved using standard descent methods. Hence, we develop probability-one homotopy algorithms to find a solution. These results easily extend to provide computationally tractable algorithms for fixed-architecture, robust control design, which appear to have some advantages over the bilinear matrix inequality (BMI) approaches resulting from extensions of the LMI framework for robustness analysis.