Browsing by Author "Ge, Yuzhen"

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    Homotopy algorithms for H²/H∞ control analysis and synthesis
    Ge, Yuzhen (Virginia Tech, 1993-12-02)
    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 a H∞ constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H∞ model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations—input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization—are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods. Although there are numerous algorithms for solving Riccati equations, there still remains a need for algorithms which can operate efficiently on large problems and on parallel machines and which can be generalized easily to solve variants of Riccati equations. This thesis gives a new homotopy-based algorithm for solving Riccati equations on a shared memory parallel computer. The central part of the algorithm is the computation of the kernel of the Jacobian matrix, which is essential for the corrector iterations along the homotopy zero curve. Using a Schur decomposition the tensor product structure of various matrices can be efficiently exploited. The algorithm allows for efficient parallelization on shared memory machines. The linear-quadratic-Gaussian (LQG) theory has engendered a systematic approach to synthesize high performance controllers for nominal models of complex, multi-input multioutput systems and hence it is a breakthrough in modern control theory. Homotopy algorithms for both full and reduced-order LQG controller design problems with an H∞ constraint on disturbance attenuation are developed. The H∞ constraint is enforced by replacing the covariance Lyapunov equation by a Riccati equation whose solution gives an upper bound on H² performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H² performance. The algorithms are based on two minimal parameter formulations: Ly, Bryson, and Cannon's 2 X 2 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 with a minimal parameter formulation applied to the upper bound minimization gives excellent results for mixed-norm synthesis.
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    Homotopy algorithms for the H² and the combined H²/H∞ model order reduction problems
    Ge, Yuzhen (Virginia Tech, 1993-04-04)
    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 a H∞ constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H∞ model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods.
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    Studies of one-dimensional unimodal maps in the chaotic regime
    Ge, Yuzhen (Virginia Tech, 1990-04-15)
    For one-dimensional uninmodal maps hλ(x) a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period doubling fixed point g(x) which depends on the details of the map hλ(x) and the scaling constant α. The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. It is conjectured that the asymptotic behavior of the partial sum of the measure as the number of levels goes to 00 is universal for the class of maps that have the same order of maximum. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequence Q and the scaling constant of Q is found to be approximately 1. We also study two three-dimensional volume-preserving quadratic maps. There is no period doubling bifurcation in either case. We have also developed an algorithm to construct the symbolic alphabet for some given superstable symbolic sequences for one-dimensional unimodal maps. Using this symbolic alphabet and the approach of cycle expansion the topological entropy can be easily computed. Furthermore, the scaling properties of the measure of constant topological entropy are studied. Our results support the conjectures that for the maps with the same order of maximum, the asymptotic behavior of the partial sum of the measure as the level of the binary goes to infinity is universal and the corresponding 'fatness' exponent is universal. Numerical computations and analysis are also carried out for the clipped Bernoulli shift.