Showing papers on "Homotopy analysis method published in 1996"

Multiparameter homotopy methods for finding DC operating points of nonlinear circuits

Abstract: This paper introduces multiparameter homotopy methods for finding dc operating points. The question of whether adding extra real or complex parameters to a single-parameter homotopy function can lead to improved solution paths is investigated. It is shown that no number of added real parameters can lead to local fold avoidance, but that generic folds may be efficiently avoided by complexifying the homotopy parameter and tracing a closed curve in complex parameter space around the critical fold value. A combination of real 2-parameter homotopy and complex parameter homotopy is shown to be sufficient for avoiding real fork bifurcations and enumerating all real, locally connected branches. Additionally, the potential of complex parameter homotopy methods for finding all circuit solutions is explored. Results from algebraic geometry indicate that if an analytic homotopy function with a single complex parameter is irreducible, then there exist regular paths through the complex parameter plane connecting any solution of H(x,/spl lambda/')=0 to any other solution of H(x,/spl lambda/')=0. Thus, in principle at least, complex parameter homotopy can be used to find all circuit solutions.

Solving geometric constraints by homotopy

Abstract: Numerous methods have been proposed in order to solve geometric constraints, all of them having their own advantages and drawbacks. In this article, we propose an enhancement to the classical numerical methods, which, up to now, are the only ones that apply to the general case.

The implementation of linear programming algorithms based on homotopies

Abstract: A fundamental homotopy-based linear programming algorithm, which utilizes Euler-predictor and Newton-corrector steps with restarts, is formulated and investigated numerically on problems representative of linear programs that arise in practice. A rich array of refinements of this basic algorithm are possible within the homotopy framework. Such refinements are needed in any practical implementation and are discussed in detail. Implications for the design of integrated large-scale mathematical programming software are also briefly considered.

Probability-one homotopy algorithms for full- and reduced-order H2/H∞ controller synthesis

Abstract: 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 H 2 performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H 2 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 overparametrization 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 2 /H∞ synthesis. The nonmonotonicity of homotopy zero curves is demonstrated, proving that algorithms more sophisticated than standard continuation are necessary.

An efficient numerically robust homotopy algorithm for H2 model reduction using the optimal projecton equations

Abstract: Homotopy approaches have previously been developed for synthesizing #2 optimal reduced-order models. Some of the previous homotopy algorithms were based on directly solving the optimal projection equations, a set of two Lyapunov equations mutually coupled by a nonlinear term involving a projection matrix r, that characterize the optimal reduced-order model. These algorithms are numerically robust but suffer from the curse of large dimensionality. Subsequently, gradient-based homotopy algorithms were developed. To make these algorithms efficient and to eliminate singularities along the homotopy path, the basis of the reduced-order model was constrained to a minimal parameterization. However, the resultant homotopy algorithms sometimes experienced numerical ill-conditioning or failure due to the minimal parameterization constraint. This paper presents a new homotopy approach to solve the optimal projection equations for #2 model reduction. The current algorithm avoids the large dimensionality of the previou...

Application of homotopy methods to nonlinear control problems

Abstract: The purpose of this article is to give a concise introduction to the use of homotopy methods for treating nonlinear control problems. The theoretical foundations are briefly sketched, especially some important topics concerning linearisation along trajectories. The main contribution is that for a certain control the deviation of the actual state from the target state is decaying exponentially with time. The proof of this statement contains an explicit design instruction for the required control. This design instruction is discussed in some detail and explicitly carried out for an example system. We conclude with a short comparison of two usual nonlinear control methods with the proposed method by means of the example system.

Homotopy Algorithm for Fixed Order Mixed Design

Abstract: Recent developments in the field of robust multivariable control have merged the theories of HQQ and H2 control. This mixed HilH^ compensator formulation allows design for nominal performance by HI norm minimization while guaranteeing robust stability to unstructured uncertainties by constraining the H^ norm. A key difficulty associated with mixed H^H^ compensation is compensator synthesis. A homotopy algorithm is presented for synthesis of fixed order mixed H^H^ compensators. Numerical results are presented for a four disk flexible structure to evaluate the efficiency of the algorithm.

Nonlinear control using homotopy methods

Abstract: Homotopy and continuation methods are commonly used tools in numerical analysis to solve nonlinear algebraic equations and optimisation problems. In this paper they are used to generalise the feedback controller design based on the well-known linearisation around an operating point. The result is a nonlinear feedback control law with guaranteed closed loop stability properties. The advantages are very unrestrictive conditions on the plant to be controlled and the possibility to upgrade a reliable linear controller.

A Homotopy Method for Equilibrium Programming under Uncertainty

Abstract: We consider a homotopy method for solving stochastic Nash equilibrium models. The algorithm works by following, via a predictor-corrector method, the one-dimensional manifold of the homotopy constructed to connect the systems of equations describing the solution set of the scenario equilibrium model (no nonanticipativity constraints) and the stochastic equilibrium model. The predictor and corrector phases of this homotopy method require the usual solutions of large linear systems, a computationally expensive task, which we render less difficult through our use of Jacobi techniques designed to take advantage of the problem's near separability across scenarios.

Computational experience with general equilibrium problems

Abstract: We report on computational experience with an implementation of three algorithms for the general economic equilibrium problem. As a result we get that the projection algorithm for variational inequalities increases the size of solvable models by a factor of 5–10 in comparison with the classical homotopy method. As a third approach we implemented a simulated annealing heuristic which might be suitable to estimate equilibria for very large models.

Development of adaptive homotopy continuation method algorithm for adaptive IIR filtering

Abstract: An exhaustive solution search method, called homotopy continuation method, is modified and applied for finding the global minimum of nonquadratic error surfaces of adaptive IIR filters. First, the homotopy continuation method is modified to solve a set of nonlinear polynomials which have time-varying coefficients. Next, it is shown that adaptive IIR filtering can be formulated as solving a set of approximated polynomials. The validity of the approach is demonstrated using an example involving the estimation of the coefficients of an all pole filter.

Convergence Theory of Probability-one Homotopies for Model Order Reduction

Abstract: 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.

Homotopy Solutions of Kepler's Equations

Abstract: Kepler's Equation is solved using an integrative algorithm developed using homotropy theory. The solution approach is applicable to both elliptic and hyperbolic forms of Kepler's Equation. The results from the proposed algorithm compare quite favorably with those from existing iterative schemes.

Analysis of nonlinear traveling waves by frequency-domain perturbation method

Abstract: We discuss a numerical method for solving the nonlinear transmission lines by the frequency-domain perturbation method. To improve the convergence, we introduce two new methods of the compensation and the homotopy techniques, which also help to make the iteration stable. This kind of transmission lines is widely used in the communication circuits such as GaAs integrated circuits, and varactor diode circuits.