We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

We present new Fortran 77 subroutines which implement the Schur method and the matrix sign function method for the solution of the continuous­time matrix algebraic Riccati equation on the basis of LAPACK subroutines. In order to avoid some of the well­known difficulties with these methods due to a loss of accuracy, we combine the implementations with block scalings as well as condition estimates and forward error estimates. Results of numerical experiments comparing the performance of both methods for more than one hundred well­ and ill­conditioned Riccati equations of order up to 150 are given. It is demonstrated that there exist several classes of examples for which the matrix sign function approach performs more reliably and more accurately than the Schur method. In all cases the forward error estimates allow to obtain a reliable bound on the accuracy of the computed solution.

Diss.--Mathematik--Chemnitz--Technische Universität, 1998. / Bibliogr. p. 103-105.

No description available.

The theory of time scales is introduced by Stefan Hilger in his PhD thesis in 1998 in order to unify continuous and discrete analysis. In our thesis, by making use of the time scale calculus we study the oscillation of nonlinear matrix differential equations of second order. the first chapter is introductory in nature and contains some basic definitions and tools of the time scales calculus, while certain well-known results have been presented with regard to oscillation of the solutions of second order matrix equations and some new oscillation criteria for the same type equations have been established in the second chapter.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 3 uppsatser.

University, Diss., 2002--Marburg.

University, Diss., 2004--Konstanz.

State dependent Riccati equation (SDRE) plays an important role in nonlinear controller design. For autonomous nonlinear systems that can be expressed in linear form with state-dependent coefficients (SDC), SDRE-based controllers guarantee local asymptotic stability of the closed-loop system, under pointwise stabilizability and detectability conditions. Moreover, the optimal control for a quadratic cost function, when it exists, corresponds to an SDRE-based control design for a specific SDC parameterization of the associated nonlinear system. Unfortunately, the implementation of the SDRE-based controllers is computationally expensive. Various techniques have been developed for solving the SDRE, which are either computationally expensive or lack acceptable precision. In this dissertation, a modified state-dependent differential Riccati equation (MSDDRE) is proposed for approximating the solution of the SDRE, which is easy to implement with moderate computation power and its solution can be made arbitrarily close to that of the SDRE. Therefore, it can be used for real-time implementation of near-optimal controllers for nonlinear systems in state-dependent linear form. The proposed technique is then extended to SDRE-based filter design and its application to SDRE-based output feedback control technique. The proposed technique is also extended to state-dependent H-inf; robust control design for a constant noise attenuation bound, when the solution exists. To reduce the design conservativeness, the technique is further extended to state-dependent H-inf; robust control design with adaptive noise attenuation bound, using gain-scheduling technique and linear matrix inequality (LMI) optimization, to approximate H-inf; optimal control with state-dependent noise-attenuation bound. Local asymptotic stability of the closed-loop system is proven for all proposed techniques. Simulation results further confirm the validity of the development and demonstrate the efficiency of the proposed techniques.

In this paper we develop new and improved results in the numerical solution of the coupled algebraic Riccati equations. First we provide improved matrix upper bounds on the positive semidefinite solution of the unified coupled algebraic Riccati equations. Our approach is largely inspired by recent results established by Liu and Zhang. Our main results tighten the estimates of the relevant dominant eigenvalues. Also by relaxing the key restriction our upper bound applies to a larger number of situations. We also present an iterative algorithm to refine the new upper bounds and the lower bounds and numerically compute the solutions of the unified coupled algebraic Riccati equations. This construction follows the approach of Gao, Xue and Sun but we use different bounds. This leads to different analysis on convergence. Besides, we provide new matrix upper bounds for the positive semidefinite solution of the continuous coupled algebraic Riccati equations. By using an alternative primary assumption we present a new upper bound. We follow the idea of Davies, Shi and Wiltshire for the non-coupled equation and extend their results to the coupled case. We also present an iterative algorithm to improve our upper bounds. Finally we improve the classical Newton's method by the line search technique to compute the solutions of the continuous coupled algebraic Riccati equations. The Newton's method for couple Riccati equations is attributed to Salama and Gourishanar, but we construct the algorithm in a different way using the Fr\'echet derivative and we include line search too. Our algorithm leads to a faster convergence compared with the classical scheme. Numerical evidence is also provided to illustrate the performance of our algorithm.