8,547 research outputs found
On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations
An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included
Numerical Solutions of Matrix Differential Models using Cubic Matrix Splines II
This paper presents the non-linear generalization of a previous work on
matrix differential models. It focusses on the construction of approximate
solutions of first-order matrix differential equations Y'(x)=f(x,Y(x)) using
matrix-cubic splines. An estimation of the approximation error, an algorithm
for its implementation and illustrative examples for Sylvester and Riccati
matrix differential equations are given.Comment: 14 pages; submitted to Math. Comp. Modellin
Approximate Kalman-Bucy filter for continuous-time semi-Markov jump linear systems
The aim of this paper is to propose a new numerical approximation of the
Kalman-Bucy filter for semi-Markov jump linear systems. This approximation is
based on the selection of typical trajectories of the driving semi-Markov chain
of the process by using an optimal quantization technique. The main advantage
of this approach is that it makes pre-computations possible. We derive a
Lipschitz property for the solution of the Riccati equation and a general
result on the convergence of perturbed solutions of semi-Markov switching
Riccati equations when the perturbation comes from the driving semi-Markov
chain. Based on these results, we prove the convergence of our approximation
scheme in a general infinite countable state space framework and derive an
error bound in terms of the quantization error and time discretization step. We
employ the proposed filter in a magnetic levitation example with markovian
failures and compare its performance with both the Kalman-Bucy filter and the
Markovian linear minimum mean squares estimator
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
An approximating method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory
In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical equation) that is derived from the theory of 1-st order partial differential equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati equation from the Hamilton-Jacobi equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained
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