Friday March 26, 2010, 11.00-12.00, Room 3721, Lindstedtsvägen 25

Elias Jarlebring
K.U. Leuven, Belgium

The transfer function can be used in a number of ways to analyze
dynamical systems. If the state equation of a continuous-time linear
time-invariant (LTI) system has a delayed term, then the transfer
function will contain a corresponding exponential term. In this talk
we will present two results on LTI systems with delays, here called
time-delay systems.

We present methods to compute the H2-norm of the transfer function of a time-delay system. The H2-norm of LTI systems without delay can be computed from the solution of the Lyapunov equation. We show that this extends to systems with delays, where instead, the so-called delay Lyapunov equation has to be solved. It turns out that the delay Lyapunov equation can be solved explicitly for time-delay systems with a single delay and we propose a new numerical scheme for the general case.

The method called Arnoldi is a very popular method for standard eigenvalue problems. Because of the exponential term in the denominator of the transfer function of time-delay systems, the characteristic equation is a nonlinear eigenvalue problem. We also present a natural generalization of Arnoldi for this nonlinear eigenvalue problem. The generalization is done in such a way that many attractive properties of Arnoldi are preserved. In particular the method is such that an arbitrary number of characteristic roots can be computed efficiently in a robust way.

We present methods to compute the H2-norm of the transfer function of a time-delay system. The H2-norm of LTI systems without delay can be computed from the solution of the Lyapunov equation. We show that this extends to systems with delays, where instead, the so-called delay Lyapunov equation has to be solved. It turns out that the delay Lyapunov equation can be solved explicitly for time-delay systems with a single delay and we propose a new numerical scheme for the general case.

The method called Arnoldi is a very popular method for standard eigenvalue problems. Because of the exponential term in the denominator of the transfer function of time-delay systems, the characteristic equation is a nonlinear eigenvalue problem. We also present a natural generalization of Arnoldi for this nonlinear eigenvalue problem. The generalization is done in such a way that many attractive properties of Arnoldi are preserved. In particular the method is such that an arbitrary number of characteristic roots can be computed efficiently in a robust way.

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