Joris Vanbiervliet

Characterizing and Computing the

It is widely known that the solutions of Lyapunov equations can be used to compute the H2 norm of linear time-invariant (LTI) dynamical systems. In this paper, we show how this theory extends to dynamical systems with delays. The first result is that the H2 norm can be computed from the solution of a generalization of the Lyapunov equation, which is known as the delay Lyapunov equation. From the relation with the delay Lyapunov equation we can prove an explicit formula for the H2 norm if the system has commensurate delays, here meaning that the delays are all integer multiples of a basic delay. The formula is explicit and contains only elementary linear algebra operations applied to matrices of finite dimension. The delay Lyapunov equations are matrix boundary value problems. We show how to apply a spectral discretization scheme to these equations for the general, not necessarily commensurate, case. The convergence of spectral methods typically depends on the smoothness of the solution. To this end we describe the smoothness of the solution to the delay Lyapunov equations, for the commensurate as well as for the non-commensurate case. The smoothness properties allow us to completely predict the convergence order of the spectral method.

{\cal H}_{2}

This paper concerns the stability optimization of (parameterized) matrices

Norm of Time-Delay Systems by Solving the Delay Lyapunov Equation

A(x)

The Smoothed Spectral Abscissa for Robust Stability Optimization

, a problem typically arising in the design of fixed-order or fixed-structured feedback controllers. It is well known that the minimization of the spectral abscissa function

Using spectral discretisation for the optimal ℋ 2 design of time-delay systems

\alpha(A)

Smooth stabilization and optimal H2 design

gives rise to very difficult optimization problems, since

Explicit expressions for the ℋ 2 norm of time-delay systems based on the delay Lyapunov equation

\alpha(A)

A NONSMOOTH OPTIMISATION APPROACH FOR THE STABILISATION OF TIME-DELAY SYSTEMS

is not everywhere differentiable and even not everywhere Lipschitz. We therefore propose a new stability measure, namely, the smoothed spectral abscissa

A direct method for the H_2 norm of time-delay systems

\tilde\alpha_{\epsilon}(A)

Nonlinear eigenproblems and delay systems

, which is based on the inversion of a relaxed