Explicit expressions for the ℋ2 norm of time-delay systems based on the delay Lyapunov equation

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. We establish, in the main result, 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.