Operator-ADI balanced truncation of delay differential systems

Abstract

We develop a balanced truncation method for linear delay differential systems based on a finite-rank alternating-direction implicit (ADI) iteration for the associated operator Lyapunov equations. The method acts directly on the infinite-dimensional history-state realization and does not discretize the delay interval during Gramian approximation or balancing. The finite-rank reachability and observability factors are represented exactly in Takenaka--Malmquist coordinates, reducing the required computations to finite-dimensional linear algebra. These factors yield a finite-dimensional, delay-free reduced model. For real system data, the implementation remains in real arithmetic. Under suitable conditions on the ADI shifts and a gap in the Hankel singular values, we prove convergence in $\mathcal H_\infty$ to the reduced transfer function obtained by exact balanced truncation of the infinite-dimensional system. The construction extends to finitely many discrete delays. Numerical experiments validate the method against independent reference computations. The complete MATLAB implementation used in the experiments is provided as supplementary material.