Time-limited H2-optimal model order reduction of linear systems with quadratic outputs

An important class of dynamical systems with several practical applications is that of linear systems with quadratic outputs. In time-limited model order reduction, the main goal is to ensure that the reduced-order model preserves the output response of the original system over a specified limited time range. To quantify the error within the specified time interval, this paper derives a mathematical expression for the time-limited H2-norm of linear systems with quadratic outputs, serving as a measure of the reduced model's accuracy over this period. Subsequently, the necessary conditions for achieving a local optimum of the time-limited H2 norm error are derived. The inherent inability to satisfy these optimality conditions within the Petrov-Galerkin projection framework is also discussed. An iterative fixed-point iteration algorithm based on these optimality conditions is proposed that ensures high fidelity within the desired time interval. Efficient implementation of the proposed algorithm for large-scale problems is also discussed. The efficacy of the proposed algorithm is demonstrated through two benchmark numerical examples. The results show that the proposed algorithm achieves higher accuracy than time-limited balanced truncation, as measured by the time-limited H2 norm error.