Cross-Gramian-based computationally efficient model reduction of nonlinear systems using unique transformation

The increasing demand for real-time computation in system deployment has made model-order reduction of large-scale systems an integral part of analysis and design tasks. Specifically, for nonlinear systems, projection-based methods often fail to capture nonlinear behaviors effectively due to their dependence on linearization around equilibrium points. In this work, we present a methodology for nonlinear model-order reduction using cross-Gramians by utilizing a unique solution to a single Sylvester equation, in contrast to the two Lyapunov equations required by conventional balanced truncation. The approach involves perturbing the system with a set of representative input signals, collecting state and output responses, and constructing the Gramians through time-domain integration. Thereafter, the singular value decomposition is used to obtain dominant subspaces and project the model into a lower order. The proposed novel approach effectively balances the accuracy and computational complexity of the model by retaining important nonlinear dynamics and drastically decreasing the dimension of the system. Numerical studies conducted on some standard systems validate that the Gramian-based reduction strategy offers a powerful and effective framework for the analysis and modeling of nonlinear dynamical systems. The presented approach is useful for computationally efficient real-time implementation of nonlinear models while exhibiting lower reduction error.