Model order reduction using dual $\nu$-gap metrics: A multi-objective optimization approach

This study investigated the problem of model order reduction by employing the $\nu$-gap metric and introduced a metaheuristic optimization framework to address its inherent nonconvexity. The $\nu$-gap metric quantifies the closed-loop distance between two systems, making it useful for generating low-order models that preserve closed-loop stability characteristics. However, minimizing the conventional $\nu$-gap alone may result in poor dynamic fidelity. To address this limitation, a modified $\nu$-gap metric based on frequency response matching was developed to explicitly capture frequency-wise discrepancies associated with time-domain behavior. By jointly considering the conventional and modified $\nu$-gap metrics, a dual $\nu$-gap–based multi-objective model reduction framework was formulated, in which stability-related and time-domain fidelity objectives are treated in a complementary manner. The resulting optimization problem is highly nonconvex and constrained due to the winding number condition. A multi-objective particle swarm optimization framework was therefore employed as a numerical tool to generate Pareto-optimal reduced-order models. Benchmark studies on large-scale systems demonstrated that the proposed framework enables a systematic exploration of trade-offs between stability preservation and time-domain fidelity, yielding practically meaningful reduced-order models with tunable performance characteristics.