Model Order Reduction of a DC Motor System with a Buck Converter via a Gramians-Based Truncation Algorithm

This paper presents a study on the feasibility of reducing the model order of a DC motor system controlled by a high-order DC-DC buck converter, with the objective of minimizing the complexity of the original model while preserving its essential dynamic characteristics for precise controller design. The primary goal is to develop and apply the Gramians-Based Truncation (GBT) algorithm to reduce the original fourth-order system to third-order, second-order, and first-order models. By computing the controllability and observability Gramians through the solution of Lyapunov equations, the authors perform a system balancing procedure based on singular values to identify and eliminate states with negligible contributions. The implementation of the GBT algorithm in Matlab yielded norm reduction errors of 12.134639 and 12.135958 for the third-order and second-order models, respectively. These results demonstrate the capability of these reduced models to preserve the time-domain and frequency-domain response characteristics of the original system. In contrast, the first-order model exhibits a substantially higher error (410.183959) and fails to maintain consistency in the input–output response, particularly during the startup phase and in applications requiring accurate phase signal processing. The findings confirm the viability of the GBT method in simplifying complex dynamic models and underscore the importance of selecting an appropriate reduction order to balance model accuracy with implementation simplicity in engineering applications. These results pave the way for further research on improving reduction techniques to optimize phase information preservation and better meet the demands of modern control systems.