Development of stability-preserving time-limited model reduction framework for 2-D and 1-D models with error bound

关键词:  Model reduction · Weighted Gramian · Limited interval Gramian · Minimal realization · Approximation error · Error bound

Key words:  Model reduction · Weighted Gramian · Limited interval Gramian · Minimal realization · Approximation error · Error bound

摘要:

Due to their complex structure, 2-Dmodels are challenging towork with; additionally, simulation, analysis, design, and control get increasingly difficult as the order of the model grows. Moreover, in particular time intervals, Gawronski and Juang’s timelimited model reduction schemes produce an unstable reduced-order model for the 2-D and 1-D models. Researchers revealed some stability preservation solutions to address this key flaw which ensure the stability of 1-D reduced-order systems; nevertheless, these strategies result in large approximation errors. However, to the best of the authors’ knowledge, there is no literature available for the stability preserving time-limited-interval Gramian-based model reduction framework for the 2-D discrete-time systems. In this article, 2-D models are decomposed into two separate sub-models (i.e., two cascaded 1-D models) using the condition of minimal rank-decomposition. Model reduction procedures are conducted on these obtained two 1-D sub-models using limited-time Gramian. The suggestedmethodology works for both 2-D and 1-Dmodels. Moreover, the suggested methodology gives the stability of the reduced model as well as a priori error-bound expressions for the 2-D and 1-D models. Numerical results and comparisons between existing and suggested methodologies are provided to demonstrate the effectiveness of the suggested methodology.

Abstract:

Due to their complex structure, 2-Dmodels are challenging towork with; additionally, simulation, analysis, design, and control get increasingly difficult as the order of the model grows. Moreover, in particular time intervals, Gawronski and Juang’s timelimited model reduction schemes produce an unstable reduced-order model for the 2-D and 1-D models. Researchers revealed some stability preservation solutions to address this key flaw which ensure the stability of 1-D reduced-order systems; nevertheless, these strategies result in large approximation errors. However, to the best of the authors’ knowledge, there is no literature available for the stability preserving time-limited-interval Gramian-based model reduction framework for the 2-D discrete-time systems. In this article, 2-D models are decomposed into two separate sub-models (i.e., two cascaded 1-D models) using the condition of minimal rank-decomposition. Model reduction procedures are conducted on these obtained two 1-D sub-models using limited-time Gramian. The suggestedmethodology works for both 2-D and 1-Dmodels. Moreover, the suggested methodology gives the stability of the reduced model as well as a priori error-bound expressions for the 2-D and 1-D models. Numerical results and comparisons between existing and suggested methodologies are provided to demonstrate the effectiveness of the suggested methodology.