Stable Reduced Model Approximation of Fractional Order Systems Using Improved Balanced Residualization Method

This proposal proposes a new technique for reduced approximation of LTI fractional order systems (FOSs). The proposed algorithm is decomposed into two phases. First, the FOSs are converted into integer-order systems. For non-commensurate FOSs, the Oustaloup approximation is used, and for commensurate FOSs, a simple mathematical replacement is used. The second step focuses on creating low-order models by residualizing a balanced realization’s fast modes. This approach employs balanced residualization to get the reduced system's denominator polynomial and a constant coefficient in the numerator to assure stability and zero steady-state error. The remaining numerator coefficients of the simplified model are produced using a straightforward procedure that matches a few of Markov parameters (MPs) and time moments (TMs). The simplified integer-order model is converted into its corresponding commensurate fractional model via inverse substitution. Additionally, it is demonstrated numerically that some current methods fall short of the stability promise. The offered approach preserves the complex system's stability, MPs, TMs, and other critical attributes in the reduced model. Several performance measures and step and frequency responses assess the strategy's success. Additionally, the percentage enhancement of the proposed reduced model error indices concerning other methods is computed and tabulated for better comprehension. The time domain characteristics of the reduced-order approximations were compared to the original system. The bar charts were drawn to better visualize the proposed results. The need for order reduction in engineering applications employing FOS is highlighted in this work.