Approximation of Fractional-Order Systems Using Balanced Truncation with Assured Steady-State Gain

• Published in: Circuits, systems, and signal processing Vol. 42; no. 10; pp. 5893 - 5923
• Main Author: Duddeti, Bala Bhaskar
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2023; Springer Nature B.V
• Subjects: Approximation; Circuits and Systems; Electrical Engineering; Electronics and Microelectronics; Engineering; Instrumentation; Integers; Mathematical analysis; Performance measurement; Signal,Image and Speech Processing; Steady state; System effectiveness; Time response; Balanced truncation method; Model order reduction; Fractional-order systems; Gain adjustment factor; Oustaloup approximation
• ISSN: 0278-081X; 1531-5878
• DOI: 10.1007/s00034-023-02393-4

This research presents a novel and stable approach for reducing the order of fractional-order systems using the advantage of a balanced truncation method and gain adjustment factor. The method can be applied to both commensurate and non-commensurate fractional systems. First, the fractional-order system is turned into an integer-order system using the Oustaloup approximation for non-commensurate systems and a simple mathematical replacement for commensurate systems. Then, balanced truncation is used to create a lower-order model. A gain factor is added to the reduced model to improve its accuracy in the steady state without changing how it works in the dynamic condition. This proposed method differs from other balanced truncation-based reduction methods that enhance steady-state response by using time moments, Pade equations, and Routh tables. These extra steps are optional with this method. The proposed strategy makes the step and Bode responses of the reduced systems more like Oustaloup’s approximate higher-order model. The method’s effectiveness is evaluated using various performance metrics such as integral square error, root mean square error, and H ∞ norm, as well as time response characteristics. Finally, inverse substitution converts the reduced integer-order model back to a commensurate fractional system. All simulations for the examples discussed in the study were conducted in MATLAB.