Stability and Stabilization of Fractional-order Systems with Different Derivative Orders: An LMI Approach

TL;DR

This paper develops new stability and stabilization criteria for fractional-order LTI systems with different derivative orders using LMIs, simplifying analysis and controller design.

Contribution

It introduces a single-order equivalent system and LMI-based conditions for stability and stabilization of fractional systems with varying orders, enhancing practical applicability.

Findings

New stability conditions easier to verify

LMI-based stabilization method applicable to systems with multiple fractional orders

Numerical examples confirm theoretical results

Abstract

Stability and stabilization analysis of fractional-order linear time-invariant (FO-LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single-order equivalent system for the given different-order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional-order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The linear matrix inequality based procedure of developing stabilizing output feedback control is preserved in spite of the complexity of assuming the most complete linear controller model, with direct feedthrough parameter. The proposed stability and stabilization theorems are applicable to FO-LTI systems with different fractional orders in one or both of and intervals. Eventually, some numerical examples are presented to confirm the obtained analytical results.