Stability and stabilization of Nabla discrete fractional-order systems: An order-independent LMI approach without conservatism

Abstract

This paper investigates the stability and controller design of linear time-invariant (LTI) Nabla discrete fractional-order systems (NDFOSs). A necessary and sufficient stability condition is established, which, for the first time, precisely characterizes the stability region of NDFOSs through a strict linear matrix inequality (LMI) formulation involving only real matrices. Compared with the existing conservative Lyapunov and region approximation methods with order-dependent constraints, the proposed approach avoids such conservatism and is applicable to all orders α > 0. To address uncertainties in the system matrix and facilitate controller design, two less conservative stability theorems are developed for NDFOSs with orders 0 < α ≤ 1 and α > 1. These theorems eliminate reliance on the fractional power of the system matrix. Building upon this theoretical framework, two distinct state feedback controller design approaches are presented: a simple LMI-based method for direct implementation, and an optimized iterative approach that achieves minimal conservatism by directly solving equality-constrained LMIs. Finally, the effectiveness of the proposed methods is validated through numerical examples with varying system parameters and orders. © 2026 The Franklin Institute. Published by Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.