Stability Region Analysis for Nabla Linear Time Invariant Fractional Order Systems

This article considers the stability of nabla linear time invariant (LTI) fractional order systems with the order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,+\infty)$ </tex-math></inline-formula>. First, the stable criterion is developed, by using the nabla Laplace transform. Compared with the existing case of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,1)$ </tex-math></inline-formula>, our work introduces a wide range of dynamic behaviors for future applications. Second, many essential properties are discussed for the developed criterion, including the changing trend of the stable/unstable region regarding the order, the containment relationship between the imaginary axis, the negative semi-axis and the stable region, the evolution of the modulus with the absolute value of argument for the point lying in the critical stable region. Third, the linear matrix inequality (LMI) condition is tentatively derived to evaluate the stability. Finally, the elaborated results are supported by three illustrative numerical examples.