Practical stability and robust state-feedback for tempered ϑ -Caputo fractional systems

This paper addresses the problem of practical stability for nonlinear tempered fractional-order systems with respect to a general function ϑ. We introduce the concept of Practical Generalised Mittag-Leffler Stability (PGMLS) for systems governed by tempered ϑ-Caputo fractional derivatives. A sufficient Lyapunov-based theorem is derived to establish PGMLS for a general class of nonlinear tempered fractional-order systems. The theoretical framework is then applied to stabilise uncertain nonlinear systems, where the stability analysis is translated into a tractable Linear Matrix Inequality (LMI) formulation. This provides a systematic method for designing observer gains that ensure practical stability. The efficacy of the proposed control scheme is validated through numerical simulations, confirming the bounded convergence of both state and fault estimation errors.