Observability of Time-Varying Fractional Dynamical Systems with Caputo Fractional Derivative

Modeling dynamical systems with real-life data having time-dependent disturbances is better captured with time-varying systems. The qualitative properties of such a system in a fractional sense are hardly examined. Observability is one property where the system’s initial states are determined based on the output of some observation system. In this paper, we investigate the observability of time-varying fractional dynamical systems. A state-transition matrix represents the solution of the time-varying fractional dynamical systems. The observability results of linear and nonlinear systems are obtained using the Gramian matrix technique and the Banach contraction mapping theorem respectively. The obtained theoretical results for the observability of the time-varying fractional dynamical systems are compared with those of the time-invariant fractional dynamical system (FDS). Several numerical examples are provided to validate the theoretical results. Also, a numerical example to study the observability of a fractional spring–mass system is provided to verify the applicability of this study.