Interval estimation for nabla fractional order linear time-invariant systems

In this paper, we provide a framework to achieve interval estimation for nabla Caputo fractional order linear time-invariant (LTI) systems in the presence of bounded model uncertainties. Interval observers based on fractional order positive systems theory are designed by possessing desired stable and positive error dynamics. Specifically, the basic concepts and conditions for guaranteeing stability and positivity of the considered systems are derived systematically by finding the system responses. Using the developed criteria and the structure of Luenberger-type observers, a classic interval observer is designed directly which further extends the system classes of interval estimation. Besides, due to the possible absence of a gain matrix which ensures positivity requirement, a more general interval observer design scheme is given by exploiting the coordinate transformation technique. Finally, some simulated cases including fault detection and fractional order circuits related scenarios are developed to validate the usefulness and practicality of the framework.