Fractional-Order Observer for Integer-Order LTI Systems

We devise an observer for integer-order LTI systems resorting to a fractional-order estimation error dynamics. For this purpose, we derive a class of fractional-order systems associated with the original integer-order LTI system and present necessary and sufficient conditions for their observability and controllability. These systems serve to compare the integer-order with the fractional-order dynamics by means of eigenvalue locations. As a result, we obtain an observer that shows a very fast convergence immediately after initialization. The algebraic decay of fractional-order systems results in a rather poor convergence of the estimation for large times. To overcome this, we propose two strategies: (i) We reinitialize the observer in short intervals such that the observer converges faster and (ii) we propose a fractional-order impulsive observer which yields the exact state in fixed time.