  Fractional-order dynamical systems are increasingly being used to model and
describe processes demonstrating long-term memory or complex interlaced
dependencies amongst the spatial and temporal components of a wide variety of
cyber-physical systems. Notable examples include networked control systems or
neurophysiological signals such as electroencephalogram (EEG). As a result, the
estimation of states of a fractional-order dynamical system becomes an
important problem to be solved. To this effect, this paper addresses the
problem of minimum-energy state estimation for discrete-time fractional-order
dynamical systems (DT-FODS). Specifically, we exploit a finite-dimensional
approximation of the infinite-dimensional DT-FODS to design a minimum-energy
state observer. The state and output equations are affected by an additive
noise that is considered to be deterministic but unknown. The resulting
estimation error is proved to be exponentially input-to-state stable with
respect to the disturbances and to a signal that is decreasing with the
increase of the accuracy of the adopted approximation model. An illustrative
example shows the effectiveness of the proposed method on real-world
neurophysiological EEG signals.
