Controllability and Observability of Linear Discrete-Time Fractional-Order Systems

Abstract

In this paper we extend some basic results on the controllability and observability of linear discrete-time fractional-order systems. For both of these fundamental structural properties we establish some new concepts inherent to fractional-order systems and we develop new analytical methods for checking these properties. Numerical examples are presented to illustrate the theoretical results.

Discrete-Time Fractional-Order Systems: Modeling and Stability Issues

“…This theorem can be proved by induction [34]. The first part of the solution of (40) represents the free response of the system and the last part takes the role of the convolution sum corresponding to the forced response.…”

“…To our knowledge, no such results have been brought in the literature up to now for FOS with non commensurate fractional-order systems in continuous time. The next section of this chapter constitutes a contribution to the study of structural properties of non-commensurate FOS, using a different approach, that is discrete-time representation [34].…”

“…This extension induces noticeable changes: there appears a neccessity of interpretation of the initial conditions [31] and a neccesity to take into account the history (or memory) of the system from the initial instant, since we have to deal, instead of the classical state, with a so-called "'pseudo state"', that is an expanded state [32,33]. This infinite-dimensional system, as we expose it in the following sections, has been studied, from the point of view of its structural properties, i.e., controllability, observability [34,35] and, to some extent, of its stability [36].…”

Controllability, reachability and minimum energy control of fractional discrete-time linear systems with multiple delays in state

“…The problems of controllability and reachability of dynamic systems, standard and fractional, have been recently considered in many papers (see [3][4][5][6][7][8][9][10][11][12][13], for example). A survey of recent results in theory of controllability of dynamical systems is given in [14].…”

On the controllability of fractional dynamical systems

“…The analysis of controllability and observability of continuous and discrete time fractional order systems modeled by fractional state space equations was provided by Bettayeb and Djennoune (2008) as well as Guermah et al (2008), respectively. Klamka (2010) addressed the minimum energy control problem of infinite-dimensional fractional-discrete time linear systems and established necessary and sufficient conditions for exact controllability.…”