Left invertibility of discrete systems with finite inputs and quantised output

Abstract

The aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems which evolve in discrete time within a continuous state-space; quantized outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. Our main results are obtained under some contractivity hypotheses. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system. Our main result is a necessary and sufficient condition for left invertibility and uniform left invertibility for joint contractive systems. In addition, an algorithm is proposed to recover inputs from output strings. A few examples are presented to illustrate the application of the proposed method.

Left invertibility of discrete-time output-quantized systems: the linear case with finite inputs

“…a set that the state has to exit to guarantee an invertibility property. In [11] a necessary and sufficient condition for left invertibility of joint contractive systems is given, but here we state the same condition in a more abstract setting: H and S are an attractor and a quantization set, not the attractor and the quantization set of the system (3). That's because in the following we will use these results for another attractor and quantization set, i.e.…”

“…, e d depends only on the partition P, and any small "disturbance" of the rate of the partition P allows the application of the Theorem 7. Further details on this point are given in [11]. ♦ 6.…”

“…Clearly x(k) ∈] − 1, 1[ for every k ∈ N, so system (11) is not ULDI. Nonetheless system (11) is ULI in 1 step. Consider indeed the quantization set (defined in Definition 4)…”

Self-similar control systems and applications to zygodactyl bird's foot

“…This opens the way to the wide theoretical background of fractal geometry and, in particular, to the branch devoted to the investigation of self-similar structures. By establishing a relation between discrete control systems and fractals, well-known concepts and results coming from self-similar dynamics (like the attractor of an iterated function systems or the celebrated Open Set Condition) are used to describe the topology of the reachable set and other properties of the dynamical systems -see for instance [9] for an investigation on the left invertibility of discrete control systems via Iteration Function Systems.…”

Left invertibility of output-quantized systems: An application to cryptography

“…In [28], the left invertibility problem for a switched system is discussed. Left invertibility setting in relation with output-quantized systems, and results about contractive systems are given in ( [9]). …”