Left invertibility of discrete systems with finite inputs and quantized output

Abstract

Abstract-The aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems that 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. We restrict to the case of contractive dynamics for fixed inputs. 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 left invertibility property of the system. Our main results are a necessary and sufficient condition for a given system to be left invertible with probability one on the space of inputs (i.e. for almost all input sequences), and necessary and sufficient conditions for left invertibility and uniform left invertibility under some weak additional hypotheses. 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

“…This is a contractive system if |a| < 1 and an expansive system if |a| > 1. If |a| < 1 the invertibility problem can be solved with the methods of section 3 (see [10]). The next Theorem shows a necessary condition for the ULI of a system of type (6): if it is not satisfied we construct inductively a pair of strings that gives rise to the same output.…”

“…Theorem 3. [10] Denote with ∂S the boundary of S. Suppose that H ∩ ∂S = ∅. Then there exists a (computable) k such that V IG k = V IG k ∩ S. ♦ If instead the system (3) is joint expansive, then the map x(k) → x(k +1) admits an inverse for every u ∈ U.…”

“…The main tools used in the paper are the theory of Iterated Function Systems (IFS), and a theorem of Kronecker. The use of IFS in relation with left invertibility is described in [10]. The Kronecker's theorem has to do with density in the unit cube of the fractional part of real numbers.…”

“…Invertibility of nonlinear systems is discussed in [21]. More recent work has addressed the left invertibility for switched systems ( [28]), and for quantized contractive systems ( [10]).…”

“…[10] Denote with ∂S the boundary of S. Suppose that H ∩ ∂S = ∅. Then there exists a (computable) k such that…”