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… Show more
“…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.…”
Section: Output-quantized Linear Systems Of Dimensionmentioning
confidence: 99%
“…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.…”
Section: Background: Attractors and Left Invertibilitymentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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]).…”
Section: Introductionmentioning
confidence: 99%
“…[10] Denote with ∂S the boundary of S. Suppose that H ∩ ∂S = ∅. Then there exists a (computable) k such that…”
This paper studies left invertibility of discrete-time linear outputquantized systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility, i.e. injectivity of I/O map is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set and is much easier to detect. The condition under which left invertibility and left D-invertibility are equivalent is that the elements of the dynamic matrix of the system form an algebraically independent set. Our main result is a method to compute left D-invertibility for all linear systems with no eigenvalue of modulus one. Therefore, we are able to check left invertibility of output-quantized linear systems for a full measure set of matrices. Some examples are presented to show the application of the proposed method.
“…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.…”
Section: Output-quantized Linear Systems Of Dimensionmentioning
confidence: 99%
“…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.…”
Section: Background: Attractors and Left Invertibilitymentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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]).…”
Section: Introductionmentioning
confidence: 99%
“…[10] Denote with ∂S the boundary of S. Suppose that H ∩ ∂S = ∅. Then there exists a (computable) k such that…”
This paper studies left invertibility of discrete-time linear outputquantized systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility, i.e. injectivity of I/O map is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set and is much easier to detect. The condition under which left invertibility and left D-invertibility are equivalent is that the elements of the dynamic matrix of the system form an algebraically independent set. Our main result is a method to compute left D-invertibility for all linear systems with no eigenvalue of modulus one. Therefore, we are able to check left invertibility of output-quantized linear systems for a full measure set of matrices. Some examples are presented to show the application of the proposed method.
This paper studies left invertibility of discrete-time linear I/O quantized linear systems of dimension 1. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility, i.e. injectivity of I/O map, is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership in sets of a given partition, left D-invertibility considers only distances, and is very easy to detect. Considering the system x + = ax + u, our main result states that left invertibility and left D-invertibility are equivalent, for all but a (computable) set of a's, discrete except for the possible presence of two accumulation point. In other words, from a practical point of view left invertibility and left D-invertibility are equivalent except for a finite number of cases. The proof of this equivalence involves some number theoretic techniques that have revealed a mathematical problem important in itself. Finally, some examples are presented to show the application of the proposed method.
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