An upper bound of the reachability index for a special class of positive 2-D systems

Bailo, Esteban; Gelonch, Josep; Romero‐Vivó, Sergio

doi:10.13001/1081-3810.1289

Cited by 4 publications

(3 citation statements)

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“…The analysis of local reachability index for positive two-dimensional systems described by the Fornasini-Marchesini state-space model is a complex task from the mathematical point of view. Different quadratic upper bounds on I LR have been consecutively derived in the literature (see [1] and the references therein, [9] and [28]), but being at most (n + 3 (⌊n/2⌋) 2 in the greatest event (see [2]). In this contribution, using the composition table introduced in [2] and limiting the set of entries on the table where a vertex can be deterministically reached, a new family of systems is constructed whose local reachability index can be cubic under suitable conditions, and largely exceeds the aforementioned upper bounds.…”

Section: Discussionmentioning

confidence: 99%

“…In this paper, we focus our attention on positive two-dimensional (2-D) systems described by the Fornasini-Marchesini state-space model (see [1] and [26]) which is as follows:…”

Section: Introductionmentioning

confidence: 99%

“…This index plays an important role because it enables us to determine in a finite number of steps whether a system is locally reachable or not as well as to know how many separations sets at most are needed to achieve the states. Previous studies such as [10], [11], [16] and references therein, [20] and [21] have studied and reported that the reachability index for positive 1-D systems is always bounded by the dimension n. However, several attempts in two dimensional case (see [1], [2] and [9] and references therein) have been made to obtain a general formula depending on n on the local reachability index for any positive 2-D systems. So far, despite the use of graph-theoretic techniques, to solve this question seems to be a nontrivial goal to accomplish due to its computational complexity (see [30]).…”

Section: Introductionmentioning

confidence: 99%

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A first cubic upper bound on the local reachability index for some positive 2-D systems

Bailo

Gelonch

Romero‐Vivó

2019

RACSAM

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The calculation of the smallest number of steps needed to deterministically reach all local states of an nth-order positive 2-D system, which is called local reachability index (I LR ) of that system, was recently tackled by means of the use of a suitable composition table. The greatest index I LR obtained in the previous literature was n + 3 (⌊n/2⌋) 2 for some appropriated values of n. Taking as a basis both a combinatorial approach of such systems and the construction of suitable geometric sets in the plane, an upper bound on I LR depending on the dimension n for a new family of systems is characterized. The 2-D influence digraph of this family of order n ≥ 6 consists of two subdigraphs corresponding to a unique source s. The first one is a cycle involving the first n 1 vertices and is connected to the another subdigraph through the 1-arc (2, n 1 + n 2 ), being the natural numbers n 1 and n 2 such that n 1 > n 2 ≥ 2 and n − n 1 − n 2 ≥ 1. The second one has two main cycles, a cycle where only the remaining vertices n 1 + 1, . . . , n appear and a cycle containing only the vertices n 1 + 1, . . . , n 1 + n 2 − 1. Moreover, the last vertices are connected through the 2-arc (n 1 + n 2 − 1, n). Furthermore, if n ≥ 12 and is a multiple of 3, for appropriate n 1 and n 2 , the I LR of that family is at least

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Section: Discussionmentioning

confidence: 99%

“…In this paper, we focus our attention on positive two-dimensional (2-D) systems described by the Fornasini-Marchesini state-space model (see [1] and [26]) which is as follows:…”

Section: Introductionmentioning

confidence: 99%