hal-01698372
https://ensta-bretagne.hal.science/hal-01698372
https://ensta-bretagne.hal.science/hal-01698372/document
https://ensta-bretagne.hal.science/hal-01698372/file/jusVol11No2paper06%281%29.pdf
[UNIV-BREST] Université de Bretagne occidentale - Brest (UBO)
[INSTITUT-TELECOM] Institut Mines Télécom
[ENSTA-BRETAGNE] ENSTA Bretagne
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-UBS] Université de Bretagne Sud
[ENSTA-BRETAGNE-STIC] Département STIC
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[ENIB] Ecole Nationale d'Ingénieurs de Brest
[LAB-STICC] Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
[TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes
[INSTITUTS-TELECOM] composantes instituts telecom
Computability of the Avoidance Set and of the Set-Valued Identification Problem
Welte, Anthony
Jaulin, Luc
Ceberio, Martine
Kreinovich, Vladik
[INFO.INFO-RB] Computer Science [cs]/Robotics [cs.RO]
[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]
[SPI.AUTO] Engineering Sciences [physics]/Automatic
[INFO.INFO-AI] Computer Science [cs]/Artificial Intelligence [cs.AI]
ART
avoidance set
set-valued uncertainty
computability
forbidden region
In some practical situations, we need to find the avoidance set, i.e., the set of all initial states for which the system never goes into the forbidden region. Algorithms are known for computing the avoidance set in several practically important cases. In this paper, we consider a general case, and we show that, in some reasonable sense, the corresponding general problem is always algorithmically solvable. A similar algorithm is possible for another general system-related problem: the problem of describing the set of all possible states which are consistent with the available measurement results. 1 Formulation of the Problem In control, we usually deal with robots (or other controlled devices) whose states s are described by tuples of real numbers s = (s 1 , · · · , s d). The dynamics of such devices is usually described by a system of differential equations ds i dt = f i (s(t)), for a known computable functions f i (s). In most practical situations, we can use these equations to compute, for each initial state s 0 at the starting moment t 0 , and for each moment of time t < t 0 , the state s(s 0 , t) of the system at the moment t; see, e.g., [2, 4, 10, 12]. Often in control, we have a set S of states that a robot (or other controlled device) needs to avoid. Because of this necessity: • once we know how the states change in time, i.e., once we know the algorithm s(t, s 0) that describes how the state s at moment t depends on t and on the initial state s 0 , • we need to find the set S 0 of all the initial states for which the trajectory avoids the forbidden set S for all moments of time from the starting moment t 0 to a given future moment T. In other words, we want to find the avoidance set S 0 = {s 0 : s(t, s 0) ̸ ∈ S for all t ∈ [t 0 , T ]}. There exist algorithms for solving this problem in some specific situations; see, e.g., [3, 7, 9]. In this paper, we analyze the general problem of computing the avoidance set, and we show that this problem is, in some reasonable sense, algorithmically computable.
2017
2018-02-01
en
Journal of Uncertain Systems
World Academic Press