Motion Game Control Under Temporary Failure of Control Unit
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MOTION GAME CONTROL UNDER TEMPORARY FAILURE OF CONTROL UNIT UDC 517.977
K. A. Chikrii
Abstract. The paper deals with the game problem of bringing the trajectory of a dynamic system to a given set under a failure of the control system on a time interval of known length but at the instant of time unknown a priori. Two approaches are proposed on the basis of the method of resolving functions, which make it possible to develop the sufficient conditions for the game termination in a certain guaranteed time starting from the given initial positions. Keywords: conflict-controlled process, multi-valued mapping, Aumann integral, Pontryagin condition, Cauchy formula, Minkowski geometrical difference. INTRODUCTION Consider a special linear nonstationary game of approach with a given cylindrical terminal set. The process develops so that control units of the first player fail at some a priori unknown time for a time necessary for repair and known in advance. The process continues until the trajectory reaches the terminal set. It is necessary to determine the guaranteed game completion time and control of the first player that ensures this result. The technique of constructing resolving functions [1, 2] is used to establish two types of the sufficient game completion conditions. In one case, the shear of trajectory caused by the failure is eliminated at once on the time interval that follows immediately the interval of control units failure, on the assumption that this is possible, and then the process continues and resolving function is accumulated. In the other case, the whole bundle of trajectories generated by various controls of the second player on the emergency interval is brought to the terminal set. Some condition, relating the time of failure elimination on which the parameters of the bundle and dimension of the objective set depend, should be satisfied in this case. A similar problem was considered in [3] based on direct Pontryagin’s methods [4]. This study is close to the studies [4–18]. SCHEME WITH IMMEDIATE ELIMINATION OF THE CONSEQUENCES OF FAILURE Let a conflict-controlled process z& = A ( t ) z + b Q ( t )u - v, z Î R n , z ( t 0 ) = z 0 , t ³ t 0 ³ 0 , u ÎU ( t ) , v ÎV ( t ) ,
(1)
be specified, where A ( t ) is a matrix function of order n, whose elements are measurable functions summable on any finite interval, the control domains U ( t ) and V ( t ) , U ( t ) Î K ( R p ) , V ( t ) Î K ( R q ) , p, q Î N , are measurable compact-valued mappings for t Î [ t 0 , + ¥ ) , and N is the set of natural numbers. Function b Q ( t ) has the form V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2014, pp. 130–136. Original article submitted September 2, 2013. 1060-0396/14/5003-0439
©
2014 Springer Science+Business Media New York
439
ì 0, t Î [ Q, Q + h], b Q (t ) = í î 1, t Î [ Q, Q + h], where Q ³ t 0 is the time of failure of the control units, h, h > 0 , is the time for failure
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