Download e-book for kindle: Control Under Lack of Information by Andrew N. Krasovskii, Nikolai N. Krasovskii

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By Andrew N. Krasovskii, Nikolai N. Krasovskii

ISBN-10: 146122568X

ISBN-13: 9781461225683

ISBN-10: 1461275830

ISBN-13: 9781461275831

The mathematical idea of keep an eye on, basically constructed over the past a long time, is used for fixing many difficulties of sensible significance. The potency of its functions has elevated in reference to the refine­ ment of desktop innovations and the corresponding mathematical gentle­ ware. Real-time keep watch over schemes that come with computer-realized blocks are, for instance, attracting ever extra consciousness. the speculation of keep an eye on offers summary versions of managed platforms and the methods discovered in them. This concept investigates those versions, proposes equipment for solv­ ing the corresponding difficulties and shows how you can build keep watch over algorithms and the tools in their desktop cognizance. the standard scheme of regulate is the subsequent: there's an item F whose nation at whenever fast t is defined by means of a part variable x. the article is subjected to a keep an eye on motion u. This motion is generated through a keep an eye on equipment U. the thing can be stricken by a disturbance v generated by way of the surroundings. the data at the kingdom of the method is provided to the generator U by means of the informational variable y. The mathematical personality of the variables x, u, v and yare made up our minds by way of the character of the system.

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8) u[t*[·]t*) = {u[t] E and some disturbances are realized. 8) may be arbitrary Borelmeasurable functions in t E [t*, t*). 8), respectively. 9) satisfies the Lipschitz condition in t. Consequently, it is absolutely continuous. 11) Here G is a compact set in the space {t, x}, t E [to, '19], x E R n which satisfies the following condition. 11). 12) 41 1. 2). 4. The positional quality index We consider minimax-maximin (with respect to u and v) problems for some quality index that determines the quality of the process of control.

K, X[ti], y[t i ], c:)+ < ti+I' y[tIJ = y*. 38) Here {uW, l = 1, ... 3). 38) x[tiJ are the values of the phase vector of the controlled x-object. These values x[t;] are realized at the time moments T = t;. We choose the functions -a~, V[2] p(t, x, y, c:) = {Pl(t, x, y, c:), l = 1, ... ,4; 4 Pl(t,x,y,c:) ~ O,LPl(t,X,y,c:) = I}. 40) 1=1 We imagine temporary that the motion x[t], t* ~ t ~ f) of the controlled x-object is formed in the following way: For the fixed initial position {t*, x*}, for the chosen parameter c: > 0 and for the chosen partition ~{ti}' the motion x[tJ is determined at t* ~ t ~ f) as the step-by-step solution of differential equation x[tJ = B[I](t)u(1)[t;]v(1)[t] + B[2] (t)U(2) (ti' X[ti], y[t i ], c:)+ +C[I] (t)v*[tJ + C[2] (t)v* [tJ , ti ~ t < ti+I' i = 1, ...

V,,(ti. ",'[ti]. 4). 8 where the disturbance is formed in the most unfavourable way with respect to minimization of the quality index 'Y. 34 Control Under Lack of Information The considered problem can be formalized within the framework of the antagonistic differential game for two persons. 80) holds for every position {t*, x*} E G. 78) respectively. ) is considered as that of the second one. So the informal description of the saddle point {80' (. ), 88 (-)} and the value of the game p0(t*, x*) is the following.

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Control Under Lack of Information by Andrew N. Krasovskii, Nikolai N. Krasovskii

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