Functional stability of aggregate systems

Abstract

i. If they are to be solved, the practical problems connected with the design, construction, and operation of complex systems require the creation of methods of analysis and synthesis for such systems. Here a mathematical model of such a complex system is investigated, just as in the theory of automatic control mathematical models of real control systems are studied. In this paper complex systems whose models can be represented as an aggregate are considered [i, 2]. The investigation centers on the important problem of analyzing the functional stability of an aggregate. Stability belongs not to the material object but only to some property or aspect of it. Thus, some characteristics of a system may be stable (in some sense) in relation to some disturbances and others unstable in the same sense with respect to the same disturbances. In [3] it was indicated that in investigating the stability of complex systems it is useful to study the behavior of some functionals of the process of system functioning. This is connected with the fact that every specific system can be characterized by numerical or qualitative indexes whose values in a given realization are sufficient in practice for evaluation of the result. As shown in [2], it is possible to select for aggregates a class of functionals of practical interest such that the functionals are supplementary coordinates and vary in accordance with laws similar to the laws of variation of the remaining coordinates. Hence we assume that the functionals investigated are among the supplementary coordinates. 2. Since it is extremely complicated, and often quite tropes sible, to find the ~trajectory ~ of motion of an aggregate by analytic methods, it is useful to define the permissible domains of occurrence of the supplementary coordinates of the aggregate in each fundamental state. We introduce the following notation: i = i, 2 .... are the fundamental states; F i = I, 2 .... are the domains of variation of the supplementary coordinates in each state, where the d imens ion of the s ta te F i equals mi; Ai ~ F i a r e the p e r m i s s i b l e domains of va r i a t ion of the supp lemen ta ry coord ina tes (it is poss ib le that A i = A where A is the empty set , for some i); zi iS the vec tor for the s u p p l e m e n t a r y coord ina tes in the i th fundamenta l s ta te ; m i is the d imens ion of the vec tor . Moreover , we a s s u m e that t r a n s i t i o n s between s ta tes a r e defined in. some way, and that the laws of va r i a t ion of the s u p p l e m e n t a r y coord ina tes a r e defined in each fundamenta l s ta te . DefiJaition Io The aggregate functions stably with probability 1 e relative to the given domains A i, i = = I, 2 ..... if there exist domains A0i ~=2 Ai, possibly possessing some additional properties such that z(to) E UAoi, implies. P (z (0 d [.j A,., t >i to} > ~ ~.