Analysis of degeneration of complex dynamic system with human components, страница 2

where  describes  human operator’s ability to predict development of information process,  is time estimation of informative perception, that depends on adaptive abilities of human operator,  is value of reactive delay, determined with a training of human operator. There is integrator for accumulating abilities. Median values of parameters are , , . Mathematical model of training human operators has interval parameters, which vary not more than in 25 percents.

Consider a complex continuous-time dynamic system that has interval parameters only in a state matrix in the following form

; ; ,                                                                       (8)                                                                                                                                                    

with state , external influence , output ;  is state matrix of system with interval parameters,  and  are input and output matrices respectively: ; , .

Interval state matrix  of system (8) consists of interval scalar components:

, ;.          (9)

We will use two representations of interval state matrix. The first representation holds:

,                                                                                                                   (10)

with , , where  

 and  are corner realizations of interval matrix .

And second representation of interval matrix  is decomposed with median and symmetric interval components in the form

,                                                                                                              (11)

where  is median component of matrix  with the fixed scalar components , determined by

,                                                                              (12)

with elements  in the following form

,                                                           (13)

and  is symmetric interval matrix components of interval matrix  with symmetric corner realizations  and  satisfying equations

,                                          (14)

.                                       (15)

IV.  FUNCTIONALS OF DEGENERATION OF COMPLEX SYSTEM WITH INTERVAL PARAMETERS

Obviously, functionals of degeneration of complex dynamic system, that is described by interval model, are interval values [1] and determined by the following form

,                                                                            (16)

where   is   interval criterion matrix of the complex system such that

,                                                                                                             (17)

with median component of interval criterion matrix  with fixed scalar components, symmetric interval matrix component  of  the interval criterion matrix ; ,  are maximum and - interval singular conditioning values of matrix  respectively.

Interval values of functionals of degeneration (16) are calculated with interval arithmetic [1].

Functionals of degeneration can be decomposed because of interval into two components:

,                                                                                                 (18)

where  is median component of interval functionals of degeneration  and  is symmetric interval component of interval functionals of degeneration . Its relative estimation can be calculated by the following form

,                                                                                                              (19)

and absolute estimation of interval functionals of degeneration such that

.                                                                                                        (20)

Form interval criterion matrix  of complex dynamic system (8) in the form (17) for different model description external influence [7].

Interval criterion matrix  of system (8) for a case of external multifrequency harmonic influence  with a frequency   can be represented as

,                                                                                                           (21)

where matrix  satisfies to matrix equation of Sylvester [5, 6] with interval matrix components in the following form

,                                                                                                   (22)