Розв’язання задач гарантованої чутливості для дискретних систем

Анотація

UDC 681.5.07

PrOBLEM SOlving guaranteed SENSITIVITY

For Descrete SYSTEMS.

L. Pantalienko

Annotation. The algorithms for solving problems of guaranteed sensitivity of discrete equations dependent settings. Considered objectives of the presence of constraints on the function of the sensitivity of linear and nonlinear types. Based on practical methods of numerical stability assessment conducted initial field sensitivity for features.

Key words: discrete system, stability, parameters, the guaranteed sensitivity, sensitivity functions.

Managing the real system, as a rule, in the face of uncertainty under the influence of different factors of technical nature, the environment and operating conditions. Some of these factors are basic information about the properties of the system are taken into account in the model because of its vector phase state. But whatever was not a full set of vector always there are really a lot of uncontrolled additional factors that could significantly affect the operation of the facility. This, for example, inaccurate given external perturbation errors when the program management, measurement error and so on. In this regard naturally require the system to be insensitive (tolerant) to changing parameters.

To specify terms of efficiency of the system introduced restrictions on dynamic sensitivity function - derivatives of the state of the parameter . Problems associated with sensitivity analysis in such a setting, belongs to a class of problems guaranteed and limited sensitivity .

In contrast to known methods  of numerical solution of problems of guaranteed sensitivity is based on algorithms for practical stability  structured areas of initial

conditions. In the case of nonlinear discrete systems previously conducted its linearization in the vicinity of any settlement movement.

         The aim of research ─ development of numerical methods for solving problems of guaranteed sensitivity based algorithms practical stability.

Materials and methods of research. The work used mathematical methods for stability analysis, sensitivity and optimization algorithms.

Explore the system of linear discrete equations

, ,                                        (1)

if restrictions on the function of sensitivity  , .

Warehouse for system (1) the sensitivity equations

                   , ,          ,                  (2)

and the area of initial conditions served in a structured form: .

To build constructive algorithms assess the initial area to consider permissible sensitivity functions set consider acceptable set linear and nonlinear types:

 , ;                          (3)

 , ,                                            (4)

де  − scalar continuously differentiated function, and set  − convex, closed and contain zero point ().

In the case of restrictions (3), (4) provided stability for sensitivity functions become appropriate form

,

, , ;                                       

,

                   , ,                

and is the same for all modes of settlement.

         From these positions may be considered discrete nonlinear system of equations

 , ,                         

having linearization in the vicinity of any settlement movement, such as zero. As a result, we obtain a linear system with permanent disturbances:

, ,                                 

where ,  − known matrix,  − vector function dimension , determining the error of approximation ().

         Criteria of stability for system relative sensitivity functions

 ,

made according to form

,

, , ;                              

,

         , , .                

where …, ,.

Results. For discrete parametric systems performed numerical solution of problems of guaranteed sensitivity methods practical stability for structured areas of initial conditions.

Conclusions

Based on numerical methods of stability carried guaranteed sensitivity of solving problems for discrete systems. Obtained the best assessment of areas of initial conditions in the presence of constraints on the function of sensitivity.