Research of direct and inverse problems of sensitivity using methods of practical stability by direction
DOI:
https://doi.org/10.31548/energiya2(72).2024.154Abstract
The general statements of the problems of construction of extreme ranges of tolerances on parameters for a linear system of differential equations in the presence of dynamic restrictions on the spread of the state vector are considered. It was found that such sensitivity problems are closely related to the problems of estimating the maximum area of stability in the corresponding function space.
In order to obtain upper estimates of the sought values, the problem of estimating tolerances from the positions of stability in a single direction is formalized. The cases of linear and nonlinear restrictions on the spread of the phase coordinate vector with initial linear initial conditions relative to the vector of the system parameters are studied. Necessary and sufficient conditions are formulated in the form of criteria for estimating tolerances on parameters covering the case of fixed initial conditions.
The tasks of estimating the maximum volume of the stability region with respect to the deviations of the state vectors and system parameters are considered. For the case of compatible admissible constraints, the paper gives the corresponding numerical calculations for the upper estimates. In particular, for fixed initial conditions, the maximum set of tolerances for system parameters with compatible dynamic restrictions on fluctuations of state vectors and parameters are determined according to these evaluation criteria.
From the standpoint of practical stability in direction, direct sensitivity problems are investigated, numerical estimates of the spread of the phase coordinate vector of the system in the presence of linear and fixed initial conditions relative to the parameter vector are given.
It has been demonstrated that in the given specification of the problem of maximum estimation of tolerances it is completely covered by problem statements of practical stability by direction. Thus, to calculate areas of guaranteed sensitivity, the concept of practical stability in a single direction is extended to the space of sensitivity functions at the initial moment of time.
The above algorithms are used to estimate the range of tolerances for the adjustment parameters of the discrete model of the induction acceleration system with the given limitations on the spread of the quality criterion; the variation of the quality indicator within the tolerance field is presented in the form of a linear form relative to the spread of the parameter vector.
Key words: practical stability, single direction, tolerances on parameters, sensitivity, induction acceleration system
References
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