Simulation of fractional aircraft control systems by spectral method in Faber-Schauder function system

Mathematica modeling, numerical technique and program complexes


Аuthors

Rybin V. V.*, Tsvetaev V. E.*

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: dep805@mai.ru

Abstract

Today a variety of Computer Mathematics Systems (CMS) and expansion packs are applied for the spectral method of non-stationary control systems computer simulation. Modern technologies associated with fractal approach in different applied areas, and in radio engineering, radiodetection and theory of dynamic systems in particular, produce new element base, where mathematical models contain fractional integrating and derivative operators.

For example, control systems with PID control realize control laws, increasing processing speed and stability margin compared to the similar systems realizing classical control laws. Technical implementation of such derivatives and integrals can be realized with several methods: by Gruenwald’s approximate dependencies, or continued fractions, or Fourier transform and spectral transform.

Spectral method is already propagated to control systems for models containing fractional integrating and derivative components, and the expansion packs MLSY_SM, Spektr_SM+Simulink+Matlab, Spektr_SM+VisSim+Mathcad CMS have been modified for such systems’ simulation. This software commplex does not contain program packs in the Faber-Schauder function system.

The presented paper considers the development of the MLSY_SM SCM Mathcad expansion pack for non-stationary uninterrupted non-stationary control systems with integer number and fractional order spectral method in the Faber-Schauder functions system analysis.

Derivation of spectral algorithms with fractional integrating and derivative components of an arbitrary order and of some other spectral characteristics is based on using a symbolic processor of Computer Mathematics Systems Mathcad. Software implementation of the expansion pack MLSY_SM_SH+Mathcad contains software modulus developed using the derived symbolic algorithms. The structure of the pack and its program modules calls are described in Appendix. The expansion pack itself is used for analyzing and parametric synthesizing of the control system for homing missile. Mathematical model of the missile uses a target coordinator (mounted on a gyro-platform) for measuring the sight line rotational speed. Differential equation, describing the target coordinator, contains a fractional order derivative.

As a result of the performed work the expansion pack MLSY_SM_SH of Computer Mathematics Systems Mathcad applied for investigation on a stochastic model homing system which uses a fractional target seeker was developed. The dependence of the root mean missing value from differential operator fractional order of target coordinator and navigation constant of command generation block was studied. Optimal values of a fractional parameter and navigation constants were selected. A comparative analysis of homing missile system’s classical and fractional models was carried out.

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Keywords:

non-stationary control systems, spectral form of mathematical description, Faber-Schauder function system, Computer Mathematics Systems, fractional integrating and derivative elements

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