Б. О. Джолдошев а из Института автоматики и информационных технологий нан кр, г. Бишкек; «Cинтез кибернетических автоматических систем с использованием эталонной модели»

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Б. О. Джолдошев а из Института автоматики и информационных технологий нан кр, г. Бишкек; «Cинтез кибернетических автоматических систем с использованием эталонной модели»

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1. Introduction


There are a lot of methods of design of robust control that are developed with increasing interest and some of them become classical. Commonly all of them are dedicated to defining the ranges of parameters (if uncertainty of parameters takes place) within which the system will function with desirable properties, first of all, will be stable [1, 2]. Thus there are many research efforts which successfully attenuate the uncertain changes of parameters in small (regarding to magnitudes of their own nominal values) ranges. But no one existing method can guarantee the stability of designed control system at arbitrarily large ranges of uncertainly changing parameters of plant. The approach that is offered in the present work relies on the results of catastrophe theory [3, 4, 5, 6, 7], uses nonlinear structurally stable functions, and due to bifurcations of equilibrium points in designed nonlinear systems allows to stabilize a dynamic plant with ultimately wide ranges of changing of parameters.

It is known that the catastrophe theory deals with several functions which are characterized by their stable structure. Today there are many classifications of these functions but originally they are discovered as seven basic nonlinearities named as ‘catastrophes’:



(fold);

(cusp);

(swallowtail);

(butterfly);

(hyperbolic umbilic);

(elliptic umbilic);

(parabolic umbilic).

A part of the catastrophe which does not contain parameters ki is called as ‘germ’ of catastrophe. Adding any of them to dynamic system as a controller will give effect shown below. On the example of the catastrophe ‘elliptic umbilic’ added to dynamical systems we shall see that:



  1. new (one or several) equilibrium point appears so there are at least two equilibrium point in new designed system,

  2. these equilibrium points are stable but not simultaneous, i.e. if one exists (is stable) then another does not exist (is unstable),

  3. stability of the equilibrium points are determined by values or relations of values of parameters of the system,

  4. what value(s) or what relation(s) of values of parameters would not be, every time there will be one and only one stable equilibrium point to which the system will attend and thus be stable.

Let us consider the cases of second-order systems (2) and examples (possible applications) of design of control systems (3): double pendulum’s stable oscillations.


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