A method for forming a family of full-order modal regulators with an elliptic domain of coefficients belonging

The problem of synthesis of regulators consists of two sequentially solved problems: 1) find the regulator (if it exists) such that it provides stability and a given quality of control to a closed system; 2) if possible, find not one regulator, but some set: a set of controller settings on the one hand guarantees that some inaccuracies in the controller settings will not lead to a violation of the specified control quality, and on the other hand, it will allow you to set the task of finding the optimal controller settings from an acceptable set. Currently, a large number of methods for the synthesis of regulators have been developed, taking into account both the features of the task of the mathematical model of the control object and the features of the task of the control objectives: as examples, we can cite works on the synthesis of PI and PID regulators, on modal control, on the synthesis of stabilizing feedbacks in the presence of structural and parametric uncertainty in the model of the control object, etc. Thus, the first of these tasks can be attributed to the solved ones. Two approaches can be attributed to the solution of the second problem: the D-partitioning method and the Yula-Kucher parametrization. The D-partitioning method allows you to build stability regions of a closed control system in the space of one or two parameters (most often, the parameters of the controller). It is known that the D-partitioning method can also be used if it is required to provide the specified root quality indicators: root stability margin and oscillation. A significant disadvantage of the D-partitioning method is that in the case of several variables, only two independent variables should be left as variable, and the rest should be artificially fixed. The parametrization of Yula-Kucera reduces the problem of forming stabilizing regulators to solving the polynomial Bezu equation. However, these methods do not solve the problem of forming a family of regulators that provide a given quality of control. The modal control method allows synthesizing a controller for control objects of arbitrary order; this method assumes that the control object is described by a linear differential equation of the n-th order (where n is any non-negative integer) without delay. The modal regulator is also sought in the form of a linear differential equation. The quality of control is given in the form of a region S on a complex plane that determines the desired location of the poles of the transfer function of a closed system. It has been repeatedly proved in the literature that a modal regulator of the order n-1 and higher provides any given location of the poles of the transfer function of a closed system, and thereby guarantees stability and specified root quality indicators for a closed system. The (n-1)-th order regulator is called a full-order modal regulator. In this article, a method for forming a family of full-order modal regulators with an elliptic domain of coefficients that guarantee a given control quality is developed. This method is a generalization of D-partitioning. It should be noted that there are no analytical methods for solving this problem, in this regard, it is of interest to obtain estimates (from above and from below) of the ellipsoid radius in the space of the regulator coefficients. The paper proposes an algorithm for calculating an estimate from above of the radius of the elliptical region of setting the coefficients of the regulator. The effectiveness of the method is illustrated by an example. The expansion of the range of permissible values of the regulator coefficients can be obtained by combining two (or more) ellipsoids with different orientation of the semi-axes. Therefore, the proposed method can be used to form in the space of the coefficients of the regulator of the boundary of the stability region and a given control quality.

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