Synthesis method of modal regulator for control object with interval uncertainty of coefficients

As a result of the identification of technological processes, the model of the control object is most often restored in the form of a linear differential equation (of a given order), the coefficients of which are determined with accuracy to belong to certain intervals. In fact, such interval uncertainty of coefficients means that the control object is described by the whole family of models. In most of the classical schemes of synthesis, the regulator is calculated for the model with exactly the desired coefficients. In the case of a control object with interval uncertainty of coefficients, the regulator can be calculated according to classical schemes for a model with average values from the given intervals. In this case, after the closure of the control object (described by the family of models) by the synthesized controller, uncertainty appears in the transfer function of the closed system. Since the properties of stability and control quality of the system are determined by the location of the poles of its transfer function, the question arises: at what size of the interval uncertainty in the control object the closed system will still retain the properties of stability (robust stability) and control quality (robust control quality)? In the modal control scheme, the control quality is set as an area S on the complex plane, which determines the desired location of the poles of the transfer function. Therefore, the questions of research (verification) of robust stability and robust control quality can be considered from a single point of view: is it necessary to check whether the roots of a given family of polynomials in the domain S? In the literature devoted to robust theory focuses on the problem of robust stability, and the problem of robust quality control fades into the background and is still not solved. The aim of the study is to develop a scheme for the synthesis of a modal controller for the case of interval uncertainty of the coefficients in the object model and a method for the study of robust control quality for a closed control system. This paper generalizes a theorem known as the" zero elimination principle" and the problem of checking the robust quality of control is reduced to the problems of constructing a" geometric image "of a family of polynomials with interval uncertainty of coefficients and subsequent verification of the belonging of a given point to a "geometric image". The geometric image of such a family represents a convex linear shell on the complex plane. The technology of construction of convex linear shells is described in many works. In the article the method of synthesis of a robust modal regulator is offered, the criterion of robust quality of control is received. The synthesis method is brought to computational procedures and can be implemented on a computer. The synthesis technology is illustrated by an example. Practical significance: this method of synthesis allows to calculate the regulator settings in the conditions of interval uncertainty of coefficients in the control object model.

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