Determination of the Area of Robust Stability of the System on the Basis of V. L. Kharitonov’s Theorem

The parameters of the object of regulation during operation due to various reasons may vary. These changes can lead to a change in the performance indicators of the automatic system, as well as its stability. This article proposes an approach to determine the range of acceptable values of the parameters of the control object of an automatic system with a PID controller, in which the system will remain stable. Thus, the problem arises of analyzing an automatic control system given not only by a single model with clearly defined parameters, but by a family of models belonging to a given set — the task of robust regulation. The search for ranges in which the parameters of the regulated object can change is based on the solution of the nonlinear programming problem in this paper. The conclusion of the objective function and constraint system using the theorem of V. L. Kharitonova on the robust stability of linear systems. The main idea is that each parameter of the regulatory object can be changed by some value hi1 in the direction of decrease and by hi2 — in the direction of increase. Replacing the notation used in the theorem of V. L. Kharitonov, the lower and upper boundaries of the change of parameters by the sum and difference of the nominal values of the parameters and the corresponding hi1, hi2, we get a system of restrictions. Moreover, for the stability of Kharitonov polynomials, it is most convenient to use the Lienar-Shipar criterion. The larger the values of hi1, hi2, the wider the ranges of variation of the parameters, and the smaller the inverse of the sum of these values. Based on this statement, the objective function is formed. It should be noted that the condition for the considered automatic system on which the proposed approach is based is sufficient, but not necessary, since the coefficients of the polynomial are interdependent. An example with the help of which the proposed approach is demonstrated is considered. This approach can also be applied to other linear systems for which theconditions of V. L. Kharitonova. 

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