Abstract
The paper deals with the problem of synthesis of a stable characteristic polynomial families describing the control systems' dynamics in conditions of the interval uncertainty. Investigation is based on the system mathematical model in the form of its root locus portrait generated by the polynomial free term variation that is named in the paper as the "free root locus portrait". The root loci of the Kharitonov's polynomials family (subfamily) is picked out of the whole polynomial family and is considered for carrying out the investigation. Specific regularities of the interval root locus portrait have been discovered. On the basis of these regularities main properties of the system root locus portrait have been defined. A stability condition has been formulated that allows to calculate the polynomial free term variation interval ensuring the polynomial family hurwitz stability. This stability condition is applicable to the class of polynomials having their free root locus poles lying within the left half-plane of roots or, in other words, being stable when their free term is equal to zero. The stable family is being synthesized by setting up (adjusting) the given initial family that is supposed to be unstable, i.e. the proposed method of synthesis allows to turn stable (hurwitz) the given nonhurwitz interval polynomial family. The setting up criterion is specified in terms of proximity i.e. as the nearest distance from the "unstable" system roots to the "stable" ones as measured along the root trajectories. The stable polynomial could be selected as the nearest to the given unstable one with or without consideration of the system quality requirements. In the course of the setting up procedure new boundaries of only the polynomial free term variation interval (stability interval) are calculated that allows to ensure system stability without modification of its root locus portrait configuration. A numerical example of the polynomial setting up procedure has been given.
Highlights
The issues of assuring the acceptable dynamic characteristics of the plants operation in conditions of uncertainty are currently among the most important ones that drive forward the development of the automatic control theory. [1, 2]
The task of attaining stability of the interval family of control system characteristic polynomials has been solved in the paper
The method and algorithm have been developed for setting up the interval polynomial so that it gets stable in cases when the stability verification showed that the initial polynomial was unstable
Summary
The issues of assuring the acceptable dynamic characteristics of the plants operation in conditions of uncertainty are currently among the most important ones that drive forward the development of the automatic control theory. [1, 2]. Solutions for the tasks of parametric synthesis and analysis of uncertain systems (interval dynamic systems (IDS) in particular) with application of the main provisions of the root locus theory are offered in [16,17,18,19,20]. The contents of this paper are devoted to the root locus approach to the above described problem solution in application to the interval polynomial. It represents further development of works [16,17,18,19,20]. The technique that has been developed can be used for setting up the values of the parameter variation intervals limits for ensuring the IDS stability in the cases when the stability test had shown that the given system was unstable
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