Synthesis of Hurwitz Polynomial Families Using Root Locus Portraits
А. А. Несенчук · Automation Control and Intelligent Systems · 2019
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.