Parametric Synthesis of a Robust Controller Based on the Method of Dominant Poles

In the paper a linear control system described by its characteristic polynomial with interval coefficients including parameters of controller linearly is considered. Problem of the research is finding parameters of a controller guaranteeing dynamic characteristics of a system despite interval parametric uncertainty of its object. It is proposed to base a controller synthesis on root quality indices: minimal stability degree and maximal oscillability degree. Desired values of these indices will be provided with the help of dominant poles method. Applying this method consists in placing a pair of complex-conjugate dominant poles; all other poles — unrestricted poles — will be placed by defining a right border of their allocation area on a complex plane. To apply dominant poles method, a feature of stability degree and oscillability degree to be determined by images of certain vertices of a parametric polytope was used. To synthesize a controller, it is proposed to divide its parameters in two groups: dependent ones and unrestricted ones. The first group of controller parameters is to provide desired allocation of dominant poles in one of vertices of parametric polytope (a dominant vertex). Unrestricted parameters of a controller are to provide desired distance between dominant poles and allocation area of unrestricted poles. To find coordinates of a dominant vertex and verifying vertices providing unrestricted poles allocation, an interval extension of basic phase equation of a root locus theory was developed. This resulted in interval phase inequalities, whose solution allows finding coordinates of desired vertices of characteristic polynomials coeffi cients polytope. Knowing a dominant vertex polynomial and dominant poles allows expressing dependent parameters of a controller from unrestricted ones. Obtained expressions allow placing unrestricted poles in a desired area of a complex plane. To do this, a D-partition by unrestricted parameters of a controller is performed in all verifying vertices of parametric polytope of a system. After choosing values of unrestricted parameters from intersection of all stability domains obtain during D-partition, dependent parameters of a controller can be calculated. An example of synthesizing a PID-controller guaranteeing desired values of dynamics characteristics for an interval control system of the fourth order is provided.

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