Finding verifying vertices of a coefficients polytope of characteristic polynomial for analyzing a robust oscillability of a control system with interval parameters

Abstract

The paper is dedicated to extending a root locus theory to systems with interval parameters, which are described with a characteristic polynomial with interval coefficients. On a basis of angular equation of a root locus a set of interval inequalities, including departure angles of root locus edge branches, determining oscillability degree of a control system, was derived. Solving the system of inequalities for low-order control systems resulted in a set of vertices of a characteristic polynomial coefficients polytope, projections of which on a complex plane determine an oscillability of such systems. Examples of application are provided.

Synthesis of a linear controller for a control system with interval parameters on a base of D‐partition in vertices of a coefficient's polyhedron of characteristic polynomial

“…As established in Khozhaev et al [38], to find the coordinates of the vertex whose image determines the degree of robust system oscillation, it is necessary, employing and to form and solve the interval double angular inequality as follows: Based on , we can write the interval double angular inequality for the IDS of an arbitrary order. Solving the obtained double angular inequalities for each , we can find the coordinates of the check vertices.…”

“…When finding these coordinates, the following issues should be taken into account: It is necessary to check the subdistributive property of interval calculations [40,41], that is, if it is possible to write double angular inequalities and in several ways, utilizing only those expressions that give a narrower interval for each value . If coordinates of the checking vertex contain three neighboring interval of the ICP coefficient with the same limits (maximum or minimum), then the image of the vertex cannot be on the boundary of the ICP root location domain. The Proof of Remark 2 is given in literature [37,38]. …”

“…As established in Khozhaev et al [38], to find the coordinates of the vertex whose image determines the degree of robust system oscillation, it is necessary, employing 8 and 9 to form and solve the interval double angular inequality as follows:…”

“…neighboring interval of the ICP coefficient with the same limits (maximum or minimum), then the image of the vertex cannot be on the boundary of the ICP root location domain. The Proof of Remark 2 is given in literature [37,38].…”

“…The disadvantage of the proposed solution is in the necessity to apply the D-partition method for all 2 m vertices of the coefficient polyhedron (m is the number of the interval ICP coefficients), which is a rather laborious procedure. In spite of efforts to reduce the number of checking vertices in several articles [22,23,[28][29][30][31][32][33][34][35][36][37][38], the analysis has shown that approaches proposed in these papers are still insufficient, since they give an excessive number of check vertices too. In this regard, it is essential and crucial to cope with the issue of determining the minimum set of check vertices and carrying out the vertex D-partition by desired controller parameters.…”