Analyzing Robust Stability of an Interval Control System on the Basis of Vertex Polynomials

In the paper, a characteristic polynomial of an interval control system, whose coefficients are unknown or may vary within certain ranges of values, is considered. Parametric variations cause migration of interval characteristic polynomial roots within their allocation areas, whose borders determine robust stability degree of the interval control system. To estimate a robust stability degree, a projection of a polytope of interval characteristic polynomial coefficients on a complex plane must be examined. However, in order to find a robust stability degree it is enough to examine some vertices of a coefficient polytope and not the whole polytope. To find these vertices, which fully determine a robust stability degree, it is proposed to use a basic phase equation of a root locus method. Considering the requirements to placing allocation areas of system poles an interval extension of expressions for angles included to the phase equation. The set of statements, allowing to find a sum of pole angles intervals in the case of degree of oscillating robust stability, were formulated and proved. From these statements, a set of double interval angular inequalities was derived. The inequalities determine ranges of angles of all root locus edge branches departure from every pole. Considered research resulted in a procedure of finding coordinates of verifying vertices of a coefficients polytope and vertex polynomials according to these vertices. Such polynomials were found for oscillating robust stability degree analysis of interval control systems of the second, the third and the forth order. Also, similar statements were derived for aperiodical robust stability degree analysis. Numerical examples of vertex analysis of oscillating and aperiodical robust stability degree were provided for interval control systems of the second, the third and the fourth order. Obtained results were proved by examining root allocation areas of interval characteristic polynomials examined in application examples of proposed methods.

1. Haritonov V. L. Asimptoticheskaya ustojchivost’ semejstva sistem linejnyh differencial’nyh uravnenij (Asymptotic stability of a linear differential equations family), Differenc. Uravneniya, 1978, vol. 14, no. 11, pp. 2086—2088 (in Russian).

2. Anderson B. D. O., Kraus F., Mansour M., Desgupta S. Easily testable sufficient conditions for the robust stability of systems with multi-affine parameter dependence, editors: M. Mansour, Balemi, Truol, Robustness of dynamic systems with parameter uncertainties, Birkhauser, Basel, Switzerland, 1992, pp. 81—92.

3. Kawamura T., Shima M. Robust stability analysis of characteristic polynomials whose coefficients are polynomials of interval parameters, Journal of Mathematical System, Estimation and Control, 1996, vol. 6, no. 4, pp. 1—12.

4. Polyak B. T., Shcherbakov P. S. Robastnaya ustojchivost’ i upravlenie (Robust stability and control), Moscow, Nauka, 2002, 303 p. (in Russian).

5. Rimskij G. V. Kornevye metody issledovaniya interval’nyh system (Root methods of interval systems examining), Minsk, Institut tekhnicheskoj kibernetiki NAN Belarusi, 1999, 186 p. (in Russian).

6. Nesenchuk A. A. Analiz i sintez robastnyh dinamicheskih sistem na osnove kornevogo podhoda (Analysis and synthesis of robust dynamics systems on a base of root approach), Minsk, OIPI, 2005, 232 p. (in Russian).

7. Vadutov O. S., Gayvoronskiy S. A. Primenenie rebernoj marshrutizacii dlya analiza ustojchivosti interval’nyh polinomov (Application of edge routing to stability analysis of interval polynomials), Izv. AN. TiSU, 2003, no. 6, pp. 7—12 (in Russian).

8. Gayvoronskiy S. A. Opredelenie rebernogo marshruta dlya analiza robastnoj sektornoj ustojchivosti interval’nogo polinoma (Determination of the edge routing for the analysis of the robust sector stability of an interval polynomial), Izvestiya RAN. Teoriya i sistemy upravleniya, 2005, no. 5, pp. 11—15 (in Russian).

9. Gusev Yu. M., Efanov V. N., Krymskij V. G. Analiz i sintez linejnyh interval’nyh dinamicheskih sistem (sostoyanie problemy). Analiz s ispol’zovaniem interval’nyh harakteristicheskih polinomov (Analysis and synthesis of linear interval dynamic systems (problem condition). Analysis with the help of interval characteristic polynomials), Tekhnicheskaya Kibernetika, 1991, no. 1, pp. 3—30 (in Russian).

10. Gayvoronskiy S. A. Vershinnyj analiz kornevyh pokazatelej kachestva sistemy s interval’nymi parametrami (Vertex analysis of root quality indexes of the system with interval parameters), Izvestiya Tomskogo politekhnicheskogo universiteta, 2006, vol. 309, no. 7, pp. 6—9 (in Russian).

11. Uderman E. G. Metod kornevogo godografa v teorii avtomaticheskih system (Root locus method in control theory, Moscow, Nauka, 1972. 448 p. (in Russian).