On Robust Stability of Differential-Algebraic Equations with Structured Uncertainty
We consider a linear time-invariant system of differential-algebraic equations (DAE), which can be written as a system of ordinary differential equations with non-invertible coefficients matrices. An important characteristic of DAE is the unsolvability index, which reflects the complexity of the internal structure of the system. The question of the asymptotic stability of DAE containing the uncertainty given by the matrix norm is investigated. We consider a perturbation in the structured uncertainty case. It is assumed that the initial nominal system is asymptotically stable. For the analysis, the original equation is reduced to the structural form, in which the differential and algebraic subsystems are separated. This structural form is equivalent to the input system in the sense of coincidence of sets of solutions, and the operator transforming the DAE into the structural form possesses the inverse operator. The conversion to structural form does not use a change of variables. Regularity of matrix pencil of the source equation is the necessary and sufficient condition of structural form existence. Sufficient conditions have been obtained that perturbations do not break the internal structure of the nominal system. Under these conditions robust stability of the DAE with structured uncertainty is investigated. Estimates for the stability radius of the perturbed DAE system are obtained. The text of the article is from the simpler case, in which the perturbation is present only for an unknown function, to a more complex one, under which the perturbation is also present in the derivative of the unknown function. We used values of the real and the complex stability radii of explicit ordinary differential equations for obtaining the results. We consider the example illustrating the obtained results.
About the Authors
Alexey D. Kononov, Postgraduate, Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation, e-mail: firstname.lastname@example.org
34A09, 34D20, 37C75
1. Bobylev N.A., Emel’janov S.V., Korovin S.K. Estimates for perturbations of stablematrices. Automation and Remote Control, 1998, vol. 59, no 4, pp. 467-475.
2. Gantmacher F.R. Teoriya matrits [The theory of matrices]. Moscow, Nauka Publ.,1988. 548 p.
3. Kryzhko I.B. An inequality for the perturbations of the spectrum of matrix.Dal’nevost. Mat. Zh., 2000, vol. 1, no 1, pp. 111–118. (in Russian)
4. Molchanov A.P., Morozov M.V. Sufficient Conditions for Robust Stability of LinearNonstationary Control Systems with Periodic Interval Constraints. Automationand Remote Control, 1997, vol. 58, no 1, pp. 82-87.
5. Molchanov A.P., Morozov M.V. Robust absolute stability of nonstationary discretesystems with periodic constraints. Automation and Remote Control, 1995, vol. 56,no 10, pp. 1432–1437.
6. Morozov M.V. Robust stability conditions for linear nonstationary control systemswith periodic interval constraints. Probl. Upr., 2009, no 3, pp. 23–26. (in Russian)
7. Polyak B.T.Robastnaja ustojchivost’ i upravlenie [Robust stability and control].Moscow, Nauka Publ., 2002. 273 p.
8. Shcheglova A.A., Kononov A.D. On Robust Stability of Systems of Differential-Algebraic Equations. Izv. Irkutsk. Gos. Univ. Ser. Mat.[The Bulletin of IrkutskState University. Series Mathematics], 2016, vol. 16, pp. 117–130.
9. Shcheglova A.A., Kononov A.D. Robust stability of differential-algebraic equationswith an arbitrary unsolvability index. Automation and Remote Control, 2017, vol.78, no 5, pp. 798–814. https://doi.org/10.1134/S0005117917050034
10. Shcheglova A.A. The solvability of the initial problem for a degenerate linear hybridsystem with variable coefficients. Russian Mathematics, 2010, vol. 54, no 9, pp.49-61. https://doi.org/10.3103/S1066369X10090057
11. Byers R., Nichols N.K. On the stability radius of a generalized state-space system.Lin. Alg. Appl., 1993, vol. 188-189, pp. 113–134. https://doi.org/10.1016/0024-3795(93)90466-2
12. Davison E.J., Qiu L. The stability robustness of generalized eigenvalues. Proc. ofthe 28th IEEE Conf., 1989, vol. 3, pp. 1902-1907.
13. Du N.H., Linh V.H., Chyan C.-J. On data-dependence of exponential stabilityand stability radii for linear time-varying differential-algebraic systems. Journal ofDifferential Equations, 2008, vol. 245, pp. 2078-2102.
14. Du N.H., Linh V.H., Mehrmann V. Robust stability of differential-algebraicequations. Differential-Algebraic Equations Forum I, Springer, 2015, pp. 63-95.
15. Du N.H., Linh V.H., Mehrmann V., Thuan D.D. Stability and RobustStability of Linear Time-Invariant Delay Differential-Algebraic Equations. SIAMJournal on Matrix Analysis and Applications, 2013, vol. 34, pp. 1631-1654.https://doi.org/10.1137/130926110
16. Du N.H. Stability radii of differential-algebraice quations with structuredperturbations. Systems Control Letters, 2008, vol. 60, pp. 596–603.https://doi.org/10.1016/j.sysconle.2011.04.018
17. Du N.H., Liem N.C., Thuan D.D. Stability radius of implicit dynamic equationswith constant coefficients on time scales. Systems Control Letters, 2011, vol. 60.pp. 596–603. https://doi.org/10.1016/j.sysconle.2011.04.018
18. Du N.H., Linh V.H. Stability radii for linear time-varying differential-algebraicequations with respect to dynamic perturbations. Journal of DifferentialEquations, 2006, vol. 230. pp. 579-599.
19. Hinrichsen D., Kelb B., Linnemann A. An algorithm for the computation ofthe structured complex stability radius. Automatica, 1989, vol. 25, pp. 771–775.https://doi.org/10.1016/0005-1098(89)90034-4
20. Hinrichsen D. Robust stability of positive continuous timesystems. Numer. Funct. Anal. Optim., 1996, vol. 17, pp. 649–659.https://doi.org/10.1080/01630569608816716
21. Linh V.H., Thuan D.D. Spectrum-Based Robust Stability Analysis of Linear DelayDifferential-Algebraic Equations. Numerical algebra, matrix theory, differential-algebraic equations and control theory: Festschrift in honor of Volker Mehrmann,Springer, 2015, pp. 533-557. https://doi.org/10.1007/978-3-319-15260-8_19