On robust stability of systems of differential-algebraic equations

A. A. Shcheglova   ; A. D. Kononov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 16 (2016), pp. 117-130

Résumé

We consider linear time-invariant systems of ordinary differential equations with degenerate matrix before the derivative of the desired vector function. Such systems are called differential-algebraic equations (DAE). The unsolvability measure with respect to the derivatives for some DAE is an integer that is called the index of the DAE. The analysis is carried out under the assumption of existence of a structural form with separated differential and algebraic subsystems. This structural form is equivalent to the input system in the sense of solution, and the operator transforming the DAE into the structural form possesses the left inverse operator. The finding of the structural form is constructive and do not use a change of variables. In addition the problem of consistency of the initial data is solved automatically. We prove that regularity of the matrix pencil is sufficient for existence of the structural form in the time-invariant case. We show the connection between matrix pencil index, unsolvability index of DAE, and the order of linear differential operator transforming the DAE into the structural form. The approach uses the concept of $r$-derivative array equations, where $r$ is the unsolvability index. The existence of a nonsingular minor of order $n(r+1)$ in the matrix describing derivative array equations is necessary and sufficient for existence of this structural form ($n$ is the dimension of DAE under consideration). In the paper we investigate the problem of asymptotic stability of DAE in case of perturbation which is defined by means of matrix norm. The perturbation introduced into the system of DAE does not break its intrinsic structure and is closely connected with location of the minor mentioned above in the matrix describing derivative array equations. The sufficient conditions of robust stability for index-one and index-two systems are obtained. We use the values of real and complex stability radii obtained for system of ordinary differential equations solved with respect to the derivatives. We consider the example illustrating the obtained results.

Détail

Keywords: differential-algebraic equations, robust stability.

Bibliographie

[1] Gantmakher F. R., The theory of matrices, Nauka, M., 1988 (in Russian) | MR

[2] Polyak B. T., Robust stability and control, Nauka, M., 2002 (in Russian)

[3] Trenogin V. A., Functional analysis, Nauka, M., 1980 (in Russian) | MR

[4] Shcheglova A. A., “The transformation of a linear algebraic-differential system to an equivalent form”, Analytical Mechanics, Stability and Motion Control, Proceeding of the IX International Chetaev Conference (Irkutsk, June 2007), v. 5, 298–307 (in Russian)

[5] Shcheglova A. A., “Existence of solution to initial problem for a degenerat time-varying linear hybrid system”, Russian Mathematics, 2010, no. 9, 49–62 | DOI | MR

[6] R. Byers, N. K. Nichols, “On the stability radius oa a generalized state-space system”, Linear Algebra Appl., 188–189 (1993), 113–134 | DOI | MR | Zbl

[7] C. J. Chyan, N. Y. Du, V. H. Linh, “On data-dependence of exponential stability and the stability radii for linear time-varying differential-algebraic systems”, J. Differ. Equ., 245 (2008), 2078–2102 | DOI | MR | Zbl

[8] N. Y. Du, V. H. Linh, “Stability radii for linear time-varying differential-algebraic equations with respect to dynamics perturbations”, J. Differ. Equ., 230 (2006), 579–599 | DOI | MR | Zbl

[9] N. Y. Du, “Stability radii of differential-algebraic equations with structured perturbations”, Syst. Control Lett., 57 (2008), 546–553 | DOI | MR | Zbl

[10] N. Y. Du, D. D. Thuan, N. C. Liem, “Stability radius of implicit dynamic equations with constant coefficients on time scales”, Syst. Control Lett., 60 (2011), 596–603 | DOI | MR | Zbl