Computing the regularization of a linear differential-algebraic system. (English) Zbl 1325.93028
Summary: We study the regularization problem for linear differential-algebraic systems. As an improvement of former results we show that any system can be regularized by a combination of state-space and input-space transformations, behavioral equivalence transformations and a reorganization of variables. The additional state feedback which is needed in earlier publications is shown to be superfluous. We provide an algorithmic procedure for the construction of the regularization and discuss computational aspects.
MSC:
| 93B17 | Transformations |
| 93C05 | Linear systems in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
References:
| [1] | Berger, T., On differential-algebraic control systems (2014), Institut für Mathematik, Technische Universität Ilmenau. Universitätsverlag Ilmenau: Institut für Mathematik, Technische Universität Ilmenau. Universitätsverlag Ilmenau Ilmenau, Germany, (Ph.D. thesis) |
| [2] | Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (1989), North-Holland: North-Holland Amsterdam · Zbl 0699.65057 |
| [3] | Kunkel, P.; Mehrmann, V., Differential-Algebraic Equations. Analysis and Numerical Solution (2006), EMS Publishing House: EMS Publishing House Zürich, Switzerland · Zbl 1095.34004 |
| [4] | Lamour, R.; März, R.; Tischendorf, C., (Differential Algebraic Equations: A Projector Based Analysis. Differential Algebraic Equations: A Projector Based Analysis, Differential-Algebraic Equations Forum, vol. 1 (2013), Springer-Verlag: Springer-Verlag Heidelberg, Berlin) · Zbl 1276.65045 |
| [5] | Riaza, R., Differential-Algebraic Systems. Analytical Aspects and Circuit Applications (2008), World Scientific Publishing: World Scientific Publishing Basel · Zbl 1184.34004 |
| [6] | Willems, J. C., System theoretic models for the analysis of physical systems, Ric. Autom., 10, 71-106 (1979) · Zbl 0938.37525 |
| [7] | Willems, J. C., Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Automat. Control, AC-36, 259-294 (1991) · Zbl 0737.93004 |
| [8] | Willems, J. C., The behavioral approach to open and interconnected systems, IEEE Control Syst. Mag., 27, 46-99 (2007) · Zbl 1395.93111 |
| [9] | Polderman, J. W.; Willems, J. C., Introduction to Mathematical Systems Theory. A Behavioral Approach (1998), Springer-Verlag: Springer-Verlag New York |
| [10] | Campbell, S. L.; Kunkel, P.; Mehrmann, V., Regularization of linear and nonlinear descriptor systems, (Biegler, L. T.; Campbell, S. L.; Mehrmann, V., Control and Optimization with Differential-Algebraic Constraints. Control and Optimization with Differential-Algebraic Constraints, Advances in Design and Control, vol. 23 (2012), SIAM: SIAM Philadelphia), 17-36 · Zbl 1317.93071 |
| [11] | Benner, P.; Losse, P.; Mehrmann, V.; Voigt, M., Numerical linear algebra methods for linear differential-algebraic equations, (Ilchmann, A.; Reis, T., Surveys in Differential-Algebraic Equations III. Surveys in Differential-Algebraic Equations III, Differential-Algebraic Equations Forum (2015), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 117-170 · Zbl 1343.65100 |
| [12] | Berger, T.; Reis, T., Regularization of linear time-invariant differential-algebraic systems, Systems Control Lett., 78, 40-46 (2015) · Zbl 1320.93030 |
| [13] | Golub, G. H.; van Loan, C. F., (Matrix Computations. Matrix Computations, Johns Hopkins Series in the Mathematical Sciences, vol. 3 (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, MD) · Zbl 0865.65009 |
| [14] | Beelen, T. G.J.; Van Dooren, P. M., An improved algorithm for the computation of Kronecker’s canonical form of a singular pencil, Linear Algebra Appl., 105, 9-65 (1988) · Zbl 0645.65022 |
| [15] | Berger, T.; Ilchmann, A.; Trenn, S., The quasi-Weierstraß form for regular matrix pencils, Linear Algebra Appl., 436, 4052-4069 (2012) · Zbl 1244.15008 |
| [16] | Van Dooren, P. M., The computation of Kronecker’s canonical form of a singular pencil, Linear Algebra Appl., 27, 103-140 (1979) · Zbl 0416.65026 |
| [17] | Beelen, T. G.J.; Van Dooren, P. M., A pencil approach for embedding a polynomial matrix into a unimodular matrix, SIAM J. Matrix Anal. Appl., 9, 77-89 (1988) · Zbl 0646.65041 |
| [18] | Berger, T.; Reis, T., Controllability of linear differential-algebraic systems—a survey, (Ilchmann, A.; Reis, T., Surveys in Differential-Algebraic Equations I. Surveys in Differential-Algebraic Equations I, Differential-Algebraic Equations Forum (2013), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 1-61 · Zbl 1266.93001 |
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