Classical Results on the Stability of Linear Time-Invariant Systems, and the Schwarz Form

The paper deals with the classical results of Routh and Hurwitz, Biehler and Kharitonov, concerning the stability of a linear time invariant differential equation. It is shown that these results follow directly from a Schwarz matrix representation of stable systems.

[1]  F. Gantmacher,et al.  Applications of the theory of matrices , 1960 .

[2]  P. Parks A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  C. Chen,et al.  A matrix for evaluating Schwarz's form , 1966 .

[4]  S. Barnett,et al.  Canonical forms for time-invariant linear control systems: a survey with extensions Part I. Single-input case , 1978 .

[5]  F. R. Gantmakher The Theory of Matrices , 1984 .

[6]  J. P. Lasalle Liapunov’s Direct Method , 1986 .

[7]  S. Bhattacharyya,et al.  Robust control , 1987, IEEE Control Systems Magazine.

[8]  B. Ross Barmish,et al.  New Tools for Robustness of Linear Systems , 1993 .

[9]  G. Meinsma Elementary proof of the Routh-Hurwitz test , 1995 .

[10]  A. Borobia,et al.  Three coefficients of a polynomial can determine its instability , 2001 .

[11]  J. Ackermann,et al.  Robust control , 2002 .

[12]  Olga Holtz Hermite–Biehler, Routh–Hurwitz, and total positivity , 2003, math/0512591.

[13]  Miguel V. Carriegos,et al.  Canonical forms for single input linear systems , 2003, Syst. Control. Lett..

[14]  Robert Shorten,et al.  Hurwitz Stability of Metzler Matrices , 2010, IEEE Transactions on Automatic Control.

[15]  E. J. Routh A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion , 2010 .

[16]  M. Tyaglov Sign patterns of the Schwarz matrices and generalized Hurwitz polynomials , 2012 .

[17]  M. Tyaglov On the spectra of Schwarz matrices with certain sign patterns , 2012, 1201.0738.