A unified structural interpretation of some well-known stability-test procedures for linear systems

A number of well-known stability-test procedures for continuous-and discrete-time systems are re-examined in a unified manner, leading to well-defined network-theoretic interpretations. The representation and network interpretation are based on the fact that the stability of any linear system (scalar or multivariable) is equivalent to the stability of a related all-pass system, which in turn can always be synthesized as a cascade of (scalar or matrix) two-pair all-pass (lossless) networks. The original system of interest is stable if and only if each all-pass two-pair is stable (and hence "lossless bounded real"). As a result of this interpretation, a number of related issues, such as enumeration of unstable poles, prematured terminations, and singularity situations can all be approached in a unified manner, based only on "two-pair extraction formulas." In addition, the network interpretation also leads to direct test procedures for testing relative stability, and the stability of multi-input, multi-output systems.

[1]  A. Cohn,et al.  Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise , 1922 .

[2]  E. A. S Guillemin,et al.  Synthesis of Passive Networks , 1957 .

[3]  D. Youla,et al.  On the factorization of rational matrices , 1961, IRE Trans. Inf. Theory.

[4]  Dragoslav D. Šiljak,et al.  Generalization of Hurwitz, Nyquist, and Mikhailov stability criteria , 1965 .

[5]  D. C. Youla,et al.  Interpolation with positive real functions , 1967 .

[6]  M. Marden.  Geometry of Polynomials , 1970 .

[7]  K. Åström.  Introduction to Stochastic Control Theory , 1970 .

[8]  D. Siljak.  Algebraic criteria for positive realness relative to the unit circle , 1973 .

[9]  A. Gray,et al.  Digital lattice and ladder filter synthesis , 1973 .

[10]  B. Anderson,et al.  A simple test for zeros of a complex polynomial in a sector , 1974 .

[11]  G. Hostetter.  An improved test for the zeros of a polynomial in a sector , 1975 .

[12]  A. Gray,et al.  A normalized digital filter structure , 1975 .

[13]  Brian D. O. Anderson,et al.  Stability of matrix polynomials , 1977 .

[14]  S. Mitra,et al.  Cascaded lattice realization of digital filters , 1977 .

[15]  S. Barnett,et al.  Routh’s Algorithm: A Centennial Survey , 1977 .

[16]  E. Jury.  Stability of multidimensional scalar and matrix polynomials , 1978, Proceedings of the IEEE.

[17]  Y. Kamp,et al.  The Nevanlinna–Pick Problem for Matrix-Valued Functions , 1979 .

[18]  Suguru Arimoto,et al.  Generalized Rouche's theorem and its application to multivariate autoregressions , 1980 .

[19]  Y. Genin,et al.  On the role of the Nevanlinna–Pick problem in circuit and system theory† , 1981 .

[20]  S. Gutman,et al.  A general theory for matrix root-clustering in subregions of the complex plane , 1981 .

[21]  S. Ahn,et al.  Stability of a matrix polynomial in discrete systems , 1982 .

[22]  E. Jury,et al.  Stability independent and dependent of delay for delay differential systems , 1984 .

[23]  P.P. Vaidyanathan,et al.  Low passband sensitivity digital filters: A generalized viewpoint and synthesis procedures , 1984, Proceedings of the IEEE.

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

[25]  P. P. Vaidyanathan,et al.  A unified approach to orthogonal digital filters and wave digital filters, based on LBR two-pair extraction , 1985 .

[26]  Sanjit K. Mitra,et al.  A general theory and synthesis procedure for low-sensitivity active RC filters , 1985 .

[27]  Sanjit K. Mitra,et al.  A general family of multivariable digital lattice filters , 1985 .