Ho, Ming-Tzu; Datta, Aniruddha; Bhattacharyya, S. P. Generalizations of the Hermite-Biehler theorem. (English) Zbl 0949.15037 Linear Algebra Appl. 302-303, 135-153 (1999). The authors generalize the Hermite-Biehler theorem which characterizes the Hurwitz stability of a real polynomial \(\delta(s).\) Let \(\delta(s)\) be a real polynomial of degree \(n\) and let \(l\) and \(r\) denote the numbers of roots in the left half and the right half of the complex plane, respectively. The signature \(l-r\) of \(\delta(s)\) determines \(l\) and \(r\) since \(l+r=n.\) Let \(i\) be the square root of \(-1\) and let \(\omega\) be a real number. Then \(\delta(i\omega)=p(\omega)+iq(\omega).\) The authors find a formula expressing the signature of \(\delta(s)\) involving \(p\) and \(q\) and zeros of these functions. Reviewer: E.Ellers (Toronto) Cited in 1 ReviewCited in 11 Documents MSC: 15A63 Quadratic and bilinear forms, inner products 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory Keywords:Hermite-Biehler theorem; generalized interlacing; stability; root distribution PDFBibTeX XMLCite \textit{M.-T. Ho} et al., Linear Algebra Appl. 302--303, 135--153 (1999; Zbl 0949.15037) Full Text: DOI References: [1] Gantmacher, F. R., The Theory of Matrices (1959), Chelsea Publishing Company: Chelsea Publishing Company New York · Zbl 0085.01001 [2] Kharitonov, V. L., Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differential’nye Uravneniya, 14, 2086-2088 (1978) [3] S.P. Bhattacharyya, Robust Stabilization Against Structured Perturbations, Lecture Notes in Control and Information Sciences, vol. 99, Springer, Berlin, 1987; S.P. Bhattacharyya, Robust Stabilization Against Structured Perturbations, Lecture Notes in Control and Information Sciences, vol. 99, Springer, Berlin, 1987 · Zbl 0699.93074 [4] Chapellat, H.; Mansour, M.; Bhattacharyya, S. P., Elementary proofs of some classical stability criteria, IEEE Trans. Education, 33, 3 (1990) [5] S.P. Bhattacharyya, H. Chapellat, L.H. Keel, Robust Control the Parametric Approach, Prentice-Hall, Englewood Cliffs, NJ, 1995; S.P. Bhattacharyya, H. Chapellat, L.H. Keel, Robust Control the Parametric Approach, Prentice-Hall, Englewood Cliffs, NJ, 1995 · Zbl 0838.93008 [6] M. Mansour, Robust stability in systems described by rational functions, in: C.T. Leondes (Ed.), Control and dynamic systems, vol. 51, Academic Press, New York, 1992, pp. 79-128; M. Mansour, Robust stability in systems described by rational functions, in: C.T. Leondes (Ed.), Control and dynamic systems, vol. 51, Academic Press, New York, 1992, pp. 79-128 · Zbl 0781.93078 [7] Stojic, M. R.; Siljak, D. D., Generalization of hurwitz nyquist and mikhailov stability criteria, IEEE Trans. Automat. Contr., AC-10, 250-254 (1965) [8] M.T. Ho, A. Datta, S.P. Bhattacharyya, A new approach to feedback design part I: Generalized interlacing and proportional control, Department of Electrical Engineering, Texas A & M University, College Station, TX, Tech. Report TAMU-ECE97-001-A; M.T. Ho, A. Datta, S.P. Bhattacharyya, A new approach to feedback design part I: Generalized interlacing and proportional control, Department of Electrical Engineering, Texas A & M University, College Station, TX, Tech. Report TAMU-ECE97-001-A [9] M.T. Ho, A. Datta, S.P. Bhattacharyya, A new approach to feedback design part II: PI and PID controllers, Department of Electrical Engineering, Texas A & M University, College Station, TX, Tech. Report TAMU-ECE97-001-B; M.T. Ho, A. Datta, S.P. Bhattacharyya, A new approach to feedback design part II: PI and PID controllers, Department of Electrical Engineering, Texas A & M University, College Station, TX, Tech. Report TAMU-ECE97-001-B This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.