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The Hermite-Biehler Theorem and Its Generalization

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Structure and Synthesis of PID Controllers

Abstract

The classical Hermite-Biehler Theorem and our generalization of it will be described in this chapter. These results will be crucial in our characterization of stabilizing PID controllers.

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Notes and References

  1. Gantmacher F. R., The Theory of Matrices, New York; Chelsea Publishing Company, 1959.

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  2. Chapellat H., Mansour M. and Bhattacharyya S. P., “Elementary Proofs of Some Classical Stability Criteria,” IEEE Transactions on Education, Vol. 33, No. 3, Aug. 1990.

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  3. Kharitonov V. L., “Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations,” Differential’nye Uravneniya, Vol. 14, 2086–2088, 1978.

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  4. Bhattacharyya S. P., Robust Stabilization Against Structured Perturbations, Vol. 99, Lecture Notes in Control and Information Sciences, Springer-Verlag, 1987.

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  5. Ho M. T., Datta A. and Bhattacharyya S. P., “Generalizations of the HermiteBiehler Theorem,” Linear Algebra and its Applications (to appear).

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  6. Ho M. T., Datta A. and Bhattacharyya S. P., “An Elementary Derivation of the Routh-Hurwitz Criterion,” IEEE Transactions on Automatic Control, Vol. AC-43, No. 3, 405–409, March 1998.

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  7. Stojic M. R. and Siljak D. D., “Generalization of Hurwitz, Nyquist, and Mikhailov Stability Criteria,” IEEE Trans. on Automat. Contr., Vol. AC-10, 250–254, July, 1965.

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© 2000 Springer-Verlag London

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Datta, A., Ho, MT., Bhattacharyya, S.P. (2000). The Hermite-Biehler Theorem and Its Generalization. In: Structure and Synthesis of PID Controllers. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-3651-4_3

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  • DOI: https://doi.org/10.1007/978-1-4471-3651-4_3

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84996-889-8

  • Online ISBN: 978-1-4471-3651-4

  • eBook Packages: Springer Book Archive

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