Abstract
We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H∞ Armijo line-search problem with no more computation than is required to compute a single H∞ norm.
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Research supported in part by NSF under Grant ECS-85-52465, ONR under Grant N00014-86-K-0112, an IBM faculty development award, and Bell Communications Research.
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Boyd, S., Balakrishnan, V. & Kabamba, P. A bisection method for computing the H∞ norm of a transfer matrix and related problems. Math. Control Signal Systems 2, 207–219 (1989). https://doi.org/10.1007/BF02551385
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DOI: https://doi.org/10.1007/BF02551385