A bisection method for computing the H_infinity norm of a transfer matrix and related problems

S. Boyd, V. Balakrishnan, and P. Kabamba

Mathematics of Control, Signals, and Systems, 2(3):207-219, 1989

Abstract: Inspired by recent work of Byers [1] we establish a simple connection 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 of this connection, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem and the work of Willems [2].

This result yields a simple bisection algorithm to compute the H_infinity norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and of course 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, e.g., the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H_infinity Armijo line search problem with no more computation than is required to compute a single H_infinity norm.

References:

R. Byers. A Bisection Algorithm for Measuring the Distance of a Stable Matrix to the Unstable Matrices. Technical Report, North Carolina State Univ. at Raleigh, 1987. To appear in SIAM Journal on Scientific and Statistical Computing.
J. C. Willems. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Automatic Control, AC-16:621--634, 1971.

Keywords: H_infinity norm, Hamiltonian, Sturm test, Armijo line search.

Download PDF Bibtex entry