On computing the infinity norm

Abstract

existing methods. The main computation work involved in the algorithm be its transfer matrix. Assume that { A , B, C, D } has no (purely) includes one Cholesky decomposition [step I], one singular value imaginary eigenvalues which are both controllable and observable. Then decomposition [step 21, and one matrix-pseudoinverse operation [step 31. the matrix-valued function G is bounded on the imaginary axis, and the All these operations are well known, and can be performed very infinity norm of G is, by definition, efficiently by the existing packages such as LINPACK, EISPACK, etc. Even when balancing is not required, this algorithm would hopefully IlGIIm=sup llG(b,ll provide us with an efficient method for obtaining a minimal realization U E R from a TFM.2) When evaluating F ( C 3 in step 2), we need the inverse of 6(Cl); however, this matrix-inverse operation can be avoided if we note the identity b(cb) = 0. By choosing two polyno_mials x(z) and y(z) such that x(z)6(z) + y(z)b(z) = 1, we then have b(Cb)-' = x(cb).3) It is known [ 11 that the controllability Gramian and the observability Gramian are the fundamental invariants under the bilinear transformation s = zl/z + 1. Hence, in (5), if ( A , B, C, D) has diagonal Gramians, then (a, B, e, d) will also have the same diagonal Gramians which, in turn, implies that (a, B, e, 8) is actually the balanced realization for w.

Erweiterung der Methode zur direkten Berechnung der H∞ -Norm auf diskrete Systeme und auf Systeme mit mehreren Ein- oder Ausgängen Extending the method for direct calculation of the/H∞ - norm to discrete systems and to systems with multiple inputs or outputs

“…Die ersten von Boyd, Balakrishnan, Kabamba [2] und Röbel [3] entwickelten Verfahren für kontinuierliche Systeme wurden von Hinrichsen, Kelb, Linnemann, Pritchard [4,5] und Bruinsma, Steinbuch [6] verbessert. Dies sind iterative Verfahren, die auch für Mehrgrößensy-steme angewendet werden können.…”