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Convex Analysisの二,三の進展について

The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and  for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.

All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds†

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

The basic reproduction number ℛ0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation...

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The construction of next-generation matrices for compartmental epidemic models.

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Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

In this paper we give essentially complete results concerning various algebraic and topological aspects of feedback stabilization. In particular, we give necessary and sufficient conditions for a given transfer function matrix to have a right-coprime or a left-coprime factorization, and exhibit a large class of transfer function matrices that have both. We give the most general set of feedback stability criteria available to date, and derive a characterization of all compensators that stabilize a given plant. We define what is meant by "proper" and "strictly proper" in an abstract setting and show that (i) every strictly proper plant can be stabilized by a proper compensator, and (ii) every compensator that stabilizes a strictly proper plant must be proper. We then define a topology for unstable plants and compensators, and show that it is the weakest topology in which feedback stability is a robust property.

Hans Schneider

Papers

Algebraic and topological aspects of feedback stabilization

Some theorems on the inertia of general matrices

Cross-Positive Matrices

The diagonal equivalence of a nonnegative matrix to a stochastic matrix

Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley