An overview of the spectral theory of the linear-quadratic and H∞ problems is presented. Originally, the spectral theory of the linear quadratic problem emerged out of the problem of characterizing the finite-time escape of the Riccati differential equation as a condition on the spectrum of a Toeplitz-plus-Hankel operator. Later and following an independent line of thought, the spectral theory of the H∞ problem emerged as a Toeplitz-plus-Hankel operator characterization of the smallest achievable tolerance in feedback design. This common Toeplitz-plus-Hankel operator structure shared by the linear-quadratic and H∞ problems is elucidated by mapping the usual H∞ frequency-response specification into the time domain, leading to an inequality bound on a quadratic functional precisely induced by the Toeplitz-plus-Hankel operator. With this deep linear-quadratic-H∞ connection at hand, we derive a linear-quadratic solution to the H∞ problem, and show that such issues as the Adamjan-Arov-Krein problem, pole-zero cancellation in optimal H∞ compensation, etc. can be fruitfully attacked using simple linear-quadratic arguments.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics