J. William Helton

Extending H ∞ control to nonlinear systems: control of nonlinear systems to achieve performance objectives

H-infinity control originated from an effort to codify classical control methods, where one shapes frequency response functions for linear systems to meet certain objectives. H-infinity control underwent tremendous development in the 1980s and made considerable strides toward systematizing classical control. This book addresses the next major issue of how this extends to nonlinear systems. At the core of nonlinear control theory lie two partial differential equations (PDEs). One is a first-order evolution equation called the information state equation, which constitutes the dynamics of the controller. One can view this equation as a nonlinear dynamical system. Much of this volume is concerned with basic properties of this system, such as the nature of trajectories, stability, and, most important, how it leads to a general solution of the nonlinear H-infinity control problem.

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: Parametrization of the set of all solutions

We consider a general matrix version of a Pick-Loewner interpolation problem on the closed unit disk. Solutions are allowed to have a finite numberl of free poles in the open disk. We show that the smallestl for which a solution to the problem exists is the number of negative eigenvalues of an appropriately defined “Pick matrix,” and for this value ofl we obtain a linear fractional map parametrization of the class of all solutions. The idea is to adapt the Grassmannian approach involving Krein space geometry and invariant subspace representations of the authors; this was successful previously for the case where all interpolating points are inside the disk. Also an appendix includes an errata to earlier work together with simplified proofs.

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

A set

The matricial relaxation of a linear matrix inequality

S\subseteq\mathbb{R}^n

Proper analytic free maps

is called semidefinite programming (SDP) representable or semidefinite representable if

NonlinearH∞ control theory for stable plants

S

Engineering Systems and Free Semi-Algebraic Geometry

equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality (LMI). Clearly, if

The convex Positivstellensatz in a free algebra

S

Homotopy methods for counting reaction network equilibria

is SDP representable, then

Shift invariant manifolds and nonlinear analytic function theory

S