Spectral value sets arise in the modeling of linear dynamical systems with uncertain feedback. They are important because they model uncertainty inherent in the feedback which is assumed to depend linearly on the output. They are parametrized by a parameter E which bounds the norm of the uncertainty in question. Theoretical aspects of the project include analysis of properties of specific extremal points of spectral value sets for a given value of E and at points of coalescence of the spectral value set components for critical values of E, both in terms of local geometry and algebraic measures. Algorithmic aspects include the development and analysis of fast methods to compute (1) maximizers of the real part or modulus over a given spectral value set for fixed E and (2) the complex stability radius (or its reciprocal, the H-infinity norm) defined as the largest value of E such that the associated spectral value set lies inside the stability region (the left half-plane or the unit disk). They also include developing methods to design controllers for open-loop plants that result in closed-loop systems with desired stability and optimality properties, such as locally maximizing the stability radius (minimizing the H-infinity norm). Since these functions are not concave or convex and their optimizers are typically at points where they are not differentiable, methods for nonsmooth, nonconvex minimization are needed, including methods that can handle constraints efficiently.
The mathematical properties of spectral value sets and associated algorithms to compute their extremal values have both theoretical and practical importance. The broader goal of the project is to bring the tools of algorithms for optimization over spectral value sets and related problems to a wide community of scientists and engineers, for use in many different kinds of applications. The investigator's open-source software is already in use in a variety of applications, including the design of aircraft controllers, a proton exchange membrane fuel cell system, power systems, observer-based fault detection and minimally invasive surgery. All of these systems require controllers to work effectively: a complex system such as an airplane or a power plant requires automatic controllers to function safely and effectively, in addition to skilled operators who know how to use such systems. However, current methods are limited to small or moderate-sized systems, which cannot model real physical systems accurately. The new methods will allow the design of controllers for much larger systems than was previously possible, including control of discretized systems of partial differential equations, which have applications throughout the natural sciences and engineering.
