Scalable Methods for Approximating and Optimizing Robust Stability Functions

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Robust stability functions (RSFs) have broad importance in many fields in science and engineering. In the case of linear dynamical systems, described by ordinary differential or difference equations with no feedback control, RSFs are real-valued functions of system coefficient matrices. The most important examples are the pseudospectral abscissa and radius functions and the distance to instability. The principal investigator is developing and analyzing new efficient iterative methods to approximate RSFs much more quickly than is currently possible. In many applications the coefficient matrices depend on parameters which may be varied in order to optimize robust system stability. Because RSFs are not convex and are typically not differentiable at optimizers, it is essential to use nonsmooth, nonconvex optimization (NNO) methods that take this into account. The efficiency of the new methods will open the way to computing and optimizing RSFs for much larger systems than was previously possible, including discretized systems of partial differential equations. In practice most dynamical systems include feedback control. Then the RSFs of interest become more complex and an important generalization is the H-infinity norm. The PI is involved in the development of software that can be widely used in the engineering community. This award was selected for partial funding by a program which promotes the reuse of Cyberinfrastructure (CI) elements through the Office of Cyberinfrastructure at the National Science Foundation.The goal of the project is to bring new optimization tools 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 controllers and the design of winding systems for elastic web materials. 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, these computations are currently limited to small or moderate-sized systems, which cannot model real physical systems very accurately. New scalable 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. This award was selected for partial funding by a program which promotes the reuse of Cyberinfrastructure (CI) elements through the Office of Cyberinfrastructure at the National Science Foundation.