Stability Analysis of Large-Scale Nonlinear Systems using Parallel Computation

Arizona State University, Scottsdale AZ

Investigators

Abstract

As engineered systems grow in complexity, the difficulty of safe and reliable operation of these systems becomes more challenging. For example, consider the $15 billion international nuclear fusion reactor being built in Cadarache, France. Although the world has known for 60 years that it is possible to produce energy from nuclear fusion by heating plasma in a magnetic field, physicists have never been able to control the magnetic field accurately enough to produce significant amounts of power. The reason is that even the simplest models of magneto hydrodynamics involve more than 20 coupled nonlinear differential equations. Although algorithms for control have made great strides in recent years, control of systems of this complexity is still out of reach. This project will design new algorithms for control which use supercomputers and massively parallel computation in an attempt to enable the safe and reliable design of controllers for large complex systems such as describe plasma in a reactor. At the heart of the project is a new way of using convex optimization to parameterize Lyapunov functions (a measure of energy). Specifically, while the well-known sum-of-squares parameterization of positive polynomials is convex, reliable and accurate for small-scale systems, it cannot be readily adapted to supercomputers and other forms of massively parallel computation. The essence of this project, then is to look for alternative mathematical parameterizations of Lyapunov functions which are convex and furthermore are amenable to parallel computation. Such alternatives exist in classical mathematical results by Handelman, Polya and Bernstein. The scope of work is to use those results to create parallel codes, which can study multiple coupled nonlinear equations and determine the best possible Lyapunov function fit within the mathematical Language of polynomials. The project will test these algorithms on cluster and parallel graphics processor computing machines and will be able to study nonlinear differential equations with up to 20 states. These algorithms will then be applied to discretized nonlinear partial differential equation representations of the magneto-hydrodynamics of plasma in a nuclear fusion reactor to obtain a function of energy which can then be used to design and test magnetic and radio frequency controllers which will reduce or eliminate magneto hydrodynamic instabilities. The algorithms developed can also be applied to any large nonlinear system, implying they can be used to improve understanding and control in applications such as chemical reactors, gene regulatory networks and communication satellites.