Reachability and robust design in dynamic systems

Abstract

Systems of engineering interest usually evolve in time. Models that capture this dynamic behavior can more accurately describe the system. Dynamic models are especially important in the chemical, oil and gas, and pharmaceutical industries, where processes are intrinsically dynamic, or taking into account dynamic behavior is critical for safety. Especially where safety is concerned, uncertainty in the inputs to these models must be addressed. The problems of forward reachability and robust design provide information about a dynamic system when uncertainty is present. This thesis develops theory and numerical methods for approaching the problems of reachability and robust design applied to dynamic systems. The main assumption is that the models of interest are initial value problems (IVPs) in ordinary differential equations (ODEs). In the case of reachability analysis, the focus is on efficiently calculated enclosures or "bounds" of the reachable sets, since one motivating application is to (deterministic) global dynamic optimization, which requires such information. The theoretical approach taken is inspired by the theory of differential inequalities, which leads to methods which require the solution of an auxiliary IVP defined by parametric optimization problems. Major contributions of this work include methods and theory for efficiently estimating and handling these auxiliary problems. Along these lines, a method for constructing affine relaxations with special parametric properties is developed. The methods for calculating bounds also are extended to a method for calculating affine relaxations of the solutions of IVPs in parametric ODEs. Further, the problem of ODEs with linear programs embedded is analyzed. This formulation has further application to dynamic flux balance models, which can apply to bioreactors. These models have properties that can make them difficult to handle numerically, and this thesis provides the first rigorous analysis of this problem as well as a very efficient numerical method for the solution of dynamic flux balance models. The approach taken to robust design is inspired by design centering and, more generally, generalized semi-infinite programming. Theoretical results for reformulating generalized semi-infinite programs are proposed and discussed. This discussion leads to a method for robust design that has clear numerical benefits over others when the system of interest is dynamic in nature. One major benefit is that much of the computational effort can be performed by established commercial software for global optimization. Another method which has a simple implementation in the context of branch and bound is also developed.