Date of Award
Doctor of Philosophy (PhD)
Chemical and Biomolecular Engineering
Joseph K Scott
Marc R Birtwistle
Eric M Davis
Systems of nonlinear ordinary differential equations (ODEs) are used to model an incredible variety of dynamic phenomena in chemical, oil and gas, and pharmaceutical industries. In reality, such models are nearly always subject to significant uncertainties in their initial conditions, parameters, and inputs.
This dissertation provides new theoretical and numerical techniques for rigorously enclosing the set of solutions reachable by a given systems of nonlinear ODEs subject to uncertain initial conditions, parameters, and time-varying inputs. Such sets are often referred to as reachable sets, and methods for enclosing them are critical for designing systems that are passively robust to uncertainty, as well as for optimal real-time decision-making. Such enclosure methods are used extensively for uncertainty propagation, robust control, system verification, and optimization of dynamic systems arising in a wide variety of applications.
Unfortunately, existing methods for computing such enclosures often provide an unworkable compromise between cost and accuracy. For example, interval methods based on differential inequalities (DI) can produce bounds very efficiently but are often too conservative to be of any practical use. In contrast, methods based on more complex sets can achieve sharp bounds, but are far too expensive for real-time decision-making and scale poorly with problem size.
Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative while maintaining high efficiency by exploiting redundant model equations that are known to hold for all trajectories of interest (e.g., linear relationships among chemical species in a reaction network that hold due to the conservation of mass or elements). These linear relationships are implied by the governing ODEs, and can thus be considered redundant. However, these advances are only applicable to a limited class of system in which pre-existing linear redundant model equations are available. Moreover, the theoretical results underlying these algorithms do not apply to redundant equations that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear redundant equations, etc.
This dissertation continues a line of research that has recently achieved very promising bounding results using methods based on differential inequalities. In brief, the major contributions can be divided into three categories: (1) In regard to algorithms, this dissertation significantly improves existing algorithms that exploit linear redundant model equations to achieve more accurate and efficient enclosures. It also develops new fast and accurate bounding algorithms that can exploit nonlinear redundant model equations. (2) Considering theoretical contributions, it develops a novel theoretical framework for the introduction of redundant model equations into arbitrary dynamic models to effectively reduce conservatism. The newly developed theories have more generality in terms of application. For example, complex nonlinear constraints that involve states, time derivatives of the system states, and time- varying inputs are allowed to be exploited. (3) A new differential inequalities method called Mean Value Differential Inequalities (MVDI) is developed that can automatically introduce redundant model equations for arbitrary dynamic systems and has a second-order convergence rate reported the first time among DI-based methods.
Shen, Kai, "Advances in Reachability Analysis for Nonlinear Dynamic Systems" (2019). All Dissertations. 2380.