(485a) Rigorous Convex Enclosures of the Reachable Sets of Nonlinear ODEs Under Uncertainty

Detailed models of chemical processes are indispensible for solving various design and control problems. Such problems are typically formulated in terms of optimization problems where the process model appears in the constraints, and it is well known that failing to account for model uncertainty in a rigorous manner can result in sub-optimal or even infeasible solutions. For design and control problems, this naturally corresponds to either suboptimal process designs or control schemes, or to designs and control schemes which violate operational constraints when implemented. Depending on the nature of the violated constraints, the results of such faulty designs range from low quality products to costly equipment failures and serious safety hazards. For linear process models, the effects of model uncertainties on the model solutions can often be easily computed and appropriately accounted for in process design. On the other hand, when the process model under consideration has significant nonlinearities, elucidating the effects of model uncertainties on the model solutions can be extremely difficult. Furthermore, minor uncertainties can potentially have a dramatic impact on the model solutions.

In this work, computational methods are presented which can provide guaranteed convex enclosures of the reachable set of solutions obtainable by a nonlinear dynamic process model subject to uncertainty. Specifically, nonlinear ordinary differential equations (ODEs) are considered with uncertain initial conditions and uncertain parameters appearing in the ODE right-hand side functions. Because the proposed enclosures are guaranteed to contain all ODE solutions obtainable from the given uncertainty set, these methods are most useful for problems which require a "worst-case" view of uncertainty, rather than a probabilistic one. Given an uncertainly set, these enclosures can be used independently to verify that the solutions of a particular model do not enter undesirable regions, or they can be used in conjunction with global optimization algorithms to construct conservative restrictions of the feasible sets of robust optimization problems formulated as semi-infinite programs or min-max problems. In either application, a key feature of these enclosures is that they can be infinitely refined by partitioning the uncertainty set.

The methods presented apply to uncertainty sets expressed as n-dimensional intervals. Given such an uncertainty set, the first class of methods presented provide rigorous interval enclosures of the possible ODE solutions point-wise in the independent variable. This is done by using the theory of differential inequalities and interval arithmetic to construct an auxiliary system of ODEs which describes upper and lower bounds on the possible ODE solutions as functions of the independent variable. Computing bounds in this manner is very efficient since it requires only a single numerical integration. However, it is well known that previous methods based on this theory can only provide very weak bounds when the ODEs do not have special structure, which is often the case in practice. To address this issue, the approaches proposed here leverage known physical information in a novel manner to greatly strengthen the computed interval bounds. In particular, very effective techniques have been achieved for nonlinear ODEs with right-hand side functions that can be expressed by a constant matrix pre-multiplying a vector of nonlinear rate functions. This structure is ubiquitous in dynamic models of chemical reaction kinetics, even in the case of non-mass-action rate laws, and very tight bounds can be achieved for such systems.

The second method proposed computes convex polyhedral enclosures of the ODE solutions obtainable from the given uncertainty set. This method is more computationally intensive than the first, but is capable of providing tighter enclosures when the set of obtainable solutions cannot be accurately represented by an interval in state space. This method involves computing convex and concave relaxations of the ODE solutions with respect to the uncertain model parameters and initial conditions, for each fixed value of the independent variable. This is done by a novel relaxation theory developed by the authors. These relaxations are then used to describe a convex enclosure of the reachable set of ODE solutions, and supporting hyperplanes to this convex enclosure are computed. For each hyperplane, the desired normal is first chosen and then an appropriate intercept is obtained from the solution of a convex dynamic optimization problem. A major advantage of this approach is that, through the use of convex and concave relaxations of the ODE solutions, the resulting polyhedral enclosure is valid even if the reachable set of solutions is itself nonconvex.