(682d) Rapid and Accurate Uncertainty Propogation for Nonlinear ODEs Using Nonlinear Solution Invariants

In this talk, we will present a new technique for computing tight interval enclosures of the solutions of nonlinear ordinary differential equations (ODEs) subject to a range of parameters and initial conditions. ODEs can be used to model a wide variety of dynamic chemical processes, and these models nearly always involve uncertain parameters or inputs. Quantifying the effects of such uncertainties on the model solutions provides essential information for many applications, including state and parameter estimation, controller synthesis, robust MPC, safety verification, and fault detection. Moreover, bounding the solutions of ODEs plays an essential role in all known methods for deterministic global optimization of dynamic models.

Despite substantial recent progress, for many systems of practical complexity, state-of-the-art ODE bounding methods still provide an unsatisfactory compromise between computational efficiency and the accuracy of the computed enclosures. In particular, classical methods based on interval computations are very efficient but often produce bounds that are too conservative to be useful. In contrast, modern methods based on high-order set enclosures (e.g., polytopes, zonotopes, and Taylor models) can provide highly accurate enclosures, but are much more computationally demanding and do not scale favorably in the number of uncertain model parameters. These issues severely limit the use of existing bounding methods for online control applications, and are also the predominant factors limiting the effectiveness of global dynamic optimization algorithms. Therefore, there is a critical need for an alternative approach that can provide bounds that are simultaneously sharp and highly efficient.

In 2013, Scott and Barton reported a fast, interval-based algorithm capable of producing very accurate bounds for a limited class of systems that obey affine solution invariants (i.e., affine equations satisfied by the system states for all time and all realizations of the uncertain parameters). These invariants are redundant with the original model, and can be exploited through a bounds-tightening procedure to greatly reduce the conservatism of fast interval bounding methods. More recently, we have made the crucial observation that arbitrary nonlinear systems (i.e., systems that do not obey any invariants) can be artificially endowed with affine solution invariants by introducing additional state variables defined as linear combinations of the original states. Our preliminary results, presented at last years AIChE meeting, show that this simple tactic can provide much more accurate bounds than standard interval methods with only a minor increase in CPU time. However, no general-purpose method has so far been proposed for identifying artificial invariants that will be effective in reducing conservatism. Moreover, this approach has so far been limited to invariants that are affine in the states, which proves to be a serious limitation for many nonlinear systems of interest.

In this talk, we will present an extension of this approach to permit the construction and exploitation nonlinear solution invariants, rather than only affine invariants. This is of interest first because many important systems naturally obey nonlinear invariants (e.g., Hamiltonian systems), and second because it provides much greater freedom in the design of effective artificial invariants for arbitrary systems that do not have pre-existing invariants. We will first outline our new theory and algorithms that enable the use of nonlinear invariants, and then present several case studies showing how nonlinear invariants can be constructed and exploited in order to produce highly accurate bounds using only fast interval computations.