(599h) Improved Bounds on the Solutions of Nonlinear Dynamic Systems Using Centered-Form Differential Inequalities

In this talk, we will present and compare several new techniques for computing interval enclosures of the solutions of nonlinear ordinary differential equations (ODEs) subject to a range of parameters and initial conditions. Bounding algorithms of this type are broadly used for uncertainty propagation, set-based state estimation, fault detection, and the global solution of optimal control problems. However, despite significant recent progress, state-of-the-art ODE bounding methods still provide an unsatisfactory compromise between computational efficiency and accuracy for many systems of practical complexity. In particular, classical methods based on interval computations are very efficient but often produce bounds that are extremely conservative. In contrast, methods based on higher-order set enclosures such as polytopes, zonotopes, and Taylor models often provide highly accurate enclosures, but are much more computationally demanding. Together, these issues severely limit the use of existing bounding methods for online control applications, and are also primarily responsible for the inefficiency of current global dynamic optimization algorithms.

To address this problem, this talk will present several new approaches based on extensions of the classical theory of differential inequalities (DI), which furnishes simple interval enclosures at very low computational cost. One of the primary drawbacks of the standard DI method is that it computes bounds assuming that, at every point in time, the uncertain quantities (i.e., all model parameters and state variables) can vary independently of one another (because they are bounded in a Cartesian product of intervals). This greatly simplifies computations, but also leads to severe overestimation of the true solution set in many cases. However, this problem is not unique to dynamic systems – a simpler form of the problem also in occurs when using interval arithmetic to bound the range of a given function, where it is known as the ‘dependency problem.’ Moreover, several well known strategies have been developed that partially resolve the dependency problem using derivative information while largely maintaining the computational efficiency of standard interval arithmetic. These techniques are broadly known as ‘centered forms’ and include the mean-value form and slope forms of several varieties.

In this work, we extend these centered-forms to dynamic problems by combining them with the theory of differential inequalities in a novel way. The resulting bounding methods are most similar to the method of Chachuat and Villanueva (Comp. & Chem. Eng., 2012), which combines differential inequalities with Taylor models (a form of enclosure for the range of a function that is represented by a polynomial approximation with a rigorous bound on the approximation error). However, the proposed methods use only first order derivative (or slope) information, whereas Taylor models use at least second order information. Thus, the proposed methods are expected to be more computationally efficient in general. We will present detailed numerical comparisons for several different centered-form ODE bounding methods, along with comparisons to existing state-of-the-art bounding methods, on a test set of nonlinear models of chemical reaction systems, mechanical systems, and aerospace systems. Our results indicate that a significantly better compromise between efficiency and accuracy can be achieved in many cases. Moreover, the use of centered-forms can also enhance the effectiveness of another bounds tightening technique recently developed by the authors based on the use of solution invariants. We conclude with a discussion of key problem features that appear to dictate the effectiveness of the new approaches.