We present a new technique for computing tight, pointwise-in-time interval enclosures of the solutions of nonlinear ordinary differential equations (ODEs) subject to a range of parameters and initial conditions. The computation of these state bounds is a central step in algorithms for solving dynamic optimization problems with nonlinear ODEs embedded to guaranteed global optimality. Moreover, these bounds can be used for propagating uncertainty through nonlinear dynamic models, for nonlinear state-estimation for process control, and for robust dynamic optimization problems arising in, e.g., robust model-predictive control, fault-detection, and safety verification problems.
Although state bounding methods have been extensively developed in the past several years, there remains for many problems an unsatisfactory trade-off between the computational cost of the methods and the accuracy (i.e., tightness) of the resulting bounds. This is especially true for highly nonlinear models over large parameter ranges, and is the predominant factor limiting the effectiveness of global dynamic optimization algorithms for problems of practical size.
In this work, we build upon a recently published state bounding method that exploits affine solution invariants present in many models of interest (e.g., chemical reaction kinetics). In this method, solution invariants are used to define a bounds-tightening procedure that is then incorporated into a very inexpensive state bounding method based on differential inequalities and interval arithmetic. For many problems, this scheme provides bounds that are substantially tighter than those produced by differential inequalities alone, while maintaining the low cost of the original method. The drawback, however, is that this method applies only to models with affine solution invariants, and even for such models, no further improvements to the bounds can be made after all existing invariants have been exploited.
To overcome this limitation, we present a somewhat surprising result: Significant and often dramatic improvements in state bounds can be obtained for arbitrary models by applying the aforementioned bounding technique to a lifted (i.e., higher-dimensional) system constructed by defining one or more new state variables in terms of the existing ones. These definitions constitute solution invariants for the lifted model, while the right-hand side functions of the new variables are derived by direct differentiation. While these new equations are redundant with the original model in real number arithmetic, they provide alternative enclosures in interval arithmetic that can effectively contribute to the bounding method. The use of such lifted systems provides new and unexplored degrees-of-freedom for the design of effective bounding methods, with the potential to profoundly impact global dynamic optimization capabilities. We present some preliminary insights into the optimal design of such systems with illustrative examples.