Computing Interval Bounds On the Solutions of Nonlinear Index One DAEs

A method is presented for computing tight interval bounds on the solutions of nonlinear semi-explicit, index one differential-algebraic equations (DAEs) subject to given intervals of permissible initial conditions and parameters. Differential-algebraic equations are used to model a wide variety of important dynamic chemical processes through the combination of material and energy balances, thermodynamic relationships, reaction kinetics and empirical correlations. Moreover, such models often contain parameters which are not known precisely, such as physical constants, factors in empirical correlations, process disturbances and model uncertainties. Accordingly, one often requires information about an entire family of possible solutions, parameterized by these uncertainties, rather than a single nominal solution.

Assuming that the uncertain or unknown parameters of interest lie within known intervals, the proposed method computes time-varying upper and lower bounds on the solutions of the given DAEs attainable with parameters in these intervals. Similar methods for bounding the solutions of ordinary differential equations (ODEs) have been studied extensively, with applications in uncertainty analysis, state and parameter estimation, safety verification, fault detection, global optimization, validated numerical integration, and controller synthesis. On the other hand, producing guaranteed bounds on the solutions of DAEs has received much less success and remains an open challenge.

The proposed approach for bounding DAE solutions combines concepts from differential inequalities and interval Newton-type methods. The first key result is an interval inclusion test which verifies the existence and uniqueness of DAE solutions over a given time step, and provides a crude interval enclosure. This test combines a well-known interval inclusion test for solutions of ODEs (used in standard Taylor methods) with an interval inclusion test for solutions of systems of nonlinear algebraic equations from the literature on interval Newton methods. The second key result is a set of sufficient conditions, in terms of differential inequalities, for two time-varying trajectories to bound the differential state variables; i.e., those state variables whose time derivatives are given explicitly by the DAE equations. Using this result, refined, time varying bounds on the state variables over the given time step are computed using a technique which simultaneously applies differential inequalities to the differential variables and an interval Newton-type method to the algebraic variables.

               

Based on these key results, an efficient numerical implementation using interval computations and a standard numerical integration code is presented. Strengths and shortcomings of the proposed algorithm are discussed in the context of a DAE model for a simple distillation process.