Exponential Enclosure Techniques for the Computation of Guaranteed State Enclosures in ValEncIA-IVP

Verified integration of initial value problems for sets of ordinary differential equations can be performed by numerous approaches. The most important ones are based on either interval or Taylor model arithmetic and enclose with certainty the sets of reachable states at a given point of time. Commonly, such tools are based on a Taylor series expansion of the solution of differential equations in time and sometimes in the initial conditions and uncertain parameters. However, the use of high series expansion orders prevents one from applying these tools in real-time environments. This becomes necessary if model-based predictive control strategies are implemented, exploiting an online evaluation of state equations over a finite time horizon. Therefore, it is desirable to reduce the computational effort as far as possible by finding a trade-off between the simplicity of the descriptions of verified state enclosures on the one hand and the amount of overestimation that is contained in the solutions on the other. In ValEncIA-IVP, such enclosures are defined either by interval boxes computed by a simple iteration scheme on the basis of a non-verified numerical approximate solution or by means of exponential enclosures. In this paper, the technique of exponential state enclosures is extended to a novel iteration scheme based on complex-valued interval arithmetic. This iteration reduces overestimation significantly for oscillatory linear state equations. Simulation results for different practically motivated systems with an extension to nonlinear models conclude this paper.