Guaranteed Bounds for Uncertain Systems: Methods Using Linear Lyapunov-like Functions, Differential Inequalities and a Midpoint Method

In general, models of biological or technical applications are represented by nonlinear systems. Moreover, these systems contain multiple uncertain or unknown parameters. These uncertainties are the reason for some numerical and analytical problems in finding guaranteed bounds for the solution of the state space representation. Unfortunately, several industrial applications are demanding exactly these guaranteed bounds in order to fulfil regulations set by the state authorities. To get an idea of the solution of systems with uncertainties the numerical integration of the system's differential equations has to be done with randomly selected values for the unknown parameters. This computation is done several times, in some circumstances more than a thousand times. This approach is well known as the Monte-Carlo method, but this stochastic approach cannot deliver guaranteed bounds for the domain of the system's solution. Thus, we developed a method to find guaranteed bounds which uses linear Lyapunov-like functions to solve this problem. In this work we combine this method with a theory first introduced by Midler. Differential inequalities are used by Mutter to obtain guaranteed bounds. Intersecting the results of both methods provides improved and tight bounds for the original uncertain system. Another approach is shown using a midpoint method providing guaranteed bounds. We achieve guaranteed and finite simulation bounds as a result of our approaches. The results can be used as an initial interval for further methods based on interval arithmetic. An example of a bioreactor with two state variables is shown in this paper to illustrate the methods.