Consistency tests in guaranteed simulation of nonlinear uncertain systems with application to an activated sludge process

In this paper, interval arithmetic simulation techniques are presented to determine guaranteed enclosures of the state variables of both continuous and discrete-time systems with uncertain but bounded parameters. In nonlinear uncertain systems axis-parallel interval boxes are mapped to complexly shaped regions in the state space that represent sets of possible combinations of state variables. The approximation of each region by a single interval box causes an accumulating overestimation from time-step to time-step, usually called the wrapping effect. The algorithm presented in this paper minimizes the wrapping effect by applying consistency techniques based on interval Newton methods. Subintervals that do not belong to the exact solution at a given time can be eliminated in order to give a tighter but still conservative approximation of the exact solution. Additionally, efficient splitting and merging strategies are employed to limit the number of subintervals. The proposed algorithm is applied to the simulation of an activated sludge process in biological wastewater treatment.