Bounding the Dynamic Behavior of an Uncertain System via Polynomial Chaos-based Simulation

Parametric uncertainty can represent parametric tolerance, parameter noise or parameter disturbances. The effects of these uncertainties on the time evolution of a system can be extremely significant, mostly when studying closed-loop operation of control systems. The presence of uncertainty makes the modeling process challenging, since it is impossible to express the behavior of the system with a deterministic approach. If the uncertainties can be defined in terms of probability density function, probabilistic approaches can be adopted. In many cases, the most useful aspect is the evaluation of the worst-case scenario, thus limiting the problem to the evaluation of the boundary of the set of solutions. This is particularly true for the analysis of robust stability and performance of a closed-loop system. The goal of this paper is to demonstrate how the polynomial chaos theory (PCT) can simplify the determination of the worst-case scenario, quickly providing the boundaries in time domain. The proposed approach is documented with examples and with the description of the Maple worksheet developed by the authors for the automatic processing in the PCT framework.