Assessment of severity of nonlinearity for distributed parameter systems via nonlinearity measures

The performance of controlled distributed parameter systems (DPSs) not only depends on the controller, but also on the dynamic nature of the process itself. One of the primary factors affecting DPS control is process nonlinearity. In many situations, the extent and severity of nonlinearity is the crucial characteristic in deciding whether linear system analysis and controller synthesis methods are adequate. However, due to their spatio-temporal coupling, traditional nonlinearity measures cannot be directly applied to nonlinear DPSs. In this study, a nonlinearity measure method to quantify the severity of nonlinearity for a class of DPSs is proposed. First, time/space separation and model reduction are carried out using proper orthogonal decomposition (POD). Thus, an optimal linear time-invariant model with a low-order is obtained through the optimization of a spatio-temporal error while full state feedback is incorporated in order to stabilize the linear model. Finally, nonlinearity quantification for DPSs is calculated using the obtained stable linear time-invariant system. The complexity of the calculations for nonlinearity measures is greatly reduced after the model reduction using POD. This method easily estimates the extent to which the process behavior deviates from linearity, which aids in determining whether a linear system analysis and controller synthesis methods are adequate. The nonlinearity quantification indicates that DPSs with smaller values are better approximated by a linear model than DPSs with larger values in the target time/space domain. The effectiveness of the proposed method is illustrated using two numerical examples.