Identification and Response Prediction of Switching Dynamic Systems Using Interval Analysis

Introduction

Modeling, parameter identification/characterization and response prediction of uncertain switching dynamic systems is of the most challenging problems in control engineering especially in diagnosis and process safety analysis in which one wishes to calculate accurately system’s response to determine whether the system is in healthy operational state/condition or to guarantee that it never enters in forbidden operating zones. This is classically performed with the aide of a mathematical model of the system whilst the latter usually describes the system’s behavior in a simplified manner. This leads to a difference between the system’s response and the predicted value called modeling error. There exist three major approaches to deal with this problem that are the deterministic, probabilistic, and set-membership approaches. In the deterministic approach, the model parameters are constant scalars and are adjusted in such way that a certain optimization criterion e.g. the power of the error is minimized. However, this approach does not provide any information on modeling error in prediction phase. An alternative is to use the probabilistic approach in which the modeling error is characterized by a probability density function (PDF). In this case, the PDF is used to calculate a probabilistic interval for the system’s response at any instant. Though, it suffers from three major shortcomings:

• • It is not always an easy task to characterize modeling error by a certain PDF,

• • The properties of the PDF may be modified during iterative algorithms, and

• • The existence of the system’s response in the probabilistic interval at any instant is not guaranteed.

In the third approach, the model parameters are considered time-varying but bounded variables which are characterized by an interval of real numbers. Since the model parameters are intervals, the predicted system’s response at any instant is not anymore a real number but an interval of real numbers. The set of predicted intervals at different instances generates a tube through time called wrapping envelope. If the model parameters are characterized properly, it is guaranteed by the inclusion property of interval analysis that the wrapping envelope includes the system’s response (Neumaier, 2001). This fundamental property is the main motivation of exploring this approach in modeling, identification and response prediction of uncertain dynamic systems.