Wavelet analysis is an emerging field of mathematics that has provided new tools and algorithms suited for the type of problems encountered in process monitoring and control. In this paper, a review is presented for the applications of wavelet transform in advanced control systems, particularly in Model Predictive Control, Intelligent Control, Robust Control, Adaptive Control, Nonlinear Control, Process Modeling and Control, Process Identification and Control, Process Monitoring, Diagnosis and Control, Statistical Process Control and Optimal Control. The underlying principles in each of these control strategies have also been briefly discussed.
Over the past few years, wavelets and wavelet-based analysis have found their way into many different fields of science and engineering. The wavelet transform is a tool that provides descriptions of functions or signals in the time-frequency plane (Daubechies, 1992). Using traditional mathematical tools, one can study the properties of a phenomenon either in time domain or frequency domain. Although the Fourier transform and its inverse allow a passage from one domain to another domain, it does not give a simultaneous view of the phenomenon in both domains. Wavelets, on the other hand, are the basis functions, which are localized in both time and frequency domains. A basis, which preserves the time-frequency information of the signal, is of significant advantage in control system synthesis and analysis. Wavelets, because of their time-frequency localization and multiresolution properties offer an efficient framework for representation and characterization of signals, especially that non-stationary in nature. Wavelets can also be useful in obtaining process models that describe the process behavior on different time scales. These properties are very attractive from process identification and controller design prospective and recently have been shown to be of significant advantage for some of the process engineering problems (Bakshi and Stephanopoulos, 1993; Palavajjhala et al., 1994).
Wavelets, or analyzing wavelets, are the building blocks on wavelet transform, just as trigonometric functions of different frequencies are the building blocks used in Fourier transforms. Wavelets are generated by the dilation and translation of a single prototype function called the mother wavelet. The mother wavelet is an absolutely integrable function denoted by y(t). Thus, a family of wavelets can be obtained from the mother wavelet as: where s and t represent the dilation and translation parameters respectively (s, t Î R, s ¹ 0). The factor is used to ensure that the energy of the dilated and translated versions is same as that of the mother wavelet. For the wavelets to be useful analyzing functions, the mother wavelet must integrate to zero over the whole real line (i.e., _¥ to ¥ in time, t). |
