Optimization and control of process systems typically requires an accurate process model. Fundamental first principles models can be difficult to develop if the underlying process is not well understood. The resulting fundamental model may also have numerous unknown parameters and severe complexity. At the other extreme, a linear approximation of a system may be easy develop, but such linear models can be insufficient to capture the process system characteristics requisite for optimization and control. In many cases, a methodology that can provide a simple system model while still accurately representing the complex nonlinear behavior of actual process systems is desired. Local modeling is such a method for the development of such system representations.
Many different approaches have been proposed for using local models to approximate nonlinear systems. For a detailed review, see [Murray-Smith and Johansen(1997)]. In many cases, the different approaches can be distinguished by the choice of model weights. These weights are based on the current system's location within a partitioned space. Some authors have used exponential functions to characterize the valid space of a local model [Banerjee et al.(1997)Banerjee, Arkun, Ogunnaike, and Pearson,Balle et al.(1997)Balle, Juricic, Rakar, and Ernst,Johansen and Foss(1993a)]. Fuzzy logic rules have been used to chose model weights for combining local linear models [Rueda(1996),McGinnity and Irwin(1997)]. These applications use fuzzy membership functions to partition the operating space into different model regimes. The basic modeling methods are very similar no matter the approach: determine where the local models are to be used in the operating space and then devise a method to mesh the local models together.
One obvious extension to such modeling work is the development of closed-loop control using local controllers. Work has been presented for control of nonlinear systems using fuzzy logic to choose between local controllers [Chak and Feng(1994),Logan and Pachter(1997),Logan and Pachter(1994)]. Artificial neural networks have also been used to piece together linear models and controllers in attempts to control nonlinear systems [Pottmann and Jorgl(1993)]. These applications provide methods for implementation of nonlinear control on systems that may not be able to use techniques such as Input-Output Linearization (IOL).
We propose a framework where local models can be easily used to approximate a true nonlinear system. Most published results use either exponential functions or fuzzy logic rules for representing model regimes. In the method presented here, individual model weights are based on the distance of the current operating point from the operating points of the nearest local models. This is similar to using fuzzy logic rules with pyramid shaped membership functions. Some other applications do not adequately represent system dynamics as process changes are made. This type of implementation effectively uses discrete switching for jumping between models. Using the proposed method, a range of nonlinear system dynamics are well-modeled. The current operating point is taken as a filtered value of the known system inputs, which realistically slows the model response and prevents discrete jumps from one model to another. This method does not constrain local models to be linear system models. In some cases, low-order nonlinear empirical models can greatly improve model validity, but also be simple to develop. The linear model mixing methodology is used with multiple IMC controllers to provide adequate control in many different situations. This system can also make use of existing models created at different steady-state operating points. Given a system with two standard operating points (for different product grades) and models at these two points, minimal work and modeling effort would be required to blend the models to create a more accurate system model.