Cstr Control Using Multiple Models

Almost every real process exhibits nonlinear behavior in a full operating range. Local Model Networks are networks which are composed of locally accurate models, where output is interpolated by smooth locally active validity functions. This divide-and-conquer strategy is a general way of coping with complex systems. The architecture of LMN benefits from being able to incorporate a priori knowledge and conventional system identification methodology. The LMN structure also gives transparent and simple representation of the nonlinear system. Contrary to the black box representation of the nonlinear process by the neural networks, the conventional design methods can be utilized for nonlinear controller design. The idea of the LMN approach is to split the whole operating region into several sub-regions where in each region sub-region the process has close to linear behavior. For each region a local linear model is developed to approximate the non-linear dynamics. The global model of the process is a linear combination of the local models. In an initial off-line identification phase the local models and the validity function parameters have to be identified. Several methods can be used to obtain LMN parameters. The Expectation Maximization (McLachlan and Krishnan, 1997) algorithm is usually used for the Gaussian process models although it requires a priori knowledge of complexity of the system or more precisely the number of local models. Another development is the local linear model tree LOLIMOT (Nelles, 1997). It is based on the idea to approximate a nonlinear map with piece-wise linear local models. The algorithm systematically bisects partitions of input space. Local models that do not fit sufficiently well are replaced by two or more smaller models in the expectation that they will fit the nonlinear target function better in their region of validity. Another training algorithm discussed in (Johansen and Foss, 1995) uses two loops for structure optimization and parameter estimation to iteratively increase the number of models and thus preventing from overparametrization. If linear local models are employed and the parameters of the validity function are fixed, the parameters of local models can be obtained using the standard least-squares method.