Chemical Engineering Science 63 (2008) 5129--5140 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces Use of bifurcation analysis for development of nonlinear models for control applications M.P. Vega, E.L. Lima, J.C. Pinto ∗ Programa de Engenharia Química / COPPE, Universidade Federal do Rio de Janeiro, Cidade Universitária-CP: 68502, Rio de Janeiro-RJ 21945-970, Brazil ARTICLE INFO ABSTRACT Article history: Received 22 January 2008 Received in revised form 28 June 2008 Accepted 1 July 2008 Available online 5 July 2008 Keywords: Process control Nonlinear dynamics Polymerization Model reduction Hybrid neural model Nonlinear model predictive control Nonlinear system identification poses challenging questions because a closed general theory is not avail- able for this field. Particularly, nonlinear models based on neural networks (NN) may present incompatible general dynamic process behavior, leading to improper closed-loop responses, even when they allow for satisfactory one step ahead prediction of process dynamics, as required by traditional validation methods. It is shown here that performing detailed bifurcation and stability analysis may be very helpful for the adequate development and implementation of nonlinear models and model based controllers. The study of many parameters that are defined a priori during the training of the NN shows that the spurious dynamic behavior is related mostly to the use of incomplete data sets during the learning process. This is an indication that, for each kind of process, the number, range and distribution of the data points in the operation region of interest are of paramount importance for proper training of the nonlinear model. Strategies to improve the quality of the training procedure are provided and analyzed both theoretically and experimentally, using the solution polymerization of styrene in a tubular reactor as a case study. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction The complexity of mathematical models used to describe actual processes strongly depends on its final use. Models are generally required to allow for best process description with the minimum mathematical complexity; however, these two objectives are nor- mally exclusive. This is one of the reasons that make process model- ing such a difficult task. In the case of model based process control, simplicity is a very important required characteristic, as the model has to be solved many times at each sampling interval. One typical example is the nonlinear model predictive control (NMPC), where an optimization problem based on the internal model has to be solved iteratively at each sampling interval (Henson, 1998; Allg ¨ ower and Zheng, 2000; Camacho and Bordons, 2004). When all the important characteristics of the process to be mod- eled are known, building a phenomenological model by applying mass, energy and momentum balances to the process is normally an easy task. However, difficulties may emerge during the solution of the resulting system of equations. A disadvantage of the first principles modeling method is that the resulting dynamic model may be too complex to be employed for model based process con- trol design. For this reason, a number of different model reduction techniques have been proposed in the literature (see, for examples, ∗ Corresponding author. Tel.: +55 21 562 8337; fax: +55 21 590 7135. E-mail address: pinto@peq.coppe.ufrj.br (J.C. Pinto). 0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.07.007 Benallou and Seborg, 1986; Pinto and Biscaia, 1988; Levine and Rouchon, 1991; Hwang, 1991; Duchene and Rouchon, 1996). Empirical linear modeling constitutes a well-established disci- pline (see, for example, Luyben, 1990); however, as pointed out by Pearson and Ogunnaike (1997), a well-developed theory for nonlin- ear system identification is not available yet. The neural network (NN) approach has proved to be a useful tool and is the most popu- lar framework for empirical nonlinear model development, although estimating the huge number of parameters frequently present in the model may be regarded as a major problem to be solved (see, for examples, Cybenko, 1989; Su and McAvoy, 1997). A very interesting and natural approach to the modeling prob- lem is building a hybrid model, where certain amounts of both phe- nomenological and empirical information are used. This information can be introduced into the model in different ways. The most nat- ural way to introduce phenomenological information is using first principle mass, energy and momentum balances. Empirical models are then used to describe model parameters and unknown functions (Pottmann and Henson, 1997). Another possible alternative is us- ing a fundamental model to describe the process characteristics, and utilizing a nonlinear empirical model to represent the residual be- tween the plant and the model. In the last decade, the hybrid neural modeling approach has been proved to be an effective alternative for modeling complex chemical processes (Bhat and McAvoy, 1990; Willis et al., 1991; Psichogios and Ungar, 1991, 1992; Thompson and Kramer, 1994; Sayer et al., 1997; Cubillos and Lima, 1998). This is par- ticularly true in the polymerization field (see, for examples, Doyle III et al., 2003; Nogueira et al., 2003; Chang et al., 2007).