Control of an anaerobic digester through normal form of fold bifurcation A. Rincon, F. Angulo * , G. Olivar Department of Electrical and Electronics Engineering and Computer Science, Universidad Nacional de Colombia, Sede Manizales, CeiBA Complexity, Manizales, Colombia article info Article history: Received 17 September 2008 Received in revised form 15 April 2009 Accepted 23 April 2009 Keywords: Adaptive control Anaerobic digester Bifurcations Feedback linearization Normal form abstract Nonlinear dynamics is ubiquitous in engineering systems. As some parameters are varied bifurcations arise in the state variables. Generically, when one parameter changes, Hopf and fold bifurcations are found. Other ones can also be present due to special systems characteristics, such as symmetries. Know- ing in advance the significant bifurcation scenario, a novel approach to control can be considered. We compute the normal form corresponding to such a bifurcation and we take this model as the nominal model of the plant. Then we design a nonlinear control which takes advantage of the precise bifurcation scenario. This general method is applied, in this paper, to an anaerobic digester. We will control the pro- cess with an adaptive controller. Specifically, we want to compare with the case that the nominal plant is considered as a linear model, such as it is typical in adaptive control techniques. Our proposed method has more benefits in signal con- trol effort, faster convergence rate and low error. This paper shows how the combination of appropriated nonlinear dynamic techniques such as bifurca- tions and normal forms, and nonlinear control, can give rise to an improvement of the traditional methodology. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Nonlinear dynamics has been a hot topic in the last decades, and it is still a very active research area. In the last years we have seen many applications of nonlinear dynamics to science and engi- neering systems, such as power electronics, biological systems, traffic flow, and so on. The main characteristics from nonlinear dynamics regard to existence of limit cycles, quasiperiodicity or chaos in the state space; coexistence of several attractors with their basins of attraction, and bifurcations as some parameters are varied, to name but a few [1]. Different mathematical tools for analyzing the dynamics, corresponding to nonlinear phenom- ena, have been described in the literature, such as the reduction to the center manifold, or the normal form theory [2]. But still there is not much cross-fertilization between nonlinear dynamics and control, and only a few papers combine tools from both re- search fields. In this paper, we propose a control strategy, partially with the aim to contribute to reduce the gap between these areas. We make specific use of the normal form theory from nonlinear dynamics, and adaptive control from nonlinear control [3]. Both techniques are well-known in the corresponding research fields, but rarely have been put together appropriately. Thus our strategy can be described by the following steps: first, we obtain a nonlin- ear model for the system; second, we perform bifurcation analysis in order to know which are the nonlinear characteristics of the phenomena that we want to study; third, we compute the corre- sponding nonlinear normal form for the bifurcation scenario; and forth, we design a nonlinear control taking advantage of the knowledge of the bifurcation scenario. We want to emphasize that this strategy can be applied to many engineering systems, but the details depend very much on the specific system dynamics and the nonlinear control which is chosen. Thus, generically, if we vary a parameter of a nonlinear dynamical system, Hopf and fold bifurcations will occur, and each one has its own normal form [2]. Also, different specific nonlinear controllers can be chosen. For example, in the application which we describe in the following sections, we are interested in the nonlinear phenomena mainly caused by a fold bifurcation, and the nonlinear control we chose is an adaptive control. Explicitly showing the details for each pos- sible bifurcation and each nonlinear control in a general system would be a lengthy and exhaustive task, and no doubt, it would not fit into the standards of this journal. Instead, our aim is to clearly show the steps of our strategy and apply it to a standard problem in chemical engineering processes, such as a wastewater treatment plant. Moreover, we want to emphasize specifically the advantage of considering the nonlinear normal form as the model of the plant, instead of the linear model, which is usually the case in standard control [3]. 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.04.006 * Corresponding author. Tel.: +57 6 8879400; fax: +57 6 8879498. E-mail addresses: arincons@unal.edu.co (A. Rincon), fangulog@unal.edu.co (F. Angulo), golivart@unal.edu.co (G. Olivar). Journal of Process Control 19 (2009) 1355–1367 Contents lists available at ScienceDirect Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont