Estimation in nonlinear constrained systems with severe disturbances
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Aluminium is a metal playing an important role in all modern life of today, and it will continue to play an important role in many years to come. As with every business and everyone who participates in the global economy, climate change is a challenge also shared by the aluminium industry. In order for the aluminium industry to meet their goals in reducing greenhouse gas emissions and increase energy e¢ ciency in aluminium production, not only new technology and knowledge will play an important role, but also the speed in which the organization is able to utilize new technology and knowledge is crucial. An important factor for succeeding in reducing green-house gas emissions and increase energy efficciency, is e.g. the use of new advanced process control systems and increased process knowledge. On the way towards new advanced state of the art process control systems, an extension of the current control paradigm may be preferred. In this work, the so-called correlation line, which is very well known in the community of aluminium production, is investigated and discussed, and it is shown that, given constant alumina concentration, the correlation line is unique. Based on the correlation line, a control structure is proposed, aiming for constant mass of aluminium .uoride in the Hall-Heroult cell. It is shown that the control structure reduces process variations and energy consumption without loss of production. New advanced process control systems imply utilizing state of the art process control systems as e.g. Nonlinear Model Predictive Control(NMPC). There is, however, a growing understanding that the real challenges in advanced model-based control applications like NMPC are the state estimation problem and organizational issues. The state estimation problem - because by not being able to measure all the states, the quality of the estimates determine to a large extent the performance of the NMPC application. Further, the quality of the estimates is dependent of e.g. the accuracy of the model (or system knowledge), the accuracy of the measurements, the estimation algorithm and the tuning of the estimator. Also, but not that obvious, the knowledge about how the uncertainty/inaccuracy/noise is entering and in.uencing the system, measurements and control inputs is also of great importance. One might believe that by defeating these obstacles, the success of an NMPC application is guarantied. However, this might not be the case. Those responsible for commissioning an NMPC application in an organization, e.g. in the process industry, will meet an organization that at the current moment most likely do not have the knowledge to understand what NMPC is all about, which is resistant to changes and might not share the assumptions on which the model are founded. Further, they might not agree in the control actions the NMPC might take, because it does not line up with the knowledge embodied in the organization. These aspects makes the organization the important second challenge, because the organization is the real user of the NMPC application and the degree of success is dependent on their ability to make use of it. This thesis addresses these two extremities regarding modeling and state estimation by 1) enhancing the toolbox used for modeling and state estimation in constrained nonlinear systems with severe disturbances, and 2) to see the process involved in applying modeling and estimation from an organization point of view. Item 2) is addressed by looking at possibilities that opens up by the use of modeling and simulation in the context of learning, knowledge creation and knowledge sharing. These issues are addressed by analyzing the role of modeling and simulation in the context of conserving knowledge. The theory is applied to analyze the experience gained from the work processes involved in producing and implementing the content described in the papers Kolås and Støre (2008) and Kolås (2007a). E.g., it is concluded that if the knowledge is complex, the use of modeling and simulation serve as a simplification of complexity, suitable for other parts of the organization that is only interested in the behavior and not the detailed chemistry/physics. In a situation like this, the simpli.ed knowledge could act as a source for learning and knowledge creation. Item 1) is addressed by comparing the Extended Kalman filter (EKF) and the Unscented Kalman .lter (UKF). Several authors have experienced shortcomings applying the EKF to systems with severe nonlinearity and/or constraints. In this work we investigate the use of an alternative to the EKF, the UKF, and a broad overview of di¤erent UKF algorithms is given. Further, an extension to the ensemble of UKF algorithms is proposed, and finally, the issue of how to add constraints using the UKF approach is addressed. The performance of the constrained approach is compared with EKF and a selection of UKF algorithms on nonlinear process systems with multimodal probability density functions. The conclusion is that with an algebraic reformulation of the correction part, the reformulated UKF shows very strong performance on our selection of nonlinear constrained process systems. Subsequently, noise modeling is studied based on a hypothesis that it is important to model noise correctly. In practice this implies a critical view on the dominating .additive noise paradigm. as a means to model uncertainty. Alternative concepts of modeling the noise are investigated, and it is shown that modeling noise by introducing it in the system auxiliary variables and control inputs may have a positive impact on estimation performance. Finally, an algorithm for computing the dynamic eigenvalues for linear time-varying (LTV) systems is proposed. The algorithm can be used to analyze the stability for some types of LTV systems, e.g., for some types of systems, the algorithm could be applied to analyze the stability of the EKF and UKF.