R. Bakker

Learning Chaotic Attractors by Neural Networks

An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorithm learns to short-term predict the time series. At the same time a criterion, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored that tests the hypothesis that the reconstructed attractors of model-generated and measured data are the same. Training is stopped when the prediction error is low and the model passes this test. Two other features of the algorithm are (1) the way the state of the system, consisting of delays from the time series, has its dimension reduced by weighted principal component analysis data reduction, and (2) the user-adjustable prediction horizon obtained by error propagationpartially propagating prediction errors to the next time step. The algorithm is first applied to data from an experimental-driven chaotic pendulum, of which two of the three state variables are known. This is a comprehensive example that shows how well the Diks test can distinguish between slightly different attractors. Second, the algorithm is applied to the same problem, but now one of the two known state variables is ignored. Finally, we present a model for the laser data from the Santa Fe time-series competition (set A). It is the first model for these data that is not only useful for short-term predictions but also generates time series with similar chaotic characteristics as the measured data.

Optimum NOx abatement in diesel exhaust using inferential feedforward reductant control

Abstract To adequately control the reductant flow for the selective catalytic reduction of NO x in diesel exhaust gas a tool is required that is capable of accurately and quickly predicting the engines fluctuating NO x emissions based on its time-dependent operating variables, and that is also capable of predicting the optimum reductant/NO x ratio for NO x abatement. Measurements were carried out on a semi-stationary diesel engine. Four algorithms for non-linear modelling are evaluated. The models resulting from the algorithms gave very accurate NO x predictions with a short computation time. Together with the small errors this makes the models very promising tools for on-line automotive NO x emission control. The optimum reductant/NO x ratio (to get the lowest combined NO x +reductant emission of the exhaust treating system) was best predicted by a neural network.

Neural networks for prediction and control of chaotic fluidized bed hydrodynamics: A first step

A neural-network-based model that has learnt the chaotic hydrodynamics of a fluidized bed reactor is presented. The network is trained on measured electrical capacitance tomography data. A training algorithm is used that does not only minimize the short-term prediction error but also the information needed to synchronize the model with the real system. This forces the model to focus more on learning the longer term dynamics of the system, expressed in the average multi-step-ahead prediction error and dynamic invariants such as correlation entropy and dimension. The availability of the model is an important step towards control of chaos in gas-solid fluidized beds.

Optimum deNOx performance using inferential feedforward reductant flow control

To adequately control the reductant flow for the catalytic removal of NO x from diesel exhaust gases a tool is required that is capable of accurately and quickly predicting the engines NO x emissions based on its operating variables, and that is also capable of predicting the optimum ammonia/NO x ratio for NO x removal. Two algorithms for non-linear modelling are evaluated: (1) neural networks and (2) the split & fit algorithm of- Bakker et al. [1,2]. Measurements were carried out on a semi-stationary diesel engine. Results of the split & fit algorithm and the neural network were compared to (3) the traditionally used engine map and (4) a linear fit. Both the neural network and the split & fit algorithm gave excellent NO x predictions with a short computation time (0.3 ms), making them very promising tools in real-time automotive NO x emission control. With regard to the estimation of the optimum NH 3 /NO x ratio, the neural network predicts the effect of NH 3 /NO x ratio on the final NO 2 emission very well.

Inferential NOx emission prediction in feedforward reductant control for SCR

To adequately control the reductant flow for the selective catalytic reduction of NOx in diesel exhaust gas a tool is required that is capable of accurately and quickly predicting NOx emissions from the engines operating variables. Two algorithms for non-linear modelling are evaluated: neural networks (Solla et al., Adv. in Neural Information Processing Systems 12 (MIT Press, Five Cambridge Center, Cambridge, MA, 2000)) and the split & fit algorithm (Bakker et al., submitted for publication to NIPS). Measurements were carried out on a transient automotive diesel engine and a semi-stationary diesel engine. Both algorithms gave excellent predictions with a short computation time (0.03–0.13 ms). This makes them very promising tools in automotive catalytic NOx emission control.