Probabilistic Analysis of Bimodal State Distributions in SCR Aftertreatment Systems

Sensor selection for the control of modern powertrains is a recognised technical challenge. The key question is which set of sensors is best suited for an effective control strategy? This paper addresses the question through probability modelling and Bayesian analysis. By quantifying uncertainties in the model, the propagation of sensor information throughout the model can be observed. The specific example is an abstract model of the slip behaviour of Selective Catalytic Reduction (SCR) DeNOx aftertreatment systems. Due to the ambiguity of the sensor reading, linearization-based approaches including the Extended Kalman Filter, or the Unscented Kalman Filter are not successful in resolving this problem. The stochastic literature suggests approximating these nonlinear distributions using methods such as Markov Chain Monte Carlo (MCMC), which is able in principle to resolve bimodal or multimodal results. However, the most effective methods are Hamiltonian solvers, which again struggle with the strongly nonlinear system behaviour, often getting stuck in just one of the possible solutions. This paper proposes a solution that uses hierarchical stochastic modelling to add another dimension to the parameter space. This is an implementation of the parallel tempering approach, a less intrusive alternative to the “Wormhole” technique. The result is that the chain sampler moves from a low precision distribution which encapsulates both modes to a high precision solution consisting of individual modes. Using this technique, it is possible to solve the observation problem, and thus determine how effective a given sensor configuration is in any operating condition. The same hierarchical model can also be used to identify the sensor accuracy using a machine learning approach.