Development and identification of hierarchical nonlinear mixed effects models for the analysis of dynamic systems: identification and application of hierarchical nonlinear mixed effects models for the determination of steady-state and dynamic torque responses of an SI engine

Abstract

Multi-level or hierarchical models present various features for dealing with data grouped at several levels. The majority of applications of hierarchical models use clustered data that is static in nature and collected over a long period of time. The purpose of this study is investigating hierarchical models for application with highly dynamic systems. Steady-state data are conventionally employed for engine torque mapping purposes. The data takes much time to collect and the dynamics of the system are routinely ignored. This valuable information could be used for better control of the system.In this study, an innovative transient spark-sweep approach is developed for collecting dynamic torque data more efficiently. The means of data collection implies a structure for which a multi-level model is best suited. A multi-model augmented D-optimal design is created, and the experimental data collected. Spark excitation is applied at speed/load points using Amplitude Modulated Pseudo Random Signal (AMPRS), and the torque response over the operating space is thus obtained. Conditional first-order linearization is used within the identification process for determining the hierarchical model parameters. The level-1 Nonlinear Auto Regressive eXogenous (NARX) models are separately determined using an Iterative Generalized Least Square (IGLS) method and the results are employed for initialisation of the covariance matrix and the model level-2 parameters. A novel gradient optimiser was established to facilitate the dynamic hierarchical model identification. Additionally, the uncertainty associated with model selection was mitigated using a multi-model approach. The model identified is evaluated and compared with experimental dynamic and steady-state data. It shows behaviour, both dynamic and steady state, providing prediction over a wider extrapolated spark range than conventional approaches. The new approach is eight time faster than current state-of-the-art approaches.