## Abstract |

### Observer Design and Model Augmentation for Bias Compensation with Engine Applications

Control and diagnosis of complex systems demand accurate knowledge of certain quantities to be able to control the system efficiently and also to detect small errors. Physical sensors are expensive and some quantities are hard or even impossible to measure with physical sensors. This has made model-based estimation an attractive alternative.

Model-based estimators are sensitive to errors in the model and since the model complexity needs to be kept low, the accuracy of the models becomes limited. Further, modeling is hard and time consuming and it is desirable to design robust estimators based on existing models. An experimental investigation shows that the model deficiencies in engine applications often are stationary errors while the dynamics of the engine is well described by the model equations. This together with fairly frequent appearance of sensor offsets have led to a demand for systematic ways of handling stationary errors, also called bias, in both models and sensors.

In the thesis systematic design methods for reducing bias in estimators are developed. The methods utilize a default model and measurement data. In the first method, a low order description of the model deficiencies is estimated from the default model and measurement data, resulting in an automatic model augmentation. The idea is then to use the augmented model for estimator design, yielding reduced stationary estimation errors compared to an estimator based on the default model. Three main results are: a characterization of possible model augmentations from observability perspectives, an analysis of what augmentations that are possible to estimate from measurement data, and a robustness analysis with respect to noise and model uncertainty.

An important step is how the bias is modeled, and two ways of describing the bias are introduced. The first is a random walk and the second is a parameterization of the bias. The latter can be viewed as an extension of the first and utilizes a parameterized function that describes the bias as a function of the operating point of the system. The parameters, rather than the bias, are now modeled as random walks, which eliminates the trade-off between noise suppression in the parameter convergence and rapid change of the offset in transients. This is achieved by storing information about the bias in different operating points. A direct application for the parameterized bias is the adaptation algorithms that are commonly used in engine control systems.

The methods are applied to measurement data from a heavy duty diesel engine. A first order model augmentation is found for a third order model and by modeling the bias as a random walk, an estimation error reduction of 50\,\% is achieved for a European transient cycle. By instead letting a parameterized function describe the bias, simulation results indicate similar, or better, improvements and increased robustness.

*Erik Höckerdal*

**2008**

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Senast uppdaterad: 2019-08-05