## Abstract |

### Model Error Compensation in ODE and DAE Estimators with Automotive Engine Applications

Control and diagnosis of complex systems demand accurate information of the system state to enable efficient control and to detect system malfunction. 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 observers are sensitive to errors in the model and since the model complexity has to be kept low to enable use in real-time applications, the accuracy of the models becomes limited. Further, modeling is difficult and expensive with large efforts on model parametrization, calibration, and validation, and it is desirable to design robust observers based on existing models. An experimental investigation of an engine application shows that the model have stationary errors while the dynamics of the engine is well described by the model equations. This together with frequent appearance of sensor offsets have led to a demand for systematic ways of handling operating point dependent stationary errors, also called biases, in both models and sensors.

Systematic design methods for reducing bias in model based observers are developed. The methods utilize a default model, described by systems of ordinary differential equations (ODE) or differential algebraic equations (DAE), and measurement data. A low order description of the model deficiencies is estimated from the default model and measurement data, which results in an automatic model augmentation. The idea is then to use the augmented model in observer design, yielding reduced stationary estimation errors compared to an observer based on the default model. Three main results are: a characterization of possible model augmentations from observability perspectives, a characterization of augmentations 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 analyzed. 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. By utilizing a parameterized function, a memory is introduced that enables separate tracking of aging and operating point dependence. This eliminates the trade-off between noise suppression in the parameter convergence and rapid change of the offset in transients. Direct applications for the parameterized bias are online adaptation and offline calibration of maps commonly used in engine control systems.

The methods are evaluated on measurement data from heavy duty diesel engines. A first order model augmentation is found for an ODE of an engine with EGR and VGT. By modeling the bias as a random walk, the estimation error is reduced by 50\,\% for a certification cycle. By instead letting a parameterized function describe the bias, better estimation accuracy and increased robustness is achieved. For an engine with intake manifold throttle, EGR, and VGT and a corresponding stiff ODE, experiments show that it is computationally beneficial to approximate the fast dynamics with instantaneous relations, transforming the ODE into a DAE. A main advantage is the possibility to use more than 10 times longer step lengths for the DAE based observer, without loss of estimation accuracy. By augmenting the DAE, an observer that achieves a 55\,\% reduction of the estimation error during a certification cycle is designed.

*Erik Höckerdal*

**2011**

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Senast uppdaterad: 2021-11-10