LiU startsida
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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Last updated: 2021-11-10
