State and parameter estimation plays
an important role in many
different engineering fields. Estimation of systems described by linear
and nonlinear differential equations has been very well studied in
the literature. Work in the past decade has been geared toward efficiently
extending these algorithms to constrained systems. Of recent interest
is the evaluation of state estimation techniques for differential-algebraic
equation (DAE) systems. The algebraic equations in these studies are
exact, an example being the mole fractions adding to unity. However,
there are situations where algebraic equations can be of both certain
and uncertain types. In this paper, we propose a modified extended
Kalman filter (EKF) approach that can handle uncertainties in both
differential and algebraic equations, and equality constraints. We
also show the importance of this work in estimation of mole fraction,
temperature, and pressure profiles in a water gas shift reactor. The
impact of location and type of measurements on the estimation accuracy
is also studied.
Growing
complexity of processes necessitates the use of information
from sensors along with first-principles mathematical models to ensure
safe and optimal operations. Use of sensors in complex processes requires
identifying optimal location of sensors that can maximize information
from a process. Classical sensor placement approaches for nonlinear
systems that use state estimation schemes usually incorporate linearized
models around the steady-state operating point. However, such approaches
face difficulties when abnormalities or disturbances drift the system
away from the normal operating point. Therefore, use of models that
can appropriately track the behavior of the system in the sensor placement
framework are of interest. However, the computational complexity of
the detailed models makes such approaches intractable. In this work,
we develop a sensor placement framework that combines genetic algorithms
and the extended Kalman filter to obtain optimal sensor locations.
Within this framework, we have investigated the applicability of simplified
models by comparing the results of sensor placement for simplified
and detailed models. The effect of the simplified models on the estimation
accuracy and the optimal sensor network is further evaluated by analyzing
the sensitivity to different parameters. Results show that an appropriate
simplified model can not only significantly reduce the computational
time of the sensor placement algorithm, but also yield a senor network
with similar characteristics as the sensor network obtained using
the detailed model. Further, information loss in using simplified
models in sensor placement may be partially compensated through tuning
of the filter parameters, resulting in acceptable, optimal sensor
placement solutions.
Systematic scaling analysis of model equations can be valuable as a tool for developing computationally tractable simulations of physical systems. The scaling analysis methods in literature pose difficulties in the calculation of scale and reference values, when nonlinear terms are involved in the model equations. Further, existing methods involve trial and error procedures in the scaling process. In this paper, a systematic approach for handling nonlinear terms is suggested, which results in appropriate scale and reference values that render the dimensionless variable variations to be of order one. Further, trial and error procedures are avoided through a new approach wherein a set of nonlinear algebraic equations are solved to identify the scale and reference values. The proposed scaling approach is common to any given model equations with fixed parameters. However, it is to be noted that the proposed procedure may not handle situations when model
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