  State estimation incorporates the feedback in optimization based advanced
process control systems and is very important for the performance of model
predictive control. We describe the extended Kalman filter, the unscented
Kalman filter, the ensemble Kalman filter, and a particle filter for
continuous-discrete time nonlinear systems involving stochastic differential
equations. Continuous-discrete time nonlinear systems is a natural way to model
physical systems controlled by digital controllers. We implement the state
estimation methods in Matlab, illustrate and evaluate their performance using
simulations of the modified four-tank system. This system is non-stiff and the
state estimation methods are implemented numerically using an explicit
numerical integration scheme. We evaluate the accuracy of the state estimation
methods in terms of the mean absolute percentage error over the simulation
horizon. Each method successfully estimates the states and unmeasured
disturbances of the simulated modified four-tank system. The key contribution
is an overview and comparison of state estimation methods for
continuous-discrete time nonlinear stochastic systems. This can guide efficient
implementations.
