Computation and sampling time requirements for real-time implementation of observers is studied. A common procedure for state estimation and observer design is to have a system model in continuous time that is converted to sampled time with Euler forward method and then the observer is designed and implemented in sampled time in the real time system. When considering state estimation in real time control systems for production there are often limited computational resources. This becomes especially apparent when designing observers for stiff systems since the discretized implementation requires small step lengths to ensure stability. One way to reduce the computational burden, is to reduce the model stiffness by approximating the fast dynamics with instantaneous relations, transforming an ordinary differential equations (ODE) model into a differential algebraic equation (DAE) model. Performance and sampling frequency limitations for extended Kalman filter (EKF)'s based on both the original ODE model and the reduced DAE model are here analyzed and compared for an industrial system. Furthermore, the effect of using backward Euler instead of forward Euler when discretizing the continuous time model is also analyzed. The ideas are evaluated using measurement data from a diesel engine. The engine is equipped with throttle, exhaust gas recirculation (EGR), and variable geometry turbines (VGT) and the stiff model dynamics arise as a consequence of the throttle between two control volumes in the air intake system. The process of simplifying and modifying the stiff ODE model to a DAE model is also discussed. The analysis of the computational effort shows that even though the ODE, for each time-update, is less computationally demanding than the resulting DAE, an EKF based on the DAE model achieves better estimation performance than one based on the ODE with less computational effort. The main gain with the DAE based EKF is that it allows increased step lengths without degrading the estimation performance compared to the ODE based EKF.
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Figure 1
Schematic of the diesel engine model
Figure 2
(Color online) DAE simulations with and without control volume adjustment for a step length of 50 ms using backward Euler, compared to high end simulation of the default ODE using the Matlab variable step length solver, ode23t, with absolute and relative tolerances of 10$^{-\text{6}}$. The better dynamic behavior of the model with increased volume is seen in the areas highlighted with solid ellipses, representing open throttle operation. Also, there is only minor loss in simulation accuracy for the larger volume during closed throttle operation, indicated by dotted ellipses. (a) $\dot{p}_{\text{im}}=~\frac{R_{\text{a}} ~~~~T_{\text{im}}}{V_{\text{im}}}(W_{\text{th}}(\cdot)~+~W_{\text{egr}}(\cdot)~-~W_{\text{ei}}(\cdot))$; (b) $\dot{p}_{\text{im}}=~\frac{R_{\text{a}}T_{\text{im}}}{V_{\text{im}}+~V_{\text{ic}}}(W_{\text{th}}(\cdot)~+~W_{\text{egr}}(\cdot)~-~W_{\text{ei}}(\cdot))$.
Figure 3
(Color online) Estimates of EKF observers based on ODE and DAE models with (a) 3 ms and (b) 10 ms step lengths and using forward Euler for the prediction. Using the ODE based EKF and a step size of 10 ms, results in a high frequency estimation error, especially prominent for high pressures corresponding to open throttle operation. It can also be noted that also the ODE based EKF with 10 ms step length would predict the states well if a low pass filter was applied.
Figure 4
(Color online) Estimated PDF's of the estimation errors for $p_{\text{im}}$, $p_{\text{ic}}$, and $W_{\text{c}}$ from observers based on ODE and DAE models with different step lengths. The PDF's of the estimation errors for simulation step lengths of (a) 3 ms are similar for both the DAE and the ODE based EKF's. For step lengths of (b) 10 ms the ODE based EKF estimation error PDF's are wider than those of the DAE based EKF which agrees with the noisy estimates of the ODE based EKF observed in Figure
Figure 5
(Color online) Normalized RMSE as function of step length for DAE and ODE EKF's. Both (a) forward and (b) backward Euler results are presented. The RMSE is normalized by the ODE RMSE with 3 ms step length for each variable. Presented discretization step lengths are $T_{\rm~s}$ = 3, 10, 15, 20, 30, 50, 100, and 125 ms, where the last five only applies to the DAE observers due to stability issues of the ODE observers. Note that the scale is logarithmic.
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