Modified Extended Kalman Filtering for Nonlinear Stochastic Differential Algebraic Systems Swapnil S Bhase * Mani Bhushan ** Sachin Kadu *** Sulekha Mukhopadhyay *,**** * Homi Bhabha National Institute, Mumbai 400094, India ** Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India (e-mail: mbhushan@iitb.ac.in) *** Reactor Projects Division, Bhabha Atomic Research Centre, Mumbai 400085, India **** Chemical Engineering Division, Bhabha Atomic Research Centre, Mumbai 400085, India Abstract: The extended Kalman filter (EKF) is one of the most widely used nonlinear filtering technique for a system of differential algebraic equations (DAEs). In this work we propose an alternate EKF approach for state estimation of nonlinear DAE systems that addresses shortcomings of the EKF approaches available in literature (Becerra et al., 2001; Mandela et al., 2010). The proposed approach is based on the idea that since the algebraic equations are assumed to be exact, the error covariance matrix of only the differential states needs to be directly propagated during the prediction step. The error covariance matrix for algebraic states and cross covariance matrix between the errors in differential and algebraic states, which are required to incorporate effect of prior algebraic state estimates on the update step, can be computed from the differential state error covariance matrix alone using the linearized algebraic equations. The update step of the proposed work also follows a similar philosophy and ensures that the covariance update is not approximate. The efficacy of the proposed EKF approach is evaluated using benchmark case studies of a Galvanostatic charge process and a drum boiler. 1. INTRODUCTION Modeling of various physical and chemical processes often gives rise to nonlinear differential algebraic equations (DAEs). For a process involving different time scales, the fast rate phenomena are usually modeled using quasi- steady state approximation to yield algebraic equations that are coupled with differential equations. State estimation of DAEs has received relatively less at- tention compared to the estimation of systems described by ordinary differential equations (ODEs). For system with linear DAE models, Nikoukhah et al. (1992) applied Kalman filter (KF) approach for state estimation. In case of nonlinear DAE systems, Becerra et al. (2001) presented an EKF approach in square-root form. In their approach, an implicit stochastic differential equation (SDE) model is derived from the linearization of DAEs to propagate the error covariance matrix of differential states. However, they assumed that only the differential states are measured and hence their method cannot incorporate the informa- tion available via measurements of algebraic states during the EKF update step. To address this issue, Mandela et al. (2010) developed an alternate EKF approach that can incorporate measurements of both differential and algebraic states. In their work, an augmented state vector consisting of both differential and algebraic state variables is considered. The corresponding error covariance matrix of the augmented state is propagated using an implicit SDE model which is derived from the linearization of DAEs. Their approach enabled incorporation of algebraic states since the augmented prior covariance matrix was considered during the update step. However, the update step uses an approximate method to update the aug- mented covariance matrix to avoid singular augmented covariance matrix. Some other approaches have been re- ported for estimation of a DAE system, such as unscented Kalman filter (Mandela et al., 2010), ensemble Kalman filter (Puranik et al., 2012), particle filter (Haßkerl et al., 2016) and iterative EKF (Purohit and Patwardhan, 2018). Although these approaches provide improved performance, they are computationally more complex than the existing EKF approaches. In the current work, we propose an alternate EKF ap- proach which avoids the drawbacks of approaches avail- able in literature. In particular, our approach can incor- porate measurements of both differential and algebraic states as well as does not use an approximate method to obtain error-covariance matrix during the update step. The approach is based on the key idea that since the differential equations are assumed to be stochastic while the algebraic equations are considered to be exact, it is more appropriate to propagate the error covariance matrix of only the differential states during the prediction step. The error covariance matrix of algebraic states and the cross-covariances of differential and algebraic state errors can be computed from the error covariance matrix of the Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Copyright lies with the authors 2375