A Computationally Efficient and Robust Implementation of the Continuous-Discrete Extended Kalman Filter

Abstract

Abstract-We present a novel numerically robust and computationally efficient extended Kalman filter for state estimation in nonlinear continuous-discrete stochastic systems. The resulting differential equations for the mean-covariance evolution of the nonlinear stochastic continuous-discrete time systems are solved efficiently using an ESDIRK integrator with sensitivity analysis capabilities. This ESDIRK integrator for the meancovariance evolution is implemented as part of an extended Kalman filter and tested on a PDE system. For moderate to large sized systems, the ESDIRK based extended Kalman filter for nonlinear stochastic continuous-discrete time systems is more than two orders of magnitude faster than a conventional implementation. This is of significance in nonlinear model predictive control applications, statistical process monitoring as well as grey-box modelling of systems described by stochastic differential equations.

High-order accurate continuous-discrete extended Kalman filter for chemical engineering

“…The latter MDE solver is implemented with only local error control. Jørgensen et al (2007) have shown its superiority to the built-in Matlab ODE solver ODE15s in terms of efficiency of computation. However, our numerical experiments in Section 3.2 say that the accuracy of ESDIRK3(4) may compromise the reliability of state estimation in chemical and other engineering.…”

“…Thus, the continuousdiscrete stochastic state-space model (1), (2) is best suited for state estimation in chemical systems and widely used in chemistry research and industrial applications (see, for instance, Wilson et al (1998); Soroush (1998); Dochain (2003); Rawlings (2002, 2005); Jørgensen (2007); Rawlings and Bakshi (2006); Romanenko and Castro (2004); ). Concerning state estimation algorithms, we have to remark that, at present, there exist a great variety of different methods starting from a rigorous probabilistic approach solving Kolmogorov's (Fokker-Planck's) forward equation (as discussed, for instance, in Jazwinski (1970); Maybeck (1982)) till approximate approaches including various nonlinear modifications and implementations of the well-known Kalman filter (see Lewis (1986); Singer (2002Singer ( , 2006; Julier et al (2000); Julier and Uhlmann (2004); Ito and Xiong (2000); Nørgaard et al (2000); Haykin (2008, 2009); Arasaratnam et al (2010); Frogerais et al (2012); Jørgensen et al (2007); Kulikov and Kulikova (2014); Rawlings and Bakshi (2006); Romanenko and Castro (2004); ; Schneider and Georgakis (2013)) as well as optimization based approaches usually referred to as the moving horizon estimation (studied by Jang et al (1986); Rao et al (2001); Rawlings (2002, 2005); Rawlings and Bakshi (2006) and so on). Undoubtedly, the extended Kalman filter (EKF) still remains among the most popular and widely used numerical techniques for practical state estimation in nonlinear stochastic systems because of its implementation simplicity and good performance.…”

Practical implementation of extended Kalman filtering in chemical systems with sparse measurements

“…The above-mentioned obstacle can be overcome by means of the variable-stepsize ODE solvers with automatic error control, i.e., for example, by commonly used MATLAB codes [13,Section 12.2]. However, it is shown in [16,23,29,32] that usually EKF techniques grounded in general purpose ODE solvers, including MATLAB software, lose the EKF implementations based on specially created methods for solving MDEs (2.1), (2.2). This is because all MATLAB codes and general purpose ODE integrators are not intended for retaining the positive semi-definiteness of computed covariance matrix and treat the MDEs as conventional ODE systems.…”

A computationally efficient Kalman filter based estimator for updating look-up tables applied to NOx estimation in diesel engines

“…Charalampidis andPapavassilopoulos (2011), Jørgensen, Homsen, Madsen, andKristensen (2007) and Singer and Sea (1971); the latter make an interesting study on the total number of operations required for the updating phase. Chandrasekar, Kim, and Bernstein (2007) present an interesting methodology when system order is extremely large by using the finite-horizon optimization technique for obtaining reduced-order systems.…”