Variants of extended Kalman filtering approaches for Bayesian tracking. (English) Zbl 1353.93113
Summary: We provide a tutorial for a number of variants of the Extended Kalman Filter (EKF). In these methods, so called, sigma points are employed to tackle the nonlinearity of problems. The sigma points exactly represent the mean and the variance of the state distribution function in a dynamic state equation. The initially developed EKF variant, that is, Unscented Kalman Filter (UKF) (also called sigma point Kalman filter) shows enhanced performance compared with that of conventional EKF in the literature. Another variant, which is not well known, is Central Difference Kalman Filter (CDKF) whose way to approximate the nonlinearity is based on the Sterling’s polynomial interpolation formula instead of the Taylor series. Endeavor to reduce the computational load resulted in the development of square root versions of both UKF and CDKF, that is, square root unscented Kalman filter and Square Root Central Difference Kalman Filter (SR-CDKF). These SR-versions are supposed to be numerically more stable than their original versions because the state covariance is guaranteed to be positive definite by avoiding the step of matrix decomposition. In this paper, we provide the step-by-step algorithms of above-mentioned EKF variants with their pros and cons. We apply these filtering methods to a number of problems in various disciplines for performance assessment in terms of both Mean Squared Error (MSE) and processing speed. Furthermore, we show how to optimize the filters in terms of MSE performance depending on diverse scenarios. According to simulation results, CDKF and SR-CDKF show the best MSE performance in most scenarios; particularly, SR-CDKF shows faster processing speed than that of CDKF. Therefore, we justify that SR-CDKF is the most efficient and the best approach among the Kalman variants including the EKF for various nonlinear problems. The motivation of this paper targets at the contribution to the disseminative usage of the Kalman variants approaches, particularly, SR-CDKF taking advantage of its estimating performance and high processing speed.
MSC:
| 93E11 | Filtering in stochastic control theory |
| 93E10 | Estimation and detection in stochastic control theory |
| 93E03 | Stochastic systems in control theory (general) |
| 93C55 | Discrete-time control/observation systems |
Keywords:
central difference Kalman filter (CDKF); extended Kalman filter; sigma-point Kalman filter; square-root CDKF; square root UKF; unscented Kalman filterReferences:
| [1] | KalmanRE. A new approach to linear filtering and prediction problems. Transactions of the ASME-Journal of Basic Engineering1960; 82(Series D):35-45. |
| [2] | OndelO, BoutleuxE, BlancoE, ClercG. Coupling pattern recognition with state estimation using Kalman filter fault diagnosis. IEEE Transactions on Industrial Electronics2012; 59(11):4293-4300. |
| [3] | KhanesarMA, KayscanE, TeshnehlabM, KaynakO. Extended Kalman filter based learning algorithm for type‐2 fuzzy logic systems and its experimental evaluation. IEEE Transactions on Industrial Electronics2012; 59(11):4443-4455. |
| [4] | MarinaHGD, PeredaFJ, Giron‐SierraJM, EspinosaF. UAV attitude estimation using unscented Kalman filter and TRIAD. IEEE Transactions on Industrial Electronics2012; 59(11):4465-4474. |
| [5] | MitsantisukC, OhishiK, KatsuraS. Estimation of action/reaction forces for the bilateral control using Kalman filter. IEEE Transactions on Industrial Electronics2012; 59(11):4383-4393. |
| [6] | ToscanoR, LyonnetP. A Kalman optimization approach for solving some industrial electronics problems. IEEE Transactions on Industrial Electronics2012; 59(11):4456-4464. |
| [7] | NamK, OhS, FujimotoH, HoriY. Estimation of sideslip and roll angles of electric vehicles using lateral tire force sensors through RLS and Kalman filter approaches. IEEE Transactions on Industrial Electronics2013; 60(3):988-1000. |
| [8] | SmidlV, PeroutkaZ. Advantages of square‐root extended Kalman filter for sensorless control of AC drives. IEEE Transactions on Industrial Electronics2012; 59(11):4189-4196. |
| [9] | HuangR, PatwardhanSC, BieglerLT. Robust stability of nonlinear model predictive control with extended Kalman filter and target setting. International Journal of Robust and Nonlinear Control2013; 23(11):1240-1264. · Zbl 1271.93117 |
| [10] | ZhifengG, GuixinY, ChangqingZ, TianzhangS. Design of the multi‐objective constrained nonlinear robust excitation controller with extended Kalman filter estimates of all state variables. International Journal of Robust and Nonlinear Control2015; 25(6):791-808. · Zbl 1309.93165 |
| [11] | CuiP, ZhangH, LamJ, MaL. Real‐time Kalman filtering based on distributed measurements. International Journal of Robust and Nonlinear Control2013; 23(14):1597-1608. · Zbl 1286.93175 |
| [12] | UtzT, FleckC, FrauhammerJ, Seiler‐ThullD, KugiA. Extended Kalman filter and adaptive backstepping for mean temperature control of a three‐way catalytic converter. International Journal of Robust and Nonlinear Control2014; 24(18):3437-3453. · Zbl 1302.93218 |
| [13] | JulierSJ, UhlmannJK, Durrant‐WhyteHF. A new approach for filteirng nonlinear systems. Proceedings of the American Control Conference1995; 3:1628-1632. |
| [14] | LuanX, ShiY, LiuF. Unscented Kalman filtering for greenhouse climate control systems with missing measurement. International Journal of Innovative Computing, Information and Control2012; 8(3B):2173-2180. |
| [15] | JwoD, YangC, ChuangC, LeeT. Fuzzy adaptive unscented Kalman filter control of epileptiform spikes in a class of neural mass models. Nonlinear Dynamics2014. |
| [16] | ItoK, XiongK. Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control2000; 45(5):910-927. · Zbl 0976.93079 |
| [17] | NorgaardM, PoulsenN, RavnO. New developments in state estimation for nonlinear systems. Automatica2000; 36(11):1627-1638. · Zbl 0973.93050 |
| [18] | van derMerweR. Sigma‐point Kalman filters for probabilistic inference in dynamic state‐space models. Ph.D. Thesis, Portland, Oregon, 2004. |
| [19] | Van der MerweR, WanEA. The square‐root unscented Kalman filter for state and parameter‐estimation. In 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vol. 6 Salt Lake City: Utah, U.S.2001; 3461-3464. |
| [20] | KarimpourH, MahzoonM, KeshmiriM. Exploring the analogy for adaptive attitude control of a ground‐based satellite system. Nonlinear Dynamics2012; 69(4):2221-2235. |
| [21] | WonSHP, GolnaraghiF, MelekWW. A fastening tool tracking system using an IMU and a position sensor with Kalman filters and a fuzzy expert system. IEEE Transactions on Industrial Electronics2009; 56(5):1782-1792. |
| [22] | LiangY, XuS, TingL. T‐S model‐based SMC reliable design for a class of nonlinear control systems. IEEE Transactions on Industrial Electronics2009; 56(9):3286-3295. |
| [23] | CampoloD, SchenatoL, LijuanP, DengX, GuglielmelliE. Multimodal sensor fusion for attitude estimation of micromechanical flying insects: a geometric approach. Intelligent Robots and Systems, Iros 2008 IEEE/RSJ International Conference on, Nice, France, 2008; 3859-3864. |
| [24] | BazinJC, KweonI, DemonceauxC, VasseurP. UAV attitude estimation by vanishing points in catadioptric images. Robotics and Automation ICRA 2008 IEEE International Conference on, Pasadena, CA, U.S., 2008; 2743-2749. |
| [25] | TangT. A helicoopter’s formation flying model in the low airspace with two telegraphy poles and electrical wire. Nonlinear Dynamics2012; 69:399-408. |
| [26] | EustonM, CooteP, MahonyR, KimJ, HamelT. A complementary filter for attitude estimation of a fixed‐wing UAV. Intelligent Robots and Systems, IROS 2008 IEEE/RSJ International Conference on, Nice, France, 2008; 340-345. |
| [27] | FeiY, ShenX. Nonlinear analysis on moving process of soft robots. Nonlinear Dynamics2013; 73:671-677. |
| [28] | WangW, LiuZ, XieR. Quadratic extended Kalman filter approach for GPS/INS integration. Aerospace Science and Technology2006; 10:709-713. · Zbl 1195.70056 |
| [29] | WendelJ, TrommerGF. Tightly coupled GPS/INS integration for missile applications. Aerospace Science and Technology2004; 8:627-634. |
| [30] | JwoD, YangC, ChuangC, LeeT. Performance enhancement for ultra‐tight GPS/INS integration using a fuzy adaptive strong tracking unscented Kalman filter. Nonlinear Dynamics2013; 73:377-3995. |
| [31] | SuhYS. Attitude estimation by multiple‐model Kalman filters. IEEE Transactions on Industrial Electronics2006; 53(4):1386-1389. |
| [32] | KitagawaG. Non‐Gaussian state‐space modeling of nonstationary time series. Journal of American Statistical Association1987; 82(400):1032-1063. · Zbl 0644.62088 |
| [33] | BerzuiniC, BestNG, GilksWR, LarizzaC. Dynamic conditional independence models and Markov chain Monte Carlo. Journal of American Statistical Association1997; 82(440):1403-1412. · Zbl 0913.62025 |
| [34] | KotechaJH, DjurićPM. Gaussian particle filtering. IEEE Transactions on Signal Processing2003; 51(10):2592-2601. · Zbl 1369.94195 |
| [35] | LiuB, HoteitI. Nonlinear Bayesian mode filtering. International Journal of Innovative Computing, Information and Control2015; 11(1):231-245. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
