基于正交变换的改进CKF算法
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摘要
为了解决容积卡尔曼滤波(CKF)算法在处理高维问题时出现的非局部采样问题,提出基于采样点正交变换的改进CKF算法(TCKF).从数值积分近似角度导出无迹卡尔曼滤波(UKF)和CKF两种近似滤波算法,并指出CKF只是UKF的一个特例;基于多元Taylor级数展开分析,揭示CKF在克服UKF数值不稳定性问题的同时,引入非局部采样问题;对Cubature点集进行正交变换得到TCKF算法,并从理论上证明,在高维、强非线性等非局部采样问题突出的滤波模型中,TCKF具有比CKF更高的估计精度.仿真实例验证了所提出算法的有效性.
In order to solve the nonlocal sampling problem inherent in the cubature kalman filter(CKF) algorithm for high dimensional problems, a methodology based on orthogonal transformation on the cubature points is proposed. Firstly, the unscented Kalman filter(UKF) algorithm and CKF algorithm are deduced from the perspective of numeriacal integration in the gaussian filtering framwork, and it is pointed out that the CKF is virtually a spacial case of the UKF. Then, the performance of the unscened transform(UT) is analyzed based on the multi-dimensional Taylor series, it reveals that the problem of numerical instability of the UKF can be solved by using the CKF, meanwhile the nonlocal sampling problem is introduced. Finally, through the orthogonal transformation of the sampling point in the CKF algorithm, the TCKF algorithm is derived. It is proved theoretically that the TCKF algorithm has higher estimation accuracy than the CKF algorithm in the high-dimentional and strongly nonlinearity situation where local sampling problems are prominent.simulation examples verify the effectiveness of the proposed algorithm.
In order to solve the nonlocal sampling problem inherent in the cubature kalman filter(CKF) algorithm for high dimensional problems, a methodology based on orthogonal transformation on the cubature points is proposed. Firstly, the unscented Kalman filter(UKF) algorithm and CKF algorithm are deduced from the perspective of numeriacal integration in the gaussian filtering framwork, and it is pointed out that the CKF is virtually a spacial case of the UKF. Then, the performance of the unscened transform(UT) is analyzed based on the multi-dimensional Taylor series, it reveals that the problem of numerical instability of the UKF can be solved by using the CKF, meanwhile the nonlocal sampling problem is introduced. Finally, through the orthogonal transformation of the sampling point in the CKF algorithm, the TCKF algorithm is derived. It is proved theoretically that the TCKF algorithm has higher estimation accuracy than the CKF algorithm in the high-dimentional and strongly nonlinearity situation where local sampling problems are prominent.simulation examples verify the effectiveness of the proposed algorithm.
引文
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[2]Julier S J,Uhlman J K,Durrant-Whyte H F.A new method for the nonlinear transformation of means and covariances in filters and estimators[J].IEEE Trans on Automatic Control,2000,45(3):477-482.
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[4]Arasaratnam I,Haykin S.Cubature Kalman filtrs[J].IEEE Trans on Automatic Control,2009,54(6):1254-1269.
[5]Lerner U N.Hybrid bayesian networks for reasoning about complex systems[D].San Francisco:Department of Computer Science,Stanford University,2002.
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[7]Julier S J,Uhlmann J K.Unscented filtering and nonlinear estimation[J].Proc of the IEEE,2004,92(3):401-422.
[8]Van der Merwe R.Sigma-point Kalman filters for probabilistic inference in dynamic state-space models[D].Portland:Oregon Health Sciences University,Department of Computer Science,2004.
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[10]Julier S J,Uhlmann J K.The scaled unscented transformation[C].Proc of the 2002 American Control Conf.Anchorage:IEEE,2002:4555-4559.
[11]Potnuru D,Chandra K P B,Arasaratnam I.Derivative-free square-root cubature Kalman filter for nonlinear brushless DC motors[J].IET Electric Power Applications,2016,10(5):419-429.
[12]Zhan A,Bao S D,Bi W H.Low-cost adaptive square-root cubature Kalman filter for systems with process model uncertainty[J].J of Systems Engineering and Electronics,2016,27(5):945-953.
[13]Liu M,Lai J Z,Li Z M.An adaptive cubature kalman filter algorithm for inertial and land-based navigation system[J].Aerospace Science and Technology,2016,51:52-60.
[14]Teng C H,Lin S F,Jwo D J.Fuzzy adaptive cubature Kalman filter for integrated navigation systems[J].Sensors,2016,16(8):1167.
[15]Zhang W J,Wang S Y,Feng Y L.Huber-based high-degree cubature kalman tracking algorithm[J].Acta Physica Sinica,2016,65(8):354-362.
[16]Wu H,Chen S X,Yang B F.Robust range-parameterized cubature Kalman filter for bearings-only tracking[J].J of Central South University,2016,23(6):1399-1405.
[2]Julier S J,Uhlman J K,Durrant-Whyte H F.A new method for the nonlinear transformation of means and covariances in filters and estimators[J].IEEE Trans on Automatic Control,2000,45(3):477-482.
[3]Norgaard M,Poulsen N K,Ravn O.New developments in state estimation for nonlinear systems[J].Automatica,2000,36(11):1627-1638.
[4]Arasaratnam I,Haykin S.Cubature Kalman filtrs[J].IEEE Trans on Automatic Control,2009,54(6):1254-1269.
[5]Lerner U N.Hybrid bayesian networks for reasoning about complex systems[D].San Francisco:Department of Computer Science,Stanford University,2002.
[6]Xiu D X.Numerical integration formulas of degree two[J].Applied Numerical Mathematics,2008,58(10):1515-1520.
[7]Julier S J,Uhlmann J K.Unscented filtering and nonlinear estimation[J].Proc of the IEEE,2004,92(3):401-422.
[8]Van der Merwe R.Sigma-point Kalman filters for probabilistic inference in dynamic state-space models[D].Portland:Oregon Health Sciences University,Department of Computer Science,2004.
[9]Gustafsson F,Hendeby G.Some relations between extended and unscented Kalman filters[J].IEEE Trans on Signal Process,2012,60(2):545-555.
[10]Julier S J,Uhlmann J K.The scaled unscented transformation[C].Proc of the 2002 American Control Conf.Anchorage:IEEE,2002:4555-4559.
[11]Potnuru D,Chandra K P B,Arasaratnam I.Derivative-free square-root cubature Kalman filter for nonlinear brushless DC motors[J].IET Electric Power Applications,2016,10(5):419-429.
[12]Zhan A,Bao S D,Bi W H.Low-cost adaptive square-root cubature Kalman filter for systems with process model uncertainty[J].J of Systems Engineering and Electronics,2016,27(5):945-953.
[13]Liu M,Lai J Z,Li Z M.An adaptive cubature kalman filter algorithm for inertial and land-based navigation system[J].Aerospace Science and Technology,2016,51:52-60.
[14]Teng C H,Lin S F,Jwo D J.Fuzzy adaptive cubature Kalman filter for integrated navigation systems[J].Sensors,2016,16(8):1167.
[15]Zhang W J,Wang S Y,Feng Y L.Huber-based high-degree cubature kalman tracking algorithm[J].Acta Physica Sinica,2016,65(8):354-362.
[16]Wu H,Chen S X,Yang B F.Robust range-parameterized cubature Kalman filter for bearings-only tracking[J].J of Central South University,2016,23(6):1399-1405.