Data Fusion in Sensor Networks - Semantic Scholar
Adaptive Cubature Kalman Filter for Nonlinear State and Parameter Estimation Huimin Chen Dept. of Electrical Engineering University of New Orleans New Orleans, LA 70148, U.S.A.
Problem Formulation • Discrete-time state nonlinear dynamic model with known input
• Nonlinear observation model
• Recursively estimate • state • slowly time varying parameter • Solution type • • •
Minimum mean square error (MMSE) estimate for the state Least squares (of measurement residuals) for the parameter Simple dynamic model for parameter state augmentation
• Want to have efficient algorithm to recursively update both state and parameter
SSCET’12, New Orleans
2
Existing Nonlinear Filtering Techniques • Density estimate based methods • Exact: finite dimensional closed form solution only for special classes of nonlinear model • Approximate: computationally expensive • Particle filtering and various Markov chain Monte Carlo metohds • Homotopy based methods • Particle flow by solving differential equations • Point estimate based methods • Extended Kalman filter: popularly used, can include higher order terms and apply linearization iteratively • Unscented Kalman filter: can choose deterministic sigma-points for certain type of nonlinearity, essentially linear MMSE update • Cubature Kalman filter: suitable for additive white Gaussian noise SSCET’12, New Orleans
3
Cubature Kalman Filter • Prediction and update require integration of nonlinear function with Gaussian kernal
cubature points Cholesky decomposition of
• Apply linear MMSE update
SSCET’12, New Orleans
4
Recursive Least Squares Filter • Minimize (weighted) measure residuals squared
• Need parameter adaptation
• and covariance update
SSCET’12, New Orleans
5
Joint State and Parameter Update State prediction
can be replaced by applying weighed cubature points
Adaptive cubature rule
State update Parameter update
Key issue How to update estimation error covariance of the state?
tuning para.
cubature-point adaptation standard LMMSE update
SSCET’12, New Orleans
6
Overall Recursive Estimator • Recursive least squares algorithm for parameter update • Adaptive cubature points around the estimated parameter
• Nonlinearity coupled in state and parameter leads to larger spread in cubature points • State update follows LMMSE criterion • Unlike extended Kalman filter, iteration in parameter estimate does not help • No need to augment state Sequential update of state and parameter, thus computationally efficient
SSCET’12, New Orleans
7
Filter Implementation on Realistic Applications • Two nonlinear estimation problems arising from • battery state-of-charge estimation
• vehicle state estimation • They both contain • nonlinear state dynamics • unknown parameter • and have real-time processing constraint • We consider three representative point estimate based methods • dual extended Kalman filter • cubature Kalman filter by state augmentation • proposed adaptive cubature Kalman filter
SSCET’12, New Orleans
8
Battery State-of-Charge Estimation • Battery state of charge (SoC): percentage of the available capacity, depending on temperature, discharge rate, battery design, etc. • Monitoring battery SoC is crucial for battery health management and remaining useful life prediction
• Typically, SoC has to be estimated using voltage, current and temperature measurements with properly chosen battery model
SSCET’12, New Orleans
9
Battery State-of-Charge Estimation • State-of-charge (SoC) related to current and Coulombic efficiency • Simplified electrical model of battery state dynamics driven by current • Nonlinear model for hysteresis • Battery voltage • Primary goal: Estimate SoC with the measured battery voltage and current • Note: OCV(∙) has to be obtained empirically using the same type of LiIon battery
SSCET’12, New Orleans
10
Testing Results
• Note: Augmented CKF has similar accuracy to adaptive CKF, but requires more than twice the computational time in one full cycle of filter update
SSCET’12, New Orleans
11
Vehicle State Estimation • Vehicle state estimation is an important component in vehicle stability control for energy saving and driver comfort
• Tire slip, tire force, road friction, etc. can affect the dynamics of vehicle state SSCET’12, New Orleans
12
Vehicle State Estimation
Primary goal: Estimate vehicle’s body slip angle inclination
and road sideward
SSCET’12, New Orleans
13
Mountain road, no change of body slip angle
SSCET’12, New Orleans
14
Asphalt road, natural Corning, with slow over-steer
SSCET’12, New Orleans
15
Vehicle State Estimation – Testing Summary • Dual EKF diverged in 6 out of 50 testing cases. • Adaptive CKF does not overshoot in estimating the body slip angle compared with augmented CKF when no steering occurred. • Adaptive CKF can meet the real-time processing requirement with sampling rate of 100Hz. • Choice of forgetting factor does not affect filter performance much under those representative challenging scenarios. [z]
zk
3 z(k)
ModOutEstim
ModOutEstim ModOutEstim
uk-1
2 u(k-1)
xk uk-1 zk
{Vx0} [z]
Vx0 z
Pk-
xk-1
xk-1
xk-1
Measurement _Update
1 ModOutEstim
2 x(k)
xk
Conditional _Logic
Pk
Pk-
StateEstim
Vx0
{Vx0}
xk-
Enable
Enable
1 Enable
xk-
xk
U( : ) Reshape
3 P (k)
Pk-1
Initialize _x Time_Update
x (state ) 1: vx [m/s ] 2: vy [m/s ] 3: psiD [rad/s] 4: mu _est [-] 5. phi _road [m/ss ]
u (input ) 1: delta _f [rad] 2: vxw_1 [m/s ] 3: vxw_2 [m/s ] 4: vxw_3 [m/s ] 5: vxw_4 [m/s ] 6: Fz1 [N] 7: Fz2 [N] 8: Fz3 [N] 9: Fz4 [N]
z (measurement ) 1: ax [m/s 2] 2: ay [m/s 2] 3: psiD [rad/s ]
structuur van diagram volgens Figuur2-1 van: An Introduction to the Kalman Filter Greg Welch an Gary Bishop May 23, 2003
Copyright ©2006 TNO Automotive, The Netherlands project : DEMO Bodyslipangle estimation
SSCET’12, New Orleans
16
Concluding Summary • Proposed an adaptive cubature Kalman filter algorithm for nonlinear state and parameter estimation • It is competitive among point estimate based recursive algorithms for nonlinear model with additive Gaussian noise • For battery state-of-charge estimation, adaptive CKF yields better accuracy than dual EKF with comparable computational complexity • For vehicle state estimation, adaptive CKF is more robust against road condition variation compared with CKF using state augmentation • There is clear advantage in using cubature-point adaptation to avoid modeling the nonlinear coupling between state and parameter •
sequential algorithm for parameter and state update
•
one tuning parameter to approximately cover all aspects in state augmentation
•
suitable for high dimensional estimation/filtering problem
SSCET’12, New Orleans
17
Acknowledgement • Office of Naval Research: ONR-DEPSCoR N00014-09-1-1169 • Louisiana Board of Regents: LEQSF-EPS(2012)-OPT-IN-12 • Drs. X. Rong Li and Vesselin P. Jilkov @ University of New Orleans for numerous discussions on nonlinear filtering methods • Dr. Genshe Chen @ Intelligent Fusion Technology, Inc. for collaborative research and development on battery health management • Mr. Roel Leenen @ TNO for providing testing data on vehicle state estimation
SSCET’12, New Orleans
18
Problem Formulation • Discrete-time state nonlinear dynamic model with known input
• Nonlinear observation model
• Recursively estimate • state • slowly time varying parameter • Solution type • • •
Minimum mean square error (MMSE) estimate for the state Least squares (of measurement residuals) for the parameter Simple dynamic model for parameter state augmentation
• Want to have efficient algorithm to recursively update both state and parameter
SSCET’12, New Orleans
2
Existing Nonlinear Filtering Techniques • Density estimate based methods • Exact: finite dimensional closed form solution only for special classes of nonlinear model • Approximate: computationally expensive • Particle filtering and various Markov chain Monte Carlo metohds • Homotopy based methods • Particle flow by solving differential equations • Point estimate based methods • Extended Kalman filter: popularly used, can include higher order terms and apply linearization iteratively • Unscented Kalman filter: can choose deterministic sigma-points for certain type of nonlinearity, essentially linear MMSE update • Cubature Kalman filter: suitable for additive white Gaussian noise SSCET’12, New Orleans
3
Cubature Kalman Filter • Prediction and update require integration of nonlinear function with Gaussian kernal
cubature points Cholesky decomposition of
• Apply linear MMSE update
SSCET’12, New Orleans
4
Recursive Least Squares Filter • Minimize (weighted) measure residuals squared
• Need parameter adaptation
• and covariance update
SSCET’12, New Orleans
5
Joint State and Parameter Update State prediction
can be replaced by applying weighed cubature points
Adaptive cubature rule
State update Parameter update
Key issue How to update estimation error covariance of the state?
tuning para.
cubature-point adaptation standard LMMSE update
SSCET’12, New Orleans
6
Overall Recursive Estimator • Recursive least squares algorithm for parameter update • Adaptive cubature points around the estimated parameter
• Nonlinearity coupled in state and parameter leads to larger spread in cubature points • State update follows LMMSE criterion • Unlike extended Kalman filter, iteration in parameter estimate does not help • No need to augment state Sequential update of state and parameter, thus computationally efficient
SSCET’12, New Orleans
7
Filter Implementation on Realistic Applications • Two nonlinear estimation problems arising from • battery state-of-charge estimation
• vehicle state estimation • They both contain • nonlinear state dynamics • unknown parameter • and have real-time processing constraint • We consider three representative point estimate based methods • dual extended Kalman filter • cubature Kalman filter by state augmentation • proposed adaptive cubature Kalman filter
SSCET’12, New Orleans
8
Battery State-of-Charge Estimation • Battery state of charge (SoC): percentage of the available capacity, depending on temperature, discharge rate, battery design, etc. • Monitoring battery SoC is crucial for battery health management and remaining useful life prediction
• Typically, SoC has to be estimated using voltage, current and temperature measurements with properly chosen battery model
SSCET’12, New Orleans
9
Battery State-of-Charge Estimation • State-of-charge (SoC) related to current and Coulombic efficiency • Simplified electrical model of battery state dynamics driven by current • Nonlinear model for hysteresis • Battery voltage • Primary goal: Estimate SoC with the measured battery voltage and current • Note: OCV(∙) has to be obtained empirically using the same type of LiIon battery
SSCET’12, New Orleans
10
Testing Results
• Note: Augmented CKF has similar accuracy to adaptive CKF, but requires more than twice the computational time in one full cycle of filter update
SSCET’12, New Orleans
11
Vehicle State Estimation • Vehicle state estimation is an important component in vehicle stability control for energy saving and driver comfort
• Tire slip, tire force, road friction, etc. can affect the dynamics of vehicle state SSCET’12, New Orleans
12
Vehicle State Estimation
Primary goal: Estimate vehicle’s body slip angle inclination
and road sideward
SSCET’12, New Orleans
13
Mountain road, no change of body slip angle
SSCET’12, New Orleans
14
Asphalt road, natural Corning, with slow over-steer
SSCET’12, New Orleans
15
Vehicle State Estimation – Testing Summary • Dual EKF diverged in 6 out of 50 testing cases. • Adaptive CKF does not overshoot in estimating the body slip angle compared with augmented CKF when no steering occurred. • Adaptive CKF can meet the real-time processing requirement with sampling rate of 100Hz. • Choice of forgetting factor does not affect filter performance much under those representative challenging scenarios. [z]
zk
3 z(k)
ModOutEstim
ModOutEstim ModOutEstim
uk-1
2 u(k-1)
xk uk-1 zk
{Vx0} [z]
Vx0 z
Pk-
xk-1
xk-1
xk-1
Measurement _Update
1 ModOutEstim
2 x(k)
xk
Conditional _Logic
Pk
Pk-
StateEstim
Vx0
{Vx0}
xk-
Enable
Enable
1 Enable
xk-
xk
U( : ) Reshape
3 P (k)
Pk-1
Initialize _x Time_Update
x (state ) 1: vx [m/s ] 2: vy [m/s ] 3: psiD [rad/s] 4: mu _est [-] 5. phi _road [m/ss ]
u (input ) 1: delta _f [rad] 2: vxw_1 [m/s ] 3: vxw_2 [m/s ] 4: vxw_3 [m/s ] 5: vxw_4 [m/s ] 6: Fz1 [N] 7: Fz2 [N] 8: Fz3 [N] 9: Fz4 [N]
z (measurement ) 1: ax [m/s 2] 2: ay [m/s 2] 3: psiD [rad/s ]
structuur van diagram volgens Figuur2-1 van: An Introduction to the Kalman Filter Greg Welch an Gary Bishop May 23, 2003
Copyright ©2006 TNO Automotive, The Netherlands project : DEMO Bodyslipangle estimation
SSCET’12, New Orleans
16
Concluding Summary • Proposed an adaptive cubature Kalman filter algorithm for nonlinear state and parameter estimation • It is competitive among point estimate based recursive algorithms for nonlinear model with additive Gaussian noise • For battery state-of-charge estimation, adaptive CKF yields better accuracy than dual EKF with comparable computational complexity • For vehicle state estimation, adaptive CKF is more robust against road condition variation compared with CKF using state augmentation • There is clear advantage in using cubature-point adaptation to avoid modeling the nonlinear coupling between state and parameter •
sequential algorithm for parameter and state update
•
one tuning parameter to approximately cover all aspects in state augmentation
•
suitable for high dimensional estimation/filtering problem
SSCET’12, New Orleans
17
Acknowledgement • Office of Naval Research: ONR-DEPSCoR N00014-09-1-1169 • Louisiana Board of Regents: LEQSF-EPS(2012)-OPT-IN-12 • Drs. X. Rong Li and Vesselin P. Jilkov @ University of New Orleans for numerous discussions on nonlinear filtering methods • Dr. Genshe Chen @ Intelligent Fusion Technology, Inc. for collaborative research and development on battery health management • Mr. Roel Leenen @ TNO for providing testing data on vehicle state estimation
SSCET’12, New Orleans
18
