Data fusion with kalman filtering
- 1. Sensor Data Fusion Using Kalman Filters Antonio Moran, Ph.D. amoran@ieee.org
- 2. Kalman Filtering Estimation of state variables of a system from incomplete noisy measurements Fusion of data from noisy sensors to improve the estimation of the present value of state variables of a system
- 3. Kalman Filtering Given a System x = A x + B u + w . y = Cx + n where x = nx1 state vector y = mx1 measurement vector w = nx1 process noise system disturbance n = mx1 measurement noise Q = w covariance matrix R = n covariance matrix System u w n y w and n are white gaussian noises
- 4. Kalman Filtering Problem Definition ^ Find an estimate x of state vector x from incomplete and noisy measurement vector y
- 5. Kalman Filtering System x = A x + B u + w . y = Cx + n Kalman Filter (Estimator) x = A x + B u + L (y – C x) . ^^ w and n are not included. Their present values are unknown. L: filter gain. nxm matrix. Filter gain L ensures that the estimation error e converges to zero lim e(t) 0 t ∞ e = x - x^
- 7. Kalman Filtering Computing the Filter Gain L x = A x + B u + L (y – C x) . ^^ L = S CT R-1 S is a positive matrix solution of the Riccati equation: A S + SAT - S CT R-1 CT S + Q = 0 Matrixes A, C, Q and R are known Estimator:
- 8. Data Fusion Using Kalman Filter Problem Formulation Fuse the noisy signals of three sensors to improve the estimate of a variable (temperature, distance, position, etc.)
- 9. Air Temperature Sensor 1 Sensor 2 Sensor 3 Gaussian process mean value: 20 std. dev.: 1.1 Sensors noise with zero mean
- 10. Sensor 1 Sensor 2 Sensor 3 Sensors Noise and Bias std. dev.: 0.8 bias: 0.4 std. dev.: 0.8 bias: -0.4 std. dev.: 0.8 bias: 0.0
- 11. Sensor Data Fusion with Kalman Filter Sensor 1 Sensor 2 Sensor 3 Sensor Data Fusion with Kalman Filter Improved Estimation of Measured Variable
- 12. Kalman Filtering System x = A x + B u + w . y = Cx + n x = w . y1 = x + n1 y2 = x + n2 y3 = x + n3 x = air temperature Changes in temperature are modeled as a gaussian process Sensor 1 Sensor 2 Sensor 3 Noisy measurements Determining matrixes: A = 0 B = 0 C = 1 1 1 Q = 1.21 R = 0.64 0 0 0 0.64 0 0 0 0.64
- 13. Estimation Using One Sensor The Kalman filter significantly reduces the noise but does not eliminate the bias First sensor with noise and bias
- 14. Estimation Using Two Sensors Sensor fusion through Kalman filtering significantly improves the estimation reducing the effect of sensor noise and bias First and Second sensors with noise and bias
- 15. Estimation Using Three Sensors No significantly improvement with the third sensor (respect to when using first and second sensors) Three sensors. Third sensor only with noise
- 16. Estimation Error Higher error when using only the first sensor with bias. e = x - x^ Lower error when using two and three sensors.
- 17. Estimation Error Mean Value Standard Deviation e = x - x^ First Sensor First and Second Sensors Three Sensors -0.00035 0.082111 -0.00051 0.072695 -0.39896 0.097506 Higher estimation error when using only the first sensor (noise and bias). Estimation error significantly reduces with two and three sensors. Bias is almost eliminated.
- 18. Estimator Performace for Non- Gaussian Processes The estimator performs well even if the process is not gaussian. Temperature changes behave as a uniform (non-gaussian) process with zero mean
- 19. Conclusions Sensor data fusion can be easily implemented using Kalman filters. For filter design, changes in temperature are modeled as a gaussian process. However, the filter performs well even when used in other probabilistic processes. Sensor fusion through Kalman filtering significantly improves the on-line estimation reducing the effect of sensor noise and bias. Research goes on for the simultaneous estimation of several variables.