1. PSG College of Technology
KALMAN FILTER types
KARTHICK S
Assistant Professor
PSG College of Technology

2. Kalman Filter
It involves the estimation of the state of the system on the basis of noisy measurements.
Kalman filter is simply an optimal recursive data processing algorithm.
It processes all available measurements, regardless of their precision, to estimate the current state of the system with use of knowledge of the system and measurement device dynamics, the statistical description of the system noises, measurement errors, and uncertainty in the dynamics models and any available information about initial conditions of the variables of interest.

3. State Estimation
The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to
Fk is the state transition model which is applied to the previous state
Bk is the control-input model which is applied to the control vector
Vk is the process noise which is assumed to be drawn from a zero mean multivariate normal distribution with covariance Qk.

4. State Estimation contd..
At time k an observation (or measurement) of the true state is made according to
yk is the observation model which maps the true state space to the measurement.
nk is the observation noise which is assumed to be zero mean Gaussian white noise with covariance Rk.

5. State Estimation contd..
The Kalman filter is a recursive estimator. That is, only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state.

6. Discrete Kalman Filter-Algorithm

7. Discrete Kalman Filter-Algorithm contd..
Measurement Update Equations
The residual reflects the discrepancy between the predicted measurement and the actual measurement. A residual of zero means that the two are in complete agreement. 
Innovation or measurement residual:

8. Discrete Kalman Filter-Algorithm contd..

9. Merits of Kalman Filter
• Kalman filter is simply an optimal recursive data processing algorithm.
It is powerful, fast, supports estimation of past, present and future states. 
• It is interesting to note that the derivation provided by Kalman applies to arbitrary random signals, described by up to a second order average statistical properties.

10. Demerits of Kalman Filter
It is applicable only to linear or nearly linear problems.
The process and/or the measurement relationships must be linear.
The above drawbacks can be overcome by using EKF or UKF.

11. EKF
A Kalman filter that linearizes about the current mean and covariance is referred to as an Extended Kalman filter (EKF). i.e. we can linearize the nonlinear function around the current estimate to compute the state estimate even in the face of nonlinear relationships.
The Extended Kalman filter (EKF) is one of the nonlinear filter which has become a standard technique used in a number of nonlinear estimation and machine learning applications.

12. State Estimation

13. State Estimation contd..
The nonlinear function f( ) in the above difference equation relates the state at time step k-1 to the state at k.
At time k an observation (or measurement) yk of the true state xk is made accordingly as

14. State Estimation contd..
The nonlinear function h( ) in the above measurement equation relates the state at time step k to the measurement at k.
The function f can be used to compute the predicted state from the previous estimate and similarly the function h can be used to compute the predicted measurement from the predicted state.
However, f and h cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian) is computed. At each time step the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.
The Extended Kalman filter also has two distinct phases: Predict and Update.

15. EKF Algorithm

16. EKF Algorithm contd..

17. EKF Algorithm contd..

18. Drawbacks of EKF
Unlike the Kalman filter, the EKF is not an optimal estimator. In addition, if the initial estimate of the state is wrong, or if the process is modeled incorrectly, the filter may quickly diverge, owing to its linearization.
Another problem with the EKF is that the estimated covariance matrix tends to underestimate the true covariance matrix and therefore risks becoming inconsistent in the statistical sense without the addition of "stabilizing noise".
The EKF can be viewed as providing ''first-order'' approximations to the optimal terms. These approximations, however, can introduce large errors in the true posterior mean and covariance of the transformed Gaussian random variable, which may lead to sub-optimal performance and sometimes divergence of the filter.

19. Merits and Demerits of EKF
EKF may be applied to estimation of nonlinear multidimensional systems with small non-linearities.
It has also been successfully applied to the land data assimilation problem as well as in 2D hydrodynamics and oceanography.
Demerits
By using EKF, Jacobians or Hessians are calculated at each time step to estimate the state of the non-linear system.
For high dimensional systems, the calculation of matrix of partial derivatives for is a tedious one.

20. Comparison
EKF KF

21. UKF
When the state transition and observation models are highly non-linear, the extended Kalman filter (EKF) can give particularly poor performance. This is because only the mean is propagated through the non-linearity.
The Derivative Free Kalman Filter or the Unscented Kalman filter(UKF) addresses this problem by using a deterministic sampling technique known as the Unscented transformation to pick a minimal set of sample points (called sigma points) around the mean.
These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true non-linear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity.

22. UKF contd..
In addition, this technique removes the requirement to analytically calculate Jacobians, which for complex functions can be a difficult task in itself. Hence the UKF is also called as the Derivative Free Kalman Filter.

23. Unscented Transformation

24. Unscented Transformation contd..

25. Unscented Transformation contd..

26. Unscented Transformation contd..

27. State Estimation
In UKF the state is estimated using two phases namely time update phase and measurement update phase i.e., predict and update steps involved.
Unscented Kalman Filter Algorithm
Time Update Equations
As with the EKF, the UKF prediction can be used independently from the UKF update. The estimated state and covariance are augmented with the mean and covariance of the process noise as follows:

28. UKF Algorithm contd..

29. UKF Algorithm contd..

30. UKF Algorithm contd..

31. UKF Algorithm contd..

32. UKF Algorithm contd..

33. UKF Algorithm contd..

34. UKF Algorithm contd..

35. Merits of UKF
The UKF effectively evaluates both the Jacobian and Hessian precisely through its sigma point propagation, without the need to perform any analytic differentiation. Hence it is called as Derivative Free Kalman Filter.
The UKF captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity but the EKF in comparison achieves only first order accuracy.