Kalman Filters z-1I C(n) F(n+1,n) Process Equation Observation y(n) x(n) Process Noise, v1(n) z-1I C(n) s(n) F(n+1,n) Measurement noise v2(n) Process Equation Measurement Equation Models the state x(n) as the output of a linear system excited by white noise v1(n) Relates the observable output of the system y(n) to the state x(n) F(n+1,n) denotes the state transition matrix for the linear system, relating the state at time n+1 with that at time n. The fact that n,n+1 appear in the argument emphasizes the fact that a general Kalman filter setup can model non-stationary systems, in which the state transition matrix changes with time. x(n/script_y_sub_n) represents the state estimate at time n given the ‘n’ previous observations y1, y2, …. yn. Signal Model: Observation Model: The filtering problem is posed as follows – Using the set of observed data , to find for each the MMSE of the state vector , and then to estimate the clean signal from the signal model.

Nonlinear State Estimation Unscented Kalman Filters (motivation… where Extended Kalman filters fail) Nonlinear State Estimation Predicting state vector, Kalman gain, Predicting observation When propagating a GRV through a linear model (Kalman filtering), it is easy to compute the expectation E[ ] However, if the same GRV passes through a nonlinear model, the resultant estimate may no longer be Gaussian – calculation of E[ ] is no longer easy! It is in the calculation of E[ ] in the recursive filtering process that we now need to estimate the pdf of the propagating random vector F, H are (nonlinear) vector functions a Posteriori estimate Prediction of Innovation The way Kalman filtering proceeds recursively is as follows -- (a) predict the state vectors at time k using the previous state estimate and process noise -- this is represented as x_hat_superscript'-'. (a) predict the observation vector at time k using the 'predicted' state vector at time k and the measurement noise -- this is represented as y_hat_superscript'-'. (b) the actual observation at time 'k' -- yk, brings with it some additional information relative to all previous n-1 observations -- this extra information is represented by the innovations process which is given as the difference between actual observation at time k (yk) and the predicted observation of time k (predicted in (a) above). (b) this innovation term is multiplied by the 'kalman-gain' to give an accurate 'estimate' of the state at time 'k'. Hence, the superscript '-' distinguishes the 'predicted' value and the 'estimated' value. We are ultimately interested in the 'estimated' state values at any time step. P (in the expression for Kalman Gain) denotes the covariance matrix. Some notes on EKF – It is possible to extend the linear state-space Kalman filters to systems exhibiting nonlinear dynamic behavior through a linearization procedure. Most other linear techniques can’t exploit this extension - such as Wiener filtering (which is limited to linear systems). The Wiener filter is restricted to linear systems because the concept of impulse response (which is the basic notion of wiener filtering) is meaningful only in the context of linear systems

Unscented Kalman Filters (motivation… where Extended Kalman filters fail) Extended Kalman Filters (EKF) avoid this by linearizing the state-space equations and reducing the estimate equations to point estimates instead of expectations The covariance matrices (P) are computed by linearizing the dynamic equations In EKF, the state distributions are approximated by a GRV and propagated analytically through a ‘first-order linearization’ of the nonlinear system - If the assumption of local linearity is violated, the resulting filter will be unstable - The linearization process itself involves computations of Jacobians and Hessians (which is nontrivial for most systems)

The Unscented Transform Problem statement: developing a method for calculating the statistics of a random variable which has undergone a nonlinear transformation The intuition: It is easier to approximate a distribution than it is to approximate an arbitrary nonlinear transformation Consider a nonlinearity The Unscented transform chooses (deterministically) a set of ‘sigma’ points in the original (x) space which when mapped to the transformed (y) space will be capable of generating accurate sample means and covariance matrices in the transformed space An accurate estimation of these allows us to use the Kalman filter in the usual setup, with the expectation values being replaced by sample means and covariance matrices as appropriate

The Unscented Transform (illustration) 0.2812 0.0046 0.0046 0.2014 Px 1.2692 0.0060 0.0060 0.8818 0.2709 0.0510 0.0510 0.1607 Py Py_ut

Unscented Kalman Filters (UKFs) Instead of propagating a GRV through the “linearized” system, the UKF technique focuses on an efficient estimation of the statistics of the random vector being propagated through a nonlinear function To compute the statistics of , we ‘cleverly’ sample this function using ‘deterministically’ chosen 2L+1 points about the mean of x It can be proved that with these as sample points, the following ‘weighted’ sample means and covariance matrices will be equal to the true values We use this set of equations to predict the states and observations in the nonlinear state-estimation model. The expected values E[ ] and the covariance matrices are replaced with these sample ‘weighted’ versions. Sigma Points

Unscented Kalman Filters (UKFs)… Algorithm development Augment the state vector to include the process and measurement noise and initialize the mean and covariance matrix of the augmented vector. Compute the sigma points about the previously estimated mean vector Compute the predicted value of the state vector and cov. matrix at time k Predict the current observation given previous observed samples (mapped from the sigma points) 1’st step – only initialization, (no expectation calculation) – compute the expectation only if we have some prior information of statistics of state vector. - Augment the state vector with 2 terms (initialized as 0 vectors) that allow to include v and n if we don’t know the covar. matrices of these 2 before hand. (has a flavor of adaptive kalman filter – where we don’t have knowledge of the covar. Matrices. ). Using this augmented vector allows us to incorporate for systems where the process and measurement noise statistics also need to be updated recursively (along with the state vectors). In many applications, this may not be necessary (if the process and measurement noise statistics are known a priori). - At every time step k, we compute the sigma points afresh – the I’th sigma point will be the state mean vector x_hat_a + the I’th column of the square root of the (L+lambda)P. 2n’d step – compute prediction of states – this is achieved by computing the expected value (achieved by the weighted sum of ) the nonlinearity (F) at the sigma points. - compute the measurement updates using the measurement equation on the sigma points - Calculate the Kalman gain (after having calculated the appropriate covariance matrices). Reminders: Anything with superscript ‘–’ is predicted estimate. When added to the Kalman Gain * innovations, this predicted estimate changes to a ‘filtered’ estimate of the state (which is most accurate given the k observation sequences y1, y2…. yk. Compute the innovations covariance matrix, cross correlation matrix and hence the Kalman gain and update the current estimate of the state at time k Eric A. Wan and Rudolph van der Merwe, “The Unscented Kalman Filter for Nonlinear Estimation”

Unscented Kalman Filters (UKFs) vs. Particle Filters (PFs) The unscented transformation in UKFs allows the calculation of accurate sample means and covariance matrices using a few ‘cleverly’ chosen samples Particle filtering employs Monte-Carlo techniques to obtain an estimate of the statistics of the propagated random vector The two techniques are very close conceptually There is a subtle difference : in UKF based implementations, the sigma points for sampling are deterministically chosen whereas in PF based techniques, the sampling itself is done randomly Particle Filtering is more accurate but this comes at the cost of complexity (introduced due to the need for Monte-Carlo simulations) The complexity of UKF based implementations is about the same as that of regular Kalman filters