7  Optimal Smoothing

Reference: Simon (2006).

• The concept of smoothing is to obtain the best estimate of a state at given point in time, based on measurements that are available both in past and future to that time. There are different types of smoothing.

• Fixed point smoothing: Here, the timestep of a state to be updated is fixed, say \(\hat{\mathbf{x}}_k\), and the estimate will be updated with the measurement data that keeps rolling in, \(\mathbf{y}_{k+1},\mathbf{y}_{k+2}\) etc… Example: a satellite takes an image at time \(k\). In order to more accurately process the photograph at time \(k\), we need the estimate of satellite’s state (position and velocity) at \(k\). As the satellite continues to be in orbit, we get the state data in future time \(k+1,k+2\) etc… which can be used to better estimate the state at time \(k\).

• Fixed-lag smoothing: Here, the better estimation of state at time \(k\) is performed with a fixed time interval of measurement data in the future, say data till \(k+N\). That is, the better estimation of state \(\mathbf{x}_k\) is obtained using data from a fixed interval \(N\) in future i.e. \(\mathbf{y}_{k+1},\mathbf{y}_{k+2},\dots,\mathbf{y}_{k+N}\).

• Fixed-interval smoothing: This is similar to moving window convolution. With all the measurement data being in hand, the estimation of state \(\mathbf{x}_k\) is performed using data with fixed window/interval as \(\mathbf{y}_{k-N},\dots,\mathbf{y}_{k+N}\).

• For this smoothing to happen, an alternate form of Kalman filter was derived that varies from Eq. 3.3 Eq. 3.5, the derivations were given in the reference.

• All the algorithms were given in detail in the reference.

Simon, Dan. 2006. Optimal State Estimation: Kalman, h Infinity, and Nonlinear Approaches. John Wiley & Sons.