Optimal State Estimation

By Dan Simon

Chapter 9.3 - Fixed-Lag Smoothing

9.3 FIXED-LAG SMOOTHING

In fixed-lag smoothing we want to obtain an estimate of the state at time (k - N) given measurements up to and including time k, where the time index k continually changes as we obtain new measurements, but the lag N is a constant. In other words, at each time point we have N future measurements available for our state estimate. We therefore want to obtain _{k-N, k} for k = N, N + 1, …, where N is a fixed positive integer. This could be the case, for example, if a satellite is continually taking photographs that are to be displayed or transmitted N time steps after the photograph is taken. In this case, since the photograph is processed N time steps after it is taken, we have N additional measurements after each photograph that are available to update the estimate of the satellite state and hence improve the quality of the photograph. In this section we use the notation

Note that the notation has changed slightly from the previous section. In the previous section we used the notation _{k, m} to refer to the estimate of x_k given measurements up to and including time (m — 1). In this section (and in the remainder of this chapter) we use _{k, m} to refer to the estimate of x_k given measurements up to and including time m.

Let us define x_{k, m} as the statex_{k - m} propagated with an identity transition matrix and zero process noise to time k. With this definition we see that

We can therefore define the augmented system

The Kalman filter estimates of the components of this augmented state vector are given as

We see that if we can use a Kalman filter to estimate the states of the augmented system (using measurements up to and including time k), then the estimate of the last element of the augmented state vector, x_k+₁_{, N+}₁ equal to the estimate of x_k-N given measurements up to and including time k. This is the estimate that we are looking for in fixed-lag smoothing. This idea is illustrated in Figure 9.7.

Figure 9.7 This illustrates the idea that is used to obtain the fixed-lag smoother. A fictitious state variable x_{k, m} is initialized as x_{k, m} = x_k-m and from that point on has an identity state transition matrix. The a posteriori estimate of x_{k+m, m}is then equal to _{k-m, k}.

From Equation (9.10) we can write the Kalman filter for the augmented system of Equation (9.41) as follows:

where the L_{k, i} matrices are components of the smoother gain that will be determined in this section. Note that L _{k, 0} is the standard Kalman gain. The smoother gain L_kis defined as

From Equation (9.10) we see that the L_k gain matrix is given by

where the covariance matrices are defined as

The L_kexpression above can be simplified to

Prom Equation (9.10) we see that the covariance-update equation for the Kalman filter for our augmented system can be written as

Substituting for L_kfrom Equation (9.47) and multiplying out gives

This gives us the update equations for the P matrices. The equations for the first column of the P matrix are as follows:

The equations for the diagonal elements of the P matrix are as follows:

These equations give us the formulas that we can use for fixed-lag smoothing. This gives us the estimate E(x_k-N y₁, …, y_k ) for a fixed N as k continually increments. The fixed-lag smoother is summarized as follows.

The fixed-lag smoother

Run the standard Kalman filter of Equation (9.10) to obtain , L_k , and .

Initialize the fixed-lag smoother as follows:

For i = l, …, N + l, perform the following:

each

_{k, k}

_{k-N, k}

The percent improvement due to smoothing can be computed as

EXAMPLE 9.2

Consider the same two state system as described in Example 9.1. Suppose we are trying to estimate the state of the system with a fixed time lag. The discretization time step T = 0.1 and the standard deviation of the acceleration noise is 10. Figure 9.8 shows the percent improvement in state estimation that is available with fixed-lag smoothing. The figure shows percent improvement as a function of lag size, and for two different values of measurement noise. The values on the plot are based on the theoretical estimation-error covariance. As expected, the improvement in estimation accuracy is more dramatic as the measurement noise decreases. This was discussed at the end of Section 9.2.

Figure 9.8 This shows the percent improvement of the trace of the estimation-error covariance of the smoothed estimate of the state (relative to the standard Kalman filter) for Example 9.2. As the number of lag intervals increases, the estimation error of the smoother decreases and the percent improvement increases. Also, as the measurement noise decreases, the improvement due to smoothing is more dramatic.

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INTRODUCTION

TABLE OF CONTENTS

Chapter 9.3 - Fixed-Lag Smoothing

9.3 FIXED-LAG SMOOTHING

INTRODUCTION

TABLE OF CONTENTS

Chapter 9.3 - Fixed-Lag Smoothing

9.3 FIXED-LAG SMOOTHING

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