Sect_7.4.6

7.4.6 Introduction to the Kalman Filter-Smoother

SMOOTHING

Initial State

The true value of the initial state vector is generally not known. However, the smoothed estimate of should not be too different from the filtered one, which has a known stochastic behaviour described by the variance-covariance matrix Q. A certain imprecision in the initial estimate must be tolerated. This error can be seen as having the potential for minimisation. If all the elements of Q are zero, the notion of smoothing becomes meaningless because later measurements cannot have any effect on .

The filtered estimate can be considered as an apriori estimate of the true value of , with a random error v. This relation can be written in the form of an observation equation, similar to eqn (7.4-4):

(7.4.15)

v corresponds to the definition of a residual, as it is the difference between a vector of observations and a functional model of the observations. However, in this particular case of (pseudo-)observations of the parameters, the functional model is the identity, and eqn (7.4-15) may also be written in terms of state vector increments :

(7.4.16)

Thus, v and have the same magnitude, but opposite signs. In many cases, for example when a quadratic form is considered, both terms can be used. Depending on the context however, the conceptual difference between a residual and a state vector increment should not be forgotten. To avoid any confusion with the filtered estimate already obtained, the smoothed estimate of is preferably noted ₀.

Kinematic Model

The concept of a non-perfect initial estimate also has an influence on the parametric formulation of the kinematic model. Discrepancies between prediction and observations are not only due to errors in the kinematic model, but also to errors in the estimate at time t₀. Thus the kinematic relation is applied to unknown state vectors at both epochs t₀ and t:

This relation is already linear, however, to be consistent with the measurement model it must be written in terms of state vector increments for the epoch t. For the epoch t0, the full value of the parameters can be considered, because the state vector X₀ is only relevant for linear relations:

(7.4.17)

Between epochs t₀ and t, the state vector is affected by the noise w. The weight attributed to the observation eqn (7.4-17) only depends on the variance-covariance of the system noise acting between the epochs t₀ and t : P_w = Q_ww^-1.

Least Squares Filter / Smoother: Statement

According to the Least Squares principle, a minimum value for several quadratic forms should be obtained. The correspondence of the filtered and smoothed estimate of x₀ requires that ^T P (or ^T P ) be a minimum, see eqns (7.4-15) or (7.4-16). On the other hand, for an optimal adjustment of the observations, a solution minimising ^TP is desirable. Finally, for an optimal fit of the change in the state vector with the kinematic model, it is necessary to keep _TPw as small as possible.

Generally, all three quadratic forms cannot attain their absolute minimum simultaneously, as each of the conditions sets other requirements on the adjustment. The optimal solution can be defined as requiring the overall minimisation of the quadratic forms:

(7.4.18)

From eqn (7.4-18), it appears that the solution is some form of average between initial conditions, observations and kinematic model. The relative magnitude of the elements in the weight matrices P, P and P_w is of paramount importance. If a filter is to give realistic estimates, the relative weighting of the components (a process sometimes referred to as "tuning") deserves particular attention. The response of a filter for some extreme cases of weighting can be described:

P = 0, no useable initial state vector. This implies that the state vector cannot be estimated until observations allow for a first determination. This case is of little practical importance since an approximate state vector is generally required for the linearisation of the measurement model. If the trace of P tends towards infinity, more observations will be required to get rid of any biases in the initial parameters. Obviously, the effect of P diminishes as observations are added over time. After a "settlement period", the behaviour of a filter is mainly influenced by P and P_w.
P = 0, a filter could operate just as well without any measurements, its function reduced to predicting future states according to the initial values and the assumed dynamics. If the trace of P tends towards infinity, the output depends solely on the observations and reflects the full range of their variations.
P_w = 0, renders the kinematic model useless and future states become unpredictable. If the trace of P_w tends towards infinity, the predicted estimate is not considered free of errors, as the term Q^T remains in Q. However, the filter cannot accommodate unforeseen deviations from the assumed dynamics and tries, no matter what the observations indicate, to bring the estimate back to the predefined trajectory.

There is an obvious analogy here with geodetic networks in which different types of observables are combined, for example directions and distances. To obtain small residuals on the distances, the analyst can attribute a large weight to them. However, the ratio of the residuals aposteriori / apriori may indicate that some observables have been outrageously favoured. The adjustment should then be repeated with rescaled weight matrices, until consistent ratios are obtained for the different types of observables. It must be emphasised that such a procedure is meaningful only if there is an adequate number of redundant observations. In any case, it is preferable to use weights based on experience, if available, rather than try to compensate apparently discrepant variances computed from very few redundant measurements.

Similarly, after some observations have been processed, an unbalance in the weighting of the measurement and kinematic models may become evident. It is possible to include within a filter algorithm, tests of the ratios of the aposteriori / apriori variances and modify the weights accordingly, noting the danger mentioned above. Filters having this feature are sometimes called "adaptive".

Least Squares Filter / Smoother: Solution

Compared to the Basic Least Squares Filter, a new observation is introduced: the observation of the new estimate of the initial state vector. In addition, the relation involving the kinematic model is modified. All these observations can be fully described by their functional and stochastic models and combined in a system of parametric equations, using eqns (7.4-15) and (7.4-17):

(7.4.19)

The matrices can be replaced by single elements, yielding the standard form of a system of parametric equations:

where the solution is, as usual:

Following back-substitution of

(7.4.20)

the solution is:

(7.4.21)

(7.4.7)

This last relation is the filtered estimate already obtained using the Basic Least Squares Filter. The formulae for the variance-covariance of the parameters and residuals derived for the Basic Least Squares Filter still hold, as does the extension to the Kalman form. The only interest in the above development is therefore the new estimate of the initial state vector in eqn (7.4-21). The expression for the smoothed estimate of x₀ and its variance-covariance can be written in a simpler form:

(7.4.22)

(7.4.23)

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