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ECE5550: Applied Kalman Filtering
KALMAN FILTER GENERALIZATIONS
5.1: Maintaining symmetry of covariance matrices
■ The Kalman filter as described so far is theoretically correct, but has known vulnerabilities and limitations in practical implementations.
■ In this unit of notes, we consider the following issues:
1. Improving numeric robustness;
2. Sequential measurement processing and square-root filtering;
3. Dealing with auto- and cross-correlated sensor or process noise;
4. Extending the filter to prediction and smoothing;
5. Reduced-order filtering;
6. Using residue analysis to detect sensor faults.
Improving numeric robustness
■ Within the filter, the covariance matrices 
and 
must remain
1. Symmetric, and
2. Positive definite (all eigenvalues strictly positive).
■ It is possible for both conditions to be violated due to round-off errors in a computer implementation.
■ We wish to find ways to limit or eliminate these problems.
Dealing with loss of symmetry
■ The cause of covariance matrices becoming asymmetric or non-positive definite must be due to either the time-update or measurement-update equations of the filter.
■ Consider first the time-update equation:
• Because we are adding two positive-definite quantities together, the result must be positive definite.
• A “suitable implementation” of the products of the matrices will avoid loss of symmetry in the final result.
■ Consider next the measurement-update equation:
■ Theoretically, the result is positive definite, but due to the subtraction operation it is possible for round-off errors in an implementation to
result in a non-positive-definite solution.
■ The problem may be mitigated in part by computing instead
• This may be proven correct via
■ With a “suitable implementation” of the products in the 
term,
symmetry can be guaranteed. However, the subtraction may still give a non-positive definite result if there is round-off error.
■ A better solution is the Joseph form covariance update.
• This may be proven correct via
■ Because the subtraction occurs in the “squared” term, this form “guarantees” a positive definite result.
■ If we end up with a negative definite matrix (numerics), we can replace it by the nearest symmetric positive semidefinite matrix.
■ Omitting the details, the procedure is:
• Calculate singular-value decomposition: Σ = USVT .
• Compute H = VSVT .
• Replace Σ with (Σ + Σ T + H + HT )/4.
5.2: Sequential processing of measurements
■ There are still improvements that may be made. We can:
• Reduce the computational requirements of the Joseph form,
• Increase the precision of the numeric accuracy.
■ One of the computationally intensive operations in the Kalman filter is
the matrix inverse operation in 
■ Using matrix inversion via Gaussian elimination (the most
straightforward approach), is an O(m3) operation, where m is the dimension of the measurement vector.
■ If there is a single sensor, this matrix inverse becomes a scalar division, which is an O(1) operation.
■ Therefore, if we can break them measurements intom single-sensor measurements and update the Kalman filter that way, there is
opportunity for significant computational savings.
Sequentially processing independent measurements
■ We start by assuming that the sensor measurements are independent. That is, that
■ We will use colon “:” notation to refer to the measurement number. For example, z k :1 is the measurement from sensor 1 at time k.
■ Then, the measurement is
where Ck(T):1 is the first row of Ck (for example), and vk :1 is the sensor noise of the first sensor at time k, for example.
■ We will consider this a sequence of scalar measurements z k :1 ... z k:m , and update the state estimate and covariance estimates in m steps.
■ We initialize the measurement update process with 
and 
■ Consider the measurement update for the ith measurement, z k:i
= E[xk | Zk — 1 , z k :1 ... z k:i — 1] 十 Lk:i (z k:i — E[z k | Zk — 1 , z k :1 ... z k:i — 1])
■ Generalizing from before
■ Next, we recognize that the variance of the innovation corresponding to measurement z k:i is
■ The corresponding gain is 
and the updated state is
with covariance
■ The covariance update can be implemented as
■ An alternative update is the Joseph form,
■ The final measurement update gives 
and 
Sequentially processing correlated measurements
■ The above process must be modified to accommodate the situation where sensor noise is correlated among the measurements.
■ Assume that we can factor the matrix Σ = SvS , where Sv is a lower-triangular matrix (for symmetric positive-definite Σv, we can) .
• The factor Sv is a kind of a matrix square root, and will be important in a number of places in this course.
• It is known as the “Cholesky” factor of the original matrix.
• In MATLAB, Sv = chol(SigmaV,'lower');
• Be careful: MATLAB’s default answer (without specifying “lower”) is an upper-triangular matrix, which is not what we’re after.
■ The Cholesky factor has strictly positive elements on its diagonal (positive eigenvalues), so is guaranteed to be invertible.
■ Consider a modification to the output equation of a system having correlated measurements
• Note that we will use the “bar” decoration (-) frequently in this chapter of notes.
• It rarely (if ever) indicates the mean of that quantity.
• Rather, it refers to a definition having similar meaning to the original symbol.
• For example, z-k is a (computed) output value, similar in interpretation to the measured output value zk.
■ Consider now the covariance of the modified noise input 
■ Therefore, we have identified a transformation that de-correlates (and normalizes) measurement noise.
■ Using this revised output equation, we use the prior method.
■ We start the measurement update process with 
and 
■ The Kalman gain is 
and the updated state is
with covariance
(which may also be computed with a Joseph form update, for example).
■ The final measurement update gives 
and 
LDL updates for correlated measurements
■ An alternative to the Cholesky decomposition for factoring the covariance matrix is the LDL decomposition
where Lv is lower-triangular and Dv is diagonal (with positive entries).
■ In MATLAB, [L,D] = ldl(SigmaV);
■ The Cholesky decomposition is related to the LDL decomposition via
■ Both texts show how to use the LDL decomposition to perform a sequential measurement update.
■ A computational advantage of LDL over Cholesky is that no square-root operations need betaken. (We can avoid finding Dv(1)/2.)
■ A pedagogical advantage of introducing the Colesky decomposition is that we use it later on.