Hybrid Kalman / Minimax Filtering
Dan Simon
Innovatia Software
dan@innovatia.com
Copyright 1998–2007 Innovatia Software. All Rights Reserved.

Preliminaries

As discussed in a previous paper, Kalman filtering is an estimation method which minimizes the "average" estimation error. Minimax filtering minimizes the "worst-case" estimation error. But there are a couple of disadvantages to both methods of filtering.

§ The Kalman filter assumes that the dynamic system's noise properties are known exactly, but the minimax filter assumes that nothing is known about the noise properties. What if we have some incomplete knowledge of the noise statistics?

§ The Kalman filter minimizes the "average" estimation error, and the minimax filter minimizes the "worst-case" estimation error. What if we prefer to minimize some combination of these objective functions?

This paper presents an example and simulation results for a hybrid Kalman / minimax filter.

Mathematics

These questions gave rise to the concept of hybrid Kalman / minimax filtering. The concept is very heuristic but powerful. Suppose we have a dynamic system for which we want to estimate the state variables. Suppose we design a Kalman filter for the system and find the steady-state gain, denoted by K⁽²⁾. Suppose we then design a minimax filter for the same system and find the gain, denoted by K^{(1 )}. Then the hybrid filter gain is given by the following equation:
K = d K⁽²⁾ + (1-d) K^{(1 )}
where d Î [0, 1]. The parameter d is the relative weight given to H₂ performance. If d = 0, then the hybrid filter is equivalent to the minimax filter. If d = 1, then the hybrid filter is equivalent to the Kalman filter. There are two factors that must be taken into account in choosing the parameter d.

§ Although the Kalman filter may be stable and the minimax filter may be stable, a combination of the two filters may be unstable. So d must be chosen such that the resulting filter is stable.

§ The parameter d can be chosen based on the relative weight given by the filter designer to H₂ performance. This weight can be determined on the basis of the designer's confidence in the a priori noise statistics.

Example

The example considered in this paper is a phase-locked loop (PLL) for a Global Positioning System (GPS) receiver. A PLL tracks the phase of an incoming sinusoid. The phase information can then be translated into user velocity information.

This is a good example for the hybrid filter, because the noise statistics of the GPS signal is often known only approximately. Also, it is more important to minimize worst-case error rather than average error because if the phase estimation error of the PLL becomes too large, then the PLL "loses lock," the phase estimation error begins growing without bound, and all user velocity information is lost.

So suppose we have a GPS receiver with a phase discriminator that can somehow output phase information (modulo 2*pi). Our challenge is to design a filter that can give use absolute phase information such that we maintain phase lock in the noisiest possible conditions.

Our measurement is q (phase), and we will define the dynamic system such that our state variables are q (phase) and its first three derivatives. Then our system equations become

x_k+1 = Ax_k + w_k
z_k = Hx_k + v_k

where x_k+1 is the four-element state vector, w_k is the process noise, z_k is the measurement, and v_k is the measurement noise. The matrices A and H are given as

At this point our system is completely defined except for the process noise and the measurement noise. If we assume that the fourth derivative of the phase can be modelled as a white-noise process with a given variance, that gives us the process noise. The measurement noise depends on the fidelity of the phase discriminator.

Now if the two noise processes are exactly known, then the steady-state Kalman filter can be used to estimate the GPS carrier phase. If, however, nothing is known about the two noise processes, then the minimax filter can be used. But if the user has some inexact idea of the noise processes, then a hybrid filter can be used.

Simulation Results

The hybrid Kalman / Minimax PLL filter discussed above was simulated for a GPS receiver mounted on a ballistic missile test flight originating in California and ending in the South Pacific. The behavior of the filter was investigated by examining its ability to track the one GPS satellite carrier phase for the first 60 seconds of the flight. The filter ran at a rate of 50 Hz. It was assumed in this simulation that there was an unknown, constant phase bias (due to atmospheric disturbances) of 1 radian. It was further assumed that the measurement noise had a Laplacian (exponential) density, which has heavier "tails" than a Gaussian (normal) density. So the true measurement equation was
z_k = q _k + v_k + 1
but the filter was designed based on the incorrect measurement equation
z_k = q _k + v_k

What filter works best for this problem? The Kalman filter, the minimax filter, or the hybrid filter? I calculated the Kalman filter gain for the problem (which I will call K⁽²⁾) assuming a GPS carrier-to-noise ratio of 20 dB-Hz, and I also calculated the minimax filter gain for the problem (which I will call K^{(1 )}). Then I computed the maximum eigenvalue magnitude of the resulting estimator for values of d ranging from 0 to 1. The results are shown below.

Notice that the largest estimator eigenvalue exceeds 1 for values of d between about 0.01 and 0.31. This indicates that we must choose d outside of this range in order to have a stable hybrid filter. If we want the hybrid filter to be at least as stable as the pure minimax filter, then we need to choose d > 0.45.

Recall that the primary goal of our mission to maintain phase lock on the GPS carrier signal. With this goal in mind, the probability of loss of lock was determined via Monte Carlo simulations for various values of the parameter d. This probability is shown in the figure below for three different values of GPS Carrier-to-Noise Ratio (CNR). Note that a higher CNR means less measurement noise, and the Kalman filter gain was derived for CNR = 20.

The horizontal axis only goes down to d = 0.3 because the hybrid filter is unstable for smaller values of d. We can see from the above figure that the hybrid filter results in a marked improvement over the pure Kalman filter (d = 1) and the pure minimax filter (d = 0). Furthermore, the advantage becomes more pronounced as the measurement noise increases (i.e., as CNR decreases). For example, a pure Kalman filter with a CNR of 20 has a 20% chance of losing lock. But a hybrid filter with d around 0.4 or 0.5 only has a 2% chance of losing lock. This is a ten-fold improvement!

The hybrid Kalman / minimax filter takes advantage not only of the noise statistics knowledge that is inherent in the Kalman filter design, but also takes advantage of the robustness of minimax filtering. Hybrid filtering is an approach which gains the best of both worlds.

References

Introductions to Kalman filtering and minimax filtering are available on-line. Further details regarding the example given in this white paper are available in the following papers. The web site for my book about optimal filtering is at http://academic.csuohio.edu/simond/estimation, where you can download additional code and additional tutorials.

· D. Simon and H. El-Sherief, "Hybrid Kalman / Minimax Filtering in Phase-Locked Loops," Control Engineering Practice, October 1996.

· D. Simon and H. El-Sherief, "Hybrid H2/H-infinity Estimation for Phase-Locked Loop Filter Design," American Control Conference, 1994.

The following references provide a lot of good information about the Global Positioning System.

· P. Janiczek and S. Gilbert, "Global Positioning System Papers," The Institute of Navigation, Washington, DC.

· Hoffmann-Wellenhof, B. Lichtenegger, and J. Collins, "GPS: Theory and Practice," Springer-Verlag.

· E. Kaplan, "Understanding GPS: Principles and Applications," Artech House Publishers.

· B. Parkinson and J. Spilker, "Global Positioning System: Theory and Practice," American Institute of Aeronautics and Astronautics.

· D. Wells, "Guide to GPS positioning," Canadian GPS Associates, Fredericton, New Brunswick, Canada.

Some details about phase-locked loops can be found in the following collection of reprinted papers.

· W. Lindsey and C. Chie, "Phase-Locked Loops," IEEE Press, New York, NY.

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Last Revised: May 29, 2007