Fuzzy Filtering

Fuzzy Filtering
Dan Simon
Innovatia Software
dan@innovatia.com
Copyright 1998–2007 Innovatia Software. All Rights Reserved.

Preliminaries

Previous white papers by Innovatia Software have focused on the topics of Kalman filtering, H-infinity filtering, and hybrid Kalman / H-infinity filtering. This paper discusses fuzzy logic, a more heuristic approach to filtering. The application is fuzzy filtering for phase-locked loops (PLLs), although the approach is general enough to be applied to most estimation problems. See the H-infinity filtering and hybrid Kalman / H-infinity filtering papers for some basic information about PLLs.

Fuzzy Logic

Fuzzy logic works the way that humans think as opposed to the way that computers typically work. For example, consider the task of driving a car. You notice that the stoplight ahead is red and the car ahead is braking. Your mind might go through the thought process, "I see that I need to stop. The roads are wet because it's raining and there is a car only a short distance in front of me. Therefore I need to apply a significant pressure on the brake pedal." This is all subconscious (in general), but that's the way we think - in fuzzy terms. Do our brains compute the precise distance to the car ahead of us and the exact coefficient of friction between our tires and the road, and then use a Kalman filter to derive the optimal pressure which should be applied to the brakes? Of course not. We use common-sense rules and they seem to work pretty well. On the other hand, when we do finally get around to pressing the brake pedal there is some exact force that we apply, say 1.326 pounds. So although we think in fuzzy, noncrisp ways, our final actions are crisp. The process of translating the results of fuzzy reasoning to a crisp, nonfuzzy action is called defuzzification.

A Fuzzy Phase-Locked Loop

Now think about how a fuzzy phase-locked loop (PLL) might work. A PLL estimates the phase of a time-varying sinusoidal signal which is corrupted by noise. In addition to the signal being corrupted by noise, the PLL's phase measurement is noisy. The PLL's task is to find the best estimate of the true phase in spite of the noise.

As the PLL estimates the phase, it also predicts what the next phase measurement is going to be (based on its past estimates). We will call the difference between the actual phase measurement and the PLL's prediction of the measurement the error. The change in error is the change in this quantity from one time period to the next. So we can see that if the error is positive (i.e., the measured phase is larger than I expected it to be based on past estimates), then I probably need to increase my phase estimate or I may lose lock on the phase. If both the error and the change in error is positive, then I probably need to increase my estimate by a large amount because there may not be much time before I lose lock. These specifications are called fuzzy rules. Below is a sample table of fuzzy rules that might be used for PLL filtering. The inputs are Error and Change in Error, and the possible outputs are in green and represent how much to change the phase estimate.

CHANGE IN ERROR	ERROR
CHANGE IN ERROR	Negative	Zero	Positive
Negative	Large Neg.	Negative	Zero
Zero	Negative	Zero	Positive
Positive	Zero	Positive	Large Pos.

The engineering challenge is to define the fuzzy quantities. What does "zero" mean in fuzzy terms? What does "large negative" mean? Our definition of these fuzzy terms comprise what are called fuzzy membership functions. We may define a phase error of 1 radian as 30% positive and 70% large positive; we may define a phase error of 1 degree as 90% zero and 10% positive. One nice feature of fuzzy logic is that the membership functions can be refined; they can be optimized. The fuzzy system can be trained by tweaking the membership functions so that the system performs optimally. This feature will come in handy later on in this example.

Defuzzification

So suppose we have a phase error of -15 degrees and a change in error of -10 degrees. How do we actually figure out how to change our phase estimate based on the rules in the table above? Well, suppose that a phase error of "negative" is represented by the function on the left in the figure below. That is, a phase error is considered negative only if it is between -25 and 0 degrees. A phase error's level of membership increases from 0 to 1 as the error increases from -25 to -12.5 degrees. The error's level of membership decreases from 1 to 0 as the error increases from -12.5 to 0 degrees. So our phase error of -15 degrees belongs to the negative class with a level of membership equal to 0.8. Similarly, from the function in the center in the figure below, we see that our change in error of -10 degrees belongs to the zero class with a level of membership equal to 0.5. Our fuzzy rule from the table above says that if our phase error is negative and our change in error is zero, then our output should be negative. The fuzzy output corresponding to this rule is represented by the function on the right in the figure below. We want a negative output, but our output should only have a level of membership of 0.5 in the negative class. This is because 0.5 is the minimum contributor to this rule.

Defuzzification

But that's not all. Looking at the function on the left in the figure below, we see that our phase error of -15 degrees also belongs to the zero class with a level of membership equal to 0.2. As before, from the function in the center in the figure below, our change in error of -10 degrees belongs to the zero class with a level of membership equal to 0.5. Another fuzzy rule from the table says that if our phase error is zero and our change in error is zero, then our output should be zero. The fuzzy output corresponding to this rule is represented by the function on the right in the figure below. So we want a zero output, but our output should only have a level of membership of 0.2 in the zero class. Again, this is because 0.2 is the minimum contributor to this rule.

Defuzzification

So now we have two fuzzy outputs; a negative output with a 0.5 level of membership, and zero output with a 0.2 level of membership. Next we perform defuzzification to convert our fuzzy outputs to a single number. There are many ways of defuzzifying; one way is by simply computing the centroid of the fuzzy outputs. This process is illustrated in the figure below. In practice, there may be many fuzzy outputs to defuzzify. To obtain a numeric output we can take the center of gravity of all of the fuzzy outputs.

Center of Gravity

Conclusion

Innovatia Software's white paper Gradient Descent Optimization of Fuzzy Functions shows how the fuzzy rules of a PLL filter can be trained to optimize filter performance. The performance of the optimal fuzzy filter is compared with the nominal, heuristic, non-optimized filter discussed in this paper.

References

Fuzzy logic is a subject on which many excellent books and papers have been written. A couple of texts that I've found particularly useful are listed below. In particular, I gained most of my early understanding of fuzzy logic from Bart Kosko's book. I was pretty disappointed with the first half of his book, which deals with neural networks. But the second half, which discusses fuzzy logic, is excellent.

· B. Kosko, "Neural Networks and Fuzzy Systems," Prentice Hall, 1992.

· G. Klir and B. Yuan, "Fuzzy Sets and Fuzzy Logic," Prentice Hall, 1995.

Kosko has also written a popular and entertaining book on the topic which deals with fuzzy logic from a cultural and social point of view.

· B. Kosko, "Fuzzy Thinking," Hyperion, 1993.

The Institute of Electrical and Electronics Engineers (IEEE) publishes a quarterly journal called IEEE Transactions on Fuzzy Systems, which contains cutting-edge research on the topic in each issue.

There are also a lot of internet resources about fuzzy logic. Some excellent web pages on the subject are the following.

In addition, I have made some MATLAB code available for the reduction of fuzzy rule bases. Fuzzy rule base reduction may be important in those cases where a fuzzy system needs to be implemented in real time. Click here for a short description of the algorithm and instructions on how to download the MATLAB code.

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Last Revised: May 29, 2007