In the last video, we covered
an overview of a Fuzzy inference system, and the basic
mechanics of how it all works. In this video, I want to
continue that discussion by walking through a slightly
more complex example, because it's going
to allow us to talk about some important
concepts that we missed in the last video– namely, membership function
shapes, Fuzzy operators, multiple input
inference systems, and rule firing strength. So I hope you stick
around for it. I'm Brian, and welcome
to a MATLAB Tech Talk. For this video, we're going
to look at a Fuzzy inference system that determines
how much you should tip at a restaurant
given two inputs– food quality and
service quality.
Now, you might be thinking that
this is a rather silly example, but as we go through
it, keep in mind that the steps that we
take and the understanding we get from walking
through this problem will help you solve other
Fuzzy logic-based problems, whether you're designing
a Fuzzy controller or any other
decision-making process. OK, to walk through
this example, I created a simple app
in MATLAB to visualize each of the steps in
the inference process. And like we did before, we're
going to break this problem up into three distinct sections– fuzzification, inference,
and defuzzification. So let's start
with fuzzification. We have two crisp
inputs into the system– food quality and
service quality– and they range from 0 to 10. And we can think about it like
a feedback card at a restaurant.
How was your service? How was your food,
rated 0 to 10? And to fuzzify
these two values, we use the linguistic
variables– those are the human
interpretable adjectives– and their respective
membership functions. So in this example, food is
described as delicious, fine, and rancid, and the
membership functions for each are described by these
three trapezoidal functions. And remember– one way to get
membership function shapes like this is to poll
hundreds of restaurant-goers and ask them what
range of numbers they think represent
rancid food, and what range is fine food, and
what range is delicious food.
And since everyone will have a
slightly different definition of these terms, we're going
to get these overlapping membership functions where
a specific value could be a partial member
to multiple functions. Like here, food quality of 7
is 45% delicious and 55% fine. Service quality also has
membership functions. Here, I'm modeling
the three adjectives of poor, good, and excellent
with Gaussian functions. And choosing the shape of
the membership functions comes down to trying to
match the real distribution of the definitions
of these adjectives.
Here I'm just
claiming that maybe we polled a bunch of
people and just tried to fit the best curve to it. However, in many cases, the
shapes of these functions are determined through
experience and expertise of the problem. If we go over to the MATLAB
documentation for membership functions in the
fuzzy logic toolbox, you can see that there are
many different shapes that you can choose from to create the
membership function that you need. Again, for this
example, I'm using trapezoidal functions
for food quality, and Gaussian for
service quality. All right, so back
at our problem. We have our two crisp inputs. And if you recall
from the last video, we can go from a crisp
input to a fuzzy variable by determining the
degree of membership in these functions
for each input.
Like we said earlier, the
fuzzified food quality of 7 becomes 0.45, 0.55, and 0, and
the fuzzified service quality for, let's say, 8 becomes
0.38, 0.15, and 0. With these two
fuzzified variables, we can move on to inference,
or using logical statements to infer what the
output should be. In this case, we have
three logical statements. If service is poor or food
is rancid, then tip is low. If service is good,
then tip is average. And if service is excellent
or food is delicious, then tip is high. And these logical statements
are similar to what we did in the last video, but
with one major difference. We now have to deal with logical
operators, like and, not, and, in this case, or. So let's talk about
fuzzy logic operators. Fuzzy logic operators
are analogous to their binary counterparts. If we look at a binary or, it
has the following truth table– 0 or 0 become 0, since
neither of them are 1. And the other three input
combinations result in a 1, since at least one
of the inputs is a 1. You can see that
one way we could calculate the result
of a binary or is with the maximum function.
The output is always the
maximum of the two inputs. And the good thing about using
a maximum function like this is that it works when the
two input variables are not a binary 1 or 0, but
rather fuzzy variables that can range between 0 and 1. So as an example, if
we or the two fuzzy variables 0.2 and 0.8, then the
result is the maximum, or 0.8. Since at least one of these
inputs is relatively large, the output of the or is
also relatively large. And similarly, if we have
two low-degree memberships, like 0.2 and 0.1, then the
output will also be low. Now, we don't have an and or
a not operator in our problem, but let's talk
about them anyway.
Here is the truth
table for a binary and, and for a binary not. And we can calculate
them respectively with a minimum function,
and with 1 minus the input. Therefore, a fuzzy and just
becomes the minimum function of the two input variables,
and a fuzzy not is just 1 minus the input variable. Now, there are other ways
to define fuzzy operators than these three
that I've presented. Other ways include T-norms,
or triangular norms, and their counterparts,
T-conorms. On Wikipedia there's
some nice graphics that visually show what these
different norms are doing. For example, the minimum T-norm
produces this 3D contour. Notice that as long as
one of the inputs is 0, then the output is 0. And if both inputs are
1, then the output is 1– which is exactly the binary and. But with this
minimum function, you can see what the output would
be for any values for the two fuzzy inputs. Similarly, instead
of minimum, if we use the product
T-norm for a fuzzy and, then the 3D contour
would look like this. The four binary conditions of
0 0; 1, 0; 0, 1; and 1, 1 each produce the expected
output for a binary and, but in between,
the fuzzy variables would produce slightly
different values than the minimum function.
So as you can probably imagine,
there are many different ways to define each of
the fuzzy operators, and they'll each produce
slightly different results. However, for this
video, I'm going to stick with max, min, and 1
minus the input to represent or, and, and not. OK, back to our problem. Our three statements become
the following mathematical equations. Low is the maximum
of poor and rancid. Average equals good,
and high is the maximum of excellent and delicious. And for the inputs, food quality
of 7 and service quality of 8, we get low is 0, average
is 0.15, and high is 0.45. These values are the
rule firing strengths. That is, they represent how much
each particular rule impacts the final output. And this might not
be too impressive with just three
rules, but it becomes really important for
systems that have a couple dozen rules– or more– because viewing the
rule firing strengths can help you understand which rules
have the largest impact, which is helpful for troubleshooting,
or just understanding your system.
So if we go back to
the simple flowchart that we developed
in the first video, the rule base in
this inference step uses the rule firing strength
for fuzzy logical implication of the output variables. So fuzzification works
on the input variables, and then the rule base
implies the output variables. All right, so we have a
fuzzy variable indicating how much tip we should leave. That is, what degree of
low, average, and high should the tip be? And in this case,
it's 0.15 and 0.45. And to get this fuzzy variable
back into a crisp tip value– a very specific amount– we need to defuzzify it.
And once again, we need
the membership functions for the tip
adjective to do this. And to keep with my theme of
showing different membership function shapes, I choose
triangles for this one. A low tip ranges between 0%
and 15%, average is 5% to 25%, and high is 15% to 30%. Now we can use the
degree of membership in each one of these functions
to determine the tip percentage with defuzzification. And I said in the last video
that there are many ways to defuzzify a fuzzy number,
and in the MATLAB documentation, you can see a handful of methods
that the fuzzy logic toolbox software supports, including
the popular centroid method, which is what I'm going to
use for this tipping problem.
And let me quickly reiterate
what the centroid method is. We can fill each output
membership function based on the degree
of membership that is stated in the fuzzy variable. So we don't fill the low
membership function at all, since it has 0
degrees of membership. But we fill the average
function up 15% from the bottom, and then we fill
the high function 45% up from the bottom. We then combine these filled
areas into a single shape, and then the defuzzified
number is the x-coordinate of the centroid of the shape. Or in other words, if
you think of this area as a plate with uniform
thickness and density, the centroid is the
point along the x-axis about which the fuzzy set
would perfectly balance. And in this case, that
point is at 20.2%. Now, if I change the inputs into
this fuzzy inference system, you can start to get a feel for
how the input membership sets change, which then affect
the logical statements, which change the degree of membership
in the output sets, which change the centroid location,
changing the output value.
Look at this one where the
food is perfectly delicious, but the service quality
was abysmally poor. So these two
conditions average out, and the tip is exactly
the average amount of 15%. And notice that with
the centroid method, the absolute lowest
output possible is the centroid of the
lowest membership set, which, in this case, is 6.8%. And the highest possible output
is the centroid of the highest membership set, which is 23.2%. And this is important to note,
because if you were expecting the output to have a
range of 0% to 30%, then you would need to adjust these
membership functions to account for that by making them
a little bit wider. OK, so what we've
created here is a mapping between the two input
variables and the one output variable, and that mapping
is fixed and deterministic given these
particular membership functions and these rules. So theoretically, we
could systematically cycle through every single
input combination and determine what
the output is. For example, the inputs 0.1
and 0.1 produce a tip of 6.8%, so I can just put that
value in this 3D graph.
And then the next set
of inputs of 0.1 and 0.4 also produce a tip of 6.8%, and
I put that point in the graph. Moving on to 0.1 and 0.7
produces a tip of 7%. Now we can just continue
cycling through all of the possible combinations
of food quality and service quality across
their entire range, and produce this
set of points that describe the
input/output relationship for this inference system. So that's pretty cool, but
instead of this discreet set of points, we can see
that this can be defined by a continuous 3D surface. So given a food quality
and service quality input, we can use this
service to look up what the tip output should be. And rather than run the
whole fuzzy inference system real time on your
target processor over and over again each
sample time, we could replace this entire system
with a simple lookup table that is governed by this surface. It would essentially just
use the x and y-coordinates from the inputs to look
up the output tip value. And replacing a fuzzy
inference system with a lookup table like
this might be beneficial if you want to run this function
on a microcontroller that is computationally limited.
A lookup table requires
much fewer computations to execute than
fuzzifying the inputs, performing inference, and
then defuzzifying them, which is great. And as I will show you
shortly, this control surface can be generated quite
easily with the gensurf function in MATLAB. However, there is a
downside with lookup tables, because we lose the benefit of
the human interpretable rules that created this surface. And so even if you deploy a
lookup table to your processor, it's beneficial to maintain the
fuzzy inference system offline and use it as the source
for changes to the system.
For example, let me
show you how easy it is to update the rules
in our tipping function. Now, instead of using the
rather clunky app that I wrote, I recreated it with
a MATLAB script. The first part defines the input
and output membership functions in the exact same way
that we had them before. You can see that the service
functions are Gaussian, the food functions
are trapezoidal, and the tip functions
are triangular. In the inference
section, you can see that I've defined the
exact same three rules as well.
And now I can generate
that input/output mapping that we did before with
the gensurf function. And as you can see,
it looks like what we created by manually
cycling through the inputs. Now let's say that I want to
change this system by adding a fourth rule that says that if
food is fine, then tip is high. Basically, I'm saying that
the food is the most important thing, and if it's
fine, then I'm happy to tip a high amount– no matter what the
service was like. And if I add that rule
to my inference system and generate the
new surface, you can see that the
mapping has changed. The center portion
around the fine food area has been raised, just as
we would have expected. And it's this easy
and intuitive creation and manipulation of
fuzzy inference systems that make them so powerful.
Now, I know I said that
this example was silly, but we can use this to
do so many other things. For example, this is
a fuzzy controller that can balance a pole
on top of a moving cart. It's the classic inverted
pendulum problem, but this time the
parameters of the controller are generated in the form
of membership functions and fuzzy logic
rules, rather than PID or some other controller. And we're going to talk more
about this inverted pendulum problem, as well as other
practical fuzzy logic applications in the next video. So if you don't want to miss
that or any other future Tech Ta;k videos, don't forget to
Subscribe to this channel. Also, if you want to check
out my channel, Control System Lectures, I cover more control
theory topics there as well. Thanks for watching, and
I'll see you next time..
