WEBVTT
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Today, we will be starting on a new subject
on intelligent control. The subject is on
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fuzzy control. This is module 4 and we will
be having the first lecture on this module
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on fuzzy control. Before we go in depth on
how to design fuzzy controllers, we will have
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a review today.
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The topics that we will be covering today
are fuzzy logic controllers ñ Mamdani type
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and TakagiñSugeno type, some important works
in Mamdani type FLC ñ fuzzy PD/PI/PID controller,
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fuzzy Lyapunov controller, parameter optimization,
and some important works in TakagiñSugeno
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type of FLC ñ fuzzy controller with common
input matrix, linear controller using robust
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control approach and fuzzy controller using
LMI techniques.
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Mainly two types of fuzzy logic based controllers
are available in literature. The first is
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the Mamdani type of fuzzy logic controller.
The Mamdani type of fuzzy logic controllers
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are direct adaptive type, where controllers
are designed directly based on the fuzzy rule
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base. Explicit system identification is not
done in this case; whereas TakagiñSugeno
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type fuzzy logic controllers are normally
indirect adaptive type fuzzy logic controllers,
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but the system to be controlled is identified
using T-S fuzzy model and the controller is
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designed based on the identified model.
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A Mamdani type of fuzzy logic controller would
look like this. You have the process, sensors
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and crisp-to-fuzzy interface fuzzification.
Process sensors means the process output are
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fed back through sensors. Whatever the feedback
is, it is actually a crisp value and so we
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have a fuzzification model that converts from
crisp variable to fuzzy variable. So you have
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fuzzification. Then, they have a fuzzy rule
base. Using the fuzzy linguistic variables
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that are used in the rule base and the present
status of the process in terms of linguistic
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variable, you have an inference mechanism
or rule evaluation, which actually tells us
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what should be the control action in fuzzy
linguistic variable.
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That control action is defuzzified to get
a crisp control action and is fed to the actuator
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back to the process. This is a Mamdani type
of fuzzy logic controller. Here, the heart
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of this controller is this fuzzy rule base.
Maximum research in Mamdani type of fuzzy
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logic controllers is regarding fuzzy rule
base ñ how we generate the rule base and
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how do we optimize the parameters of the rule
base. Its controller is simply expressed in
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terms of a fuzzy rule base.
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Some important works areÖ. As you know, Zedeh
is the founder of fuzzy logic concepts ñ
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A fuzzy-algorithmic approach to the definition
of complex or imprecise concepts, International
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Journal of Man-Machine Studies in 1976. Mamdani
is the pioneer in terms of proposing the fuzzy
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logic controller and that is why the direct
adaptive type of fuzzy logic controllers are
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Mamdani type ñ Application of Fuzzy Algorithms
for the Control of a Dynamic Plant, IEEE,
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volume 121, number 12, 1974.
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Kickert and Mamdani ñ Analysis of a fuzzy
logic controller, Fuzzy Sets and Systems,
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volume 1, 1978. CC Lee's Fuzzy Logic in Control
Systems is actually a survey paper ñ Fuzzy
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Logic Controller, parts I and II, IEEE Transactions
on Systems, Man, and Cybernetics, volume 20,
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number 2. This is in 1990 and I would recommend
all of you to study it. Of course, this paper
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deals basically with Mamdani type of controllers
ñ you will not get anything about T-S fuzzy
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model in this paper. Then, Mizumoto's Realization
of PID controls by fuzzy control methods,
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Fuzzy Sets and Systems, volume 70, 1995.
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As I said, research issues are alwaysÖ. Formation
of the rule base. How do we form various rules?
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The papers that would be worth noting or the
ones we should go into in detail are Mann,
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Bao-Gang Hu and Gosine ñ Analysis of direct
action fuzzy PID controller structures. This
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was published in 1999. Lopez and Martin's
A simplified version of Mamdani's fuzzy controller:
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the natural logic controller, IEEE Transactions
on Fuzzy Systems, volume 14, number 1, 2006.
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Margaliot and
Langhoiz's Fuzzy Lyapunov-based approach to
design fuzzy controllers is something that
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we will be focusing on in our future classes
ñ how to design a rule base using Lyapunov-based
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function or Lyapunov-based approach, Fuzzy
Sets and Systems, volume 106 in 1999.
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Parameter optimization is another research
issue, parameter optimization of fuzzy rule
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base membership functions ñ the parameters
that are contained in a fuzzy logic controller.
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Karr and Gentry ñ Fuzzy Control of pH, Using
Genetic Algorithms, IEEE Transactions on Fuzzy
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Systems, volume 1, number 1, 1993. Homaifar
and McCormick's Simultaneous Design of Membership
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Functions and Rule Sets for Fuzzy Controllers
Using Genetic Algorithms, IEEE Transactions
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on Fuzzy Systems, volume 3, number 2, 1995.
This is one of our own works ñ Sastry, Behera
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and Nagrath's Differential evolution based
fuzzy logic controller for nonlinear process
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control, Fundamenta Informaticae in 1999.
Here also, we use another technique called
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differential evolution to optimize the parameters
of the fuzzy logic controller and we have
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implemented to a pH reactor in real time.
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As I said, the rule base formation can be
done in three types. One is using the idea
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of PI/PD/PID controller response ñ the generic
idea that we have of how the normal response
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of a PI/PD/PID controller would look like.
Another analysis is the Fuzzy Lyapunov concept
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ñ Fuzzy Lyapunov controller concept. Here,
the rule base is formed using stability notion
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and the self-organizing rule base, where all
the parameters of the fuzzy logic controller
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are generated using the optimization concept.
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Now let us look at the fuzzy PI/PD/PID controller
and how it would behave when the normal rule
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base is formed. This shows the general characteristics
of a response of a system, given a step command.
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The system is at the origin and we give a
step command ñ unity step command 1. Then,
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we would like our system to behave like this.
What we would like to see is whether we can
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now guess our rules such that our system would
follow a behavior of this kind.
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Let us denote the error, change in error as
e and e dot and the control input as u respectively.
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Looking at the output response curve, the
following rules can be formed. If e is large
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error and e dot the change in error is small
- it can be small and it can also be medium;
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then u is large. If e, the error is medium
and the derivative of the error is medium,
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then control action is medium. This is a PD
type of controller. If error is small and
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change in error is large, then u is negative
small. This means my control action should
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be negative so that the overshoot is not there.
Fuzzy PI/PID controllerÖ.
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This is how the rules are generated for a
PD/PID type of controller. This is how the
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fuzzy PI/PID controller rules are generated
using the normal notion ñ normal notion of
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a response of the system to a unity step command.
The normal type structure of the controller
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v in case of a PID controller - I write the
control equation as: the present control action
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is the previous control action plus incremental
change in control action and this incremental
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change in control action is computed from
the rule base; whereas, if it is a fuzzy PD
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controller, the control input u is directly
computed from the rule base.
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In a general fuzzy logic controller, the control
objective is to design a fuzzy controller
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using information based on some physical intuition
even if the exact system dynamics are not
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known, but the main problem is constructing
the rule base for the controller. In general,
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Mamdani type of fuzzy logic control, the rule
base is obtained using the notion of classical
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PD, PI or PID controller, but in fuzzy Lyapunov
controller, the rule base is formed using
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the notion of Lyapunov stability.
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What is Lyapunov stability? The Lyapunov stability
is: A general single input single output nonlinear
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system x dot f x, u is Lyapunov stable around
the operating point x equal to 0, if there
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exists a continuously differentiable function
V x known as a Lyapunov function, such that
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the following requirements are met: V x is
positive definite in the neighborhood of the
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origin and V dot x, the rate derivative of
the Lyapunov function, is negative definite
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in the neighborhood of the origin.
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Fuzzy Lyapunov controller: assume that the
exact system model is unknown, but we have
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some partial knowledge about the system. Then
as in classical case, we consider a Lyapunov
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function candidate V, derive an expression
for its derivative, and then obtain the fuzzy
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rule base for the control input u so that
V dot is negative definite. Everything is
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qualitative; I will just show you how it is.
Based on the rule base, a fuzzy controller
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u is obtained using general inference mechanism
and defuzzification method.
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I will just explain to you now. You see that
we will take two different structures of the
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controller for rule base formation. The first
is representation 1. My FLC rule looks like
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this: If x1 is A1 and/or x2 is A2 and so on
and/or xn is An, then my control action is
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B, where Aiís and B are linguistic variables,
like large and small; whereas in the representation
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2, I say if x1 is A1 and/or x2 is A2 and so
on and/or xn is An, then u is a function of
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x1, x2 and xn, where f is a linear function.
I will just show you. What I am trying to
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tell you here is that I would like to generate
either of these two types of rules. How do
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I generate this type of rules? How do I define
a1 and a2 apriori? This can be done in a very
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simple manner.
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Let us think of a single link manipulator.
This is my motor, this is my link whose mass
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is given by capitalÖ this is actually not
capital but small m and the angle is actually
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theta
and this angle is actually theta. I can write
the dynamic equation of this is m l square
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theta double dot plus m g l sin theta equal
to tau. My states are theta and x2 is theta
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dot ñ this is my apriori knowledge. Another
apriori knowledge is I can say my tau is directly
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proportional to normal because of the second
law of Newtonian mechanics. My acceleration
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is directly proportional to the applied force
or the way you like to know, whatever it is.
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If I write like this, if I know this kind
of knowledge, this much knowledge is sufficient
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for me to generate the rule base. This is
the interesting thing ñ I do not have to
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know exact dynamics. I do not require this
knowledge: m l square theta double dot plus
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m g l sin theta equal to tau. I will show
you just now.
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Without knowing the complete dynamics of the
system, the following statements can be made
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about the single link manipulator. The relevant
state variables are x1 is equal to theta,
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x2 is equal to theta dot, and x2 dot which
is actually theta double dot, is proportional
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to tau. Now let us take a Lyapunov function
candidate V is half x1 square plus x2 square.
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The time derivative of V is V dot isÖ. This
is my Lyapunov function for the system I can
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always think. Then if I differentiate, V dot
is x1 x dot plus x2 x2 dot. So, x1 dot is
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x2. Here, x1 dot is x2 and I can say x2 dot
is proportional to tau; hence, we can write
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qualitatively V dot, the rate derivative is
x1 x2, and x2 and x2 dot is replaced by tau
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ñ approximately. This is a qualitative statement
and not a quantitative statement because,
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we do not say that x2 dot is exactly tau ñ
no, it is just proportional to tau. But looking
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at this expression, I can always sayÖ because
there can be some constant here ñ we do not
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worry about that.
Now, I am interested in defining a rule base
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and I am trying to do a qualitative analysis
ñ not a quantitative analysis. Had I been
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doing a quantitative analysis, I would have
liked to put some kind of constant here, but
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since I am doing a qualitative analysisÖ.
What is qualitative analysis? I am trying
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to develop a fuzzy rule base from this equation,
because, the objective is that my V dot ñ
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rate derivative of the Lyapunov function ñ
should be negative definite. How do I design
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a fuzzy controller rule base such that this
expression is qualitatively negative definite?
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That is the objective. What I should write
is find FLC rules such that V dot is qualitatively
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negative definite; this is the objective.
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V dot is x1 x2 plus x2 tau. This can be made
negative definite if the rule base is formed
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as follows. If I say x1 is negative and x2
is negative, this quantity is positive; if
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this is positive, this has to be negative,
so tau has to be negative because x2 is negative.
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x1 is negative and x2 is negative, making
this quantity positive. Hence, this quantity
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has to be negative and more than this ñ qualitatively.
To make it negative, since x2 is already negative,
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tau has to be positive, so tau is positive
big. This is one way the first rule is formed.
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The second rule is x1 is positive and x2 is
positive. That means this quantity is again
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positive and this quantity has to be negative
now, because, this is positive. To make it
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negative, x2 is already positive, so tau has
to be negative. Then tau is negative big.
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Similarly, if x1 is negative and x2 is positive,
then this quantity is negative, so V dot is
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required to be negative. Now if my control
action tau is 0, then also this is negative.
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tau is 0 ñ I can make tau as 0 because this
is already negative. Similarly, if x1 is positive
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and x2 is negative, then this is already negative.
Hence again, tau can be made 0 and my control
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action is 0.
You see how we are generating rules in such
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a way that the rate derivative of the Lyapunov
function V is a negative definite qualitatively.
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Once I formulate the rules, the next thing
I have do is that I have to find out the parameters
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and I have to optimize the parameters. Parameter
optimization is the next thing, but the most
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important is the rule base formation. This
process of rule base formation is very comprehensive
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and in fact, it is very interesting for fuzzy
logic controller. We will be having at least
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one class on this particular topic ñ how
to generate the rule base using Lyapunov concept.
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Now, parameter optimization: parameter optimization
is that, once I have formulated the rules,
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how do we fix the parameters or how do we
tune the parameters? My rules areÖ. For example,
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here, If x1 is negativeÖ. But when I say
x1 is negative, how do I define this x1 is
22:15.299 --> 22:22.299
negative? This is my x1and I have to fuzzify
x1. Negative can be like this, the negative
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also can be like this, this is one type and
this is another type. From this side, this
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is negative. How do I define this negative?
Where should I put it? For example, the membership
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function for negative here is1. In this particular
case, this is my membership function and maximum
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membership function 1 is here ñ for negative;
similarly, for x2 also. How do I the fuzzification
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of each variable such that my performance
is optimized?
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Normally, the first case is heuristically
updated. We take the help of heuristics ñ
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some kind of a trial and error method. Nowadays,
the normal practice has been genetic algorithms
23:45.020 --> 23:52.020
or evolutionary computation. What are the
GAs? The genetic algorithms perform parallel
23:52.340 --> 23:58.620
search to find out optimal parameters, where
each local search does either a hill climbing
23:58.620 --> 24:05.620
or a gradient search. In this lecture series,
we will show three types ofÖ. One is a simple
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genetic algorithm, another is univariate marginal
distribution algorithm and another is differential
24:23.070 --> 24:30.070
evolution. We will be covering all these three
algorithms to optimize the parameters of a
24:33.010 --> 24:40.010
fuzzy logic controller. Now, we go to the
next part of our topic, which is called TakagiñSugeno
24:45.309 --> 24:49.520
type of fuzzy logic controller.
24:49.520 --> 24:56.520
Here, we will outline various research issues
that are involved in designing TakagiñSugeno
25:01.600 --> 25:07.870
type. Actually, I would not say that this
is TakagiñSugeno type of fuzzy logic controller.
25:07.870 --> 25:14.870
We will say FLC using T-S fuzzy model. What
is the meaning of that? We want to design
25:21.980 --> 25:28.980
a fuzzy logic controller using T-S fuzzy model,
TakagiñSugeno Fuzzy model. That means any
25:29.649 --> 25:36.649
nonlinear system can be represented by or
approximated by T-S fuzzy model.
25:38.299 --> 25:44.080
In general, Takagi-Sugeno type of fuzzy logic
controller is an example of indirect adaptive
25:44.080 --> 25:49.809
control. The main steps are identifying the
nonlinear systems in terms of T-S fuzzy model
25:49.809 --> 25:56.809
and designing the controller based on the
identified T-S fuzzy model. We have a T-S
25:57.020 --> 26:04.020
Fuzzy model of the plant. I utilize the T-S
Fuzzy model to design my controller. In different
26:04.460 --> 26:11.460
control approaches using T-S Fuzzy model,
we will be discussing these three techniques.
26:12.559 --> 26:17.880
One is controller design with common input
matrix, then linear controllers using robust
26:17.880 --> 26:23.820
control approach and then controller design
using LMI techniques.
26:23.820 --> 26:30.820
All these three approaches are very much prevalent
in control research. I will just give you
26:34.090 --> 26:41.090
a hint of these different types of control
research or what are the different control
26:41.669 --> 26:48.669
problems these three types of systems would
define. What I am saying is thatÖ what is
26:49.279 --> 26:55.520
the meaning of common input matrix and using
that, controller design and linear controller
26:55.520 --> 26:59.240
using robust control; these are all the different
control research issues.
26:59.240 --> 27:06.240
These are the important works in the T-S fuzzy
model. First of all, Takagi and Sugeno proposed
27:07.240 --> 27:12.799
the T-S fuzzy model. That is in IEEE Transactions
on Systems, Man, and Cybernetics way back
27:12.799 --> 27:19.730
in 1985 ñ Fuzzy identification of systems
and its application to modeling and control.
27:19.730 --> 27:26.730
One of our works is among the other works
that are relevant in terms of fuzzy controller
27:29.480 --> 27:36.480
using T-S model ñ On Identification and Stabilization
of nonlinear plants using Fuzzy Neural Network,
27:40.960 --> 27:47.960
IEEE Conference on Systems, Man, and Cybernetics,
2005.That was the work of Zak. In this work,
27:51.820 --> 27:57.429
the second work, we are taking about common
input matrix.
27:57.429 --> 28:04.429
Zak's Stabilizing fuzzy system models using
linear controller appeared in Transactions
28:07.789 --> 28:14.789
on Fuzzy Systems in 1999. In this, you see
how to design linear controllers using the
28:17.120 --> 28:23.500
fuzzy T-S model, but in this case, Zak's approach,
the controller parameters, the controller
28:23.500 --> 28:30.500
gains are fixed. That means he has a fixed
gain controller. We have another paper by
28:32.169 --> 28:39.169
Prem, Indhrani and I ñ Variable Gain Controllers
for Nonlinear Systems using T-S Fuzzy model.
28:40.640 --> 28:47.640
That is in IEEE Transactions on Systems, Man,
and Cybernetics, part b in 2006. In this,
28:51.740 --> 28:58.740
we are also designing linear controllers,
but the gains are varying ñ variable gain
28:59.590 --> 29:00.269
controller.
29:00.269 --> 29:07.269
Tanaka proposed a notion of Stability and
Stabilizability of Fuzzy-Neural-Linear control
29:09.669 --> 29:16.669
systems using the linear matrix inequality
approach ñ LMI. That work appeared in IEEE
29:19.169 --> 29:25.610
Transactions on Fuzzy Systems, 1995. Lam,
Leung and Tam A Linear Matrix Inequality Approach
29:25.610 --> 29:30.870
for the Control of Uncertain Fuzzy Systems
appeared in IEEE Control Systems Magazine
29:30.870 --> 29:36.960
in 2002. You already know that, this of course,
is being solved using linear matrix inequality
29:36.960 --> 29:43.960
ñ LMI. Now, we will go a little deeper into
understanding these problems ñ what are the
29:45.380 --> 29:51.350
control problems and how the control problem
is formulated for each case.
29:51.350 --> 29:58.350
Let us consider a class of discrete nonlinear
dynamical systems described by x k plus 1
29:58.929 --> 30:05.929
is f x k u k and y k is h x k u k. You see
that this is the complete nonlinear plant
30:06.950 --> 30:13.950
and these are vector equations. x is an n-dimensional
state vector, u is a p-dimensional input vector,
30:19.380 --> 30:25.789
and y is an m-dimensional output vector. So,
f is a vector and h is a vector. The above
30:25.789 --> 30:31.399
system can be effectively modeled by fuzzy
merging of equivalent linear systems in different
30:31.399 --> 30:36.820
operating regions using the TakagiñSugeno
fuzzy model. What does it mean?
30:36.820 --> 30:43.539
A T-S fuzzy model is composed of r rules where
the j th rule has the following form. What
30:43.539 --> 30:50.539
you say is if x1 k is F1 j and so on and xn
k is Fn j, then my x k plus 1 is a linear
30:54.450 --> 31:01.450
system Aj x k plus Bj u k and y k is Cj x
k plus Dj u k, where x is the system's n states
31:04.730 --> 31:11.730
x1 to xn and the rule here jÖ. This is the
j th rule. So I have r rules. Given a current
31:16.570 --> 31:23.570
state vector x k and input vector u k, the
T-S fuzzy model infers x k plus 1 as x k plus
31:26.240 --> 31:33.240
1 is sigma j equal to 1 to r muj Aj x k plus
Bj u k upon sigma j equal to 1 r muj.
31:36.899 --> 31:43.899
What is this muj? muj is actually sigma muj
xi product i equal to 1 to n. muj is muj x1,
32:04.789 --> 32:11.789
muj x2 until muj because when I say x1 k is
F1 j, given the crisp value of x1, I get a
32:13.360 --> 32:19.899
specific membership function here. Similarly,
given a crisp value of x and k, I get a specific
32:19.899 --> 32:26.899
muj xn. The muj associated with the entire
rule is normally computed by the product principle;
32:32.350 --> 32:39.240
that is, I multiply each membership function
and find out what is muj.
32:39.240 --> 32:46.240
This is our muj, the product principle. y
k is similarly muj into associated with the
32:49.029 --> 32:56.029
rule j and my y is Cj x k plus Dj u k and
you sum over j equal to 1 to r, r rules and
33:02.480 --> 33:09.480
divided by sum j equal to 1 to r muj. So,
muj i xi is the membership function of the
33:10.620 --> 33:13.889
fuzzy term Fi j. I explained this.
33:13.889 --> 33:20.850
Pay a little attention here because, we will
be discussing more and more about this kind
33:20.850 --> 33:27.510
of structure; because, we will be going deeper
into the control system aspect in this particular
33:27.510 --> 33:34.510
model on fuzzy control. We had this rule and
we have r such rules.
33:36.000 --> 33:43.000
Given these r rules, now we can write the
overall fuzzy system, the fuzzy dynamics that
33:43.760 --> 33:50.760
the nonlinear system we described. What was
the nonlinear system? The nonlinear system
33:51.190 --> 33:58.190
was x k plus 1 is f x k u k and y k is h x
k u k. This is my nonlinear system and in
34:19.030 --> 34:24.080
the T-S fuzzy model, the nonlinear system
would look like this. You represented the
34:24.080 --> 34:31.080
same nonlinear system as x k plus 1 and here
instead of this nonlinearity, we represent
34:32.590 --> 34:38.380
in a very convenient format, which is A bar
x k plus B bar u k. It looks linear. It is
34:38.380 --> 34:44.870
not linear, but looks linear: A bar x k plus
B bar u k ñ very convenient notation, and
34:44.870 --> 34:51.870
y k is C bar x k plus D bar u k, where A bar
is equal to j equal to 1 to r sigmaj Aj, B
34:57.440 --> 35:04.440
bar is j equal to 1 to r sigmaj Bj, C bar
is sigmaj Cj and D bar is sigmaj Dj, where
35:08.290 --> 35:15.290
sigmaj is a normalized membership function
and it is muj over sigmaj equal to 1 to r
35:16.930 --> 35:18.750
muj.
35:18.750 --> 35:25.750
You must know that from j equal to 1 to r,
sigmaj is 1. This is always satisfied and
35:27.070 --> 35:34.070
always true. The overall system looks linear,
but it is not linear; this is nonlinear. Why?
35:35.590 --> 35:42.590
The overall system is nonlinear since A bar
is a function of sigmaj and this sigmaj is
35:45.260 --> 35:52.260
a function of x k because, the sigmaj defines
the fuzzy membership function of a state variable
35:56.900 --> 36:01.280
x k. Hence, the system is nonlinear.
36:01.280 --> 36:06.720
Continuous time T-S fuzzy model: The continuous
time counterpart of the overall fuzzy system
36:06.720 --> 36:13.720
isÖ. Just like we said this is discrete time,
the continuous time is also similar. x dot
36:16.520 --> 36:23.520
is A bar x plus B bar u, y is c bar x plus
D bar u, where again A bar is sigmaj Aj, j
36:28.310 --> 36:35.310
equal to 1 to r and B bar is sigmaj Bj, j
equal to 1 to r. C bar is sigmaj Cj, j equal
36:37.590 --> 36:44.590
to 1 to r and D bar is Ö. You should know
how you are finding this. We wrote down x
36:45.670 --> 36:52.670
dot is sigma i equal to 1 to r muj into Aj
x plus Bj u
divided by sigma i equal to 1 to r muj.
37:15.100 --> 37:22.100
When I divide this muj by this quantity and
represent by sigmaj, where sigmaj is muj by
37:27.090 --> 37:34.090
sigma i equal to 1 to r muj, then this representation
becomes x dot is simply sigma
j equal to 1 to r sigmaj Aj x plus Bj u. You
37:58.220 --> 38:05.220
can easily see now that if I write in terms
of x dot equal to A bar x, so sigmaj Aj j
38:05.850 --> 38:12.850
equal to 1 to r. Similarly, B bar is sigmaj
Bj j equal to 1 to r sigmaj Bj. Be very clear
38:24.510 --> 38:31.510
that once I say T-S fuzzy model, then my system
dynamics in continuous time looks like x dot
38:32.240 --> 38:37.710
equal to A bar x plus B bar u, which looks
very similar to linear system but they it
38:37.710 --> 38:43.810
is not linear because A bar is a function
of sigmaj and sigmaj is a function of x and
38:43.810 --> 38:49.930
hence, this is nonlinear. Similarly, we also
talked about discrete time system which also
38:49.930 --> 38:56.360
looks linear, but it is not linear. x k plus
1 is A bar x k plus B bar u k, where A bar
38:56.360 --> 39:03.250
and B bar are functions of x k because they
are function of sigmaj.
39:03.250 --> 39:10.230
Once we understood what the T-S fuzzy model
is, the next step is to derive the T-S fuzzy
39:10.230 --> 39:17.230
model. We can derive the T-S fuzzy model by
direct system identification or by linearization
39:18.610 --> 39:25.610
of an actual nonlinear plant. I will just
explain to you, how we linearize a plant.
39:26.650 --> 39:33.650
Given a nonlinear system, you see that x dot
is F x, u which is x plus x square plus u
39:36.030 --> 39:43.030
and I want to linearize this. We can use Taylor's
series expansion but I can only apply that
39:46.070 --> 39:51.470
when x is equal to 0 and u is equal to 0.
If I am trying to linearize around x is equal
39:51.470 --> 39:58.470
to 0 and u is equal to 0, then I can write
the expression as this approximation A x plus
40:01.380 --> 40:08.380
B u, where A is dow F upon dow x and B is
dow F upon dow u.
40:08.390 --> 40:15.390
If I am linearizing around x is equal to 0,
u is equal to 0, it means origin. But if x
40:15.420 --> 40:22.420
is not equal to 0, then you can follow the
book - Systems and Control by Zak; there are
40:25.960 --> 40:32.010
other books also, where there is a method
how to linearize a nonlinear system around
40:32.010 --> 40:39.010
some other points that are not the origin.
There is a formula here. Given x dot equal
40:40.200 --> 40:47.200
to f x plus g x u, we can find out the A matrix.
In that case, Ai transpose denotes the i th
40:51.310 --> 40:58.310
rule of A where Ai is computed by this formula
and B is g x0. There are various ways ñ you
41:00.920 --> 41:07.920
just have to learn how to linearize a nonlinear
system around various points. This is not
41:09.040 --> 41:16.040
a difficulty. You just try to understand that
we can linearize nonlinear systems around
41:17.270 --> 41:22.690
various operating points.
41:22.690 --> 41:27.900
Using two rules of T-S fuzzy model, we can
sayÖ. This is a scalar differential equation
41:27.900 --> 41:34.900
ñ x dot is x plus x square plus u. We can
write two rules for this. If x is equal to
41:41.720 --> 41:48.720
0, x dot is x plus u and if x is equal to
1, x dot is equal to 2 x plus u, where the
41:50.270 --> 41:57.270
corresponding matrices A1 is 1 and B1 is 1,
A2 is 2 and B2 is 1. These are all scalar
41:58.900 --> 42:05.260
values because, the differential equation
is scalar. I just demonstrated this for a
42:05.260 --> 42:10.470
scalar differential equation. You can also
do it for a vector differential equation.
42:10.470 --> 42:14.350
In that case, these will come.
42:14.350 --> 42:21.350
This is one approach. The other approach is
that we directly use fuzzy neural network.
42:25.530 --> 42:32.030
From the input-output data of the system using
a fuzzy neural network, we can also estimate
42:32.030 --> 42:39.030
these parameters very easily. Using gradient
descent algorithm or various kinds of algorithms,
42:39.790 --> 42:46.790
you can do that. What we try to do in this
case is that we represent a neural network
42:52.330 --> 42:59.330
where these rules are encoded in terms of
neural network parameters. Then, the neural
43:00.720 --> 43:07.720
network parameters are updated using input-output
data using the gradient descent rule.
43:14.240 --> 43:21.240
We are now very clear that we can write any
nonlinear system using either a discrete time
43:21.350 --> 43:27.970
T-S Fuzzy model or a continuous time T-S fuzzy
model. This is a discrete time T-S fuzzy model
43:27.970 --> 43:32.950
and this is my continuous time T-S fuzzy model,
where we have already defined what is A bar
43:32.950 --> 43:39.950
and what is B bar. Now I assume that the system
will have a common input matrix. When we say
43:42.070 --> 43:49.070
common input matrix, it means that for all
fuzzy zones, for every rule, the associated
43:56.380 --> 44:03.380
control matrixÖ because for each rule in
T-S fuzzy model, we have a linear system dynamics,
44:04.060 --> 44:09.280
that is, x dot is Ai x plus Bi u.
44:09.280 --> 44:16.280
If this control matrix Bi is the same for
all rules, then this is called common input
44:19.880 --> 44:26.880
matrix. This is what I say: Bj is equal to
B for all j, B is a constant matrix. In that
44:28.260 --> 44:33.020
case, this is called common input matrix.
What is the utility of this common input matrix?
44:33.020 --> 44:38.970
Suppose we design individual linear controllers
for individual subsystems, the control action
44:38.970 --> 44:45.970
corresponding to the j th subsystem is denoted
by uj k. What do we do? If I have a common
44:46.510 --> 44:53.510
input matrix, I compute what is uj for individual
subsystems such that the individual subsystem
44:57.860 --> 44:59.650
is stable.
44:59.650 --> 45:06.650
Once I compute uj, my control action u k,
which is a fuzzy blending of all control actions
45:06.990 --> 45:13.990
is sigma j uj k over j equal to 1 to r. This
ensures that individual subsystems are excited
45:14.700 --> 45:20.260
by their respective control inputs, which
is uj k. That means if I am giving to the
45:20.260 --> 45:27.260
actual plant uk, to the actual plant I am
actuating the control signal uk, it means
45:28.560 --> 45:34.950
apparently that each individual subsystem
if they are they are in reality they are being
45:34.950 --> 45:40.590
excited by the control action uj. This is
a very important notion. This is a theorem
45:40.590 --> 45:46.260
that we will prove in one of the coming classes.
45:46.260 --> 45:51.170
Making use of common input matrix, various
theorems have been presented to make the overall
45:51.170 --> 45:58.170
system stable; in fact, we have done extensive
work in this area. For example, suppose the
45:58.670 --> 46:04.470
individual control input has a form uj k is
minus Kj x k. For discrete time T-S fuzzy
46:04.470 --> 46:09.520
model, the overall system can be made stable
if there exists a common input matrix B for
46:09.520 --> 46:15.630
all subsystems and the individual gain matrices
Kjís are designed such that Aj dash is equal
46:15.630 --> 46:19.020
to Aj minus B Kjís have singular values less
than unity.
46:19.020 --> 46:26.020
If this is the case, then we can say my T-S
fuzzy model, my controller which is this one
46:29.060 --> 46:36.060
my controller where uj k is designed by this
formula where uj k is minus Kj x k and then
46:40.250 --> 46:47.250
my system is stable provided Aj minus B Kj
have singular values less than unity for each
46:48.260 --> 46:54.240
subsystem. How many subsystems do you have?
You have r rules and that means you have r
46:54.240 --> 47:00.900
subsystems. For r subsystems, each of these
quantities has maximum singular values and
47:00.900 --> 47:05.510
if less than unity, then the system is stable.
The proof and other things will be shown in
47:05.510 --> 47:08.690
the following classes.
47:08.690 --> 47:15.240
Similarly, another interesting result about
this common input matrix is - for continuous
47:15.240 --> 47:20.060
time T-S fuzzy model, the overall system can
be made stable, if there exists a common input
47:20.060 --> 47:26.970
matrix B for all subsystems and the individual
gain matrices Kjís are designed such thatÖ
47:26.970 --> 47:33.970
This is called Hermitian part of the matrices
Aj dash. If this term has stable Eigen values,
47:39.010 --> 47:46.010
where Aj dash is Aj minus B Kj; that is, if
I am designing a controller uj is minus Kj
47:49.990 --> 47:56.990
x and my overall controller is sigma j uj,
sigmaj equal to 1 to r; this is my overall
48:01.120 --> 48:08.120
controller for a continuous time system; then
the system is stable provided the Hermitian
48:10.320 --> 48:17.320
part of this Aj dash, which is this one ñ
the Hermitian part has stable Eigen values.
48:19.090 --> 48:26.090
You can look at Takagi and Sugeno's Fuzzy
identification of systems and its application
48:27.710 --> 48:33.000
to modeling and control and you get the idea
of T-S fuzzy model. For the common input matrix,
48:33.000 --> 48:37.560
you can get the idea from this paper ñ On
Identification and Stabilization of nonlinear
48:37.560 --> 48:44.560
plants using Fuzzy Neural Network, IEEE Conference
on Systems, Man, and Cybernetics in 2005.
48:44.910 --> 48:51.570
Now, we go to the linear controller using
robust control method. In this, the T-S fuzzy
48:51.570 --> 48:55.710
model is written in terms of single linear
plant and the rest of the linear models are
48:55.710 --> 49:01.560
expressed as a disturbance to this. The norm
bound on the disturbance is computed. Based
49:01.560 --> 49:06.770
on the norm bound, the controller is designed,
which makes the overall system Lyapunov stable.
49:06.770 --> 49:13.250
What does it mean? I have the i th rule as
I said earlier. If x1 t is F1 i and so on
49:13.250 --> 49:20.250
and xn t is fn i, then x dot t is Ai x t plus
Bi u t and mui is the associated membership
49:21.670 --> 49:28.670
function with the i th rule. Then, we saw
that the overall fuzzy dynamics x dot t is
49:29.980 --> 49:36.980
sigmaj Aj x t plus Bj u t, where sigma j is
defined like this.
49:39.520 --> 49:46.520
This overall fuzzy T-S model can be expanded,
x dot is as a nominal plant A x t plus B u
49:48.130 --> 49:55.130
t and these are we can say disturbance term
and this disturbance term can be again further
49:55.800 --> 50:02.800
categorized into three categories. This is
B h2 u t, B h1 x t and f x t and these two
50:05.900 --> 50:12.900
take care of this part and this one takes
care of this part and then we define norm
50:17.700 --> 50:22.670
bounds on these quantities. By defining the
norm bounds, we can design the controller
50:22.670 --> 50:25.430
around this nominal plant.
50:25.430 --> 50:31.030
There are various methods to the design controller,
which we will not discuss in detail. Computing
50:31.030 --> 50:35.920
the norm bounds of f, h1 and h2, controllers
are designed that makes the T-S fuzzy model
50:35.920 --> 50:42.920
Lyapunov stable. This is the problem formulation.
What is the problem? Given a T-S fuzzy model,
50:44.480 --> 50:51.480
express this T-S fuzzy model as x dot t around
a nominal plant and then disturbance. Then,
50:53.980 --> 51:00.980
using the robust control theory and Lyapunov
stability theorem, we can design the controller.
51:02.030 --> 51:08.390
For reference, we have a paper in IEEE Transactions
on Systems, Man, and Cybernetics, 2006. Zak
51:08.390 --> 51:14.240
also has a paper on this ñ Stabilizing Fuzzy
system models using linear controllers, IEEE
51:14.240 --> 51:19.350
Transactions on Fuzzy Systems, 1999. You can
refer these papers for more, and of course,
51:19.350 --> 51:26.350
we will be discussing this aspect in this
class later. We talked about common input
51:27.740 --> 51:32.460
matrix and we talked about robust control
theory ñ how to design controller using the
51:32.460 --> 51:35.910
T-S fuzzy model. Now, we will talk about LMI
technique.
51:35.910 --> 51:42.910
Again in LMI technique, as we said, the rule
is given; given rule i, we have a T-S fuzzy
51:43.570 --> 51:49.460
model. This is our T-S fuzzy model. You remember
this; because you should learn this model
51:49.460 --> 51:56.460
by heart. T-S fuzzy model means x dot t is
sigmai equal to 1 to r sigmai Ai x t plus
51:57.490 --> 52:03.290
Bi u t; you should learn this by heart because,
we have to be very clear. When we design a
52:03.290 --> 52:09.270
control system, the model should be very clear
to us ñ what it means. This is not a linear
52:09.270 --> 52:15.330
model; although it looks linear, this is a
nonlinear model. This approximates the system.
52:15.330 --> 52:22.160
The nonlinear system x dot is f x and u. This
is a nonlinear system. It approximates any
52:22.160 --> 52:29.160
nonlinear system. Now, given this T-S fuzzy
model, for each rule, for each subsystem,
52:37.120 --> 52:42.480
I compute a control action u t which is minus
Ki x t.
52:42.480 --> 52:49.480
If I do that, my overall control action u
t is given as minus sigmai Ki x t, i equal
52:49.940 --> 52:56.940
to 1 to r. The individual control action ui
was minus Ki x t for individual and the overall
53:00.570 --> 53:07.570
was this one. Then, the closed loop system
is Ai x t plus Bi into u. u is minus this
53:14.040 --> 53:21.040
quantity. So minus will come here and sigmai
Ki x t k, sorry, this is j, so sigmaj Kj x
53:25.730 --> 53:32.730
t, j equal to 1 to r. If you put this quantity
like this, after simplification, you get this
53:34.410 --> 53:41.410
quantity. x dot t is sigmai square Hii where
Hii is Ai minus Bi Ki and 2 sigmai sigmaj
53:48.040 --> 53:55.040
Hij plus Hji by 2 x t where Hij is Ai minus
Bi Kj. We can rewrite this expression in this
53:59.510 --> 54:02.020
form.
54:02.020 --> 54:09.020
Once we do that, we take a Lyapunov function
because, we have to analyze this particular
54:12.050 --> 54:19.050
x dot equal to we can write this term plus
this term into x t. I want to investigate
54:19.660 --> 54:24.370
the stability of this system. The best way
to investigate stability of the system, you
54:24.370 --> 54:31.370
take a Lyapunov function V equal to x transpose
Px. V dot is x dot transpose Px plus x transpose
54:31.420 --> 54:38.420
Px dot and x dot is given by this expression
V. The actual dynamics x dot t is given by
54:39.420 --> 54:40.280
this expression.
54:40.280 --> 54:47.280
If I replace x dot here, I get V dot finally
in this particular format. You see if I put
54:48.700 --> 54:55.700
that, I get sigmai square x transpose Hii
transpose P plus PHi x. Similarly, here I
54:58.650 --> 55:05.650
can write x transpose Hij plus Hji by 2 transpose
P, P Hij plus Hji by 2 x. What does it mean?
55:07.510 --> 55:13.490
You know that V dot has to be negative definite.
That means this quantity has to be negative
55:13.490 --> 55:18.180
definite and these quantities also have to
be negative definite.
55:18.180 --> 55:25.180
That gives us the condition that Hii transpose
P plus P Hii has to be negative definite.
55:26.690 --> 55:33.690
Similarly, this Hij plus Hji by 2 transpose
P plus P Hij plus Hji by 2 also has to be
55:35.140 --> 55:40.990
negative definite. This also can be negative
or equal to 0 because, we have already said
55:40.990 --> 55:47.010
that this is negative definite. The above
expressions are basic stability conditions
55:47.010 --> 55:53.920
and these are actually LMI equations ñ linear
matrix inequality. You see that this is a
55:53.920 --> 56:00.860
linear matrix inequality equation. The controller
parameter Kiís are hidden in these expressions.
56:00.860 --> 56:06.550
These can be further re-expressed in different
suitable forms and the controller parameters
56:06.550 --> 56:11.640
Kiís can be obtained by solving these expressions.
There are various methods ñ we will not be
56:11.640 --> 56:18.640
discussing now. I am just presenting how to
stabilize the fuzzy state feedback controller
56:23.150 --> 56:28.150
using linear matrix equality.
56:28.150 --> 56:35.150
In the beginning, we said the individual control
action uj is Kj x. Then we said u is sigmaj
56:39.090 --> 56:46.090
uj, j equal to 1 to r, where sigmaj is the
normalized membership function associated
56:48.730 --> 56:53.000
with rule j. This is my overall control action
u.
56:53.000 --> 57:00.000
If I give this overall control action, then
I said that my overall system dynamics for
57:02.010 --> 57:09.010
the closed loop, this is closed loop system
dynamics, becomes like this. That results
57:14.670 --> 57:21.670
in using Lyapunov stability theory to linear
matrix inequality equations. If I solve, then
57:23.730 --> 57:29.730
I properly find out, finally what should be
my kj. This is minus kj.
57:29.730 --> 57:35.820
Similarly, for a discrete time case, we can
follow the same method. The linear matrix
57:35.820 --> 57:42.320
inequality equations would look like this:
Hii transpose P Hii minus P less than 0, and
57:42.320 --> 57:49.320
this quantity is less than or equal to 0.
This gives you an idea of how the T-S fuzzy
57:51.080 --> 57:54.870
model based controller can be designed using
LMI.
57:54.870 --> 57:58.440
Some of the important works on this are by
Wang, Tanaka and Griffin ñ An Approach to
57:58.440 --> 58:03.840
Fuzzy Control of Nonlinear Systems: Stability
and Design Issues, IEEE Transactions on Fuzzy
58:03.840 --> 58:08.320
Systems, 1996; then Kim and Kim's Stability
Analysis and Synthesis for an Affine Fuzzy
58:08.320 --> 58:14.820
System via LMI and ILMI: Discrete case, IEEE
Transactions on Systems, Man, and Cybernetics,
58:14.820 --> 58:21.690
2001; and Lam, Leung and Tam's A Linear Matrix
Inequality Approach for the Control of Uncertain
58:21.690 --> 58:25.120
Fuzzy Systems, IEEE Control Systems Magazine,
2002.
58:25.120 --> 58:31.960
Other works were by Tanaka, Ikeda, Wang ñ
Fuzzy Regulators and Fuzzy Observers: Relaxed
58:31.960 --> 58:37.960
Stability Conditions and LMI-based Designs,
IEEE Transactions on Fuzzy Systems, May 1998;
58:37.960 --> 58:43.810
then Kim and Lee's New Approaches to Relaxed
Quadratic Stability Condition of Fuzzy Control
58:43.810 --> 58:50.810
Systems, IEEE Transactions on Fuzzy Systems,
October 2000; and Fang, Liu, Kau, Hong and
58:52.450 --> 58:56.400
Lee's A New LMI-based Approach to Relaxed
Quadratic Stabilization of T-S Fuzzy Control
58:56.400 --> 58:59.190
Systems, IEEE Transactions on Fuzzy Systems,
June 2006.
58:59.190 --> 59:06.190
Finally, we end this class by saying what
we did in this class; we had an overview of
59:08.290 --> 59:14.070
different fuzzy control systems. We said that
fuzzy rule base can be generated using the
59:14.070 --> 59:21.070
concept of PD/PI/PID type of response or using
the notion of Lyapunov stability concept;
59:21.490 --> 59:27.380
tuning of fuzzy controller parameters through
optimization using genetic algorithms, univariate
59:27.380 --> 59:33.450
marginal distribution algorithm or differential
evaluation ñ any kind of evolutionary computation
59:33.450 --> 59:40.450
approach we can use to optimize the FLC parameters.
When we express a nonlinear system using T-S
59:40.780 --> 59:47.780
fuzzy model, the controllers can be designed
in three different cases, when each subsystem
59:48.070 --> 59:53.730
has a common input matrix, or in a generic
case, we use the robust control theory to
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design the controller, or we can also use
linear matrix inequality approach to design
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the fuzzy controller. Thank you very much.
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