Fuzzy Lyapunov controller computing with words, this is the lecture 4 of this module 4 on
fuzzy control. As I said in the last class that given the mamdani type of fuzzy logic
controller, the membership functions are optimized using genetic algorithms.
But the most important aspect of mamdani type of controller is to generate a proper rule
base. Here, of course you can also generate rule base using genetic algorithm but today,
we will be learning how to generate a rule base using the concept of Lyapunov function.
So this is the topic today, fuzzy Lyapunov controller computing with words.
Topics to be covered today are classical Lyapunov synthesis; Fuzzy Lyapunov synthesis; rotational
proof mass actuator system, this is the physical system on which you will be testing our algorithm;
construction of rule base for RTAC, the short form is RTAC here, RTAC using fuzzy Lyapunov
synthesis: regulation problem; controller design for regulation; construction of rule
base for RTAC using fuzzy Lyapunov synthesis: tracking problem; controller design for tracking;
and finally, simulation results.
Fuzzy Lyapunov synthesis: an introduction- the classical Lyapunov synthesis method is
extended to the domain of computing with words. This new approach is used to design fuzzy
controller. Assuming minimal knowledge about the plant be controlled, the proposed method
enables us to systematically derive the fuzzy rules that constitute the rule base of the
controller. The objective is in this method, how do we obtain fuzzy rule base given the
system, some prior in knowledge of the system or approximate model of the system, how do
we generate a rule base? This approach is demonstrated by designing
Mamdani-type and Takagi-Sugeno-type fuzzy controllers for nonlinear plants.
The most difficult aspect in the design of fuzzy controllers is the construction of the
rule base. Fuzzy Lyapunov synthesis is based on extending the classical Lyapunov synthesis
method to design of fuzzy controllers. It enables to systematically derive the fuzzy
rule base. Based on classical Lyapunov synthesis method, we construct a Lyapunov function candidate
V and then determine the conditions required to make it a Lyapunov function of the closed
loop system. Since we assume fuzzy knowledge about the plant to be controlled, the derived
conditions can be stated as fuzzy if then rules.
So now let us go to the system description. Consider the following single-input, single-output
system. x dot equal to f x, u, y is h x, u where f and h are nonlinear functions. This
is n dimensional state space, so f is a n dimensional vector; and u and y they are the
input and outputs of the system respectively; x n dimensional system x1, this is x1 x2 until
xn, Rn is the state vector of the system. The control objective is to stabilize the
system around some operating point x0. Without loss of generality we may assume x0 equal
to 0.
So Lyapunov stability theory which we have been talking again and again, I am just revisiting
that, the system x dot equal to f x, u is Lyapunov stable around the operating point
x equal to 0. If there exists a continuously differentiable function V x which is known
as Lyapunov function such that the following requirements are met. V x is positive definite
in the neighborhood of 0 and v dot x is negative definite in the neighborhood of 0. If we can
ensure that then the system is Lyapunov stable.
So fuzzy Lyapunov synthesis- assume that the exact system model is unknown but we do have
some partial fuzzy knowledge about the system. As in classical case we consider a Lyapunov
function candidate V, derive an expression for its derivative and then obtain the fuzzy
rule base for the control input u, so that v dot is negative definite. Based on the rule
base, a fuzzy controller u is obtained using general inference mechanism and defuzzification
method.
What is the objective here is that we prescribe a Lyapunov function for the system and then
we construct, find the right derivative of the Lyapunov function, substitute the dynamics
of the system in that and now we create a rule base in such a way this Lyapunov function
is qualitatively negative definite, not quantitatively. So what is fuzzy Lyapunov synthesis is we
are using a qualitative negative definite of v dot. This is the difference qualitative negative
definiteness of v dot, instead of a quantitative negative.
The actual Lyapunov function approach to test the stability of the system is to find quantitavely
v dot is negative definite. But now we are talking about a concept qualitatively, if
we can make v dot to be negative definite.
FLC structures of the rule base, so normal we have two kinds of we can formulate any
fuzzy logic controller of mamdani type. We can have two categories of fuzzy rule base.
The representation 1 is if x1 is A1 and or x2 is A2 and or xn is An then u is B where
Ai’s and B are linguistic variables large or small. So x1 x2 and xn they are either
state variables or error in state variables.
Representation 2 is if x1 is A1 and or x2 is A2 and so on or xn is An then u is a definite
function here the B, u is B are the fuzzy linguistic variable where in the second representation
u is an exact function. u is f(x1 x2 … xn) where f is a linear function; this can be
also nonlinear function, f can be also linear function here, normally is a linear function.
So fuzzy Lyapunov synthesis: motivation- what is the motivation here? In a general fuzzy
logic controller, the control problem is to design a fuzzy controller using information
based on some physical intuition even if the exact system dynamics is not known.
But the main problem is constructing the rule base for the controller. Rule base is obtained
using the notion of classical PD or PID controller in general for mamdani type of fuzzy logic
controller. But in a fuzzy Lyapunov synthesis a rule base is obtained using the notion of
Lyapunov stability. So this is the prime motivation. I can tell you how normally we do a fuzzy
rule base, if this is my general characteristic of a second order system. So what I would
like, when I describe this response, this is my t, this is my y, my output. So you see
that if I am here at this zone, so at this zone I can say error, error at this zone is
very large.
I would like when I am here I should come very fast to this line, which is my desired
command signal 1.0. What I will say, when my error is small and the derivative of error,
the rate of change in error is either small or large whatever it is, when it is small
I would like it to come fast. My controller must be very large. I can write for this kind
of PD or PID type of controller the rule base is if e is small in this zone and e dot is
small or large then u is large. By simple commonsense I can write.
Similarly, if I say e is small here and if e is small in this zone and e dot is also
small then u is small because I have already reached almost I am reaching the set point.
But if e is small and e dot is large, so obviously I must reduce the control action, so that
is, in that case I have to say u is negative small, instead of positive small. Like that
the normal rule base is done in case of PD or PID type of controller.
But today we will see a very systematic method of generating rule base. Actually this entire
class today is to let you know or we will learn how to generate this rule base systematically
using Lyapunov stability theory. Before that, I will just explain to you one physical system
on which will be applying this thing, this is called rotational translational proof mass
actuator.
You can actually see this is my rigid surface to which a cart is bound using a spring. Inside
the cart there is the pendulum which rotates by the action of the torque and the pendulum
link length is l and there is an external disturbance f that is applied on the cart.
See the objective is to control this pendulum the way we like.
I may like to keep this pendulum at an angle theta or vertically downward vertically upward,
whatever is the desired thing or it has to follow certain trajectory, so whatever may
be the requirement this is a typical control system because it is nonlinear and also it
a bench mark problem.
The RTAC system, consider the following example rotation translational proof mass actuator
RTAC. The RTAC system combines a translational oscillator with a rotational proof mass actuator.
This particular portion now oscillates by the action of the spring. So this called combines
a transitional oscillator. This oscillates in the x direction with a rotational proof
mass actuator. This is rotation. The oscillator consists of a cart connected to fixed wall
by a linear spring. The cart is constrained to move in the x-direction only. The proof
mass actuator is attached to the cart and is controlled by an applied torque N. F is
disturbance force that perturbs the cart. F is a disturbance force that gives such a
disturbance input.
The cart position and actual angle is xc and theta respectively, that is the cart which
you saw here this is xc and this theta already which have already shown. Mass of the cart
is given 1.36 Kg; proof mass is 0.096 that is the mass of this cart is M and proof mass
is small m. So small m is 0.096 Kg; distance of the link length of the pendulum is 0.059
meter; Moment of inertia of the proof mass is I is 0.00022 Kg meter square; Spring constant
k is given 186.3 Newton per meter.
Now we define some normalized variables which is zeta is xc root over M plus m upon I plus
mL square; tau is t k upon M plus m; and u is this is my actual input and M plus m upon
k into 1 minus mL square; K is the spring constant; and omega is 1 upon k M plus m by
I plus mL square F.
Given that, you see that we can write, we can derive the dynamics of the RTAC system,
you can easily see that it has four states. x1 is zeta is actually represented xc normalized;
this is xc dot; the velocity of the cart normalized which is x2; and theta is the pendulum angle
x3; theta dot is the angular velocity of the pendulum which is x4; and control variable
is u which is also normalized; n the torque; and disturbance is w, so this one w is the
disturbance.
If we look at x dot this is the fourth order system and the expressions are x dot equal
to x2, the advantage is that we have not kept any parameters associated, this is simplified
x2, you can see x dot is in this, this I can say, this is simply a function of x, this
is another function of x may be g x, and this is d x.
You can say this is a function vector with four dimensions, with function vector with
another four dimensions and another function vector with four dimensions. You can see that
system is highly nonlinear if silent is defined as mL upon root over I plus mL square into
M plus m, you can easily derive this equation, I will not discuss further on this. All that
I am trying to tell you is that RTAC system has a nonlinear dynamics which has type x
dot is f x plus g x plus g x into u plus d x into w, where w is the external disturbance;
u is actual control input.
The fuzzy Lyapunov synthesis of RTAC- the system dynamics now can be written as, as
we said x dot is f x plus g x u plus d x w. Without knowing f x, g x and d x following
linguistic statements can be formulated. Even if I do not know what is f x g x d x exactly,
I can still write the state of the system is described by x1 the cart’s position;
x2 equal to x1 dot; and x3 is actuator’s angle; and of course x4 is x3 dot. That is
x3 is theta, x4 is theta dot, x1 is xc normalized, x1 is xc normalized and x2 is x1 dot. This
is another important thing that I can easily tell you, x4 dot is proportional to the control
input u that you can easily see here.
Because you see that this is the my pendulum and if I assume that there is no friction
and other things here, then my pendulum, the acceleration of this pendulum, angular acceleration
obviously is directly proportional to the torque applied N and since my input u is normalized
value of N and hence I can always say that x4 dot, the acceleration of the pendulum is
proportional to the control input u. Similarly, x2 dot the velocity of the cart
is proportional to the negative of x1, you can easily see that because the spring is
connected so it is like this, spring and this is the block mass. You can see that x2 dot
the velocity is always, because of spring action, it will be always in this direction
the velocity. If I move x in this direction, x2 the position in this direction, the velocity
is in the negative direction. So x2 dot in this is proportional to minus x1.
The control objective is to find u such that the system is stable around x equal to 0.
This is the regulation problem. Let V is half x1 square plus x2 square plus x3 square plus
x4 square be the Lyapunov function candidate. Then using S-1, the assumption that we have
four states V dot is x1, x1 dot and x1 dot is x2. Similarly, this is 2x2 x2 dot, so x2
x2 dot plus x3 x3 dot and x3 dot is x4 and similarly, this is x4 into x4 dot. Before
this is the second step, the first step V dot if you look at here is x1 x1 dot plus
x2 x2 dot plus x3 x3 dot plus x4 x4 dot, because we know already that x2 is x1 dot and x4 is
x3 dot, so this second step comes.
Now you see that, we have already said x2 dot is proportional to minus x1. Obviously,
qualitatively if I put x2 dot is minus x1, these two terms qualitatively cancels off,
where as only these two terms remain, x3 x4 is remained and I know already that x4 dot
is proportional to u, so instead of x4 dot I put u.
You see that, what is the advantage here is that I am designing a rule base but while
designing rule base I am not utilizing the system dynamics. I am finding how to what
V dot is equal to x3 x4 plus x4u. While finding this equation, I am not using the system dynamics.
This is very important, system dynamics is not used.
What is used is the simple physics of the system is that, here we use x2 equal to x1
dot obviously the velocity x2 and x1 is positioned and similarly, here we use the notion that
the x4 is x3 dot, so x4 is theta dot and x3 is theta, which is very simple and also we
used other rule from the systemic connections.
We know already that the acceleration of the pendulum is proportional to the input and
x2 dot is proportional to minus x1, just by constructional feature. Using these two rules,
we found V dot to be qualitatively x3 x4 plus x4 into u. Now, even if you do not know the
system dynamics, looking at this figure we can create a rule base for which this V dot
can be qualitatively negative definite.
So classical Lyapunov synthesis have design of view that will guarantee V dot is negative
definite. Same idea is applicable here but with the domain of computing with words means,
this is qualitatively not quantitative. What we are trying to do is we are applying Lyapunov
synthesis to the system dynamics partially known system dynamics for generation of rule
base.
Now we take the example, when I say so V derived V dot equal to approximately qualititatively
x3 x4 plus x4 into u. This can be made negative definite if the rule base is formed as follows.
You see that I will take care that you understand is because this is what you should learn how
do generate a rule base, you see x3 is negative. If x3 is negative and x4 is also negative,
that means this quantity is positive. Now x4 is negative and if u would be come negative
then this whole thing is positive, so no way you can make the system is stable. Obviously,
the rule base should be u to be positive, that is the rule.
I repeat if x3 is negative and x4 is negative, in that case x3 x4 is positive, this implies
x4 into u should be negative to make these two quantities to be qualitatively negative.
So x4 u should be negative, x4 is already known as negative, so u has to be positive.
This is first rule. If x3 is negative and x4 is negative then u is positive. Similarly,
if x3 is positive and x4 is positive. x3 is positive and x4 is positive means again x3
and x4 is positive in this case. And this implies again x4 and u should be negative
and since x4 is positive so u has to be negative which is negative. The second rule is if x3
is positive and x4 is positive then u is negative. I hope you are able to follow what I am presenting
you.
Now let us see the other possible combination. If x3 is negative and x4 is positive, in this
case x3 x4 is negative. If x3 x4 is negative I must ensure that x4 u, since it already
negative this should be 0 is sufficient, if I make x4 is 0 because x3 x4 is negative is
very good. This is simple because there are other ways also is x3 x4 is negative I can
say x4 u is also negative much better. But this is simplest one x4.
x3 is negative and x4 is positive then u is 0, you can also think another possibility,
because x4 is positive u is negative, this is also possible. Similarly if x3 is positive
and x4 is negative then again x3 x4 in this case x3 and x4 sorry x3 negative, x3 is positive
and x4 is negative, x3 x4 is negative. So this implies you can make x4 u is 0 or x4
u again negative. Since x4 is negative so u is 0 or u is positive, but in this case
we have only considered this 0.
I wish this is the most important part of this lecture, first of all using partial knowledge
of the system and without taking the help of system dynamics we first of all found out
the rate derivative of the Lyapunov function is approximately qualitatively equal to x3
x4 plus x4 u and using this expression we computed set of rule base such that V dot
can be negative definite.
What we have not until now found out in this is the parameters negative, what are the parameter
associated with negative. I am assuming this fuzzy linguistic variable to be Gaussian functions,
which we have taken in this case. Gaussian function is normally represented by two parameters-
one is mean and another is sigma. If I say sigma is 1 then these linguistic variables
they are all represented by one parameter that is the mean and the u is positive or
negative that we have defined here, we have taken them to be single term.
Standard Gaussian membership functions are used to defined the linguistic terms in the
rule base as mu positive of x is e to the power minus x minus x whole square, this is
the parameter that we have to identify. Similarly, mu negative membership function of negative
of the variable x is e to the power minus x plus x whole squared. And mu z x is 0 e
to the power minus x square.
So x represents mean, x stands for state variable, p, n, z denotes positive, negative and zero
respectively. Using the product for AND inference and center of gravity method for defuzzification
the fuzzy controller is given as we have four rules. So for four rules the u if you look
at here what I will do is that in this I find out the membership function of this, the membership
function of this. So this is mu x3 into mu x4 so that is the membership function for
my control action and this is the fuzzy singleton which I define as au, the fuzzy singleton, therefore my control
action is au.
mu negative x3, mu negative x4 into au is the output of the first rule. Here, again
to let to know I will again explain, here how we are actually doing this I do not have
to do every thing again you can easily understand, what is control action from this rule base?
Obviously, I find out what is mu and x3, and mu and x4 the membership function of this
into u is positive and this is a singleton and that is why this is au.
au is the singleton value associated with this control action au, simply multiplied.
This is the output of this controller, similarly you can easily see this is output of the rule
base, this rule and the second rule output will be mup x3 into mup x4 into this is negative,
so minus au. That is what we doing here.
So mu and x3, mu and x4 au, mup x3, mup x4 minus au I will not have been considered this
because we are assuming. Thus you see that this is 0 if I simply say the control action
is 0, because control action is 0 this is sufficient for me to make this rate derivative
to be negative definite. Simply making that what advantages I am gaining is that, I am
not reducing the parameters that has to be optimized or heuristically that has to be
tuned. This is my control action, by using center of gravity method you just add all
these individual membership functions associated with control action.
So there are 4 and finally, you get the value by taking mup x is e to the power minus x
minus x whole square in mu and x is e to the power minus x plus x whole square, you can
write u to this function. This is your control law. This is nothing to do because we have
taken Gaussian function. Finally you can simplify this u, but you also can take any other function
triangular function or whatever function you want to like, but if you look at this control
action what are the parameters that you do not know, this one is au, another is ax3,
another is ax4, all other things x3 x4, these quantities that you are getting feedback from
the system. So these three parameters they have to be found and we can do any kind of
optimization algorithm.
And particularly we have used genetic algorithm, particularly univarite marginal distribution
algorithm. We have already described in the previous classes what univarite marginal distribution
algorithm is. This has been used to minimize this cost function
and by minimizing that, the optimal parameters are found to be the mean of the linguistic
variable for x3 is 0.73 and for x4 is 0.25, this either positive or negative. So this
is the mean and au is control action singleton is 1.4. This was done, so this is design is
over, now we can simulate the system which will show the results at the end of the section.
But here, before that I will derive again another kind of rule base for the same system.
The u instead of earlier case u was simply a linguistic variable, but now I am assuming
this is a state variable technique, where u is minus k1x3 minus k2x4. Of course this
is just to simplify but actually I should have taken all the x1x2 I am not taking, just
to simplify because if I am able to control using only two variables, I can only expect
that using x1 and x2 I can improve the performance. So putting expression of u in V dot is already
we have found out this earlier because we considering same RTAC system.
So for RTAC system giving the physical principles V dot qualitatively comes out to be x3x4 plus
x4u. Now this V dot can be written as x3x4 and then x4u is this quantity which is minus
k1x3 minus k2x4, and this can be again simplified, this is 1 minus k1 into x3x4 minus k2x4 whole
square. So how do I now derive the rule base for this kind of structure?
I just have to make sure that I have to select k1 and k2 in such a way that this quantity
is negative definite. This is what, if x3 is negative and x4 is negative then u is,
you see that x3 is negative and x4 is negative, so this quantity is positive. Since x4 is
negative, the x4 square is positive. All that I have to make sure that 1 minus k has to
be negative as well as k2 is also a positive. So k2 is positive and 1 minus k1, because
this is positive, this has to be negative, 1 minus k1 is negative means k1 is greater
than 1. This is my first rule.
The second rule is if x3 is positive and x4 is positive, this quantity is positive, this
quantity is positive, to make it again negative definite, again I will have the same principle
whatever I got here I must be that, k2pp greater than 0, k1pp is greater than equal to 1. Same
condition, only this nn stands for the rule base when x3 is negative and x4 is negative
and here pp stands for x3 is positive and x4 is positive.
In the second rule you saw that, the same condition came but these are different parameters
you have to remember k1pp. If x3 is positive and x4 is positive then u is equal to minus
k1pp x3 minus k2pp x4 where I showed you that this has to be positive and k2 has to be positive.
Now let us go to the third one, x3 is negative AND x4 is positive THEN u is minus k1np x3
minus k2np x4. How do I find out what is k1 and k2? It is simple; k1np you see that, x3
is negative and x4 is positive here, that makes this is negative. This is negative means
this has to be positive, 1 minus k1 has to be positive, so k1 should be less than 1.
That is k1np is less than 1 and here this has to be again positive quantity k2x4 because
this is negative is already there, this x4 is positive so this is always positive whether
x4 is negative or positive. To make it positive k2 also has to positive, so this is what k2np
this has to be positive.
Similarly, if x3 is positive and x4 is negative, again this is negative, this has to be again
positive, and again k1 should be less than 1. So k1pn is less than 1 and k2pn has to
be again here this is positive, so k2 has to be always positive. You can easily see
that this k2 always in every case is positive, only in this case, the first one it is positive
and the second that is negative. What you got an idea that, what should be this value
k1 and k2, at least the magnitude?
Now heuristically you have to tune these values what should be k1nn k2nn k1pp k2pp k1np k2np
and k1pn and k2pn heuristically, or I can use the genetic algorithm or we have done
that univarite marginal distribution algorithm to optimize this parameter.
You see that considering the same membership function and using product for AND inference
and center of gravity method for defuzzification, the fuzzy controller is given is like this.
Since mup is this and mun is this, which we have already done in the previous case, we
can write u as like this, we have a controller and by simplifying that we have actually ten
parameters to identify.
Two parameters x3x4, the mean for linguistic variable x3 positive, for x3 positive and
x3 negative the mean is x3 and actually in case of one case mean is positive and in one
case mean is negative. That is what we are doing that when it is negative, mean is negative,
so the sign changes that is why you saw that here.
In this case this is positive x when it is negative mean is negative, so it is positive
quantity and mean is positive here in case of positive, so it is negative here. So ax3
is 0.25 here, doing optimization using univarite marginal distribution ax3 is 0.25 and ax4
is 0.12 and the parameters, the controller parameters that is k1 and k2 for all the different
fuzzy rules, four rules are there and for each rule we have two parameters k1 and k2
and we have got these values 1.1, 1.4, 0.6, 0.8, 0.35, 0.2, 0.4, 0.25. These parameters
we derived using univarite marginal distribution algorithm.
You see that simulation results x1 converges to 0 with time, what we started with we gave
some initial disturbance and using the initial disturbance we are showing now, here x1, x1
is the position of that cart, you give some little disturbance to the cart finally it should come to the
rest position.
What is the meaning of PD? This is the FLC controller where we have used the PD type
of rule base that is if x3 is positive and x4 is positive then u is negative, in that
kind of rule base. This is PD; it is actually the rule base that we formulated. That is,
we are denoting we are saying this is PD and this we are saying state feedback because
you see that u is a function of states. By doing that you see that, we have got the response
that, this is in second unit, after 40 seconds almost the oscillation comes to an end.
And here x3 is plotted which is the angular position, that also should come to, because
the cart was disturbed, although the pendulum was in the beginning initial position, but
it will start oscillating and that oscillation will gradually slow down and it equals in
same amount of time this is reduced to 0, both the PD type of rule base and state feedback
type of rule base and this is the control action and this is control action you see
that the control action is also fairly smooth both for PD and as well as state feedback.
Saying that what we did is the regulation that is given the RTAC system we gave some
disturbance and we showed that this controller is able to stabilize the system. Now the tracking
controller that we would like that our pendulum should track it as a trajectory. For tracking
a desired trajectory, what we would like to have is that instead of the Lyapunov function
which we had half x1 square plus x2 square plus x3 square plus x4 square for regulation
problem, now we will consider e1 square plus e2 square plus e3 square plus e4 square be
the Lyapunov function where er is desired state trajectory minus the actual state trajectory.
So xi is the higher state and xid is the corresponding desired state trajectory.
The control objective is to find a control action such that the actuator angle will follow
any given desired trajectory. Hence the Lyapunov function has been changed from x1 square plus
x2 square plus x3 square plus x4 square to in this case e1 square plus e2 square plus
e3 square plus e4 square. I hope this is clear to you.
Now in the same manner we can also show ei dot is xi dot d minus xi dot is same as xi
plus 1 d minus xi plus 1 which is ei plus 1. What is xi plus 1? Because we have the
parameters x1 x2 x3 x4, so when I say xi 1 plus d, when I say xi is this is x1, x1 dot
d is same as x2d. Obviously, because x2 and x1 the relation is x2 is x1 dot. By that principle
you are able to see that xi dot d is xi plus 1 d which I am writing here and similarly
also I can write x3 dot d is x4 d, x3 because x3 dot is x4. So obviously x3 dot d is desired
is x4 desired.
For i equal to 1 and 3, that is what I have done, i equal to 1 and 3 and differentiating
V we had V dot is e1 e2 plus actually this is e1 e1 dot e2 e2 dot e3 e3 dot e4 e4 dot.
We already know what is e2 is e1 dot and this is retained and e4 is e3 dot. This is our
actual expression of e1 dot.
Just like this again x2 dot is approximately minus x1 and x4 dot is approximately u, thus
in the same similar logic we can simplify this V dot to be qualitatively e3e4 plus because
qualitatively this will cancel out this because e2 dot would be minus of e1. This will be
cancelling out qualitatively, what is remained is e3e4 plus e4 and e4 dot we see that x4
dot is directly proportional to u. We can write that this is e4 and e4 dot is V where
V is x4 dot d minus u.
This is very simple because e4 dot is x4 dot desired minus x4 dot and since this is x4
dot I put this x4 dot to be u, I can replace that by u and x4 d minus u I can write V and
that V is x4 dot d minus u. What we have to do? I can create a rule base
from this Lyapunov function assuming V to be my control input and from V if I know V
then I know u, actual control action, this is pseudo control action, my actual control
action would be x4 d minus u.
In general what we learnt is that for regulation the qualitative equation for right derivative
of Lyapunov function x3x4 plus x4u for tracking you found out this to be x4V dot is qualitatively
equal to e3e4 plus e4 into V where V is x4 d minus x4 dot d minus u and u is the actual
control action and V is the actual control action.
This V can be designed the same way as that u in the regulation problem; I will not be
going in detail because this and this has the same structure. The rule base either using
the PD type of controller or the state feedback controller the two rule base will have same
form, only thing that I am computing V. After I compute V, V will have the same structure
that we earlier derived the same way but given V, I find out u to be x4 dot desired minus
V. Doing that and also the parameters in the first case will have three parameters and
in the second case similarly ten parameters that have to be optimized.
By optimizing these parameters, we get this is av not au, av is 1.5, ae3 1.23, ae4 is
0.55 and for this tracking controller two you have these parameters .37, .26 and these
are the parameters for the state feedback controller for four rules. So k1nn is 1.3
and k1np is .9, k2nn is .6, k2np is .45 and so on.
Given that, the tracking result is x3 is plotted against time desired PD and state feedback
give the same result here for the tracking. Tracking is not plotted because you see that
the rule base is very small that we have taken and also we have in this state feedback control
are also we have rejected in the state feedback the input of x1 and x3 the state feedback
for x1 and x2. The tracking here is not right, some loss here otherwise tracking is properly
and the control action you can see both PD and state feedback they behave almost similarly.
Finally the summary, in this lecture following topics have been covered a classical Lyapunov
synthesis method is extended to the domain of computing with words that means qualitatively
how can we use a rate derivative of the Lyapunov function in such a way that the rule base
can be generated. This new approach is used to design two different types of controller
actually, the two different types of rule base, controller is same fuzzy logic controller
mamdani type.
Rotational proof mass actuator system is presented for simulation. Assuming minimal knowledge
about the system we have systematically derived fuzzy rules that constitute the rule base
the controller. Using rule base controller we have design for both regulation tracking
purposes. Simulation results are presented.
References in this case are some of the related papers are you can easily see Fuzzy Lyapunov
based approach to design the fuzzy controllers, Fuzzy sets and system in 1999 by Margaliot
and Langholz and same author also have a paper on fuzzy control of a Benchmark problem; computing
with words approach, IEEE transaction Fuzzy Systems, 2004. And also we have a paper some
work on the directional intelligent controls schemes for a redundant manipulator presented
in a conference held in Pune in 2005. Thank you very much. 414