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Controller design for a T-S fuzzy model, common input matrix, this is the fifth lecture in
this fourth module on fuzzy control. In previous three classes we discussed mainly on Mamdani
type of controllers. Today onwards, we will be talking mostly on T-S fuzzy model. What
is T-S fuzzy model? What is common input matrix? They may be very new to you or some of you
have some understanding but we will learn in detail what they are.
The topics that we will be covering are T-S fuzzy model representation of a nonlinear
system; identifying the parameters of local linear models; stability analysis when the
subsystem have a common input matrix; controller design; and simulation results.
Representation of a nonlinear system- if I am looking at a non linear system discrete
time, the usual way approach is x is my state space, n dimensional state space X k plus
one is f X(k) U(k) and y(k) is h X(k) u(k). This is our normal functional description,
f is n dimensional function vector, h is again m dimensional output vector. So I have m output
and n states and u is p dimensional input vector, so the above system can be effectively
modeled by fuzzy merging of equivalent linear system in different operating regions using
Takagi-Sugano (T-S) fuzzy model. What is the meaning of this Takagi-Sugano model fuzzy
model of a nonlinear system is when although the system dynamic system non linear overall
but locally system is linear. So by fuzzy merging of linear subsystems we can construct
of the approximation of the actual nonlinear function of the system. This is the typical
T-S fuzzy model.
A T-S fuzzy model can composed of m rules where j th rule has following form Rj, the
rule j, if x1(k) is F1 j and so on this x2(k) is F2 j until xn(k) is Fn j then, that means
in the j th for the zone the system is represented by linear system, in this k time we have written,
we can also be written in a continuous time, so x j (k plus 1) is Aj X(k) plus sorry this
is mistake X (k plus 1) is a Aj X(k) plus Bj U(k) and y(k) is Cj X(k) plus Dj U(k) because
X(k plus 1) and y(k) they are of the system states, we cannot put j there j is affiliated
to the coefficient, the system matrix Aj, control matrix Bj, this is your output matrix
Cj, where x is x1 x2 until xn, n dimensional vector and j equal to 1 to m that means I
have m capital M fuzzy rules, given a current state vector X(k) and input vector U(k) in
T-S fuzzy model, infers X(k plus 1) as…..
How does it matter? It is because we have to finalize what is X (k plus 1). So X (k
plus 1) is fuzzy merging or fuzzy blending of output of each system, output of each system
is…. X (k plus 1) is here is Aj X(k) is Bj U(k) this is my j th system dynamics and
I multiply with that the corresponding muj, the membership function. You know the membership
function normally is derived the minimum of the membership function associated with F1
j to until Fn j or product of that. We can select any one of them, but whatever it is
that is muj you select to one principle and then you find out the muj, the membership
function inferred for the rule. See membership function is always given a crisp value x1
(k) I know what is the membership function F1 j. Similarly, given a crisp xn(k), what
is the membership function? The membership function associated with xn(k) is computed
from Fn j but the membership function of the rule is muj and if muj either mean minimum
of these membership functions that are compute or you can say mu1 x1(k) into mu2 x2(k) until
mun xn(k) is muj. We can do that also. Once I do that and I saw that we form all the rules
from 1 to m, divided by the summation of all this membership function muj that we have
already computed j equal to 1 to m muj.
This is how we saw the first T-S fuzzy model. We have m rules this is my j th rule and for
j th rule, this is my X(k plus 1), I multiply with corresponding membership function associated
with the rule and then I sum such quantities for each rule from 1 to m and then I divide
that quantity by summation of all the membership functions, that is muj.
And similarly, this was X(k plus 1), before 1 and y(k) is similarly muj y j (k) j equal
to 1 to m until j equal to m, muj where muj in this case we have selected in this paper
as a product but you can also take as a mean; it is all up to you the designer, so muj i
xi the membership function of fuzzy term Fi j, j equal to 1 to m.
The overall fuzzy system can be simplified into…, overall fuzzy system was this one
X(k plus 1) is this quantity, this is my fuzzy dynamics
representation of the nonlinear system in terms of fuzzy dynamics. This fuzzy dynamics can be written
in terms of X(k plus 1) is A bar X(k) plus B bar U(k) where A bar is sigma j Aj, j equal
to 1 to m where sigma j is muj upon this, you can easily do that, that is very simple
if I rewrite this equation I can easily write down this, this is simply you see that, I
can write this one as Aj so I write this one as A bar X(k) B bar U(k) where this A bar
this is at sigma muj by this quantity. I can define sigma j equal to one two and
muj this quantity. So all that the muj I am dividing by this total summation which is
this and the summation is j is equal to 1 to m Aj. This quantity is represented as sigma
j, so I am writing A bar is j equal to 1 to m sigma j Aj. This is what exactly we have
done.
X (k plus 1) is A bar X(k) plus B bar U(k) where A bar is sigma j Aj and we defined here
sigma j Aj equal to 1 to m. Similarly, B bar is sigma j Bj, j equal to 1 to m, C bar is
sigma j Cj, j equal to 1 to m, D bar is sigma j Dj where j equal to 1 to m and I already
defined what is sigma j before, muj upon total summation and you must recognize that sigma
j from 1 to m, this is most important this is always true, the total sigma j has to be
1. The overall system is nonlinear, so you can easily see that this is very convenient
form although I am representing this in a state space format, it appears to be linear
but it is not linear because A bar is a function of sigma j which you see here and sigma j
is the function of X(k). Sigma j is a muj and muj comes from x j this is a function
of X(k).
Just like we did for discrete times similarly also we can say for continuous time, instead
of X(k plus 1), I will write X dot is A bar X plus B bar U where again X is n into 1 dimensional
vector and y is your whatever we wrote here 1 into one dimensional vector and u is p into
one dimensional vector.
The whole thing again is similar A bar is sigma j sigma j Aj and B bar is sigma j Bj,
C bar is sigma j Cj and so on. What we said is until now we just said what T-S fuzzy model
is. Now given an actual non linear plant dynamics can I directly write what is T-S fuzzy model?
It is actually simple. The linear model parameter, how do we find out for each rule what should
be my individual linear system? So to find out what is individual linear system one of
the methods is linear system.
The linear model parameter Ajs and Bjs can be formed by linearizing the non linear system
dynamics. The simple example is, suppose the linear dynamics given as x dot is F (x, u)
which f (x) plus g (x) into u, this one form and that can be written as x plus x square
plus u. This is your f (x) and g (x) is one here. The aim is to find A and B is that in
a neighborhood of an operating point x0, F(x, u) equal to f (x) plus g(x)u which is Ax plus
Bu. How do you find out that? When x0 equal to 0, A is doe F and doe x, x equal to 0,
u equal to 0 and B equal to doe F by doe a x equal to 0 u equal to 0 using Taylor series
expansion.
This thing we already know, given this non linear function f, how can we find out this
A and B? That is by simply differentiating the f by x. When x0 equal to 0 and if Ai transpose
the i th row of A then Ai is doe f upon doe x equal to f0 plus fi is not minus x0 doe
f and doe x like x equal to x0 upon x0 whole square into x0 B equal to g(x0). The reference
you can find out this reference system is in control, given written by Jack. There are
many other classical text books you can follow, how to linearize on non linear system. I think
we also covered this linear in the class.
Thus two rules of the T-S fuzzy model, using T-S fuzzy model our system was x dot was x
plus x square plus u. You can easily see that x plus x square plus u. That is our system.
Rule one if f equal to 0 x0 is so I differentiae this with respect to x, so I get 2x plus 1
at x equal to 0. If I differentiate this quantity 2 x plus 1 at x equal to 0, so this will give
you 1 x equal to 0 the value is 1, into x so x plus u because doe f by doe x at x equal
to 0 into x plus, actually delta x but since the operating point is 0 so delta x is same
as x. is x plus doe f upon doe u. Since the coefficient here is 1, so doe f upon doe u
is 1 and into u. So A1 is 1 and B1 is 1.
Similarly, if x equal to 1 then we implement this second one because when x0 equal to 0
then we implement this rule and while implementing that the x0 is 2 x plus u and A2 that means
A2 is 2 and B2 is 1. This is simply a scalar system, scalar differential equation. Similarly,
this same thing can be also duplicated to the vector differential equation. The linear
model parameters Aj s and Bj s can also be identified. This is the first method by simply
linearizing. If I know a non linear system dynamics and I think that is an exact model
the best way is not to waste system against identification, simply linearize the system
around various operating zones and define the fuzzy partitioning and do the linearization.
But this also can be identified using a fuzzy neural network.
From the input output data of the system, I have a given system I generate input output
data train a fuzzy neural network. When using a fuzzy neural network the elements of Aj
and Bj are the weights of the neural network. Least square cost function is used to find
the proper weights; weights are updated using the standard gradient descent algorithm as
well. So this is simple, what is that? These are my states x1(k) x2(k) then k until xi.
So these are my states and I have m rules.
And corresponding to each rule, I have a fuzzy membership linear neural network, whose input
is X(k) and U(k) and output is X(k plus 1). Similarly, corresponding to each rule I have
a linear neural network and finally, we do the defuzzification to compute what is the
actual X(k plus 1). This fuzzy neural network can also be used to derive the T-S fuzzy model
of a nonlinear system. We identify that and this is my, if these neural network weights
after identification they become difficult. For example this one is the linear model for
the rule one, this one the linear model for rule j and this one the linear model for rule
m. Earlier in the beginning of the class we said
about T-S fuzzy model, I hope that you understood by this discussion. Now we also talked in
the beginning common input matrix. What is this common input matrix?
Now you see the discrete time T-S fuzzy model was given by X(k plus 1) A bar X(k) plus B
bar U(k) and continuous time of T-S fuzzy model x0 is A bar X plus B bar U, where A
bar is sigma j Aj summation over all rules. Similarly, B bar is sigma j Bj summation over
all rules. But for this system you will have a common input matrix when Bj is B for all
j, where B is a constant matrix. If I can define the system dynamics in terms
of a common input matrix B, B is constant matrix and if I do that, then that gives us
certain advantage.
What I am trying to say is that for if x is some fuzzy zone then if my state vector in
a specific fuzzy zone j then all that I am saying is that my x dot is Aj x plus B u,
not Bj but B u and where B is a constant matrix. This is called common input and B matrix is
same for all the rules. When this is done then such a system is known as T-S fuzzy model
then common input matrix, input matrix is common for all linear subsystem. Why we are
interested in such a system? Let us take a ball beam system.
In a ball beam system, the ball and the beam system is a nonlinear system. The beam is
made to rotate by applying torque at the center of rotation. The ball is free to roll along
the beam. The ball position and beam angle are denoted by r and theta respectively.
The dynamics of ball and beam system can be represented with a state vector x1 x2 x3 x4
as r r dot and theta theta dot as x1 dot is x2; x2 dot is B x1 x4 whole square minus g
x3; x3 dot is x4; x4 dot is u; y is x1. So single output and single input, if the above
system is linearized around the different operating points all sub systems will have
a same input matrix which is B is 0 0 0 1, this is important.
While making B equal to 0 0 0 1, sorry, this linearization of the system dynamics will
always lead to B matrix for all linear subsystems the mean matrix is always 0 0 0 1. This is
a practical example, where exist a fuzzy control system for which for all linear subsystems
the control matrix is 0 0 0 1, so this is common.
Utility of common input matrix, what is the advantage of common input matrix? Of force
when B s are same that means, there must be some advantage because system is now simpler.
Suppose we design individual linear controller for individual subsystem, I have that to say
because now I have n linear subsystems. Controlling the complete nonlinear system is they involve
controlling M subsystems. Now I am saying you have the freedom to design because linear
subsystem means, for any linear system we know where we have adequate technology of
tools, adequate methodologies for designing controllers. What we are doing here is that
let us design control action for each subsystem. The control action corresponding to the j
subsystem is denoted by uj(k) that stabilizes that subsystem.
If all linear subsystems have a common input matrix B then an overall control input of
the form U(k) is sigma j uj(k), this will ensure that individual subsystems are excited
by the respective control system. This can be established by following theorem.
What I am trying to say is this is my actual plant, I give u and I get x the response right
through the plant. Now I have, these are all the apparent subsystems linear subsystems
and fuzzy merging of this linear subsystem gives us the actual plant dynamics. Controlling
this u because I give to the plant of only u. So that is obviously my input to the plant
to this system of B u but instead what I have done, I have stabilized this by giving a input
uj, but I cannot give this uj to the plant, plant is given u.
But if I compute this term u in terms of uj using this formula, it says that if I have
a common input matrix for all the subsystems then such a control action U(k) which is written
in terms of individual control action uj, implies that individual subsystems are excited
by uj(k) So this is the theorem for a class of T-S fuzzy system with common input matrix
in all fuzzy zones, the actual control action U(k) will imply the j th subsystem is excited
by the control action uj(k) for all j, where U(k) is the fuzzy blending of all individual
control action. You must recognize this uj is actually fake. We do not apply to any actual
system. This uj is simply a control action that is if I give u the actual plant which
implies this plant is experiencing a control action not u but uj(k), this is important.
If I am giving a control action U(k) to the actual plant, individual plant is not being
excited by U(k) rather uj(k). This is very important, only if all the subsystems of the
T-S fuzzy model are having a common input matrix, otherwise not. This is very important.
The fuzzy dynamics of common input matrix be represented as X(k plus 1) is that we have
already shown, the sigma Aj j equal to 1 to m which is A bar; similarly, sigma B not Bj
because common input matrix j equal to 1 to m, so we expanded that. Now you can write
this one, I can clock this j equal to 1 to m. Sigma j Aj X(k) plus B u U(k) this is the
total thing. This is my one term and you can see this term is actually 1, sigma j j equal
to 1 to m and hence this is simply B U(k) where B is the common for all linear subsystems.
Substituting of a control input u(k), what is U(k) now? U(k) is j equal to 1 to m, sigma
j uj (k) and here I can separate this term, so j equal to 1 to m, sigma j Aj and X(k).
This is my A bar and we started with, we gave a input U(k) which is summation of this individual
thing; now we will rewrite this equation then you will see that X(k plus 1) is sigma j Aj
X(k) plus here I can bring this quantity to this side and B inside, simply B I put inside
because this is anyway in common, so B inside and uj(k). So doing that what I am trying
to do is that sure, this is actually gain, this quantity is 1.
Now, what we are doing here is that now we have this total thing. That is I can now rewrite
j equal to 1 to m sigma j and Aj X(k) here and sigma j is common here and B uj(k). I
can rewrite that. What is meaning of this? The meaning of this is that, this is a fuzzy
combination of Aj X(k) plus B uj(k). Individual system is actually excited by uj when actually
I have a excited the global system by U(k).
What I showed in this theorem is that if I give U(k) to my actual system plant this implies
my linear subsystems being excited by uj, that is the meaning of this of this method.
It is clear from the above equation that j th subsystem is excited by uj(k) and it is
proved. The corollary: if j th subsystem can be stabilized by control action uj(k) then
the actual control input can be computed as U(k) is this sigma j equal to 1 to m sigma
j uj(k) and it can be analyzed whether the control action stabilize the entire T-S fuzzy
system. This approach is not valid when the subsystems have different input matrices.
This is very important.
When the subsystems have a common input matrix, T-S fuzzy model dynamics is used. So closed
loop system dynamics with common input matrix, I will explain to you what the meaning common
input matrix is. Common input matrix means that if the actual plant is excited by capital
U(k), then the actual subsystems are excited by the individual subsystem as usage, that
gives us a fairly good method or good tool for us to design linear controllers. Now for
such common input matrix will be the closed loop dynamics? T-S fuzzy model dynamics is
sigma j Aj X(k) which I have already told you, this quantity is A bar plus B U(k), so
when U(k) is this type we have already said, the closed loop system dynamics is given by
this. Simply I replaced U(k) by this particular term, this is input.
Now assume that uj (k) is minus Kj X(k) such that j th linear subsystem is stable. This
I am assuming such that j th system subsystem is stable. If j th subsystem is stable then
the closed loop system dynamics is minus, this will retain, this quantity is same and
this quantity is here because this minus is coming here, uj (k) minus k, minus term is
written here and sigma j, this quantity is retained here, instead of uj(k) minus Kj X(k).
That gives you, if you look at that that gives you the final value here because this B Kj
I can multiply and then it becomes Aj minus B Kj sigma j is common and this side X(k)
and this side X(k), so this is my closed loop dynamics.
What implies that common input matrix mix our closed loop system dynamics is very simple.
This is very important. If you denote Aj minus B Kj by Aj it is closed loop dynamics becomes,
this particular thing where Aj dash minus B Kj we can Kj in such a way that Aj’s are
stable.
Now the point is that we have designed we can find U(k) in such a way Aj dashes are
stable, but now X(k plus 1) this is the closed loop dynamics, my stability simply depends
I know already Aj dashes are stable, so now given Aj dashes are stable, these matrices
are stable, can I say that if I write this one as A bar X(k) where A bar is simply j
equal to 1 to m sigmaj Aj dash, since dash is there I will put dash also here, so a dash
bar if I say can I say if Aj dash’s are stable then A bar is stable.
You may say, it should be but it is not, if Aj dash stable does not imply directly A bar
is stable. We will find in this lecture what are the conditions for which if Aj dashes
are stable then A bar is stable. That is the point. To show the closed loop system stability
it comes with common input matrix to analyze the stability of the quantity and stability
of the quantity where Aj’s are stable. This part I will not talk, this simply says that
when the subsystems have different input matrices then the closed loops are different then if
I find out I designed ul(k) the individual system minus Kl X(k) then X(k plus 1) is given
by this particular term and you see that here we cannot represent the way we represented,
the reason is the above equation are shows that cross term in the closed loop dynamics
when j is not equal to l. Because of this cross term that is coming here because it
had it been B it is simple but it is Bj. this makes our life little tough. Now we will go
to the actual stability analysis. The stability analysis is common input matrix that we have
now simplified.
The closed loop dynamics for discrete time case is X(k plus 1) plus sigmaj Aj dash X(k)
and where Aj dash is simply A minus B Kj. So Kj is associated state feedback for j th
linear subsystem. The closed loop dynamics for continuous time case similarly, exactly
for this is discrete time, this is the closed loop dynamics of the continuous time where
Aj dash is Aj minus B K. We will now provide various theorems which we ensure stability
of this quantity, that is we would like to say because now, this can be written as A
bar X(k) and this can be written as A bar x(t). So if Aj dash is stable I can say A
bar is stable, this is the theorem.
Stability analysis theorem 2- for discrete time T-S for the system k plus 1, this is
my T-S fuzzy system common input matrix, if U(k) is given in terms of fuzzy blending of
individual control action and the control action is taken in such a way the individual
subsystems are stable then the closed loop system X(k plus 1) which is given by this
stable if the singular values of the individual Aj dash or less than unity, where Aj dash
is Aj minus U(k).
Here the proof is the induced Euclidean norm, second norm of a real matrix A is given by
A second norm induce norm is the maximum singular value of a matrix and what is that the maximum
singular value a matrix is computed as the maximum A value A transpose A and take the
square root. I compute the maximum a value of A transpose A and take the square root
then I find maximum singular value of A where alphamax is the largest singular value of
A and lambda transpose A is the largest eigen value of A transpose A.
Fact 1: Singular values of a real matrix are real and positive, this is very important.
Fact 2: Largest singular value of any matrix A is always greater than the magnitude of
the largest eigen value matrix, this is very important. I hope that you know this from
the matrix algebra. Now, let us denote A bar dash is j equal to 1 to sigmaj Aj dash if
all Aj dash has singular value less than unity then obviously the induce norm of Aj dash
which is maximum value of Aj dash is less than 1.
Since, alphamax since largest singular value is given because we are saying in the theorem
if the largest singular value is less than 1 of all individual, if largest singular value
individual matrix is less than unity, less than 1, then the system is stable. That is
what we are saying, so you see that. Now let me find out what is the largest singular value
of A bar. The largest singular value of A bar is induce norm of it, which is induce
norm A bar dash is sigma dash Aj dash A bar is simply this and then you see that using
triangular inequality, this is summation or this is summation of two products means by
triangular inequality it is always less than equal to summation of individual absolute
product, that is sigmaj Aj norm. Now I know this Aj norm is represented by alphamax Aj
dash.
And I know already that this quantity alpha max of Aj dash always less than 1, I am assuming
that. I know that let us say all Aj in this T-S fuzzy model, they have maximum singular
value less than 1 because of that I can write this because this is less than 1, so this
is less than equal to sigmaj and you know that j equal to 1 to m sigmaj is 1, so this
is less than 1.
So alphamax the largest singular value A dash is also less than 1 and we have already said
the largest singular value is always which we said earlier, we see that, largest singular
value of any matrix always greater than the magnitude of the largest eigen value of that
matrix.
That is, since we proved the largest singular value of A bar is less than 1, means largest
eigen value of A bar is also less than 1 and for discrete time case the largest eigen value
the magnitude should be less than 1. It should be unity circle. You know already that given
a discrete time system the poles should be within the unity circle. What we said in this
theorem that if my individual subsystem in the closed loop form Aj dash they have singular
value less than 1, the maximum singular value, then the overall closed loop system matrix
A bar also will have the singular value less than 1, implying that this system is stable.
Now utility of theorem two the proof of theorem 2- is based on maximum singular value not
on maximum eigen values. Utility of the theorem, making use of theorem, the overall system
stabilized if there exists common input matrix B for all subsystems; the individual gain
matrix Kj’s are designed such that Aj dash which I have already told is Aj minus B Kj
have singular value less than unity. This design technique fails if the system is in
controllable canonical form. In controllable canonical form, when all the poles are at
origin, we can get minimum value of induced norm as 1.
Stability analysis: theorem 3- Now earlier we have talked about discrete time, now we
talking about continuous time. For continuous time T-S fuzzy system x dot t is this quantity
which is our T-S fuzzy model, this is our normal fuzzy model, with again our control
x and u(t) is sigma j.
uj(t) is minus Kj X(t) the closed loop system x dot t is stable, the Hermitian part of Aj
dashes are stable where Aj dash is Aj minus B Kj. Fact 1- any matrix A can be written
as A equal to half A plus A transpose plus half A minus A transpose, half A plus A transpose
is called Hermitian part of A. This is the Hermitian part. You can easily see that, this
total multiplication, this will cancel out so this is 2 A by A, 2A by 2 is A, but this
actually is Hermitian part known about non Hermitian part.
Fact 2- If half A plus A transpose is stable then A is also stable. That is real parts
of eigen values of A are all negative. Fact 3- if A is stable that does not imply that
half of A A transpose is also stable. The reverse is not true. If this is stable then
I can say A is stable but if A is stable we cannot say A plus A transpose is still also
stable.
Proof: the matrix measured gamma corresponding to the Euclidean norm of a real matrix is
defined as that maximum eigen value of the Hermitian part of A. Since A transpose A by
2 the Hermitian part of A real symmetric matrix if eigen values are real. So if the Hermitian
part of Aj is stable that is half Aj dash plus Aj dash transpose is stable then the
matrix measure of Aj dash which is lambda maximum of this matrix is always less than
0, the maximum eigen value.
Since the matrix measure satisfies triangular inequality, I can write the measure of overall
matrix Aj bar is gamma into summation j equal to 1 to m this is the individual Aj dash.
Now this can be written in terms of, this is less than equal to I can take this using
gamma inside using triangular instability summation, so the individual is sigmaj gamma
Aj dash, because matrix measure satisfies the triangular inequality. Using that theorem
you can say that this is less than this and since I know that the matrix measure of this
is the maximum eigenvalue of this quantity which is always less than 0.
So sigmaj is always greater than 0 and this is the negative quantity and this is the positive
quantity, the summation will be always negative. Hence the proof is if this is stable Aj bar
is also stable. So the second theorem says for linear subsystem, for a linear continuous
type system, the overall fuzzy blending of the system should be stable, provided the
Hermitian part of the individual subsystems. This is my Hermitian part of the individual
subsystem that is stable.
Utility of theorem: making use of this theorem the overall system stabilized if there exists
a common input matrix B for all subsystems and the individual gain matrix Kj’s are
designed such that this has a stable Hermitian parts. We can always design this because I
can always design such that this is a stable Hermitian part, is not a difficult thing.
Limitation of theorem 3: if the subsystems are in controllable canonical form, Hermitian
part of overall matrix will be unstable. Consider a second order system like this, the Hermitian
part is like this, the characteristic equation is given. This is very simple. This implies
that half A transpose plus A is unstable is unstable system. If the subsystems are in
controllable canonical form Hermittian part of overall matrix will be unstable.
Good News: However, for second order system in controllable canonical form stability of
individual subsystems is sufficient to ensure the global stability. Suppose individual systems
are described by the following equation, second order characteristics equation of the j th
system is this, the characteristics equation of overall system is this. Subsequently, for
the subsystem to be stable the necessary as well as sufficient condition is Aj1 and Aj1
must be positive. This implies that this particular summation is also positive. Hence, the overall
system is simple on logic.
Now we will go to theorem 4: for both discrete and continuous time is fuzzy system the closed
loop system X(k plus 1) which is given like this or in continuous time x dot is sigma
Aj dash x(t) is stable, if each individual Aj dash matrix is symmetric where Aj dash
is Aj minus B Kj. Proof- a matrix A is symmetric if A is Aj transpose. Thus for a symmetric
matrix A one can write A is half A transpose A that means it does not have non Hermitian
part. If each individual Aj dash is symmetric then Aj dash is half Aj dash transpose plus
Aj for all j.
In this case the maximum eigen value of Aj dash is equal to the matrix measure of Aj.
Hence the proof of the continuous T-S fuzzy system is similar to the matrix measure approach
we used in the previous theorem.
Again for symmetric matrix, singular value of equal to magnitude of eigen values. Therefore
the proof for discrete time T-S fuzzy systems is same as that of induced norm approach.
Utility of the theorem- making use for the theorem the overall the system can be stabilized
if there exists a common input matrix B for all subsystems an individual gain matrix Kj’s
are designed such that Aj dash equal to Aj minus B Kj are symmetric. Now we will skip
these things, now we will give some simulation results.
For surge tank, the output of the system is the level, the liquid level which is controlled
by input flow. Input can be positive or negative. Discrete time dynamical equations of the surge
tank is h (k plus 1) is h k plus t and you can see this is a nonlinear dynamics and also
u(k) upon root over 3 h (k plus 1). This is a first order system; the system can be made
stable by designing stabilizing control for different fuzzy regions. You see that?
What we did they used neural network to identify T-S fuzzy model. You see that error, the system
is identified using the input output model, the output is the range for zero to ten; number
of fuzzy clusters are five, so the five rules. The T-S fuzzy model is described by following
five rules: if x(k) is around 2, then x(k plus 1) is 0.98 something plus 0.0032 u(k)
and so on; rule two- is if x(k) is around 4 x(k plus 1) is 0.991423 x(k) plus 0.003928
u(k), so like that. You can easily see although B matrix is varying but almost similar, that
is why the method we said is also applicable here. To the result of system identification
we generated data from Surge tank model, trained a fuzzy neural network and you see that the
data the desired trajectory, and the predicted trajectory, they exactly match.
Applying the global control law, the overall system becomes like this, where the individual
subsystems is controlled by state feedback and individual k a l designed for the system
meets this criteria and control law stabilize the global system. Since the overall output
h(k) is this quantity, the output here is h(k), hj (k) is the j th sub system’s output,
if each individual system track the reference input r then the overall system will also
track r. The controller thus becomes you see this is our state feed back part and this
is my part for tracking where Kj and Fj are feedback and feed forward gains for j th system
to track the reference input r.
The simulation result is presented in the following figure; the left figure shows the
output of the system and the right figure corresponding input. That means control action
and this is my height so I am tracking a am set point tracking from 4 to 1 again 4 to
1 and this is very perfect tracking, you can easily see and controller is almost is very
smooth.
And you see that the gains that the feed back gain and feed forward gain, that are obtained,
you can easily see here, they are of time bearing nature because this is a non linear
system. It cannot have a fixed gain, constant gain it has a variable gain. So variation
of state feed gain system state is also shown in the following figure.
Now we take second example of Vanderpol oscillator. Most of you have studied this in the introduction
to nonlinear system because this is very classic example of nonlinear system. x1 dot is x2,
x2 is the quantity and y is x1, single input single output system and if I linearize this
system, I have this particular form and you can easily see, my B matrix is 0 1 and hence
this is common for all subsystems.
And in the A matrix also 0 1, for all subsystems only the second row vary according to x1 x2
and x1 square. So we can linearize that and one example is if x t is the 0 then x dot
is 0 1 minus1 2, x t 0 1 ut and similarly, we have forty nine rules. Using this the controller
is designed, the tracking is achieved by placing the closed loop holes for all subsystems are
minus 3 and minus 3.5, the feed forward gain is same for all subsystems. Thus the control
action is given by the following equation.
And you see this is my state feedback controller for regulation and this is my tracking the
reference signal. So the tracking result you see the tracking is very very perfect. Very
good tracking so given set point variation and this is my control input so the simulation
result is presented in the following figure; the left figure shows the output of the system
and the right figure shows the input of the system. This is my control action and this
is my output. So variation of state feedback, you can easily see again, these gains are
variable.
This is not a flat surface, the variable gain for k1 and k2. Summary: in this lecture the
following topics have been covered- a general nonlinear system is represented by T-S fuzzy
model. Local linear model parameters of the T-S fuzzy model are identified using a fuzzy
neural network. Different theorems have been presented to ensure stability of the closed
loop system, when the subsystems have a common input matrix. In this case the global controller
has been designed as a convex combination of the local linear controllers. Simulation
results have been presented for two nonlinear systems. That is all. Thank you very much.