Hello and welcome to the tutorial. So in the
previous class we have seen this experiment
about parameter estimation using least square
methods. So there we have seen what is parameter
estimation? And how to estimate the parameters
and why do we lead the parameter estimation
and everything. So we understood and I also
discussed about various methods are available
to do this parameter estimation and I have
discussed about least square method.
So that we have learnt what is the basic principle
of least squares, how least square works and
we also have seen the mathematical formulation
and I have also talked about some examples
like how can you incorporate the least square
estimation philosophy in the problems and
other things right. So in this tutorial what
we will be learning is this like earlier lecture
was about generic parameter estimation like
any process can be modeled BE mechanical,
electrical or from any other process.
And if your models are parametric then you
can use the technique of parameter estimation
and maybe least square estimation technique
to estimate with parameters right. So I will
be discussing today about aerodynamic parameter
estimation, so I will be talking what is aerodynamic
parameter estimation and how aerodynamic parameter
estimation is very much important. So let
us start for today’s session.
.
So again the experiment is about parameter
estimation, so this parameter estimation is
nothing but aerodynamic parameters. So we
discussed about definition of parameter estimation
and few methods I have listed which can be,
you know like from there you can estimate
parameters using the flight data, using the
states of input data and output data. But
we also should know there are other ways also
to estimate aerodynamic parameter estimation
right.
So these methods are like this, first one
is analytical method, next is wind-tunnel
method, and then third one is flight test
method to which we will be learning in detail
today flight test method. So earlier part
of the lecture was about this method right.
So this is a theoretical method you do the
parameter estimation using analytical method
from the geometrical detail of the aircraft,
but what happens there like you assume some
complex phenomena, you simplify some complex
phenomena.
So the estimated parameters may not be so
accurate, but for that you have other methods
also which is picking up so fast this state
and that is based on the CFD computational
flow dynamics also all of you know about CFD.
So CFD based methods also can give you the
computed parameters right. But again to corporate
those parameters whatever you got from the
analytical methods or from the CFD based methods
you need some experimental methods right.
So these two are experimental method when
you treat these parameters from the experiment
doing some experiment is experimental method.
So as I said you need to corporate to which
rigels or the parameters then we should go
for experimental based method. So let us see
how wind-tunnel method work.
So what it does like you have a tunnel and
then you do not, usually you do not put a
full scale model you put the scaled model
inside the tunnel and you will try to simulate
the accurate mass that you try to give the
same speed or you maintain the desired flow
which is usually in the flying condition and
then wind-tunnel methods gives you the measurement
of the force and coefficients there you get
those parameters. But again since it deals
with the scaled model and then there will
be issue of the Reynolds number you cannot
exactly similar the Reynolds number in the
wind-tunnel.
And apart from those things you will have
issue with this like dynamic derivatives and
other stuff. So again we should look for some
prominent method where you will get better
estimate of the parameters and with that we
will start this flight test method where you
will collect the data during the flying condition
and from the method variables or flight variables
you will try to estimate with parameters right.
And already I have discussed which parameter
estimation in aerospace area is very important
because once you talk about designing of control
algorithm or designing of card algorithm in
aerospace application, then you will lead
to have a very accurate model even if you
want to do something for the fault, detection
and diagnosis you should have a better model.
Aerodynamic model is becoming so important
because the accuracy of aerodynamic model
will dictate the accuracy of your equation
to motion or whole aircraft model.
So I will not talk about equation of motion
here, so we will start with this aerodynamic
model okay, aerodynamic model of forces and
moment.
.
Aerodynamic model is so as you know in this
aerodynamic model like if you see we have
force lift force drag force and movement I'm
taking up pitching mode so I am giving you
the example of the longitudinal derivatives
or longitudinal forces in moment so lift drag
and pitching movement right so you can write
lift as ½ ? v2 S into CL right this part
is your dynamic pressure and plane form area
and this called lift force coefficient so
you can write this as general lift force coefficient.
Right and drag also you can define similar
½ ? v2 S CD where this coefficient CD is
your drag force coefficient correct and preaching
movement is again ½ ? v2 dynamic pressure
times S times C bar here this is cord length
and this Cm which is your preaching movement
coefficient okay so you know force and movement
can be simplified in this term so these are
basically your non dimensionalize term right
they are very important to study in this equation
of motion now with the introduction of lift
force lift drag and pitching movement coefficients.
So I will talk about like this model like
how do you model them right so now you know
what are those coefficient so.
.
CL CD and Cm so you know they all are function
of a angle of attack ?? A change in elevator
deflection and q pitch rate it can also be
a function of other variables like Reynolds
number V and all so just for simplification
I will take this three variables so CL CD
Cm they are function of C a ?? E and q right
so if you write the aerodynamic model which
already you know just for revision I am doing
it again so CL can be written like CL0 + CL
a into a + CL ?? e times ??e + CL q here we
write CL q time qC/ 2v.
To make it non dimensionalize and everything
will be in same term and CD you know it will
be written CD0 + CD a and a + CD ?? e x ?? e
+ CD q x qc/2 v right similar Cm can be written
as Cm 0 + Cm a x a+ Cm ?? e x ?? e+ Cm q c/
2v so they are called aerodynamic model yeah
so already I have written here so this these
things are your aerodynamic model force and
movement coefficients right.
So here you see that all the variables like
CL CD and Cm they are well structured right
so this is the structured model where CL 0
CL a CL ?? AND CL q they have their own physical
significant and they are called aerodynamic
parameters so if fewer diameters like CL and
CL a Cm a which is related to your a or may
be CD a that talk about the stability basically
these things about talk about the stabilities
so they can be called as stability derivatives
also your CL q CD q CD q will be very week
parameter and Cm q so that talk about the
stability so they can also called as stability
parameter or stability derivative okay.
And the parameters which are related t control
here control is ??e elevator control here
so CL ?? e CD ?? e and Cm ?? e they are called
control derivative 
now why they are derivatives so it is quite
clear from the nature of the parameter because
if you see.
.
CL a ? CL ? a v it says the change in a what
about change you observe in CL like the change
in CL because of change a this is defined
by CL a right so accordingly everything will
be like that CL ?? e change in CL with the
change in ?? e so this is called your control
derivative right so once you have the structured
model right so they again become a parametric
models.
So today I will talking about the A is said
earlier about the aerodynamic parameter estimation
and I will show you with examples and this
is also the demonstration of the technique
what we have learn in previous class right
so I will take two parameters two variables
sorry force CL and movement Cm pitching movement
coefficient.
So you now you know the model CL is CL0 +
CL a into a + CL ?? E into ?? E + CLQ X QC/2B
yes and CL0+ CM a times a CM ?? E times ?? E
CLQ QC/2V okay so it can also be written like
this yeah, CL and same I will write in a vector
form like this and if you write all the parameters
here like CL0 CL a next is your CL ?? E and
CL q and this you have CM 0 CM alpha, CM ?? E,
CM q.
1 a ?? E and QC/2Vyes so this is the same
equation actually represent as in terms of
matrix so now if you see it will look like
this y about the output force and moment co-efficient
and if I can call this as a ? matrix which
is a parameter matrix and input if I represent
by S so this is again it is again in terms
of this very well known equation right, y
= ? Xa linear equation right, so now this
had become a candidate for less square we
have seen in previous class.
How to estimate the parameters from the linear
model which what like there yeah so I will
demonstrate the principle of less square on
this example right, so now let us talk about
the flight data right because now as I said
this is output this and here this is input
right, output this is your input and this
is parameter vector matrix, and you want to
find those aero dynamic parameters so this
is the problem for this in this example.
Now how to estimate those thing before that
I will be talking about ho9w do you get these
outputs CL and CN, and how do you collect
all the inputs right so as we know there is
no sensor which can directly give you the
force and moment co-efficient during the flying
condition so CL and CN we cannot directly
measure right, so now equation comes how will
you get CL and Cm.
Because now we are talking about the system
identification or parameter identification
it is about the input and output so how will
you generate the output or how will you get
that so it means can we do something do we
have other side variables which is related
to force and moment and from that we can get
this force and movement co-efficient so can
we do something like that, so let us see again
CL and CM.
And as you know most of the sensor which is
in build in aircraft which is mounted in aircraft
they give you the information in terms of
body frame and for in your information you
know CL specially this lift force they are
in wind frame not in body frame so you need
a conversion from body frame to wind frame
right so CL how do you establish the relation
like how you will get Cl.
.
So CL as we know CL can be also written as
Cx Sin a right minus Cj Cos a right so now
they are also force co-efficient in body frame
and you know that actually you see that the
aircraft if your x is related like this if
you define XX is true also know y here and
Z so the force in this direction will be FX,
FY and FZ, right and again you know force
can be written as Q bar then represent a time
S x Cx.
So you basically get Cx from this or Cz from
this dynamic pressure times surface area right
so you know the meaning of Cx and Cz right
so just I have told you so that you can get
the standing.
.
You can align your understanding whether right
but again we do not have sensors directly
which directly measures CX and Cz then Cx
as you know it can be written as mass times
acceleration actually expression in X direction
minus trust hydrogen with FE so please remember
this is your trust force right, okay. Divided
by dynamic pressure time area but if there
is inclination angle with the incidence so
this is called engine insertion angle so that
component also will come here.
So that will be Cos s T right and this is
your engine inclination angle, so that I though
which we will be talking where the value will
be 0 so this is not a very dominant term but
to therefore the sake of completion of this
equation I have written this so that you should
remember. And see that you will get maz, m
time az minus not will become plus Fesin component
on this angle, okay.
Now we know from this equations we know that
data for ax trust force dynamic pressures,
surface area this every things are know rigjt,
because IMU will give you the details of acceleration
and from the geometrical information of the
aircraft you will get S and sensors are there
to measure dynamic pressures you will get
the dynamic pressure, right.
And you will also get az so in this equations
everything can be measured through the flight
right, so once you have measured details about
the variables you will cx and cz and by knowing
cx and cz you get CL right, so that is how
you will get the CL lift course coefficient.
Now coming to the PC moment coefficient this
one how will you do that let how will you
measure that or how will you derive from the
flight variables just write the moment equation,
so all of you know moment can be written like
this M=Iyq? right, +Ixz(p2-r2)+pr times Ix-Iz
so this is the moment equation right.
And further you can write M as 1/2?v2sccm
basically this is your chord length c we can
write c I will just use this notation as a
c and c¯ interchangeably so here I have written
c so let us write c, but here please understand
this is your chord length, okay right and
then the same equation okay, so here you can
get same by knowing all those variables. So
you know the I'm contains the accelerometer,
gyroscope, magnetometer so from the gyroscope
you will get pr and q everything and from
the geometrical details you will get Ixz,
Iy and Ix znd Iz to you see parameters you
will get.
Now this is the only thing where we should
look for it is q? because most of the aircraft
will not have the sensor which can directly
measure q? or in our aircraft we do not have
the sensor which can directly measure q?,
so how will you get this q? you know q? variable
right, so but we know like we can measure
q, q can be measured or the data is evaluable,
okay. So from any numerical difference in
technique if you have those q you can get
that q? this is not a problem.
But problem will come when your q is not accurately
measured it has some amount of data noise
some amount of noise in this signal and if
you take the derivative of that signal it
will further amplifier a noise so the signal
will become even worst, and once you do not
have the accurate q? you will not be able
to get accurate output Cn and once you do
not have the accurate output then the estimated
parameters will not accurate, so that is why
we have to be very careful the kind of data
you are selecting or we are feeding in this
algorithm those data should be very accurate
originally accurate.
So now questions will come like how can we
region ably estimate or obtain this q? from
q right, so now suppose there is small amount
of error or some amount of error in q right,
then we need to identify the error or we need
to select if there is any presence of error
so how will you detect that so you apply FFT
from the FFT it is called fast Fourier transformation
right, so what it does like it will give you
the frequency components present in the signal
so if you pass through at and then if you
see this amplitude verse frequency draw so
it will generate some spics right like this.
Wherever frequency terms are there you will
get those spics, so I think let me tell you
about the little bit about the FFT, so that
you will be able to appreciate okay, you might
not have learned it right. So before that
so suppose you have a very clean signal or
sinusoidal signal like everyone have seen
this signal right, sinusoidal signal, okay.
.
So sinusoidal signal if you see mathematically
it will look like this right, 2pft sin?t so
here this signal have only one frequency component
which is f right, and if you do this fast
Fourier transformation or Fourier spectrum
analysis of this signal through fast furrier
transformation the smart this would I was
trying to discuss here then it since it has
only one frequency component it will generate
only one spike right.
And it was add frequency of f so you will
get the spike at f with the same magnitude
or amplitude you can also see the amplitude
of the signal if it has two frequency components
then you will get two spike generate for corresponding
frequency right.
So this is the idea of your fast furrier transformation
okay so as I said if q has noise then usually
noise enters through a very high frequency
atoms so you will get to know once you do
the spectrum analysis using FFT and if you
have seen instead few glitches or few spike
at higher frequency it may not be so big in
the amplitude it will be a small maybe in
multiple spikes at higher frequencies then
this will be dominant one so it may look like
slightly bigger right.
So from here you can easily identify that
noise high frequency noise entered in the
signal here and it will look like this these
are the high frequency noise usually noise
is higher of high frequency you will be able
to identify the noise in the signal by doing
this first data formation.
Now you can easily identify these are the
frequency which we do not need ion the signal
so you can eliminate those higher frequency
noise terms but designing a suitable low pass
filter right so low pass filter what it does
like low pass filter LPF it rejects all the
higher frequency component terms right all
the higher frequency terms and it will give
you the signal which is related to low pass
like a flow frequency.
So you can reject all those higher frequency
noises ort higher frequency term by employing
the low pass filter and then you can give
the information of cut of frequency in the
low pass filter from the FFT analysis of this
Q signal right and once you got the q signal
clean then you employ any numerical difference
in technique 
to get this q dot right so now you got the
q dot so in this equation you have the information
about q dot now you already know about inertia
parameters you already got information about
p and r from the magnet like from the IMU
through a gruel scope.
So all the flight variables which are use
in this equation or known and from that we
can get this CM so that is how you will get
Cm right, okay so ion the output you got CL
and CM and you can directly measure a from
the a sensor from flight test you have the
information about the elevated deflection
from the sensor so this also it measure and
q definitely you know and C is your code line
and velocity is also measured.
So now ion the set of input and output everything
is known so what is not known or what is unknown
of this, this parameters we do not know about
Cl 0 CL a CL d and CL q and Cm or not Cm a
CM d Cm q and it is now it has taken the structure
of y = ? x right so you can think of using
a least square in this problem and then you
will get the ? so how will you do that like
from the simple technique you can y = ? x
now right so how will be the how will your
estimated parameter will look like this so
look like in this thing again least square
technique says like it minimize the summation
of the error if you remove the error part
what I discuss in earlier class.
So what you do like you multiply first multiply
with xT here or both the side right so now
your ? will become y xT times x xT so that
will be your estimated parameter okay, so
what I will do like we have gather the data
on unsure graph so I will show you the how
data has been collected so you will be able
to see all the information about the flight
variables a velocity and qrp and everything
and from there we have derived CL and CM so
what I am do like.
I will show you through the slide so that
will be able to appreciate more once you see
it visually and then from there employing
this least square technique we have estimated
this parameter vectors ? there you will get
to know about all the CLL a CM a CM d here
whatever actually parameters we have talked
so all the parameters can be obtain using
this re square technique so from the Mattel
of simulation we have done ity I will shoe
you in this slide okay.
.
So I will show you the results which we got
for this parameter estimation right so yes.
.
So this is air craft Hansa 3 air craft we
have use to gather the data and if you see
that it is a twin seated air craft research
air craft so we use this for our research
work. And then where it is 760kg as 12.74
meter square in span is 1.221 meter cord length
is 10.47 meter as the procedure 8.8. And inertia
parameters are like I X is 73kg meter square
IYY is your 907 kg meter square IZZ is 1680
meter square IXZ is 1144 kg meter square.
.
This is your geometrical parameters of this
hansa 3 aircraft yes. So these are the longitudinal
data we have gathered during the flying condition
you see here elevated has been deflected so
this is a standard signal which we say 3211
yes. So if can see the elevator which is right
at the bottom here and we tried to make it
like a 3211. 3211 says like it should be 3t
time’s upper down 2 t times if it is down
then it should go up then three t times down
and up.
So this is over flee 3 2 t t and t. so we
tried our best to get those 3211 signal through
this elevator to side the short period more
right. And then this is the deflection in
? 2 ax a1 ay az ax will ascertain alpha with
the change in the with the change in the elevator
deflection so you have seen the changes in
the alpha so it is quite following. Right
like we have a deflection here and corresponding
to here started seeing the deflection in the
table alpha velocity.
We tried to velocity for short period should
be constant but it is stably constant in the
accuracy of may be 5 or 7 meter per second
it has gone off and you see the deflection
in the acceleration data the x data it is
of 2 meters 2 meter per second square you
can see roughly and sz also changes may be
-15 to it has gone slightly lower to 0 and
there is a change in q right.
.
The digital in radian per second units are
already written here so you can see so you
saw the changes in the q and ? and other variables
are not listed in this but we collect all
those data fight data which has been discussed
during this lecture right. so I have shown
you those important variables through this
graph right next thing was which I was talking
about the longitudinal input and then output
I have talked about two derivates so in this
graph. You see like these are the four set
moment co efficient.
So Cl is your left for this co efficient cm
is your pitching moment co efficient and which
we call as a set of output data cl and cm.
So the data has been collected from the 20
second you can see here in x axis it represents
time in second. And these are the state of
input data right so these are the eqce/2v
alpha for the same order time so. Now we have
a set of input data and output data once you
use this technique or least square method
then you will get parameter like this.
.
So these are the few parameters which I have
shown you here they are usually strong parameter
cl alpha clq cl is cm alpha cmq cm e yeah.
And so this least square method gives those
parameters in one short computation so least
square method is about the single shot computation
and it is very efficient here the value of
clq might not have come correctly basically
q derivatives we have not exacted the complete
dynamics like so maybe there is a slight change
in the in cq derivatives but other variables
are quite okay in comparison to internal values.
Yeah so now in the next lecture the next tutorial
we will be learning about the same parameterized
estimation technique or aerodynamic parameterized
technical from the different methods so here
we talk about least square method right and
then in the next class we will learn about
method which is based on the black box model
right and then you will be able to appropriate
the method of method which is based on artificial
method. And then we will have the comparison
of the results from the least square method.
And then you see which one is better or they
are comparable or what is the confidence interval
of those methods so thank you so much.
