WEBVTT
Kind: captions
Language: en
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Hello and welcome to the tutorial. So in the
previous class we have seen this experiment
00:00:20.930 --> 00:00:26.981
about parameter estimation using least square
methods. So there we have seen what is parameter
00:00:26.981 --> 00:00:32.250
estimation? And how to estimate the parameters
and why do we lead the parameter estimation
00:00:32.250 --> 00:00:39.629
and everything. So we understood and I also
discussed about various methods are available
00:00:39.629 --> 00:00:44.379
to do this parameter estimation and I have
discussed about least square method.
00:00:44.379 --> 00:00:49.219
So that we have learnt what is the basic principle
of least squares, how least square works and
00:00:49.219 --> 00:00:54.449
we also have seen the mathematical formulation
and I have also talked about some examples
00:00:54.449 --> 00:01:00.840
like how can you incorporate the least square
estimation philosophy in the problems and
00:01:00.840 --> 00:01:07.990
other things right. So in this tutorial what
we will be learning is this like earlier lecture
00:01:07.990 --> 00:01:14.130
was about generic parameter estimation like
any process can be modeled BE mechanical,
00:01:14.130 --> 00:01:20.570
electrical or from any other process.
And if your models are parametric then you
00:01:20.570 --> 00:01:25.670
can use the technique of parameter estimation
and maybe least square estimation technique
00:01:25.670 --> 00:01:30.760
to estimate with parameters right. So I will
be discussing today about aerodynamic parameter
00:01:30.760 --> 00:01:35.111
estimation, so I will be talking what is aerodynamic
parameter estimation and how aerodynamic parameter
00:01:35.111 --> 00:01:40.519
estimation is very much important. So let
us start for today’s session.
00:01:40.519 --> 00:01:44.090
.
So again the experiment is about parameter
00:01:44.090 --> 00:01:48.190
estimation, so this parameter estimation is
nothing but aerodynamic parameters. So we
00:01:48.190 --> 00:01:58.729
discussed about definition of parameter estimation
and few methods I have listed which can be,
00:01:58.729 --> 00:02:03.319
you know like from there you can estimate
parameters using the flight data, using the
00:02:03.319 --> 00:02:09.300
states of input data and output data. But
we also should know there are other ways also
00:02:09.300 --> 00:02:12.239
to estimate aerodynamic parameter estimation
right.
00:02:12.239 --> 00:02:33.209
So these methods are like this, first one
is analytical method, next is wind-tunnel
00:02:33.209 --> 00:02:49.680
method, and then third one is flight test
method to which we will be learning in detail
00:02:49.680 --> 00:02:58.140
today flight test method. So earlier part
of the lecture was about this method right.
00:02:58.140 --> 00:03:03.240
So this is a theoretical method you do the
parameter estimation using analytical method
00:03:03.240 --> 00:03:08.651
from the geometrical detail of the aircraft,
but what happens there like you assume some
00:03:08.651 --> 00:03:11.920
complex phenomena, you simplify some complex
phenomena.
00:03:11.920 --> 00:03:18.720
So the estimated parameters may not be so
accurate, but for that you have other methods
00:03:18.720 --> 00:03:23.519
also which is picking up so fast this state
and that is based on the CFD computational
00:03:23.519 --> 00:03:32.549
flow dynamics also all of you know about CFD.
So CFD based methods also can give you the
00:03:32.549 --> 00:03:37.930
computed parameters right. But again to corporate
those parameters whatever you got from the
00:03:37.930 --> 00:03:43.079
analytical methods or from the CFD based methods
you need some experimental methods right.
00:03:43.079 --> 00:03:49.799
So these two are experimental method when
you treat these parameters from the experiment
00:03:49.799 --> 00:04:04.769
doing some experiment is experimental method.
So as I said you need to corporate to which
00:04:04.769 --> 00:04:10.299
rigels or the parameters then we should go
for experimental based method. So let us see
00:04:10.299 --> 00:04:14.629
how wind-tunnel method work.
So what it does like you have a tunnel and
00:04:14.629 --> 00:04:19.410
then you do not, usually you do not put a
full scale model you put the scaled model
00:04:19.410 --> 00:04:23.950
inside the tunnel and you will try to simulate
the accurate mass that you try to give the
00:04:23.950 --> 00:04:31.310
same speed or you maintain the desired flow
which is usually in the flying condition and
00:04:31.310 --> 00:04:39.000
then wind-tunnel methods gives you the measurement
of the force and coefficients there you get
00:04:39.000 --> 00:04:44.070
those parameters. But again since it deals
with the scaled model and then there will
00:04:44.070 --> 00:04:49.640
be issue of the Reynolds number you cannot
exactly similar the Reynolds number in the
00:04:49.640 --> 00:04:53.240
wind-tunnel.
And apart from those things you will have
00:04:53.240 --> 00:05:00.840
issue with this like dynamic derivatives and
other stuff. So again we should look for some
00:05:00.840 --> 00:05:06.940
prominent method where you will get better
estimate of the parameters and with that we
00:05:06.940 --> 00:05:12.040
will start this flight test method where you
will collect the data during the flying condition
00:05:12.040 --> 00:05:18.300
and from the method variables or flight variables
you will try to estimate with parameters right.
00:05:18.300 --> 00:05:22.590
And already I have discussed which parameter
estimation in aerospace area is very important
00:05:22.590 --> 00:05:27.750
because once you talk about designing of control
algorithm or designing of card algorithm in
00:05:27.750 --> 00:05:32.610
aerospace application, then you will lead
to have a very accurate model even if you
00:05:32.610 --> 00:05:37.230
want to do something for the fault, detection
and diagnosis you should have a better model.
00:05:37.230 --> 00:05:41.110
Aerodynamic model is becoming so important
because the accuracy of aerodynamic model
00:05:41.110 --> 00:05:47.420
will dictate the accuracy of your equation
to motion or whole aircraft model.
00:05:47.420 --> 00:05:52.790
So I will not talk about equation of motion
here, so we will start with this aerodynamic
00:05:52.790 --> 00:05:59.470
model okay, aerodynamic model of forces and
moment.
00:05:59.470 --> 00:06:11.960
.
Aerodynamic model is so as you know in this
00:06:11.960 --> 00:06:19.810
aerodynamic model like if you see we have
force lift force drag force and movement I'm
00:06:19.810 --> 00:06:25.050
taking up pitching mode so I am giving you
the example of the longitudinal derivatives
00:06:25.050 --> 00:06:30.760
or longitudinal forces in moment so lift drag
and pitching movement right so you can write
00:06:30.760 --> 00:06:44.510
lift as ½ ? v2 S into CL right this part
is your dynamic pressure and plane form area
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and this called lift force coefficient so
you can write this as general lift force coefficient.
00:07:02.210 --> 00:07:11.960
Right and drag also you can define similar
½ ? v2 S CD where this coefficient CD is
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your drag force coefficient correct and preaching
movement is again ½ ? v2 dynamic pressure
00:07:32.370 --> 00:07:49.550
times S times C bar here this is cord length
and this Cm which is your preaching movement
00:07:49.550 --> 00:08:05.310
coefficient okay so you know force and movement
can be simplified in this term so these are
00:08:05.310 --> 00:08:11.860
basically your non dimensionalize term right
they are very important to study in this equation
00:08:11.860 --> 00:08:20.170
of motion now with the introduction of lift
force lift drag and pitching movement coefficients.
00:08:20.170 --> 00:08:27.240
So I will talk about like this model like
how do you model them right so now you know
00:08:27.240 --> 00:08:30.740
what are those coefficient so.
.
00:08:30.740 --> 00:08:44.120
CL CD and Cm so you know they all are function
of a angle of attack ?? A change in elevator
00:08:44.120 --> 00:08:53.540
deflection and q pitch rate it can also be
a function of other variables like Reynolds
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number V and all so just for simplification
I will take this three variables so CL CD
00:09:00.360 --> 00:09:08.740
Cm they are function of C a ?? E and q right
so if you write the aerodynamic model which
00:09:08.740 --> 00:09:21.170
already you know just for revision I am doing
it again so CL can be written like CL0 + CL
00:09:21.170 --> 00:09:35.499
a into a + CL ?? e times ??e + CL q here we
write CL q time qC/ 2v.
00:09:35.499 --> 00:09:44.730
To make it non dimensionalize and everything
will be in same term and CD you know it will
00:09:44.730 --> 00:10:01.230
be written CD0 + CD a and a + CD ?? e x ?? e
+ CD q x qc/2 v right similar Cm can be written
00:10:01.230 --> 00:10:15.649
as Cm 0 + Cm a x a+ Cm ?? e x ?? e+ Cm q c/
2v so they are called aerodynamic model yeah
00:10:15.649 --> 00:10:24.709
so already I have written here so this these
things are your aerodynamic model force and
00:10:24.709 --> 00:10:30.529
movement coefficients right.
So here you see that all the variables like
00:10:30.529 --> 00:10:37.300
CL CD and Cm they are well structured right
so this is the structured model where CL 0
00:10:37.300 --> 00:10:43.910
CL a CL ?? AND CL q they have their own physical
significant and they are called aerodynamic
00:10:43.910 --> 00:10:56.740
parameters so if fewer diameters like CL and
CL a Cm a which is related to your a or may
00:10:56.740 --> 00:11:02.149
be CD a that talk about the stability basically
these things about talk about the stabilities
00:11:02.149 --> 00:11:11.389
so they can be called as stability derivatives
also your CL q CD q CD q will be very week
00:11:11.389 --> 00:11:17.870
parameter and Cm q so that talk about the
stability so they can also called as stability
00:11:17.870 --> 00:11:36.829
parameter or stability derivative okay.
And the parameters which are related t control
00:11:36.829 --> 00:11:49.490
here control is ??e elevator control here
so CL ?? e CD ?? e and Cm ?? e they are called
00:11:49.490 --> 00:12:02.399
control derivative
now why they are derivatives so it is quite
00:12:02.399 --> 00:12:06.130
clear from the nature of the parameter because
if you see.
00:12:06.130 --> 00:12:13.470
.
CL a ? CL ? a v it says the change in a what
00:12:13.470 --> 00:12:19.350
about change you observe in CL like the change
in CL because of change a this is defined
00:12:19.350 --> 00:12:27.010
by CL a right so accordingly everything will
be like that CL ?? e change in CL with the
00:12:27.010 --> 00:12:32.779
change in ?? e so this is called your control
derivative right so once you have the structured
00:12:32.779 --> 00:12:37.740
model right so they again become a parametric
models.
00:12:37.740 --> 00:12:44.619
So today I will talking about the A is said
earlier about the aerodynamic parameter estimation
00:12:44.619 --> 00:12:50.300
and I will show you with examples and this
is also the demonstration of the technique
00:12:50.300 --> 00:12:56.619
what we have learn in previous class right
so I will take two parameters two variables
00:12:56.619 --> 00:13:02.329
sorry force CL and movement Cm pitching movement
coefficient.
00:13:02.329 --> 00:13:21.220
So you now you know the model CL is CL0 +
CL a into a + CL ?? E into ?? E + CLQ X QC/2B
00:13:21.220 --> 00:13:37.779
yes and CL0+ CM a times a CM ?? E times ?? E
CLQ QC/2V okay so it can also be written like
00:13:37.779 --> 00:13:49.519
this yeah, CL and same I will write in a vector
form like this and if you write all the parameters
00:13:49.519 --> 00:14:06.610
here like CL0 CL a next is your CL ?? E and
CL q and this you have CM 0 CM alpha, CM ?? E,
00:14:06.610 --> 00:14:27.019
CM q.
1 a ?? E and QC/2Vyes so this is the same
00:14:27.019 --> 00:14:31.970
equation actually represent as in terms of
matrix so now if you see it will look like
00:14:31.970 --> 00:14:43.490
this y about the output force and moment co-efficient
and if I can call this as a ? matrix which
00:14:43.490 --> 00:14:51.509
is a parameter matrix and input if I represent
by S so this is again it is again in terms
00:14:51.509 --> 00:15:02.220
of this very well known equation right, y
= ? Xa linear equation right, so now this
00:15:02.220 --> 00:15:08.540
had become a candidate for less square we
have seen in previous class.
00:15:08.540 --> 00:15:15.459
How to estimate the parameters from the linear
model which what like there yeah so I will
00:15:15.459 --> 00:15:27.519
demonstrate the principle of less square on
this example right, so now let us talk about
00:15:27.519 --> 00:15:36.440
the flight data right because now as I said
this is output this and here this is input
00:15:36.440 --> 00:15:51.290
right, output this is your input and this
is parameter vector matrix, and you want to
00:15:51.290 --> 00:15:57.670
find those aero dynamic parameters so this
is the problem for this in this example.
00:15:57.670 --> 00:16:04.129
Now how to estimate those thing before that
I will be talking about ho9w do you get these
00:16:04.129 --> 00:16:10.480
outputs CL and CN, and how do you collect
all the inputs right so as we know there is
00:16:10.480 --> 00:16:14.499
no sensor which can directly give you the
force and moment co-efficient during the flying
00:16:14.499 --> 00:16:21.779
condition so CL and CN we cannot directly
measure right, so now equation comes how will
00:16:21.779 --> 00:16:24.670
you get CL and Cm.
Because now we are talking about the system
00:16:24.670 --> 00:16:29.209
identification or parameter identification
it is about the input and output so how will
00:16:29.209 --> 00:16:37.111
you generate the output or how will you get
that so it means can we do something do we
00:16:37.111 --> 00:16:44.040
have other side variables which is related
to force and moment and from that we can get
00:16:44.040 --> 00:16:48.300
this force and movement co-efficient so can
we do something like that, so let us see again
00:16:48.300 --> 00:16:52.459
CL and CM.
And as you know most of the sensor which is
00:16:52.459 --> 00:16:56.329
in build in aircraft which is mounted in aircraft
they give you the information in terms of
00:16:56.329 --> 00:17:03.110
body frame and for in your information you
know CL specially this lift force they are
00:17:03.110 --> 00:17:10.600
in wind frame not in body frame so you need
a conversion from body frame to wind frame
00:17:10.600 --> 00:17:15.470
right so CL how do you establish the relation
like how you will get Cl.
00:17:15.470 --> 00:17:20.579
.
So CL as we know CL can be also written as
00:17:20.579 --> 00:17:33.029
Cx Sin a right minus Cj Cos a right so now
they are also force co-efficient in body frame
00:17:33.029 --> 00:17:42.710
and you know that actually you see that the
aircraft if your x is related like this if
00:17:42.710 --> 00:17:54.140
you define XX is true also know y here and
Z so the force in this direction will be FX,
00:17:54.140 --> 00:18:03.340
FY and FZ, right and again you know force
can be written as Q bar then represent a time
00:18:03.340 --> 00:18:19.400
S x Cx.
So you basically get Cx from this or Cz from
00:18:19.400 --> 00:18:27.760
this dynamic pressure times surface area right
so you know the meaning of Cx and Cz right
00:18:27.760 --> 00:18:33.919
so just I have told you so that you can get
the standing.
00:18:33.919 --> 00:18:38.890
.
You can align your understanding whether right
00:18:38.890 --> 00:18:48.419
but again we do not have sensors directly
which directly measures CX and Cz then Cx
00:18:48.419 --> 00:18:58.030
as you know it can be written as mass times
acceleration actually expression in X direction
00:18:58.030 --> 00:19:13.549
minus trust hydrogen with FE so please remember
this is your trust force right, okay. Divided
00:19:13.549 --> 00:19:23.250
by dynamic pressure time area but if there
is inclination angle with the incidence so
00:19:23.250 --> 00:19:27.130
this is called engine insertion angle so that
component also will come here.
00:19:27.130 --> 00:19:47.980
So that will be Cos s T right and this is
your engine inclination angle, so that I though
00:19:47.980 --> 00:19:57.070
which we will be talking where the value will
be 0 so this is not a very dominant term but
00:19:57.070 --> 00:20:02.169
to therefore the sake of completion of this
equation I have written this so that you should
00:20:02.169 --> 00:20:16.410
remember. And see that you will get maz, m
time az minus not will become plus Fesin component
00:20:16.410 --> 00:20:29.400
on this angle, okay.
Now we know from this equations we know that
00:20:29.400 --> 00:20:37.240
data for ax trust force dynamic pressures,
surface area this every things are know rigjt,
00:20:37.240 --> 00:20:44.950
because IMU will give you the details of acceleration
and from the geometrical information of the
00:20:44.950 --> 00:20:51.140
aircraft you will get S and sensors are there
to measure dynamic pressures you will get
00:20:51.140 --> 00:20:56.580
the dynamic pressure, right.
And you will also get az so in this equations
00:20:56.580 --> 00:21:02.049
everything can be measured through the flight
right, so once you have measured details about
00:21:02.049 --> 00:21:08.710
the variables you will cx and cz and by knowing
cx and cz you get CL right, so that is how
00:21:08.710 --> 00:21:14.840
you will get the CL lift course coefficient.
Now coming to the PC moment coefficient this
00:21:14.840 --> 00:21:18.900
one how will you do that let how will you
measure that or how will you derive from the
00:21:18.900 --> 00:21:26.490
flight variables just write the moment equation,
so all of you know moment can be written like
00:21:26.490 --> 00:21:52.169
this M=Iyq? right, +Ixz(p2-r2)+pr times Ix-Iz
so this is the moment equation right.
00:21:52.169 --> 00:22:07.490
And further you can write M as 1/2?v2sccm
basically this is your chord length c we can
00:22:07.490 --> 00:22:12.419
write c I will just use this notation as a
c and c¯ interchangeably so here I have written
00:22:12.419 --> 00:22:24.909
c so let us write c, but here please understand
this is your chord length, okay right and
00:22:24.909 --> 00:22:34.040
then the same equation okay, so here you can
get same by knowing all those variables. So
00:22:34.040 --> 00:22:38.919
you know the I'm contains the accelerometer,
gyroscope, magnetometer so from the gyroscope
00:22:38.919 --> 00:22:47.370
you will get pr and q everything and from
the geometrical details you will get Ixz,
00:22:47.370 --> 00:22:53.889
Iy and Ix znd Iz to you see parameters you
will get.
00:22:53.889 --> 00:23:02.049
Now this is the only thing where we should
look for it is q? because most of the aircraft
00:23:02.049 --> 00:23:07.150
will not have the sensor which can directly
measure q? or in our aircraft we do not have
00:23:07.150 --> 00:23:14.960
the sensor which can directly measure q?,
so how will you get this q? you know q? variable
00:23:14.960 --> 00:23:27.040
right, so but we know like we can measure
q, q can be measured or the data is evaluable,
00:23:27.040 --> 00:23:34.250
okay. So from any numerical difference in
technique if you have those q you can get
00:23:34.250 --> 00:23:40.380
that q? this is not a problem.
But problem will come when your q is not accurately
00:23:40.380 --> 00:23:46.990
measured it has some amount of data noise
some amount of noise in this signal and if
00:23:46.990 --> 00:23:51.280
you take the derivative of that signal it
will further amplifier a noise so the signal
00:23:51.280 --> 00:23:56.750
will become even worst, and once you do not
have the accurate q? you will not be able
00:23:56.750 --> 00:24:02.529
to get accurate output Cn and once you do
not have the accurate output then the estimated
00:24:02.529 --> 00:24:08.520
parameters will not accurate, so that is why
we have to be very careful the kind of data
00:24:08.520 --> 00:24:15.210
you are selecting or we are feeding in this
algorithm those data should be very accurate
00:24:15.210 --> 00:24:19.620
originally accurate.
So now questions will come like how can we
00:24:19.620 --> 00:24:30.470
region ably estimate or obtain this q? from
q right, so now suppose there is small amount
00:24:30.470 --> 00:24:39.899
of error or some amount of error in q right,
then we need to identify the error or we need
00:24:39.899 --> 00:24:49.950
to select if there is any presence of error
so how will you detect that so you apply FFT
00:24:49.950 --> 00:24:57.200
from the FFT it is called fast Fourier transformation
right, so what it does like it will give you
00:24:57.200 --> 00:25:06.950
the frequency components present in the signal
so if you pass through at and then if you
00:25:06.950 --> 00:25:17.350
see this amplitude verse frequency draw so
it will generate some spics right like this.
00:25:17.350 --> 00:25:24.840
Wherever frequency terms are there you will
get those spics, so I think let me tell you
00:25:24.840 --> 00:25:29.740
about the little bit about the FFT, so that
you will be able to appreciate okay, you might
00:25:29.740 --> 00:25:38.250
not have learned it right. So before that
so suppose you have a very clean signal or
00:25:38.250 --> 00:25:46.659
sinusoidal signal like everyone have seen
this signal right, sinusoidal signal, okay.
00:25:46.659 --> 00:25:49.140
.
So sinusoidal signal if you see mathematically
00:25:49.140 --> 00:25:58.019
it will look like this right, 2pft sin?t so
here this signal have only one frequency component
00:25:58.019 --> 00:26:04.500
which is f right, and if you do this fast
Fourier transformation or Fourier spectrum
00:26:04.500 --> 00:26:10.299
analysis of this signal through fast furrier
transformation the smart this would I was
00:26:10.299 --> 00:26:17.139
trying to discuss here then it since it has
only one frequency component it will generate
00:26:17.139 --> 00:26:21.950
only one spike right.
And it was add frequency of f so you will
00:26:21.950 --> 00:26:28.919
get the spike at f with the same magnitude
or amplitude you can also see the amplitude
00:26:28.919 --> 00:26:38.820
of the signal if it has two frequency components
then you will get two spike generate for corresponding
00:26:38.820 --> 00:26:44.009
frequency right.
So this is the idea of your fast furrier transformation
00:26:44.009 --> 00:26:53.830
okay so as I said if q has noise then usually
noise enters through a very high frequency
00:26:53.830 --> 00:27:02.720
atoms so you will get to know once you do
the spectrum analysis using FFT and if you
00:27:02.720 --> 00:27:12.980
have seen instead few glitches or few spike
at higher frequency it may not be so big in
00:27:12.980 --> 00:27:18.460
the amplitude it will be a small maybe in
multiple spikes at higher frequencies then
00:27:18.460 --> 00:27:23.380
this will be dominant one so it may look like
slightly bigger right.
00:27:23.380 --> 00:27:29.020
So from here you can easily identify that
noise high frequency noise entered in the
00:27:29.020 --> 00:27:37.510
signal here and it will look like this these
are the high frequency noise usually noise
00:27:37.510 --> 00:27:45.389
is higher of high frequency you will be able
to identify the noise in the signal by doing
00:27:45.389 --> 00:27:52.330
this first data formation.
Now you can easily identify these are the
00:27:52.330 --> 00:27:57.690
frequency which we do not need ion the signal
so you can eliminate those higher frequency
00:27:57.690 --> 00:28:03.549
noise terms but designing a suitable low pass
filter right so low pass filter what it does
00:28:03.549 --> 00:28:11.840
like low pass filter LPF it rejects all the
higher frequency component terms right all
00:28:11.840 --> 00:28:18.419
the higher frequency terms and it will give
you the signal which is related to low pass
00:28:18.419 --> 00:28:21.379
like a flow frequency.
So you can reject all those higher frequency
00:28:21.379 --> 00:28:26.019
noises ort higher frequency term by employing
the low pass filter and then you can give
00:28:26.019 --> 00:28:32.160
the information of cut of frequency in the
low pass filter from the FFT analysis of this
00:28:32.160 --> 00:28:43.270
Q signal right and once you got the q signal
clean then you employ any numerical difference
00:28:43.270 --> 00:28:53.179
in technique
to get this q dot right so now you got the
00:28:53.179 --> 00:28:58.820
q dot so in this equation you have the information
about q dot now you already know about inertia
00:28:58.820 --> 00:29:04.289
parameters you already got information about
p and r from the magnet like from the IMU
00:29:04.289 --> 00:29:08.950
through a gruel scope.
So all the flight variables which are use
00:29:08.950 --> 00:29:14.330
in this equation or known and from that we
can get this CM so that is how you will get
00:29:14.330 --> 00:29:24.250
Cm right, okay so ion the output you got CL
and CM and you can directly measure a from
00:29:24.250 --> 00:29:29.629
the a sensor from flight test you have the
information about the elevated deflection
00:29:29.629 --> 00:29:36.280
from the sensor so this also it measure and
q definitely you know and C is your code line
00:29:36.280 --> 00:29:41.049
and velocity is also measured.
So now ion the set of input and output everything
00:29:41.049 --> 00:29:46.970
is known so what is not known or what is unknown
of this, this parameters we do not know about
00:29:46.970 --> 00:29:56.490
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
00:29:56.490 --> 00:30:06.840
of y = ? x right so you can think of using
a least square in this problem and then you
00:30:06.840 --> 00:30:14.879
will get the ? so how will you do that like
from the simple technique you can y = ? x
00:30:14.879 --> 00:30:19.490
now right so how will be the how will your
estimated parameter will look like this so
00:30:19.490 --> 00:30:26.799
look like in this thing again least square
technique says like it minimize the summation
00:30:26.799 --> 00:30:31.080
of the error if you remove the error part
what I discuss in earlier class.
00:30:31.080 --> 00:30:39.070
So what you do like you multiply first multiply
with xT here or both the side right so now
00:30:39.070 --> 00:30:56.899
your ? will become y xT times x xT so that
will be your estimated parameter okay, so
00:30:56.899 --> 00:31:02.320
what I will do like we have gather the data
on unsure graph so I will show you the how
00:31:02.320 --> 00:31:06.779
data has been collected so you will be able
to see all the information about the flight
00:31:06.779 --> 00:31:15.490
variables a velocity and qrp and everything
and from there we have derived CL and CM so
00:31:15.490 --> 00:31:17.530
what I am do like.
I will show you through the slide so that
00:31:17.530 --> 00:31:22.460
will be able to appreciate more once you see
it visually and then from there employing
00:31:22.460 --> 00:31:27.039
this least square technique we have estimated
this parameter vectors ? there you will get
00:31:27.039 --> 00:31:34.210
to know about all the CLL a CM a CM d here
whatever actually parameters we have talked
00:31:34.210 --> 00:31:38.960
so all the parameters can be obtain using
this re square technique so from the Mattel
00:31:38.960 --> 00:31:43.059
of simulation we have done ity I will shoe
you in this slide okay.
00:31:43.059 --> 00:31:50.039
.
So I will show you the results which we got
00:31:50.039 --> 00:31:54.960
for this parameter estimation right so yes.
.
00:31:54.960 --> 00:32:01.009
So this is air craft Hansa 3 air craft we
have use to gather the data and if you see
00:32:01.009 --> 00:32:07.490
that it is a twin seated air craft research
air craft so we use this for our research
00:32:07.490 --> 00:32:17.379
work. And then where it is 760kg as 12.74
meter square in span is 1.221 meter cord length
00:32:17.379 --> 00:32:29.899
is 10.47 meter as the procedure 8.8. And inertia
parameters are like I X is 73kg meter square
00:32:29.899 --> 00:32:42.039
IYY is your 907 kg meter square IZZ is 1680
meter square IXZ is 1144 kg meter square.
00:32:42.039 --> 00:32:45.809
.
This is your geometrical parameters of this
00:32:45.809 --> 00:32:51.370
hansa 3 aircraft yes. So these are the longitudinal
data we have gathered during the flying condition
00:32:51.370 --> 00:33:00.610
you see here elevated has been deflected so
this is a standard signal which we say 3211
00:33:00.610 --> 00:33:13.990
yes. So if can see the elevator which is right
at the bottom here and we tried to make it
00:33:13.990 --> 00:33:24.639
like a 3211. 3211 says like it should be 3t
time’s upper down 2 t times if it is down
00:33:24.639 --> 00:33:28.520
then it should go up then three t times down
and up.
00:33:28.520 --> 00:33:38.669
So this is over flee 3 2 t t and t. so we
tried our best to get those 3211 signal through
00:33:38.669 --> 00:33:44.630
this elevator to side the short period more
right. And then this is the deflection in
00:33:44.630 --> 00:33:53.710
? 2 ax a1 ay az ax will ascertain alpha with
the change in the with the change in the elevator
00:33:53.710 --> 00:33:58.779
deflection so you have seen the changes in
the alpha so it is quite following. Right
00:33:58.779 --> 00:34:06.070
like we have a deflection here and corresponding
to here started seeing the deflection in the
00:34:06.070 --> 00:34:10.340
table alpha velocity.
We tried to velocity for short period should
00:34:10.340 --> 00:34:17.530
be constant but it is stably constant in the
accuracy of may be 5 or 7 meter per second
00:34:17.530 --> 00:34:25.770
it has gone off and you see the deflection
in the acceleration data the x data it is
00:34:25.770 --> 00:34:34.290
of 2 meters 2 meter per second square you
can see roughly and sz also changes may be
00:34:34.290 --> 00:34:42.700
-15 to it has gone slightly lower to 0 and
there is a change in q right.
00:34:42.700 --> 00:34:46.480
.
The digital in radian per second units are
00:34:46.480 --> 00:34:53.040
already written here so you can see so you
saw the changes in the q and ? and other variables
00:34:53.040 --> 00:35:00.870
are not listed in this but we collect all
those data fight data which has been discussed
00:35:00.870 --> 00:35:08.930
during this lecture right. so I have shown
you those important variables through this
00:35:08.930 --> 00:35:15.210
graph right next thing was which I was talking
about the longitudinal input and then output
00:35:15.210 --> 00:35:23.109
I have talked about two derivates so in this
graph. You see like these are the four set
00:35:23.109 --> 00:35:26.480
moment co efficient.
So Cl is your left for this co efficient cm
00:35:26.480 --> 00:35:33.530
is your pitching moment co efficient and which
we call as a set of output data cl and cm.
00:35:33.530 --> 00:35:38.750
So the data has been collected from the 20
second you can see here in x axis it represents
00:35:38.750 --> 00:35:45.880
time in second. And these are the state of
input data right so these are the eqce/2v
00:35:45.880 --> 00:35:51.740
alpha for the same order time so. Now we have
a set of input data and output data once you
00:35:51.740 --> 00:35:56.340
use this technique or least square method
then you will get parameter like this.
00:35:56.340 --> 00:36:01.109
.
So these are the few parameters which I have
00:36:01.109 --> 00:36:11.160
shown you here they are usually strong parameter
cl alpha clq cl is cm alpha cmq cm e yeah.
00:36:11.160 --> 00:36:16.200
And so this least square method gives those
parameters in one short computation so least
00:36:16.200 --> 00:36:22.710
square method is about the single shot computation
and it is very efficient here the value of
00:36:22.710 --> 00:36:31.510
clq might not have come correctly basically
q derivatives we have not exacted the complete
00:36:31.510 --> 00:36:39.030
dynamics like so maybe there is a slight change
in the in cq derivatives but other variables
00:36:39.030 --> 00:36:48.079
are quite okay in comparison to internal values.
Yeah so now in the next lecture the next tutorial
00:36:48.079 --> 00:36:52.930
we will be learning about the same parameterized
estimation technique or aerodynamic parameterized
00:36:52.930 --> 00:36:59.250
technical from the different methods so here
we talk about least square method right and
00:36:59.250 --> 00:37:07.020
then in the next class we will learn about
method which is based on the black box model
00:37:07.020 --> 00:37:15.790
right and then you will be able to appropriate
the method of method which is based on artificial
00:37:15.790 --> 00:37:21.119
method. And then we will have the comparison
of the results from the least square method.
00:37:21.119 --> 00:37:27.829
And then you see which one is better or they
are comparable or what is the confidence interval
00:37:27.829 --> 00:37:34.630
of those methods so thank you so much.