HW7
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Tulane University *
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516
Subject
Mechanical_engineering
Date
May 24, 2024
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Uploaded by LieutenantAardvark4281
AE512 - Dynamics, Stability, and Control of Vehicles - Spring 2024
Homework 6
Due Date
: Sunday, May 12
th
, 2024 at 11:59 PM (PST)
1.
Cost function practice
(a)
Create a Matlab function
cost
p
1
.m
which computes the cost function at a given
point
Figure 1:
cost
p
1
.m
(b)
Create a Matlab function
gradient
p
1
.m
which computes the gradient of the cost
function at a given point
Figure 2:
gradient
p
1
.m
(c)
Use an initial point, numerically solve this problem using a gradient descent
method
Figure 3:
Gradientdescentsetup
Figure 4:
Resultingsolutionsover
35
steps
(d)
Plot the cost function over a given range, On the same figure, plot the sequence.
On a separate graph, plot the cost as a function of the step number
Figure 5: plots of the gradient descent results and the cost vs steps.
(e)
Using a different initial condition and repeat part C and D
Figure 6: plots of the gradient descent results and the cost vs steps.
Figure 7: plots of the gradient descent results and the cost vs steps.
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(f)
For both initial conditions, does your solution converge to a local minimum? If
it does not converge, explain why not. What improvements can be made to the
algorithm to encourage convergence?
The solution changes when the initial conditions are swapped. Both solutions converge to
different local minima rather than the same global minima.
An improvement that could
be made is to take larger steps and iterate to smaller steps as you get closer to the global
minima.
2.
Trim the RCAM model for steady state, straight and level flight
(a)
What are the mathematical constraints on the state/control of the model.
The mathematical constraints for this problem are as follows, ˙
x
= 0,
V a
= 85
m
s
,
γ
= 0, and
x
2 =
x
7 =
x
9 = 0. The latter terms represent the no constraints for the side slip and bank
angle, and
x
9 = 0 refers to the craft heading due north.
(b)
incorporate these constraints into a cost function and translate to an uncon-
strained optimization problem
Figure 8:
cost
s
traight
l
evel.m
(c)
What is the dimension of the decision vector for the optimization problem
The decision vector for this problem is a 14x1 vector.
(d)
What is a reasonable starting point for the numerical optimization?
Will this
choice of starting point influence the optimization?
A good guess would be 85 for Z(10) to represent the airspeed target given. This guess will
influence the optimization if the gradient contains local minima. If the local minima nearest
this point does not align with the global minima, the results from the optimization may not
be the most stable conditions.
(e)
Use the function fminsearch in Matlab to attempt to find the trim point for the
RCAM model at the desired flight condition.
Figure 9: Matlab code setup
Figure 10: Results from fminsearch run
3.
Trim point verification
(a)
Verify that the trim point found in problem 2 is trim point by simulating the
non-linear RCAM model with this point as an initial condition
See below charts showing the results from using this trim point in the RCAM simulation.
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Figure 11: Results From non perturbed system
(b)
Perturb the system by introducing a small, temporary input of your choosing to
the control surfaces
Below charts show perturbed results from same simulation. In this simulation, the ailerons
and the elevators slightly for 2 seconds. Thsystem appears to be stable at this point as the
figures are both stable even with the input. The Matlab code and simulink set up is attached
below.
Figure 12: Results From perturbed system