where the heliocentric orbits of Earth and Mars are modeled by circular orbits. Use the following values for their semi-major axes and true anomalies at to = 0: {Earth, Earth (to 0)} = {1.0 AU,0.0 rad} and {aMars, Mars (to = 0)}={1.524 AU, rad}. We represent the state vector L.e., z=[r', ']' € R', by the position and velocity vectors in the heliocentric inertial frame, Use non-dimensional values for all the physical quantities with the characteristic length /* = 1 AU and the characteristic time t√/ so that the non-dimensional value of becomes unity. You may use the non-dimensional unit for the answers in the problem, including plots. We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form of a set of stochastic differential equation (SDE) as follows: da = (Ax+ Bu)dt + Gdw, da f(x, u, t)dt + Gdw, - (1) (2) ิล (t-0) and Po-diag(lo We consider the following sources of uncertainties acting on the system. The orbital state at Earth depar- ture (e.g., launch dispersion) is given by ~N(o, Po), where o the position and velocity standard deviations are given by pos = vel = 3 x 10-1 (non-dimensional). Un- modeled disturbance is characterized by Brownian motion as in Eq. (2), where G is defined as: G= 02×2 Odiet/2] (4) where dist ER represents the intensity of disturbance, and use dist = 10-3 (non-dimensional). We assume open-loop control with no uncertainty, i.e., u(t) = u(t). (a): Provide the expression of the nonlinear SDE, Eq. (2), in the form specific to the considered problem. (b): Approximate the Brownian motion (use At 10-3, non-dimensional) and perform Monte Carlo sim- ulation (M =20) by propagating the nonlinear SDE for each sample drawn from the distributions defined above. Report relevant plots, including the time history of each state element of Monte Carlo trajectories, Monte Carlo position trajectories in the 2-D plot along with the reference trajectory, etc. );
where the heliocentric orbits of Earth and Mars are modeled by circular orbits. Use the following values for their semi-major axes and true anomalies at to = 0: {Earth, Earth (to 0)} = {1.0 AU,0.0 rad} and {aMars, Mars (to = 0)}={1.524 AU, rad}. We represent the state vector L.e., z=[r', ']' € R', by the position and velocity vectors in the heliocentric inertial frame, Use non-dimensional values for all the physical quantities with the characteristic length /* = 1 AU and the characteristic time t√/ so that the non-dimensional value of becomes unity. You may use the non-dimensional unit for the answers in the problem, including plots. We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form of a set of stochastic differential equation (SDE) as follows: da = (Ax+ Bu)dt + Gdw, da f(x, u, t)dt + Gdw, - (1) (2) ิล (t-0) and Po-diag(lo We consider the following sources of uncertainties acting on the system. The orbital state at Earth depar- ture (e.g., launch dispersion) is given by ~N(o, Po), where o the position and velocity standard deviations are given by pos = vel = 3 x 10-1 (non-dimensional). Un- modeled disturbance is characterized by Brownian motion as in Eq. (2), where G is defined as: G= 02×2 Odiet/2] (4) where dist ER represents the intensity of disturbance, and use dist = 10-3 (non-dimensional). We assume open-loop control with no uncertainty, i.e., u(t) = u(t). (a): Provide the expression of the nonlinear SDE, Eq. (2), in the form specific to the considered problem. (b): Approximate the Brownian motion (use At 10-3, non-dimensional) and perform Monte Carlo sim- ulation (M =20) by propagating the nonlinear SDE for each sample drawn from the distributions defined above. Report relevant plots, including the time history of each state element of Monte Carlo trajectories, Monte Carlo position trajectories in the 2-D plot along with the reference trajectory, etc. );
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
Can you help me by providing the solutions in MATLAB code?
![where the heliocentric orbits of Earth and Mars are
modeled by circular orbits. Use the following values for their semi-major axes and true anomalies at to = 0:
{Earth, Earth (to 0)} = {1.0 AU,0.0 rad} and {aMars, Mars (to = 0)}={1.524 AU, rad}.
We represent the state vector
L.e., z=[r', ']' € R',
by the position and velocity vectors in the heliocentric inertial frame,
Use non-dimensional values for all the physical quantities with the characteristic length /* = 1 AU and
the characteristic time t√/ so that the non-dimensional value of becomes unity. You may use the
non-dimensional unit for the answers in the problem, including plots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F4f8f12d4-5a22-4fa6-807c-f2e84a070f5a%2Fyj4bfe_processed.png&w=3840&q=75)
Transcribed Image Text:where the heliocentric orbits of Earth and Mars are
modeled by circular orbits. Use the following values for their semi-major axes and true anomalies at to = 0:
{Earth, Earth (to 0)} = {1.0 AU,0.0 rad} and {aMars, Mars (to = 0)}={1.524 AU, rad}.
We represent the state vector
L.e., z=[r', ']' € R',
by the position and velocity vectors in the heliocentric inertial frame,
Use non-dimensional values for all the physical quantities with the characteristic length /* = 1 AU and
the characteristic time t√/ so that the non-dimensional value of becomes unity. You may use the
non-dimensional unit for the answers in the problem, including plots.
![We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form
of a set of stochastic differential equation (SDE) as follows:
da
=
(Ax+ Bu)dt + Gdw,
da f(x, u, t)dt + Gdw,
-
(1)
(2)
ิล
(t-0) and Po-diag(lo
We consider the following sources of uncertainties acting on the system. The orbital state at Earth depar-
ture (e.g., launch dispersion) is given by ~N(o, Po), where o
the position and velocity standard deviations are given by pos = vel = 3 x 10-1 (non-dimensional). Un-
modeled disturbance is characterized by Brownian motion as in Eq. (2), where G is defined as:
G=
02×2
Odiet/2]
(4)
where dist ER represents the intensity of disturbance, and use dist = 10-3 (non-dimensional). We assume
open-loop control with no uncertainty, i.e., u(t) = u(t).
(a): Provide the expression of the nonlinear SDE, Eq. (2), in the form specific to the considered problem.
(b): Approximate the Brownian motion (use At 10-3, non-dimensional) and perform Monte Carlo sim-
ulation (M =20) by propagating the nonlinear SDE for each sample drawn from the distributions
defined above. Report relevant plots, including the time history of each state element of Monte Carlo
trajectories, Monte Carlo position trajectories in the 2-D plot along with the reference trajectory, etc.
);](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F4f8f12d4-5a22-4fa6-807c-f2e84a070f5a%2Fzgumqld_processed.png&w=3840&q=75)
Transcribed Image Text:We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form
of a set of stochastic differential equation (SDE) as follows:
da
=
(Ax+ Bu)dt + Gdw,
da f(x, u, t)dt + Gdw,
-
(1)
(2)
ิล
(t-0) and Po-diag(lo
We consider the following sources of uncertainties acting on the system. The orbital state at Earth depar-
ture (e.g., launch dispersion) is given by ~N(o, Po), where o
the position and velocity standard deviations are given by pos = vel = 3 x 10-1 (non-dimensional). Un-
modeled disturbance is characterized by Brownian motion as in Eq. (2), where G is defined as:
G=
02×2
Odiet/2]
(4)
where dist ER represents the intensity of disturbance, and use dist = 10-3 (non-dimensional). We assume
open-loop control with no uncertainty, i.e., u(t) = u(t).
(a): Provide the expression of the nonlinear SDE, Eq. (2), in the form specific to the considered problem.
(b): Approximate the Brownian motion (use At 10-3, non-dimensional) and perform Monte Carlo sim-
ulation (M =20) by propagating the nonlinear SDE for each sample drawn from the distributions
defined above. Report relevant plots, including the time history of each state element of Monte Carlo
trajectories, Monte Carlo position trajectories in the 2-D plot along with the reference trajectory, etc.
);
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