(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft. Define a =[r] and x = x R to represent the deputy orbital state and its target (= chief orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the ECI frame. Denote the relative state by dx=x-xe Table 2 summarize the initial orbital elements. Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame) Orbital element Unit Chief ad 11500 a = 10000 km Deputy semi-major axis eccentricity ed=0.15 e=0.3 inclination id=35 ic=50 degree right ascension of ascending node = 50 Se 50 degree argument of periapsis Wd=40 We = 40 degree true anomaly at epoch Ve=0 degree (b.1): Derive the error dynamics of our system in ECI frame under the influence of u. (b.2): Consider a candidate Lyapunov function V = r K,6r+v Sv, where K = K, and K, > 0. Discuss the positive definiteness of V, and derive the Lyapunov rate of this system. (b.3): Derive a stabilizing controller such that satisfies V=-8v Pôv where P > 0, and discuss the stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?). (b.4): Show the asymptotic stability of the system by applying either Theorem 1 or Theorem 2. (b.5): For diagonal K, and P, design three types of controller gains to achieve critically, under, or over damped systems, and report the relation K, and P needs to satisfy for each controller. (b.6): Perform the numerical integrations of the controlled system with each of the (i) critically- damping, (ii) over-damping, and (iii) under-damping controllers, where assume K, = 13 and scale the length and time units appropriately as discussed in class. Simulate the dynamics for at least 5 revolutions of the chief Keplerian orbits. Discuss the results with relevant plots. (b.7): having the maximum acceleration of the thruster, i.e., ||ul|2 max, changes the stability property of the controlled system, and hence the controller performance. Perform numerical simulations with max = 1 m/s² for the three controller gains and discuss the result with relevant plots.

(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft. Define a =[r] and x = x R to represent the deputy orbital state and its target (= chief orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the ECI frame. Denote the relative state by dx=x-xe Table 2 summarize the initial orbital elements. Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame) Orbital element Unit Chief ad 11500 a = 10000 km Deputy semi-major axis eccentricity ed=0.15 e=0.3 inclination id=35 ic=50 degree right ascension of ascending node = 50 Se 50 degree argument of periapsis Wd=40 We = 40 degree true anomaly at epoch Ve=0 degree (b.1): Derive the error dynamics of our system in ECI frame under the influence of u. (b.2): Consider a candidate Lyapunov function V = r K,6r+v Sv, where K = K, and K, > 0. Discuss the positive definiteness of V, and derive the Lyapunov rate of this system. (b.3): Derive a stabilizing controller such that satisfies V=-8v Pôv where P > 0, and discuss the stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?). (b.4): Show the asymptotic stability of the system by applying either Theorem 1 or Theorem 2. (b.5): For diagonal K, and P, design three types of controller gains to achieve critically, under, or over damped systems, and report the relation K, and P needs to satisfy for each controller. (b.6): Perform the numerical integrations of the controlled system with each of the (i) critically- damping, (ii) over-damping, and (iii) under-damping controllers, where assume K, = 13 and scale the length and time units appropriately as discussed in class. Simulate the dynamics for at least 5 revolutions of the chief Keplerian orbits. Discuss the results with relevant plots. (b.7): having the maximum acceleration of the thruster, i.e., ||ul|2 max, changes the stability property of the controlled system, and hence the controller performance. Perform numerical simulations with max = 1 m/s² for the three controller gains and discuss the result with relevant plots.

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

See similar textbooks

Related questions

Question

I need help with a MATLAB code. For question b.6 I have the MATLAB code shown below. How do I edit the code to answer question b.7. Please make sure the plots are reasonable.

clc;

clear all;

% Constants

mu = 398600; % Earth gravitational parameter, km^3/s^2

% Initial chief and deputy positions and velocities in ECI frame

% Assume circular orbits in equatorial plane for simplicity

a_c = 10000; % km

a_d = 11500; % km

r_c0 = [a_c; 0; 0];

v_c0 = [0; sqrt(mu/a_c); 0];

r_d0 = [a_d; 0; 0];

v_d0 = [0; sqrt(mu/a_d); 0];

% Initial relative state

delta_r0 = r_d0 - r_c0;

delta_v0 = v_d0 - v_c0;

x0 = [delta_r0; delta_v0]; % 6x1 initial relative state

% Time span

tspan = [0 3600]; % 1 hour in seconds

% Damping cases

cases = struct( ...

'name', {'Critically damped', 'Under-damped', 'Over-damped'}, ...

'Kr', {eye(3)*2.5e-3, eye(3)*0.001, eye(3)*0.01}, ...

'P', {eye(3)*0.01, eye(3)*0.0006, eye(3)*0.02} ...

);

% Simulate each case

for i = 1:length(cases)

Kr = cases(i).Kr;

P = cases(i).P;

% ODE function

f = @(t, x) dynamics(t, x, mu, r_c0, v_c0, Kr, P);

[t, X] = ode45(f, tspan, x0);

delta_r = X(:,1:3);

% Plotting

figure(i);

plot3(delta_r(:,1), delta_r(:,2), delta_r(:,3), 'LineWidth', 1.5);

grid on;

xlabel('\delta x [km]');

ylabel('\delta y [km]');

zlabel('\delta z [km]');

title(['Deputy Relative Trajectory - ', cases(i).name]);

legend(cases(i).name);

end

% Dynamics function

function dxdt = dynamics(~, x, mu, rc, vc, Kr, P)

delta_r = x(1:3);

delta_v = x(4:6);

rd = rc + delta_r;

vd = vc + delta_v;

% Control input

u = mu * (rd / norm(rd)^3 - rc / norm(rc)^3) ...

- Kr * delta_r - P * delta_v;

% Deputy acceleration

a_d = -mu * rd / norm(rd)^3 + u;

a_c = -mu * rc / norm(rc)^3;

delta_a = a_d - a_c;

dxdt = [delta_v; delta_a];

end

(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft.
Define a =[r] and x = x R to represent the deputy orbital state and its target (= chief
orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the
ECI frame. Denote the relative state by dx=x-xe Table 2 summarize the initial orbital elements.
Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame)
Orbital element
Unit
Chief
ad 11500 a = 10000 km
Deputy
semi-major axis
eccentricity
ed=0.15
e=0.3
inclination
id=35
ic=50
degree
right ascension of ascending node = 50
Se 50
degree
argument of periapsis
Wd=40
We = 40
degree
true anomaly at epoch
Ve=0
degree
(b.1): Derive the error dynamics of our system in ECI frame under the influence of u.
(b.2): Consider a candidate Lyapunov function V = r K,6r+v Sv, where K = K, and K, > 0.
Discuss the positive definiteness of V, and derive the Lyapunov rate of this system.
(b.3): Derive a stabilizing controller such that satisfies V=-8v Pôv where P > 0, and discuss the
stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?).
(b.4): Show the asymptotic stability of the system by applying either Theorem 1 or Theorem 2.
(b.5): For diagonal K, and P, design three types of controller gains to achieve critically, under, or over
damped systems, and report the relation K, and P needs to satisfy for each controller.
(b.6): Perform the numerical integrations of the controlled system with each of the (i) critically-
damping, (ii) over-damping, and (iii) under-damping controllers, where assume K, = 13 and
scale the length and time units appropriately as discussed in class. Simulate the dynamics for at
least 5 revolutions of the chief Keplerian orbits. Discuss the results with relevant plots.
(b.7):
having the maximum acceleration of the thruster, i.e., ||ul|2 max, changes
the stability property of the controlled system, and hence the controller performance. Perform
numerical simulations with max = 1 m/s² for the three controller gains and discuss the result
with relevant plots.

Expert Solution

Step by step

Solved in 2 steps with 3 images

SEE SOLUTION Check out a sample Q&A here