Q4. Given three agents with states xi, i = (1,2,3). The agents are connected together and they update their states in continuous domain based on the following rule: 3 ž,(t) = (*, (t) – x,(t). j=1 Ax, where x = [x1,x2, X3]" and with the a) Represent the state progress as a linear system i initial condition x(0) = [x1(0), x2(0), x3(0)]". %3D %3D b) Analyze the properties of A, i.e., find its eigenvalues, eigenvectors, determinant, etc. c) Find the Jordan normal form of the matrix A. Find e4t analytically. Write down the analytical solution of x(t) and calculate lim x(t). Analyze behavior of the system as time goes to infinity. t→∞ How would the state progress with time?
Q4. Given three agents with states xi, i = (1,2,3). The agents are connected together and they update their states in continuous domain based on the following rule: 3 ž,(t) = (*, (t) – x,(t). j=1 Ax, where x = [x1,x2, X3]" and with the a) Represent the state progress as a linear system i initial condition x(0) = [x1(0), x2(0), x3(0)]". %3D %3D b) Analyze the properties of A, i.e., find its eigenvalues, eigenvectors, determinant, etc. c) Find the Jordan normal form of the matrix A. Find e4t analytically. Write down the analytical solution of x(t) and calculate lim x(t). Analyze behavior of the system as time goes to infinity. t→∞ How would the state progress with time?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given three agents with states �O, � = (1,2,3). The agents are connected together and they update
their states in continuous domain based on the following rule:
![Q4. Given three agents with states xi, i = (1,2,3). The agents are connected together and they update
their states in continuous domain based on the following rule:
3
*;(t) =
Ax, where x = [x1,x2, X3]" and with the
a) Represent the state progress as a linear system i =
initial condition x(0) -
b) Analyze the properties of A, i.e., find its eigenvalues, eigenvectors, determinant, etc.
Find the Jordan normal form of the matrix A. Find e4t analytically. Write down the analytical
solution of x(t) and calculate lim x(t). Analyze behavior of the system as time goes to infinity.
[x1(0), x2(0), x3(0)]".
%3D
How would the state progress with time?
d) Simulate the system on Simulink with x1(0) = 2, x2(0) = -4, x3(0) = 1. Comment on the
result. Submit the plot of your system. (Simulate the system sufficiently long time to see the
actual behavior!)
Note: The matrix L = -A is called the Laplacian matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d986659-db5f-4b21-b2b9-a7479f2fc80a%2F7b8b1968-25c5-4a03-9959-90c4d32da750%2Ff514c2f_processed.png&w=3840&q=75)
Transcribed Image Text:Q4. Given three agents with states xi, i = (1,2,3). The agents are connected together and they update
their states in continuous domain based on the following rule:
3
*;(t) =
Ax, where x = [x1,x2, X3]" and with the
a) Represent the state progress as a linear system i =
initial condition x(0) -
b) Analyze the properties of A, i.e., find its eigenvalues, eigenvectors, determinant, etc.
Find the Jordan normal form of the matrix A. Find e4t analytically. Write down the analytical
solution of x(t) and calculate lim x(t). Analyze behavior of the system as time goes to infinity.
[x1(0), x2(0), x3(0)]".
%3D
How would the state progress with time?
d) Simulate the system on Simulink with x1(0) = 2, x2(0) = -4, x3(0) = 1. Comment on the
result. Submit the plot of your system. (Simulate the system sufficiently long time to see the
actual behavior!)
Note: The matrix L = -A is called the Laplacian matrix.
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