Facebook
Google
GitHub
Linkedin
Hi
I have a matrix A which results in the eigenvalues:
\( \lambda = -0.5 + j \) and its complex conjugate \( \bar{\lambda} = -0.5 - j \)
The eigen vectors are:
\(v = 1 + j \) and its complex conjugate \( \bar{v} = 1-j \)
In state space the following formula is used:
\(
x(t) = x_1 + x_2
x(t) = C_1 \left[\stackrel{1}{j}\right] e^{(-0.5+j) t} + C_1 \left[\stackrel{1}{-j}\right] e^{(-0.5-j) t}
\)
Euler states:
\(e^{(-0.5+j)t} = e^{-0.5t}(cos (t) + j sin(t))\)
Ignoring \(C_1 and C_2\) for now and multiplying with the eigenvectors:
\(x_1a = e^{-0.5t}(cos (t) + j sin (t))\)
\(x_1b = e^{-0.5t}(-sin (t) + cos (t))\)
\(x_2a = e^{-0.5t}(cos (t) - j sin (t))\)
\(x_2b = e^{-0.5t}(sin (t) - j cos (t))\)
I think I am correct thus far, please correct me if I am wrong.
The next steps should result in the following equation:
\(x(t) = e^{-0.5t}(C_1\left[\stackrel{cos (t)}{-sin (t)} \right] + C_2\left[\stackrel{sin (t)}{cos (t)} \right])
\)
But I can not seem to get there. I assume it has something to do with the complex eigenvectors and that I am not using them right. I hope someone can explain it to me.
I have a matrix A which results in the eigenvalues:
\( \lambda = -0.5 + j \) and its complex conjugate \( \bar{\lambda} = -0.5 - j \)
The eigen vectors are:
\(v = 1 + j \) and its complex conjugate \( \bar{v} = 1-j \)
In state space the following formula is used:
\(
x(t) = x_1 + x_2
x(t) = C_1 \left[\stackrel{1}{j}\right] e^{(-0.5+j) t} + C_1 \left[\stackrel{1}{-j}\right] e^{(-0.5-j) t}
\)
Euler states:
\(e^{(-0.5+j)t} = e^{-0.5t}(cos (t) + j sin(t))\)
Ignoring \(C_1 and C_2\) for now and multiplying with the eigenvectors:
\(x_1a = e^{-0.5t}(cos (t) + j sin (t))\)
\(x_1b = e^{-0.5t}(-sin (t) + cos (t))\)
\(x_2a = e^{-0.5t}(cos (t) - j sin (t))\)
\(x_2b = e^{-0.5t}(sin (t) - j cos (t))\)
I think I am correct thus far, please correct me if I am wrong.
The next steps should result in the following equation:
\(x(t) = e^{-0.5t}(C_1\left[\stackrel{cos (t)}{-sin (t)} \right] + C_2\left[\stackrel{sin (t)}{cos (t)} \right])
\)
But I can not seem to get there. I assume it has something to do with the complex eigenvectors and that I am not using them right. I hope someone can explain it to me.
