- #1
FrankJ777
- 140
- 6
I'm trying to get a better understanding of marginal stability and the s domain in general. According to my textbook, a system that is marginally stable has poles that are conjugates of each other on the jω axis, and their magnitude are the resonant frequency for which if a sinusoid of that frequency were applied the system would become unstable. For any other input the system remains stable.
I was hoping to see the effect of applying a resonant frequency sinusoid to a marginally stable system, by applying the transformed sinusoid to the transfer function, and inverting back to the time domain. I used, as suggested in my textbook transfer with the characteristic equation: (s^{2}+ 16) which has the roots (s-j4) (s+j4) which are complex conjugates that lie only on the imaginary axis, jω. To apply the resonant frequency sinusoid: sin(4t) \Rightarrow \frac{4}{s^{2}+a^{16}} : so
\frac{4}{s^{2}+a^{16}} \times T(s) = \frac{4}{s^{2}+a^{16}} \frac{1}{s^{2}+a^{16}} :
I expected to end up with something of the form e^{at}\times sin(4t) where a is a positive constant and the function increases as t increases, as I would have expected a marginally stable system to do when an on frequency sinusoid is applied, but I keep ending up with a function in the time domain of the form sin(4t) + \frac{1}{4}sin(4t).
Anyway, should I be able to observe the system "blow up" using an s domain method like this, or should i have gotten the answer I was looking for and maybe my algebra was unable to find the correct form?
I was hoping to see the effect of applying a resonant frequency sinusoid to a marginally stable system, by applying the transformed sinusoid to the transfer function, and inverting back to the time domain. I used, as suggested in my textbook transfer with the characteristic equation: (s^{2}+ 16) which has the roots (s-j4) (s+j4) which are complex conjugates that lie only on the imaginary axis, jω. To apply the resonant frequency sinusoid: sin(4t) \Rightarrow \frac{4}{s^{2}+a^{16}} : so
\frac{4}{s^{2}+a^{16}} \times T(s) = \frac{4}{s^{2}+a^{16}} \frac{1}{s^{2}+a^{16}} :
I expected to end up with something of the form e^{at}\times sin(4t) where a is a positive constant and the function increases as t increases, as I would have expected a marginally stable system to do when an on frequency sinusoid is applied, but I keep ending up with a function in the time domain of the form sin(4t) + \frac{1}{4}sin(4t).
Anyway, should I be able to observe the system "blow up" using an s domain method like this, or should i have gotten the answer I was looking for and maybe my algebra was unable to find the correct form?