# Marginal stabilty in s domain

1. May 20, 2015

### FrankJ777

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?

2. May 22, 2015

bump?