# Marginal stabilty in s domain

• FrankJ777
In summary, the conversation discusses the concept of marginal stability and the use of the s domain to analyze it. A marginally stable system has poles that are complex conjugates on the jω axis, and their magnitude is the resonant frequency at which the system becomes unstable. The individual was attempting to apply a resonant frequency sinusoid to a marginally stable system using the transfer function, but kept ending up with a different function in the time domain than expected. They question if they should have been able to observe the system becoming unstable using this method or if their algebra was incorrect.
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?

bump?

## What is marginal stability in s domain?

Marginal stability in s domain refers to a system that is on the borderline between being stable and unstable. In other words, it is a critical point where small changes in parameters or inputs can cause the system to become unstable.

## What causes marginal stability in s domain?

There are a few factors that can contribute to marginal stability in s domain. One common cause is when a system has complex poles on the imaginary axis, making it difficult to determine the stability. Another factor can be the presence of multiple poles that are close to each other, creating a slow and unstable response.

## How is marginal stability in s domain determined?

Marginal stability in s domain can be determined by analyzing the poles and zeros of a system. If the poles are on the imaginary axis or close to each other, the system may be marginally stable. Additionally, the location and behavior of the poles and zeros can be analyzed using the Routh-Hurwitz stability criterion.

## Why is marginal stability in s domain important?

Marginal stability in s domain is important because it can indicate the potential for a system to become unstable. It is a critical point that should be carefully considered when designing and analyzing systems, as small changes can greatly affect the stability and performance of the system.

## How can marginal stability in s domain be improved?

There are a few ways to improve marginal stability in s domain. One approach is to add additional poles or zeros to the system in order to shift the critical point. Another method is to adjust the system parameters or inputs to move the poles away from the imaginary axis. Additionally, using advanced control techniques such as feedback control can help stabilize a marginally stable system.

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