Eigenvalues and stability - question.

In summary, to determine the stability of a system with a transfer function equal to 1, you need to look at its poles rather than its eigenvalues. Additionally, an eigenvalue of zero does not necessarily indicate a stable system.
  • #1
peripatein
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0
Hi,

Homework Statement


How may I determine whether a system is stable if its input is equal to its output, hence yielding a system(transfer) function equal to 1?
Furthermore, could an eigenvalue zero characterize a stable system?
I am attaching three examples where I am asked to determine whether the systems are stable or not.


Homework Equations





The Attempt at a Solution


In the first two cases the system's function (transfer function) is 1, hence I am not sure how to determine stability. It seems, though, that the unit is somewhat stable whereas the exponential isn't. As for the bottom sample, do I have to perform a Laplace transform and find H(s) to determine the eigenvalues?
 

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  • #2
The main question is: How do I determine the stability of a system if its input is equal to its output, hence yielding a system(transfer) function equal to 1?When it comes to the eigenvalue zero, it is impossible to determine the stability of a system based on this. This is because the stability of a system is determined by its poles, not its eigenvalues. Therefore, an eigenvalue of zero does not necessarily characterize a stable system.
 

FAQ: Eigenvalues and stability - question.

What are eigenvalues and why are they important in stability analysis?

Eigenvalues are a concept in linear algebra that represent the scalars by which a given vector is scaled when multiplied by a square matrix. In stability analysis, eigenvalues are used to determine the stability of a system by examining the behavior of the system's solutions over time.

How do eigenvalues affect the stability of a system?

The eigenvalues of a system's matrix determine whether the system is stable, unstable, or marginally stable. If the eigenvalues are all negative, the system is stable. If any eigenvalue has a positive real part, the system is unstable. If the eigenvalues are all zero or have zero real parts, the system is marginally stable.

Can a system have complex eigenvalues and still be stable?

Yes, a system can have complex eigenvalues and still be stable. In this case, the system's solutions will oscillate with a certain frequency determined by the imaginary part of the eigenvalues. As long as the real parts of the eigenvalues are negative, the system will be stable.

How do eigenvalues change when the system's parameters are varied?

When the system's parameters are varied, the eigenvalues may change in value or in position in the complex plane. This can affect the stability of the system, as the eigenvalues determine the behavior of the system's solutions. A parameter change that causes the eigenvalues to move into the unstable region will make the system unstable.

Are there any limitations to using eigenvalues for stability analysis?

Eigenvalues can only be used for linear systems, meaning that the relationship between the system's inputs and outputs is described by a linear function. Additionally, eigenvalues can only determine the stability of a system in a small neighborhood around a given equilibrium point. For nonlinear systems, other techniques such as phase plane analysis may be necessary for stability analysis.

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