Lyapunov Stability

[hypn0tik]

Anyone familiar with this concept?

One question I was assigned was:

Consider x' = Ax
Show that the origin is stable if:

A =
0 ..1
-1 0

(A = 2x2 matrix).

I found the eigen values to be:

det (A-bI)= 0
-> b^2 + 1 = 0 => b = i or b = -i

However, for it to be stable, the real part of the eigen value must be strictly less than 0.

Anyone have any ideas? Am I missing anything?

 

[hypn0tik]

Anyone at all familiar with control systems?

 

[PurdueRy]

the determinent of that is 1

 

[Heisenberg]

It looks like you found the correct eigenvalues. As far as I can tell they're both purely imaginary, which means the real part would be zero.

 

[aux]

IIRC, the Lyapunov stability condition is: the eigenvalues should have non-positive real parts and the eigenvalues with zero real parts should be different.

edit: typo fixed

 

[PurdueRy]

What is the ".."?

 

[hypn0tik]

Originally posted by: Heisenberg
It looks like you found the correct eigenvalues. As far as I can tell they're both purely imaginary, which means the real part would be zero.

Yes, I understand that, but it says that I need to prove that the system is stable around the origin. However, since the real part of the eigen values are NOT less than zero, how do I go about it?

 

[hypn0tik]

Originally posted by: PurdueRy
What is the ".."?

It's there to space out the 1 so it looks some what like a matrix.

 

[Eli]

I wish I understood this thread.. lol

 

[Heisenberg]

Originally posted by: hypn0tik

Originally posted by: Heisenberg
It looks like you found the correct eigenvalues. As far as I can tell they're both purely imaginary, which means the real part would be zero.

Yes, I understand that, but it says that I need to prove that the system is stable around the origin. However, since the real part of the eigen values are NOT less than zero, how do I go about it?

Eh, I dunno. I've never heard of Lyapunov Stability, but I can find eigenvalues in my sleep. 😉 However, if what aux posted is right, then you should be okay.

 

[PurdueRy]

" continuous-time linear time-invariant system is Lyapunov stable (internally stable) if and only if all the eigenvalues of A have real parts less than or equal to 0, and those with real parts equal to 0 are nonrepeated."

Doesn't that hold in your answer...?

 

[hypn0tik]

Originally posted by: aux
IIRC, the Lyapunov stability condition is: the eigenvalues should have non-positive real parts and the eigenvalues with zero real parts should be different.

edit: typo fixed

See, the problem is the notes I am using do not discuss the case where the real part is 0.

 

[PurdueRy]

Originally posted by: hypn0tik

Originally posted by: aux
IIRC, the Lyapunov stability condition is: the eigenvalues should have non-positive real parts and the eigenvalues with zero real parts should be different.

edit: typo fixed

See, the problem is the notes I am using do not discuss the case where the real part is 0.

if all the eigenvalues of A have real parts less than or equal to 0, and those with real parts equal to 0 are nonrepeated

 

[hypn0tik]

Originally posted by: PurdueRy
" continuous-time linear time-invariant system is Lyapunov stable (internally stable) if and only if all the eigenvalues of A have real parts less than or equal to 0, and those with real parts equal to 0 are nonrepeated."

Doesn't that hold in your answer...?

Yes, it holds. Where did you find that?

Thanks.

 

[PurdueRy]

Originally posted by: hypn0tik

Originally posted by: PurdueRy
" continuous-time linear time-invariant system is Lyapunov stable (internally stable) if and only if all the eigenvalues of A have real parts less than or equal to 0, and those with real parts equal to 0 are nonrepeated."

Doesn't that hold in your answer...?

Yes, it holds. Where did you find that?

Thanks.

http://academic.csuohio.edu/simond/linearsystems/stability/lyapunov/

::shrug:: first google link

 

[aux]

Originally posted by: hypn0tik

Originally posted by: aux
IIRC, the Lyapunov stability condition is: the eigenvalues should have non-positive real parts and the eigenvalues with zero real parts should be different.

edit: typo fixed

See, the problem is the notes I am using do not discuss the case where the real part is 0.

Looks like your lecture notes are missing the zero real parts case, try google for more info.

edit: PurdueRy found a link for you 🙂

 

[hypn0tik]

Originally posted by: aux

Originally posted by: hypn0tik

Originally posted by: aux
IIRC, the Lyapunov stability condition is: the eigenvalues should have non-positive real parts and the eigenvalues with zero real parts should be different.

edit: typo fixed

See, the problem is the notes I am using do not discuss the case where the real part is 0.

Looks like your lecture notes are missing the zero real parts case, try google for more info.

Yeah, that seems to be the problem.

I tried Wiki and didn't find anything useful there either.

Thanks everyone. I appreciate the help.