Finding deriviative of lyapunov's equation (chain rule/linear algebra)

[james1234]

Hi there!

I am just trying to work though a simple deriviation presented in a txt book (feedback control of dynamic systems p 616). I am REALLY new to linear algebra and they have skipped too many steps!

they have used the chain rule for differentiation of V(x) (a lyapunov function) where V = x^TPx ; where P is a symetric positive matrix (i have not been able to find reference to a 'positive matrix'.. is a positive-definite matrix the same thing?)

Thus we have;

d$$/dt$$x^TPx = x$$\dot{}$$^TPx + x^TPx$$\dot{}$$

from there we somehow get to

x^T(F^TP + PF)x (where i believe F is the 'state' matrix usually represened as A ie X dot = Fx + bu )

if someone could point me in the right direction it would really be appreciated!

 

[CompuChip]

It doesn't have much to do with calculus, it's just plugging in the definition.
The only step that you might be missing is that (AB...XYZ)^T = Z^T Y^T X^T ... B^T A^T for some matrices / vectors A, ..., Z.

 

[james1234]

hi CompuChip

thankyou so much for for taking the time to reply!
despite all the time I have spent pooring over it I just couldn't see it!
(and yes I was missing that step!)

thanks again! :)