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

1. Feb 8, 2010

### 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 = xTPx ; 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$$/dt$$xTPx = x$$\dot{}$$TPx + xTPx$$\dot{}$$

from there we somehow get to

xT(FTP + 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!!!

Last edited: Feb 8, 2010
2. Feb 8, 2010

### 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 = ZT YT XT ... BT AT for some matrices / vectors A, ..., Z.

3. Feb 9, 2010

### 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 couldnt see it!
(and yes I was missing that step!)

thanks again! :)