Hi all, I'm currently working on a stability proof for a discrete-time controller and attempting to use Lyapunov analysis. Most of the process makes sense except the initial formulation of the difference equation (analogous to dV/dt in continuous time).
Given a Lyapunov function V=x^2 and the discrete equivalent, V(k) = x(k)^2, I've seen two methods of deriving the difference equation V(k+1)-V(k):
1.) V(k+1)-V(k) = x(k+1)^2 - x(k)^2
2.) V(k+1)-V(k) = dV(k)/dx(k)(x(k+1)-x(k)) = 2x(k)(x(k+1)-x(k)).
Which of these two methods are correct? I can see the merit in both, but they yield very different results.
Thanks!