I am working on the stability of a system for the estimation $z_i$ predicted from x using the basic observer design. The relation is as follows: \begin{equation} z_i = \frac{\eta_i }{k_i}(\dot{x} - \dot{z}_i), \label{eq:tro_1} \end{equation} Based on the Lyapunov stability theory, I selected the Lyapunov function candidate as: \begin{equation} V(z_i) = \frac{1}{2}\left(z_i - \dot{x}\right), \label{eq:tro_5} \end{equation} The derivative of $\dot{V}(z_i)$ gives: \begin{equation} \begin{aligned} \dot{V}(z_i) &= \left(z_i - \dot{x}\right)\dot{z}_i\\ &=\left(z_i - \dot{x}\right)\dot{x} - \frac{k_i}{\eta_i}({z_i} - \dot{x})\\ &= \left(z_i - \dot{x}\right)^2 \left(1 - \frac{k_i}{\eta_i}\right)\\, \label{eq:tro_6} \end{aligned} \end{equation}
Can I then say that: for $\frac{k_i}{\eta_i}>0$, $\dot{V}(z_i)$ is negative semi-definite (not necessarily negative definite) as $V(z_i)$ will be decreasing?
I am working on this and I ama somehow stuck and want to be sure my concept is correct.