Let the pendulum equation: $\ddot \theta+\sin{\theta}+b\dot\theta=cu$, that in state space ($x_1=\theta$, $x_2=\dot \theta$) representation will be:
$\begin{cases}
\dot x_1=x_2\\
\dot x_2=-\sin{x_1}-bx_2+cu
\end{cases}$
$\textbf{The aim is to stabilize the system around $\theta=\frac{\pi}{2}$}$.
My idea so is the following:
in order to go back from the stabilization technique to the stabilization around the origin, I consider the design $u=u_{SS}+u_{\delta}$, with $u_{SS}$ for which the translated system , has the origin for $x_1=\frac{\pi}{2}$, and so at the origin an equilibrium point, so:
$\begin{cases}
z_1=x_1-\frac{\pi}{2}\\
z_2=x_2
\end{cases}$$\iff$ $\begin{cases}
\dot z_1=z_2\\
\dot z_2=-\sin{z_1+\frac{\pi}{2}}-bz_2+cu_{SS}
\end{cases}$
So the $u_{SS}$ that realizes the requirement to have at the origin an equilibrium point is given by $u_{SS}=\frac{\sin{\frac{\pi}{2}}}{c}$.
$\textbf{My problem is:}$ in Khalil NonLinear Control Theory, it is not considered this $u_{SS}$, but only considering the system traslated in the coordinates $z$ it is designed then the control $u$, as it is sufficient to translate the system without considering this $u_{SS}$ to go back from the stabilization around $\theta\neq 0$ to the stabilization around $\theta=0$.
Please can you tell me if my idea is correct or I am doing some mistakes?
$\textbf{EDIT:}$ the parameter $c$ is uncertain...can this information give the explaination for my problem?
Since I do not own this book, it is hard for me to directly assess what Khalil has to say about it. Either way, linearizing a system around any point requires one to define this point as an equilibrium (whether that is stable is not important). This is (as you show) done by offsetting the state and the input accordingly: $$z_1 = x_1 - \theta_0, ~~ z_2 = x_2, ~~ u = u_{SS} + u_\delta$$ $$\dot{z_2} = -\text{sin}(z_1 +\theta_0) - bz_2 + c(u_{SS} + u_\delta)$$ Where $\theta_0$ is the chosen equilibrium point, $u_\delta$ is the parameter you control using $z_1$ as input (which you offset manually). The actual control input is indeed $u_{SS} + u_\delta$ in which $u_{SS}$ is calculated manually using your equation: $$u_{SS} = \frac{\text{sin}(\theta_0)}{c}$$ The fact that $c$ is an uncertain parameter troubles the control goal massively, as you are uncertain about the offset input which ensures the equilibrium (i.e. the actual equilibrium can be somewhere else now). This can however be solved by implementing an integral action in the controller and discarding the input offset (this should be computed iteratively with the integral action). Now what I think answers your question is the following: The input $u_{SS}$ equals zero if the offset angle equals 0. Hence it is not required to mention it.