Given the standard dynamical model of an n-DOF robotic system consists of only revolute joints:
$ M(q)\ddot{q} + C(q,\dot{q})\dot{q}+g(q)=\tau $
The robust inverse dynamics control input $\tau$ can be written as
$\tau = \hat{M}(q) u + \hat{C}(q,\dot{q})\dot{q}+\hat{g}(q)$
where the notation $\hat{(\cdot)}$ represents an estimate of the theoretically accurate value, and the error is written as $\tilde{(\cdot)}=\hat{(\cdot)}-(\cdot)$.
If the outer-loop control input is chosen as:
$u=\ddot{q}_d+K_d \dot{\tilde{q}}+K_p \tilde{q}+\delta\;$ where $\delta$ is a term to-be-designed.
then the closed-loop system can be obtained as:
$\ddot{q}=u+\underbrace{M^{-1}\tilde{M}u+M^{-1}(\tilde{C}\dot{q}+\tilde{g})}_{:= \eta(q,\,\dot{q},u)}$
or in state-space representation as,
$e=\begin{bmatrix} e_1 \\ e_2 \end{bmatrix} = \begin{bmatrix} \tilde{q}\\ \dot{\tilde{q}} \end{bmatrix}, \; \dot{e}=\underbrace{\begin{bmatrix} 0 & I \\ -K_p & - K_d \end{bmatrix}}_{:=A}e+\underbrace{\begin{bmatrix} 0 \\ -I \end{bmatrix}}_{:=B}[\delta+\eta]$
Define $E:=M^{-1}\tilde{M}$, then the uncertainty function $\eta$ can be expressed as,
$\eta =E \, \delta + E(\ddot{q}_d +K_d \dot{\tilde{q}}+K_p \tilde{q})+M^{-1}(\tilde{C}\dot{q}+\tilde{g})$
Then, according to the textbook Robot Modeling and Control (Spong & Vidyasagar), for some $\alpha < 1$, $\gamma_1$, $\gamma_2$, and $\gamma_3 (t)$, the uncertainty function can be thus bounded:
$||\eta|| \leq \alpha ||\delta || + \gamma_1 ||e || + \gamma_2 ||e||^2 + \gamma_3 (t)$
But I am really struggling to see where this inequality comes from. I understand how the first term $\alpha ||\delta||$ is designed since $||M^{-1}||$ is bounded, and the second term involving $||e||$ also seems reasonable as we have joint variables in $\eta$. But I couldn't get the idea behind the terms $\gamma_2 ||e||^2$ and $\gamma_3 (t)$.