How to sketch the phase trajectory of the system denoted by the block diagram?
I built the following Simulink model
but I couldn't figure out what to represent the rate feedback $s+a$ with.
Using the above incomplete Simulink model I generated the following phase portrait, using arbitrary initial conditions.
If you integrate $u$ twice you would get $\theta$, so in other words the second time derivative of $\theta$, denoted with $\ddot{\theta}$, would be equal to $u$. From the block diagram it can be seen that $u$ can be calculated with $u=\mathrm{sgn}((s+a)(\theta_d-\theta))$. By using that $\theta_d=0$ and $s\,\theta=\dot{\theta}$ then the previous expression can also be written as $u = \mathrm{sgn}(- \dot{\theta} - a\,\theta)$, so you have the second order differential equation
$$ \ddot{\theta} = \mathrm{sgn}(- \dot{\theta} - a\,\theta). $$
This is equivalent to the following system of first order differential equations with $x = \begin{bmatrix}\theta & \dot{\theta}\end{bmatrix}^\top$
$$ \dot{x} = \begin{bmatrix} x_2 \\ \mathrm{sgn}(- x_2 - a\,x_1) \end{bmatrix}. $$