You want to control the Roll angle, $\phi$ of a plane by changing the angle of the plane's ailerons. The differential equation relating the aileron angle $\theta$ to the roll angle $\phi$ of the plane is given by: $\ddot{\phi} + a_2\ddot{\phi} + a_1\dot{\phi} + a_0\phi = b_0\theta + b_1\dot{\theta}$. (This equation is the flight dynamics equation.) The roll angle is measured by a gyroscopic sensor, whose dynamics is given by $\tau\dot{\phi}_m + \phi_m = \phi$, where $\phi_m$ is the angle measurement provided by the gyroscope. The aileron controller reads the difference between the desired roll angle $\phi_d$ and the measured angle $\phi_m$ and commands the required aileron angle $\theta$. The controller differential equation is: $\dot{\theta} + p\theta = k(\dot{e} + ze)$, where $e = error = \phi_d - \phi_m$. (This type of controller is called a 'lead controller', as will be seen later). Draw the block diagram of this control system. Label the blocks as plant, controller, and sensor, and show the correct transfer function is each block. Show the proper directions of all the signal arrows, and label the signal lines with what signal they represent (e.g. $\phi_d$, $\phi_m$, $e$, $\theta$, and $\phi$)
