3. Consider the equation below ml+mgl sin = Fl describing the motion of aero-pendulum of length / and mass m. The thrust force F is regulated by the PID controller - F = k, ( , ) k ( ) dr k 0 where the reference angle & is considered to be constant. Consider the parameters = 7/4, m = 0.1 [kg], l = 0.5 [m], g= 10 [m/s]. g l, mrod 0 mprop 20 (a) Define the new state z = (0-0) dr and write system into first order form. 0 (b) Using the Routh-Hurwitz criteria determine where the equilibrium (t) = is stable in the parameter space (kp, ki, kd). Now consider kp as the bifurcation parameter while fixing k = 0 [Ns], i.e., the case without integral action and kd 0.1 [N/s]. Note that in the case that state z can be dropped and the Routh-Hurwitz criteria are simplified. Use the Taylor expansion of the sin function up to cubic order around = . (c) Vary the bifurcation parameter kp is varied between -10 [N] and 10 [N] with the step-size 0.1 N and solve for the equilibria numerically. (d) Using the linearization about the equilibrium determine whether the equilibria is stable or unstable and plot the bifurcation diagram.