3. Consider the equation below ml+mgl sin = Fl describing the motion of aero-pendulum of length...

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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. 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.