The linearized longitudinal motion of a helicopter near hover (see Figure 19.3) can be modeled by the normalized third order system where q = pitch rate 𝜃 = pitch angle of fuselage u = horizontal...

The linearized longitudinal motion of a helicopter near hover (see Figure 19.3) can be modeled by the normalized third order system

where q = pitch rate 𝜃 = pitch angle of fuselage u = horizontal velocity 𝛿 = rotor tilt angle (control variable). Suppose our sensor measures the horizontal velocity u as the output; that is, y = u. (a) Find the open-loop pole locations. (b) Is the system controllable?

(c) Find the feedback gain that places the poles of the system at s = −1 ± 1j and s = −2. Note that this is a measurement feedback. (d) Now assume full state feedback and find all the possible control gains that take place the clopsed loop eigenvalues at the above mentioned locations. Notes: In all these problems, plot all the state trajectories and output trajectories, assuming a non-zero initial condition in the state. Be sensible in selecting the initial conditions x(0). Plot the open loop trajectories first and then plot the closed loop trajectories larger and defend the final control design you suggest to the customer.