Physical Setup: The equations governing the motion of an aircraft are a very complicated set of six nonlinear coupled differential equations. However, under certain assumptions, they can be decoupled and linearized into longitudinal and lateral equations. Aircraft pitch is governed by the longitudinal dynamics. In this problem, we will design an autopilot that controls the pitch of an aircraft. The basic coordinate axes and forces acting on an aircraft are shown in the figure given below. Lift Drag Weight Thrust Figure 4: System to be controlled in Problem #6. We will assume that the aircraft is in steady-cruise at constant altitude and velocity; thus, the thrust, drag, weight and lift forces balance each other in the x- and y-directions. We will also assume that a change in pitch angle will not change the speed of the aircraft under any circumstance (unrealistic but simplifies the problem a bit). Under these assumptions, the longitudinal equations of motion for the aircraft can be written as follows: a = -0.313a + 56.7q +0.2328 q -0.0139a 0.426q +0.02035 (7) (8) 0 = 56.7q. (9) For this system, the input will be the elevator deflection angle ō and the output will be the pitch an- gle of the aircraft. The numerical values are taken from the data from one of Boeing's commercial aircraft. By taking the Laplace transform of the above equations, obtain the transfer function of the open-loop system, defined as, 0(s) G(s) := 8(s)