The linearized equations governing the motion of a BOEING\'S commercial aircraft

Question

The linearized equations governing the motion of a BOEING'S commercial aircraft are given by (Messner and Tilbury. Control Tutorials for MATLAB and Simulink, Addison Wesley. 1998) dn(t)/dt = -0.313 alpha(t) + 56.7q(t) + 0.232b(t) dq(t)/dt = -0.0139 alpha(t) - 0.426q(t) + 0.0203 delta (t) d theta (t)/dt = 56.7 q(t) where theta(t) represents the pitch angle. The corresponding open-loop transfer function obtained from (1) is given by theta(s)/Delta(s) = 1.151s + 0.1774/s^2 + 0.739s^2 + 0.921 s = 1.151 (s + 0.1541)/s(s62 + 0.739s + 0.921) = 1.151 G(s) In this project we design an autopilot that controls the pitch angle theta(t) of this aircraft. The autopilot is obtained by forming the closed-loop system with an unity feedback and a controller of the form KG_c (s). For simplicity we assume that K = 1.151 K^-1 so that the open-loop feedback transfer function is K G_c (s) G(s). a) Find the steady state error due to a unit ramp input of the original (K = 1, g(s) = 1) closed loop system (G(s)/1 + G(s)). Plot the closed-loop system ramp response and observe (check) the corresponding steady state error. (b) Find the value for the static gain K such that the steady state error due to the unit ramp is at most 10% (r = 0.1). For the obtained value of A plot the corresponding closed-loop system ramp response and notice the steady state error improvement. (c) For the obtained value of A find the phase and gain stability margins and observe that the phase margin is pretty pure Design the phase-lead controller G_c (s) to improve the phase stability margin such that the compensated system has the phase stability margin close to 50 degree. Find the step response of the compensated system and compare it to the step response of the uncompensated system whose static gain K is found in part(b). Comment on the transient response improvement of the compensated system. (d) Design the phase-lag controller to satisfy the stability requirement imposed in (c). Find the step response of the system compensated (controlled) by the phase-lag controller and compare it to the step response of the system compensated by the phase-lead controller. Which one has a smaller rise time? Which one do you prefer? (e) Using the SIMULINK package, build the block diagrams for the system controlled by phase-lead and phase-lag controllers, plot the step responses in both cases, and confirm the results obtained in Parts (c) and (d).

Explanation / Answer

Could you please give specific values for the design in mat lab?

MAT lab is very specifc design tool. So if we select different value it will showcase different answers.

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