Final P roject O ptionEM L 5311: Con trol Systems TheoryCon trol of a System with Structural Nonlinearit y1 Problem Ov erviewYou mu st design a disturb an ce rejection co ntroller to suppr ess a limit cycle oscillation for a nonlin ea raeroelastic system. Th is system represents an airfoil, whose m otio n is restricted to pitching andplunging, moun ted on a wind tunnel.2DynamicModelThe dynam ics associated with the plunge (y(t)) in [meters] and pitch angle (α(t))in[rad]foran y control surface deflection (β(t)) in [rad] can be expressed as a relationship between structuraldynam ics and aerodynamics as∙mmxabmxabIα¸∙¨y¨α¸+∙cy00 cα¸∙˙y˙α¸+∙ky00 kα¸∙yα¸=∙−LM¸where L(t) and M(t) denote the aerodynamic lift and momen t defined asL = ρU2bclαΩ + ρU2bclββM = ρU2b2cmαΩ + ρU2bclββwhereΩ = α +1U˙y +1.1Ub ˙α.The variables used in the above equations are defined asU =12m/s ρ =1.23kg/m3b =0.15mxα= −1.418m =13kg ky=2900N/mIα=0.070m2kg cy=27.43kg/scα=0.04m2kg/s cmα= −0.628clα=6.28 cmβ= −0.635.clβ=3.358This system has a nonlinearit y represented by kα. Specifically, kαis rep resented by a nonlinearspring with a functional dependence on the pitc h angle askα=2.82(1 − 22.1α + 1315.5α2− 8580α3+ 17289.7α4).The system ma y undergo a limit cycle oscillation during whic h the steady -state response of thestates is represented by persistent oscillations.13 Control Objectiv esDesign a single controller that yields the follow ing desired criteria over the widest possible initialconditions.1. S ta bilize the system w ith no limit cycle oscilla tion .2. Lim it the actuator motion:• The maximum co ntrol surface deflection is ±20 deg.• The maximum co ntrol surface rate is ±40 [deg/sec].3. Guarantee classical robustness of the system.• Achiev e a gain margin of at least 6 [dB ].• Achiev e a gain margin of at least 45 [deg].4. Sho w good sensitivity and comp lementary sensitivity.4 E xpectation s1. P erform 2 con trol designs• a PID controller• a LQR controller2. Prepare a Report: As the control design engineer, you must prove to th e mana gem ent thatyour con tro ller will work as desired, or why the stated criteria can not be met. To this end, asignificant portion of the grade relies on your abilit y to effectively communicate you r resultsto the management. As a manager I need ALL the details to make an informed decision, butI also need to look at the answ ers quick ly and get the "snapshot answ er" in m inutes. Yourreport should compar e the results from the two controller, yo u should provide details on thederiva tion of yo ur con tro l design, include gain and phase margins, effects of noise, effects ofposition/rate limits,