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1. (30 pts.) A system is described by the following differential equation: x+x-4x+6x=11 a. Rewrite the equation above in state-space form x =Ax+Bu. b. Determine the characteristic equation and the characteristic roots of the system. Obtain also the transfer function of the system from input u to output a c. Describe the natural modes and the overall free response of the system. Is this system stable? d. Using MATLAB, plot the free response x(f) to the initial conditions x(0) = 1, i(0) - x(0) 0 for a time span of 5 seconds. 2. (30 pts.) Two degree-of-freedom model of an aircraft's longitudinal motion is given by: 2a 0 0+ 10a 150 -55, where a is the angle of attack (in rad), 0 is the pitch angle (in rad), q is the pitch rate (0 in rad/s), and S, is the elevator deflection (in deg). a. Express the equations of the motion above in state-space form. b. Find the eigenvalues of the system matrix and determine the stability of the motion. Verify this by plotting the free response of a(t) and 0(t) to the initial conditions 0=0.1,q=a= using MATLAB. Submit the plots. a(s) c. Obtain the transfer function from input & to output a( using MATLAB. 8,(s) d. Plot and submit also a(f), 6(f). and q(f) responses to a 1°-step elevator deflection input with zero initial conditions. 3. (40 pts.) The longitudinal derivatives of Navion general aviation airplane flying at 0.158 Mach number in sea-level condition are given below. Note that for this aircraft, the only longitudinal control available in that flying condition is the elevator. a. Determine the longitudinal state-space equations of motion for the Navion in the specified flying condition. b. Determine the characteristic equation of the Navion's longitudinal motion. c. Calculate the eigenvalues of the longitudinal flight motion. Identify the natural modes of the motion, and for each natural mode, calculate its relevant parameters (time constant for a 1s--order mode; frequency and damping ratio for a 2nd-order mode). Describe also the 1 motion associated with each natural mode and determine the overall stability of the aircraft's longitudinal motion. d. Plot and submit the free responses of a and 0 with respect to the initial condition a(0) 2° and zero for the other variables. Generate plots for two different time spans, one is for the first 10s of the responses to capture the short-period mode and the other is for 300 s to capture the phugoid mode. c. Plot and submit the responses of a and 0to a -0.02 rad-step elevator input with zero initial conditions. Navion airplane properties Altitude (ft) 0 (sea level) Flight speed (Mo. ft/s) 176 Nominal pitch angle (deg) Longitudinal derivatives X, (1/s) -0.045 Z. (1/s) -0.369 M. (1/ft.s) X. (ft/s²) 6.336 Za (ft/s²) -355.52 Me (1/s²) -8.8 x_(ft/s) Zà (ft/s) -0.898 X. (ft/s) Z. (ft/s) M. (1/s) -2.05 Xs (ft/s²) Z6, (ft/s²) -53.407 Mb (1/s²) -46.408

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