Figure 1 shows a system that controls the roll angle of an airplane, with frictional losses...

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Figure 1 shows a system that controls the roll angle of an airplane, with frictional losses ignored. If proportional control were used, the two poles at s = 0 would split off at +90° angles, and the system would be marginally stable for all values of gain. Instead, we need to add a controller that draws those poles to the left, thus stabilizing the system and increasing its speed. If we place a pole far to the left and a zero closer to the origin, then the locus from the pole will move right toward the zero, and the centroid of roots rule requires that the two complex poles move toward the left. Thus, a practical PD controller is implemented. You wish to fulfill the following design criteria: 11. Acceleration error constant, Ka, equal to 10. Damping factor, <, greater than 0.65 S Or Ge 151822 S B + +1 2 Y(s) Figure Pl a. Write the closed-loop transfer function for the system. b. Find the value of Ko such that the acceleration error constant (see the chapter on performance criteria) is K = 10. I c. Now, consider that we want the pole of the controller to be far to the left. Temporarily, consider the possibility of ßco (as far to the left as possible), so s +ß ß in the denominator of the transfer function. Make this substitution in the closed loop transfer function and determine the value of a for which the damping factor is 1. d. Set ß= 8a and rewrite the open-loop transfer function in terms of the derived values for Ko, a, and B c. Use à MATLAB-generated root locus to show the variation of the roots of the characteristic equation as K, is varied. From this plot, determine the roots with the value of K, as specified in your solution for G (s). Also verify that the damping factor criterion is met Figure 1 shows a system that controls the roll angle of an airplane, with frictional losses ignored. If proportional control were used, the two poles at s = 0 would split off at +90° angles, and the system would be marginally stable for all values of gain. Instead, we need to add a controller that draws those poles to the left, thus stabilizing the system and increasing its speed. If we place a pole far to the left and a zero closer to the origin, then the locus from the pole will move right toward the zero, and the centroid of roots rule requires that the two complex poles move toward the left. Thus, a practical PD controller is implemented. You wish to fulfill the following design criteria: 11. Acceleration error constant, Ka, equal to 10. Damping factor, <, greater than 0.65 S Or Ge 151822 S B + +1 2 Y(s) Figure Pl a. Write the closed-loop transfer function for the system. b. Find the value of Ko such that the acceleration error constant (see the chapter on performance criteria) is K = 10. I c. Now, consider that we want the pole of the controller to be far to the left. Temporarily, consider the possibility of ßco (as far to the left as possible), so s +ß ß in the denominator of the transfer function. Make this substitution in the closed loop transfer function and determine the value of a for which the damping factor is 1. d. Set ß= 8a and rewrite the open-loop transfer function in terms of the derived values for Ko, a, and B c. Use à MATLAB-generated root locus to show the variation of the roots of the characteristic equation as K, is varied. From this plot, determine the roots with the value of K, as specified in your solution for G (s). Also verify that the damping factor criterion is met