Q1 Controller Design (total 20 points) A linearized model of a boom lift (assuming the boom length is constant) is given by thetaDoubledot - a*theta = u whereu(t)is the adjusted torque provided by the hydraulic system at the base. a = 0.2is a factor depending on the inertia and dimensions of the system. You want to reach an anglethetaof 1 rad (starting from zero). Design, using rootlocus techniques, a controller, and if needed pre-filter to meet the following specifications. 1) Steady state value of angle is 1 rad 2) Settling time of angle response is less than 10s 3) Overshoot of the angleis less than 5% 4) The maximummagnitudeof theuis less than 0.5 (hard but possible) Type below the answers to the following as a numbered list

2. What is the plant G(s)?(3 points) If unsure, use G = 2.01/((s- 0.45)*(2*s+0.89)) at the cost of 3 points.
4. What is the controller (and if needed pre-filter) that you ended up using?(2 points) Note:Don\'t cancel the UNSTABLE poleof the plant! In reality, the poles are not exactly known and you don\'t want to inexactly cancel an unstable pole.
6. What was the final (i.e. steady state) value of the angle you got?(3 points)
8. What was the settling time of the angle response you got?(3 points)
10. What was the overshoot of the angle response you got?(3 points)
12. What was the maximum magnitude of the u(t) you got?(3 points)

If the parameterais changed from 0.2 to 0.4, will the controller that you designed still be able to reach the desired angle in the steady state? Say Yes or No, and then explain whyin only 1 sentence(3 points) Q2 Controller Design (total 20 points) A sensitive equipment of massm=1is siting inside a box cushioned by adamperc=10and springk=100. You want to minimize the vibrations of the equipment despite the vibrations of the box itself. This is done by using an active system that applies a forceFto the equipment to counter the vibrations. The relationship between the motionyof the equipment and the motiondof the box is given by: m*y_doubledot + c*y_dot + k*y = F + c*d_dot + k*d Design apropercontrollerK(s),that takes in the difference between the desiredyand the measuredyto provide the correct forceFso that 1) (error in)y, is less than 10% ofd, fordin the range of 1 to 30 rad/s. 2) (error in)ydue to sensor noise is reduced by a factor of 10, for noise in the range of 1000 to infinity rad/s. 3) phase margin of the open loop TF is more than 30 degrees. 4) gain margin of the open loop TF is more than 6 dB. Type below the answers to the following as a numbered list

2. What should be the value ofr ,that is, the value of the desiredy. (2 points)
4. Draw the block diagram properly, and arrange it so that you can use sisotool.(4 points) I don\'t need to see your block diagram though. Just type below the exact commands to open sisotool. I want to see what you used as G and what you called K. If unsure, simply use G = (1.02*s+9.8)/(0.11*s^2 + 1.01*s+ 10.03) to open the sisotool and for the rest of the problem, at the cost of 3 points.
6. To meet thederror specification, 20log₁₀|GK|should be larger than what value?(3 points) Hint: If unsure, please use |GK| > 25dB for the rest of the problem.
8. To meet the noise error specification, 20log₁₀|GK|should be smaller than what value?(3 points) Hint: If unsure, please use |GK|
10. Type theproperK(s) you ended up using?(3 points) Be very careful in reporting this!
12. What was the phase margin you obtained?(3 points)
14. What was the gain margin you obtained?(2 points)

Q3 Controller Design (total 20 points) 2D vehicle is moving at a speed ofv=1 m/s such that its velocity vector makes an angle? with x-axis (called heading). The coordinates of vehicles center with respect to x and y axes arexandy, and the steering angle is?. The wheelbaseL can be assumed to be 1m. The vehicle starts at origin with an angle of 45 degrees (hence att=0,y=0,y?=p/4 ) and you want it to start moving along a straight liney=2 (it\'s a line parallel to x -axis), just by modulating the steering angle. Design a controller gain matrix K (assuming a perfect observer) so that given the initial conditions (y=0,y?=p/4)the output y reaches within 1.9 to 2.1 in less than 3s, with no over/undershoot. Type below the answers to the following as a numbered list

2. Write a linearized plant differential equation from the steering angle? (as input) to the coordinatey(as output). (6 points) It is a relationship between?andyand their derivatives (and no other variables!) If unsure, then use plant TF from?toyas1/(s*(s+0.1)) and proceed with the rest of the question.

Paste below acomplete MATLAB codethat I can copy into my MATLAB and run. Your code should(14 points) 1) Convert the plant into the state space form (A, B, C, D). 2) Design a controller gain K (assuming perfect observer) so that given the initial conditions (y=0,y?=p/4 ), the output y reaches within 1.9 to 2.1 range in less than 3s, with no over/undershoot. 3) Simulate and plot the response of y with the initial conditions mentioned above In personpeoplewill have to email me (agrewal6@asu.edu) their matlab file ASAP after the quiz is over! But type the exact value of your gains here! (Very important!!!!!!!)