Question 1 20 pts For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled by the first-order nonlinear differential equation mvu - Kv - Kav², where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, is the viscous friction force, and is the aerodynamic drag. Suppose m = 4500 lbs, Kf = 2.5N/(m/s), and K = 0.8N/(m/s)². (3+7 + 10 = 20 pts) 1. Define equilibrium values for all variables of interest for the desired equilibrium point where the equilibrium speed is ū=65 mph. 2. Linearize the system around this equilibrium point, defining all variables in the equation clearly, and specify the linearized transfer function (numerical values). 3. Suppose that the custom is at/n3. Suppose that the system is at equilibrium (at 65mph), and the road grade suddenly increases to 3% (see Elevation Grade Calculator (omnicalculator.com) if you need help with this term). The equations will now change to mi = u - K₁v-K₁v² - mgsin(0), and the speed will drop. Can you use linearization to design a control law of the form u=ū+ Au, Au=-kAv, Av=v- i , where k is the gain of the controller that you will choose by "experimentation" (too small a gain and it won't have much effect, too large a gain might cause the throttle to saturate in a real-world scenario) to bring the car speed back (close) to the equilibrium value of 65mph ? If so, show me a simulation for 10 min of the system where the grade abruptly changes from 0 to 3% at 5mins, and then drops back to zero at 7mins. Include 2 subplots, one where there is no feedback control. I.e.. Ava - SO/nequilibrium value of 65mph ? If so, show me a simulation for 10 min of the system where the grade abruptly changes from 0 to 3% at 5mins, and then drops back to zero at 7mins. Include 2 subplots, one where there is no feedback control, i.e., Au = 0, so that u =ū, and a second subplot where you have designed the above controller for an appropriately chosen gain k. Are you able to make the error converge to zero? Why do you think the above controller is unable to do so ? Later on in the course, we will see how to use integral control to make the "steady-state error" (i.e, the error as t → ∞o) zero. Upload Choose a File

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Question 1 20 pts For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled by the first-order nonlinear differential equation mvu - Kv - Kav², where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, is the viscous friction force, and is the aerodynam