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.

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