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In a cruise controller the throttle of a car is controlled by a servomechanism with the aim of keeping the vehicle moving at a velocity set by the driver. We model the system with the differential equation mv =u-bv where u is the velocity of the car, m = 1500 kg is the mass of the car and b = 100 Ns/m is the friction coefficient (mainly due to air resistance). We want to design a PI-controller for the cruise controller. a) Find the transfer function from reference, r, to output velocity when u = : kp(r — v) + k₁ f (r – v). b) Approximate your transfer function in a) by the second order system H(s) = ao/(s² + b₁s + b₂), with ao, b₁ and b2 constant. Choose parameters (kp, kı) for the PI-controller to get an overshoot of 10% and a rise time of 2 s. c) What is the relation between the rise time and overshoot of the system in a) and the simplified second order model assumed in b)? (Are they smaller or larger? Motivate.)

In a cruise controller the throttle of a car is controlled by a servomechanism with the aim of keeping the vehicle moving at a velocity set by the driver. We model the system with the differential equation mv =u-bv where u is the velocity of the car, m = 1500 kg is the mass of the car and b = 100 Ns/m is the friction coefficient (mainly due to air resistance). We want to design a PI-controller for the cruise controller. a) Find the transfer function from reference, r, to output velocity when u = : kp(r — v) + k₁ f (r – v). b) Approximate your transfer function in a) by the second order system H(s) = ao/(s² + b₁s + b₂), with ao, b₁ and b2 constant. Choose parameters (kp, kı) for the PI-controller to get an overshoot of 10% and a rise time of 2 s. c) What is the relation between the rise time and overshoot of the system in a) and the simplified second order model assumed in b)? (Are they smaller or larger? Motivate.)

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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In a cruise controller the throttle of a car is controlled by a servomechanism with the aim of keeping the vehicle moving at a velocity set by the driver. We model the system with the differential equation mi u bv where u is the velocity of the car, m = 1500 kg is the mass of the car and b = 100 Ns/m is the friction coefficient (mainly due to air resistance). We want to design a PI-controller for the cruise controller. a) Find the transfer function from reference, r, to output velocity when u = kp(r − v) + kï ſ (r — v). b) Approximate your transfer function in a) by the second order system H(s) = ao/(s² + b₁s + b2), with ao, b₁ and b2 constant. Choose parameters (kp, ki) for the PI-controller to get an overshoot of 10% and a rise time of 2 s. c) What is the relation between the rise time and overshoot of the system in a) and the simplified second order model assumed in b)? (Are they smaller or larger? Motivate.)
Transcribed Image Text:In a cruise controller the throttle of a car is controlled by a servomechanism with the aim of keeping the vehicle moving at a velocity set by the driver. We model the system with the differential equation mi u bv where u is the velocity of the car, m = 1500 kg is the mass of the car and b = 100 Ns/m is the friction coefficient (mainly due to air resistance). We want to design a PI-controller for the cruise controller. a) Find the transfer function from reference, r, to output velocity when u = kp(r − v) + kï ſ (r — v). b) Approximate your transfer function in a) by the second order system H(s) = ao/(s² + b₁s + b2), with ao, b₁ and b2 constant. Choose parameters (kp, ki) for the PI-controller to get an overshoot of 10% and a rise time of 2 s. c) What is the relation between the rise time and overshoot of the system in a) and the simplified second order model assumed in b)? (Are they smaller or larger? Motivate.)
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