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Your friend Bob needs your help to design a cruise control system for his car. He drives a Toyota…

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Your friend Bob needs your help to design a cruise control system for his car. He drives a Toyota Corolla that weighs 1300 kg (2866 lbs). He wants to accelerate his car from stationary to within 2% of 18 m/s (40 mph) under 5 second and then maintain the 18 m/s speed with less than 2% error. Given Bob lives on a street that has a 40 mph speed limit, he doesn’t want his car to ever go beyond 45 mph because he doesn’t want to get a speeding ticket To help Bob with his cruise control, let’s first understand how a cruise control works, which is a classic control system. The objective of any control system is to change the dynamics of the given system to be able to achieve a desired response. This is typically done by means of a feedback connection of a controller to a plant, as shown in Figure 1. The plant is a system such as a motor, a chemical plant, or an automobile we would like to control so that it responds in a certain way. The controller is a system we design to make the plant follow a prescribed input or reference signal. By feeding back the response of the system to the input, it can be determined how the plant responds to the controller. The commonly used negative feedback generates an error signal that permits us to judge the performance of the controller PI Controller Plant x(t) e(t) K c(t) V(t) He(s) = K, + Hp(s) Figure 1. Block diagram of a closed-loop negative feedback control system In classic linear control, the transfer function, i.e., the Laplace transform of its impulse response, of the plant we wish to control is available; let us cal it Hp (s). The controller, with a transfer function Hc (s), is designed to make the output of the overall system perform in a specified way. For instance, in a cruise control system, the plant is the car, and the desired performance is to automatically set the speed of the car to a desired value. In Figure 1, x(t) is the desired output. Suppose we want to keep the speed of the car at Vo miles per hour for t 0, then ?(t) Vou(t) u(t) is the actual output, i.e., the actual speed of the car. The difference between the two e(t) = x(t)-r(t) is the error signal and the input to the controller. Given the error signal, the controller generates a control signal c(t) as the plant input. The transfer function of the plant, i.e., the model for a car in motion, can be derived by the free body diagram shown in Figure 2