Consider the concept of the automobile cruise control system. The purpose of the
ID: 2084538 • Letter: C
Question
Consider the concept of the automobile cruise control system. The purpose of the cruise control system is to maintain a constant vehicle speed in the presence of external disturbances (e.g., changes in road grade). This can be accomplished via the feedback loop below, which measures the vehicle’s speed y(t), compares it to the reference speed r(t), and automatically adjusts the throttle according to the following control law (i.e., transfer function of the cruise control system):
Here, U(s) represents the force generated at the road/tire interface, V(s) represents the vehicle velocity,
m represents the vehicle mass, and bv represents the force resistant to motion (e.g., rolling resistance
and aerodynamic drag) which are assumed to vary linearly with the vehicle’s velocity v.
Consider an automobile subject to the following parameters and performance specifications:
m = 1000 kg
Rise time < 5.0 sec
SSE < 5%
b = 50 N-sec/m
%OS < 1%
r(t) = step input, 10 m/sec
Complete the following using MATLAB:
(a) Define P(s) using the tf command and the parameters defined above. Use this definition of P(s)
for the following items.
Explanation / Answer
The cruise control system of a car is a common feedback system encountered in everyday life. The system attempts to maintain a constant velocity in the presence of disturbances primarily caused by changes in the slope of a road. The controller compensates for these unknowns by measuring the speed of the car and adjusting the throttle appropriately. To model the system we start with the block diagram in Figure 3.1. Let v be the speed of the car and vr the desired (reference) speed. The controller, which typically is of the proportional-integral (PI) type described briefly in Chapter 1, receives the signals v and vr and generates a control signal u that is sent to an actuator that controls the throttle position. The throttle in turn controls the torque T delivered by the engine, which is transmitted through the gears and the wheels, generating a force F that moves the car. There are disturbance forces Fd due to variations in the slope of the road, the rolling resistance and aerodynamic forces. The cruise controller also has a human–machine interface that allows the driver to set and modify the desired speed. There are also functions that disconnect the cruise control when the brake is touched. The system has many individual components—actuator, engine, transmission, wheels and car body—and a detailed model can be very complicated. In spite of this, the model required to design the cruise controller can be quite simple. To develop a mathematical model we start with a force balance for the car body. Let v be the speed of the car, m the total mass (including passengers), F the force generated by the contact of the wheels with the road, and Fd the disturbance forcewhere vr is the desired (reference) speed. As discussed briefly in Section 1.5, the integrator (represented by the state z) ensures that in steady state the error will be driven to zero, even when there are disturbances or modeling errors. (The design of PI controllers is the subject of Chapter 10.) Figure 3.3b shows the response of the closed loop system, consisting of equations (3.3) and (3.4), when it encounters a hill. The figure shows that even if the hill is so steep that the throttle changes from 0.17 to almost full throttle, the largest speed error is less than 1 m/s, and the desired velocity is recovered after 20 s. Many approximations were made when deriving the model (3.3). It may seem surprising that such a seemingly complicated system can be described by the simple model (3.3). It is important to make sure that we restrict our use of the model to the uncertainty lemon conceptualized in Figure 2.15b. The model is not valid for very rapid changes of the throttle because we have ignored the details of the engine dynamics, neither is it valid for very slow changes because the properties of the engine will change over the years. Nevertheless the model is very useful for the design of a cruise control system. As we shall see in later chapters, the reason for this is the inherent robustness of feedback systems: even ifthe model is not perfectly accurate, we can use it to design a controller and make use of the feedback