Cruise control

This page documents the cruise control system that is used as a running example throughout the text. A detailed description of the dynamics of this system are presented in Chapter 4 - Examples. This page contains a description of the system, including the models and commands used to generate some of the plots in the text.

Introduction

Cruise control is the term used to describe a control system that regulates the speed of an automobile. Cruise control was commercially introduced in 1958 as an option on the Chrysler Imperial. The basic operation of a cruise controller is to sense the speed of the vehicle, compare this speed to a desired reference, and then accelerate or decelerate the car as required. The figure to the right shows a block diagram of this feedback system.

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A simple control algorithm for controlling the speed is to use a "proportional plus integral" feedback. In this algorithm, we choose the amount of gas flowing to the engine based on both the error between the current and desired speed, and the integral of that error. The plot on the right shows the results of this feedback for a step change in the desired speed and a variety of different masses for the car (which might result from having a different number of passengers or towing a trailer). Notice that independent of the mass (which varies by 25% of the total weight of the car), the steady state speed of the vehicle always approaches the desired speed and achieves that speed within approximately 10-15 seconds. Thus the performance of the system is robust with respect to this uncertainty.

Dynamic model

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 $F_{\text{d}}$ the disturbance force due to gravity, friction and aerodynamic drag. The equation of motion of the car is simply

$m{\frac {dv}{dt}}=F-F_{\text{d}}.$

The force $F$ is generated by the engine, whose torque is proportional to the rate of fuel injection, which is itself proportional to a control signal $0\leq u\leq 1$ that controls the throttle position. The torque also depends on engine speed $\omega$. A simple representation of the torque at full throttle is given by the torque curve

$T(\omega )=T_{\text{m}}\left(1-\beta {\biggl (}{\frac {\omega }{\omega _{\text{m}}}}-1{\biggr )}^{2}\right),$

where the maximum torque $T_{\text{m}}$ is obtained at engine speed $\omega _{\text{m}}$. Typical parameters are $T_{m}=190$ Nm, $\omega _{m}$ = 420 rad/s (about 4000 RPM) and $\beta =0.4$.

Let $n$ be the gear ratio and $r$ the wheel radius. The engine speed is related to the velocity through the expression

$\omega ={\frac {n}{r}}v=:\alpha _{n}v,$

and the driving force can be written as

$F={\frac {nu}{r}}T(\omega )=\alpha _{n}uT(\alpha _{n}v).$

Typical values of $\alpha _{n}$ for gears 1 through 5 are $\alpha _{1}=40$, $\alpha _{2}=25$, $\alpha _{3}=16$, $\alpha _{4}=12$ and $\alpha _{5}=10$. The inverse of $\alpha _{n}$ has a physical interpretation as the effective wheel radius. The figure to the right shows the torque as a function of vehicle speed. The figure shows that the effect of the gear is to "flatten" the torque curve so that an almost full torque can be obtained almost over the whole speed range.

The disturbance force $F_{\text{d}}$ has three major components: $F_{\text{g}}$, the forces due to gravity; $F_{\text{r}}$, the forces due to rolling friction; and $F_{\text{a}}$, the aerodynamic drag. Letting the slope of the road be $\theta$, gravity gives the force $F_{\text{g}}=mg\sin \theta$, where $g=9.8\,{\text{m}}/{\text{s}}^{2}$ is the gravitational constant. A simple model of rolling friction is

$F_{\text{r}}=mgC_{\text{r}}\,{\text{sgn}}(v),$

where $C_{\text{r}}$ is the coefficient of rolling friction and sgn($v$) is the sign of $v$ or zero if $v=0$. A typical value for the coefficient of rolling friction is $C_{\text{r}}=0.01$. Finally, the aerodynamic drag is proportional to the square of the speed:

$F_{\text{a}}={\frac {1}{2}}\rho C_{\text{d}}Av^{2},$

where $\rho$ is the density of air, $C_{d}$ is the shape-dependent aerodynamic drag coefficient and $A$ is the frontal area of the car. Typical parameters are $\rho =$ 1.3 k/m${}^{3}$, $C_{\text{d}}=0.32$ and $A=$ 2.4 m${}^{2}$.

Python model

The model for the system above can be built using the Python Control Toolbox. The code blocks in this section can be used to generate the plots on this page.

Package initialization:

import numpy as np
import matplotlib.pyplot as plt
from math import pi
import control as ct

Vehicle model:

def vehicle_update(t, x, u, params={}):
    """Vehicle dynamics for cruise control system.

    Parameters
    ----------
    x : array
         System state: car velocity in m/s
    u : array
         System input: [throttle, gear, road_slope], where throttle is
         a float between 0 and 1, gear is an integer between 1 and 5,
         and road_slope is in rad.

    Returns
    -------
    float
        Vehicle acceleration
    """

    from math import copysign, sin
    sign = lambda x: copysign(1, x)         # define the sign() function
    
    # Set up the system parameters
    m = params.get('m', 1600.)              # vehicle mass, kg
    g = params.get('g', 9.8)                # gravitational constant, m/s^2
    Cr = params.get('Cr', 0.01)             # coefficient of rolling friction
    Cd = params.get('Cd', 0.32)             # drag coefficient
    rho = params.get('rho', 1.3)            # density of air, kg/m^3
    A = params.get('A', 2.4)                # car area, m^2
    alpha = params.get(
        'alpha', [40, 25, 16, 12, 10])      # gear ratio / wheel radius

    # Define variables for vehicle state and inputs
    v = x[0]                           # vehicle velocity
    throttle = np.clip(u[0], 0, 1)     # vehicle throttle
    gear = u[1]                        # vehicle gear
    theta = u[2]                       # road slope

    # Force generated by the engine
    omega = alpha[int(gear)-1] * v      # engine angular speed
    F = alpha[int(gear)-1] * motor_torque(omega, params) * throttle

    # Disturbance forces
    #
    # The disturbance force Fd has three major components: Fg, the forces due
    # to gravity; Fr, the forces due to rolling friction; and Fa, the
    # aerodynamic drag.

    # Letting the slope of the road be \theta (theta), gravity gives the
    # force Fg = m g sin \theta.
    Fg = m * g * sin(theta)

    # A simple model of rolling friction is Fr = m g Cr sgn(v), where Cr is
    # the coefficient of rolling friction and sgn(v) is the sign of v (±1) or
    # zero if v = 0.
    Fr  = m * g * Cr * sign(v)

    # The aerodynamic drag is proportional to the square of the speed: Fa =
    # 1/2 \rho Cd A |v| v, where \rho is the density of air, Cd is the
    # shape-dependent aerodynamic drag coefficient, and A is the frontal area
    # of the car.
    Fa = 1/2 * rho * Cd * A * abs(v) * v
    
    # Final acceleration on the car
    Fd = Fg + Fr + Fa
    dv = (F - Fd) / m
    
    return dv

Engine model:

def motor_torque(omega, params={}):
    # Set up the system parameters
    Tm = params.get('Tm', 190.)             # engine torque constant
    omega_m = params.get('omega_m', 420.)   # peak engine angular speed
    beta = params.get('beta', 0.4)          # peak engine rolloff

    return np.clip(Tm * (1 - beta * (omega/omega_m - 1)**2), 0, None)

Input/output model for the vehicle system:

vehicle = ct.NonlinearIOSystem(
    vehicle_update, None, name='vehicle',
    inputs = ('u', 'gear', 'theta'), outputs = ('v'), states=('v'))

Input/output torque curves (plot):

# Figure 4.2a - single torque curve as function of omega
omega_range = np.linspace(0, 700, 701)
plt.subplot(2, 2, 1)
plt.plot(omega_range, [motor_torque(w) for w in omega_range])
plt.xlabel('Angular velocity $\omega$ [rad/s]')
plt.ylabel('Torque $T$ [Nm]')
plt.grid(True, linestyle='dotted')

# Figure 4.2b - torque curves in different gears, as function of velocity
plt.subplot(2, 2, 2)
v_range = np.linspace(0, 70, 71)
alpha = [40, 25, 16, 12, 10]
for gear in range(5):
    omega_range = alpha[gear] * v_range
    plt.plot(v_range, [motor_torque(w) for w in omega_range],
             color='blue', linestyle='solid')

# Set up the axes and style
plt.axis([0, 70, 100, 200])
plt.grid(True, linestyle='dotted')

# Add labels
plt.text(11.5, 120, '$n$=1')
plt.text(24, 120, '$n$=2')
plt.text(42.5, 120, '$n$=3')
plt.text(58.5, 120, '$n$=4')
plt.text(58.5, 185, '$n$=5')
plt.xlabel('Velocity $v$ [m/s]')
plt.ylabel('Torque $T$ [Nm]')

plt.suptitle('Torque curves for typical car engine');
plt.tight_layout()

Exercises

The following exercises make use of the cruise control model described here:

• Chapter 1: Exploring the performance of a cruise controller

Further Reading

• How Stuff Works: cruise control
• Wikipedia: cruise control