Adaptive Cruise Control

Abstract

An adaptive cruise control (ACC) system is an extension of the standard cruise control system. An ACC equipped vehicle has a radar or other sensor that measures the distance to other preceding vehicles (downstream vehicles) on the highway. In the absence of preceding vehicles, the ACC vehicle travels at a user-set speed, much like a vehicle with a standard cruise control system (see Figure 6-1). However, if a preceding vehicle is detected on the highway by the vehicle’s radar, the ACC system determines whether or not the vehicle can continue to travel safely at the desired speed. If the preceding vehicle is too close or traveling too slowly, then the ACC system switches from speed control to spacing control. In spacing control, the ACC vehicle controls both the throttle and brakes so as to maintain a desired spacing from the preceding vehicle.

Appendix 6.A

This Appendix contains a proof of the result stated in section 6.6.1, namely that the magnitude of the transfer function

$${ {\hat H} } (s) = \frac{{\delta _{\,\,i} }}{{\delta _{\,\,i - 1} }} = \frac{{s + \lambda }}{{h\tau \,s^3+ hs^2+ (1 + \lambda \,h)s + \lambda }}$$

(6.36)

is always less than or equal to 1 at all frequencies if and only if h ≥ 2τ. This Appendix is adapted from the original proof presented by Swaroop (1995).

Background Result:

Consider the following quadratic inequality in ω²:

$$a\omega ^4+ b\omega ^2+ c > 0$$

(6.37)

We present the conditions on a,b,c under which the above inequality holds for all values of ω².

$$\begin{array}{lll} {a\omega ^4+ b\omega ^2+ c=a\left({\omega ^4+ 2\frac{b} {{2a}}\omega ^2+ \frac{c}{a}} \right)\hfill}\\{ = a\left[ {\left( {\omega ^2+ \frac{b} {{2a}}}\right)^2+\frac{{4ac - b^2 }}{{4a^2 }}} \right]\hfill}\\\end{array}$$

Hence

$$a\omega ^4+ b\omega ^2+ c > 0$$

if either

$$(1)\,\,\,\,\,a,b,c > 0$$

(6.38)

or

$$(2)\,\,\,b < 0,\,\,a >0\,,\,c >0\,\,{\hbox{and}}\,{\hbox{4}}ac\, - b^{\rm{2}} \,{\hbox{ >}}\,{\hbox{0}}\,\,{\hbox{i}}{\hbox{.e}}{\hbox{.}}\,\,b^{\rm{2}}- {\hbox{4}}ac\,{\hbox{ < }}\,{\hbox{0}}$$

(6.39)

Calculations:

Consider the transfer function

$$H(s) = \frac{{{\hat \delta} _{\,i } }}{{{\hat \delta } } _{\,\,i - 1} } = \frac{{s + \lambda }}{{\tau hs^3+ hs^2+ (1 + \lambda h)s + \lambda }}$$

Substituting s = jω,

$$\begin{array}{llll} {H(j\omega ) = \frac{{j\omega+\lambda }}{{(\lambda- h\omega ^2 ) + j\omega (1 + \lambda h - \tau h\,\omega ^2 )}}} \\{|H(j\omega )|^2= \frac{{\omega^2+\lambda ^2 }}{{(\lambda- h\omega ^2 ) + \omega ^2 (1 + \lambda h- \tau h\,\omega ^2 )^2 }}} \\{|H(j\omega )| \leq 1 }\\\Leftrightarrow \\{\omega ^2+ \lambda ^2\leqslant(\lambda- h\omega ^2 )^2+ \omega ^2 (1 + \lambda h - \tau h\,\omega^2 )^2 }\\\Leftrightarrow \\\end{array} $$

(6.40)

$$\tau ^2 h^2 \omega ^4+ (h^2- 2\tau h - 2\tau \lambda h^2 )\omega ^2+ \lambda ^2 h^2\geqslant 0$$

(6.41)

Comparing with the background result in equations (6.38) and (6.39)

• (1) If b > 0

  $$h^2- 2\tau h - 2\lambda \,\tau h^2> 0$$

  Hence

  $$h > \frac{{2\tau }}{{1 - 2\lambda \,\tau }}$$

  This is possible for small λ if and only if h > 2τ

• (2) If b < 0 and b ² − 4ac < 0

  $$(h^2- 2\tau h - 2\lambda \,\tau h^2 )^2- 4\tau ^2 h^2 \lambda ^2 h^2< 0$$

  After simplifying

  $$\begin{array}{lll} {\lambda< \frac{{4\tau h -h^2-4\tau ^2 }}{{8\tau ^2 h - 4\tau h^2 }}}\\{\lambda< \frac{{-(2\tau- h)^2 }}{{4\tau h(2\tau- h)}}}\\\end{array} $$

  Since λ must be positive, this implies h > 2τ

Replacing the strict inequality in equation (6.37) with a simple inequality, it follows that h ≥ 2τ. From (1) and (2), h ≥ 2τ is a necessary condition. It also follows that if h ≥ 2τ is satisfied, then one can find a λ > 0 such that$\parallel {H (j(\omega)\parallel\leq1}$. Thus h ≥ 2τ is both a necessary and sufficient condition for ensuring that the transfer function Ĥ (s) has a magnitude less than or equal to 1 at all frequencies.