Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Adaptive Cruise Control

  • Rajesh RajamaniEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_72-1


This chapter discusses advanced cruise control automotive technologies, including adaptive cruise control (ACC) in which spacing control, speed control, and a number of transitional maneuvers must be performed. The ACC system must satisfy difficult performance requirements of vehicle stability and string stability. The technical challenges involved and the control design techniques utilized in ACC system design are presented.


Spacing Control Spacing Policy Adaptive Cruise Control Switching Line Preceding Vehicle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Adaptive cruise control (ACC) is an extension of cruise control. An ACC vehicle includes a radar, a lidar, or other sensor that measures the distance to any preceding vehicle in the same lane on the highway. In the absence of preceding vehicles, the speed of the car is controlled to a driver-desired value. In the presence of a preceding vehicle, the controller determines whether the vehicle should switch from speed control to spacing control. In spacing control, the distance to the preceding car is controlled to a desired value.

A different form of advanced cruise control is a forward collision avoidance (FCA) system. An FCA system uses a distance sensor to determine if the vehicle is approaching a car ahead too quickly and will automatically apply brakes to minimize the chances of a forward collision. For the 2013 model year, 29 % vehicles have forward collision warning as an available option and 12 % include autonomous braking for a full FCA system. Examples of models in which an FCA system is standard are the Mercedes Benz G-class and the Volvo S-60, S-80, XC-60, and XC-70.

It should be noted that an FCA system does not involve steady-state vehicle following. An ACC system on the other hand involves control of speed and spacing to desired steady-state values.

ACC systems have been in the market in Japan since 1995, in Europe since 1998, and in the US since 2000. An ACC system provides enhanced driver comfort and convenience by allowing extended operation of the cruise control option even in the presence of other traffic.

Controller Architecture

The ACC system has two modes of steady state operation: speed control and vehicle following (i.e., spacing control). Speed control is traditional cruise control and is a well-established technology. A proportional-integral controller based on feedback of vehicle speed (calculated from rotational wheel speeds) is used in cruise control (Rajamani 2012).

Controller design for vehicle following is the primary topic of discussion in the sections titled “Vehicle Following Requirements” and “String Stability Analysis” in this chapter.

Transitional maneuvers and transitional control algorithms are discussed in the section titled “Transitional Maneuvers” in this chapter.

The longitudinal control system architecture for an ACC vehicle is typically designed to be hierarchical, with an upper-level controller and a lower-level controller, as shown in Fig. 1.
Fig. 1

Structure of longitudinal control system

The upper-level controller determines the desired acceleration for the vehicle. The lower level controller determines the throttle and/or brake commands required to track the desired acceleration. Vehicle dynamic models, engine maps, and nonlinear control synthesis techniques are used in the design of the lower controller (Rajamani 2012). This chapter will focus only on the design of the upper controller, also known as the ACC controller.

As far as the upper-level controller is concerned, the plant model for control design is
$$\ddot{x}_{i} = u$$
where the subscript idenotes the ith car in a string of consecutive ACC cars. The acceleration of the car is thus assumed to be the control input. However, due to the finite bandwidth associated with the lower level controller, each car is actually expected to track its desired acceleration imperfectly. The objective of the upper level controller design is therefore stated as that of meeting required performance specifications robustly in the presence of a first order lag in the lower-level controller performance:
$$\ddot{x}_{i} = \frac{1} {\tau s + 1}\ddot{x}_{i\_\mathrm{des}} = \frac{1} {\tau s + 1}u_{i}.$$
Equation (1) is thus assumed to be the nominal plant model while the performance specifications have to be met even if the actual plant model were given by Eq. (2). The lag τ typically has a value between 0.2 and 0.5 s (Rajamani 2012).

Vehicle Following Requirements

In the vehicle following mode of operation, the ACC vehicle maintains a desired spacing from the preceding vehicle. The two important performance specifications that the vehicle following control system must satisfy are: individual vehicle stability and string stability.
  1. (a)

    Individual vehicle stability

Consider a string of vehicles on the highway using a longitudinal control system for vehicle following, as shown in Fig. 2. Let x i be the location of the ith vehicle measured from an inertial reference. The spacing error for the ith vehicle (the ACC vehicle under consideration) is then defined as
$$\delta _{i} = x_{i} - x_{i-1} + L_{\mathrm{des}}.$$
Here, L des is the desired spacing and includes the preceding vehicle length i − 1. L des could be chosen as a function of variables such as the vehicle speed \(\dot{x}_{i}\). The ACC control law is said to provide individual vehicle stability if the spacing error of the ACC vehicle converges to zero when the preceding vehicle is operating at constant speed:
$$\ddot{x}_{i-1} \rightarrow 0 \Rightarrow \delta _{i} \rightarrow 0.$$
  1. (b)

    String stability

Fig. 2

String of adaptive cruise control vehicles

The spacing error is expected to be non-zero during acceleration or deceleration of the preceding vehicle. It is important then to describe how the spacing error would propagate from vehicle to vehicle in a string of ACC vehicles during acceleration. The string stability of a string of ACC vehicles refers to a property in which spacing errors are guaranteed not to amplify as they propagate towards the tail of the string (Swaroop and Hedrick 1996).

String Stability Analysis

In this section, mathematical conditions that ensure string stability are provided.

Let δ i and δ i − 1 be the spacing errors of consecutive ACC vehicles in a string. Let \(\hat{H}(s)\) be the transfer function relating these errors:
$$\hat{H}(s) = \frac{\hat{\delta }_{i}} {\hat{\delta }_{i-1}}(s).$$
The following two conditions can be used to determine if the system is string stable:
  1. (a)
    The transfer function \(\hat{H}(s)\) should satisfy
    $$\left \|\hat{H}(s)\right \|_{\infty }\leq 1.$$
  2. (b)
    The impulse response function h(t) corresponding to \(\hat{H}(s)\) should not change sign (Swaroop and Hedrick 1996), i.e.,
    $$h(t) > 0\quad \forall t \geq 0.$$

The reasons for these two requirements to be satisfied are described in Rajamani (2012). Roughly speaking, Eq. (6) ensures that \(\vert \vert \delta _{i}\vert \vert _{2} \leq \vert \vert \delta _{i-1}\vert \vert _{2}\), which means that the energy in the spacing error signal decreases as the spacing error propagates towards the tail of the string. Equation (7) ensures that the steady state spacing errors of the vehicles in the string have the same sign. This is important because a positive spacing error implies that a vehicle is closer than desired while a negative spacing error implies that it is further apart than desired. If the steady state value of δ i is positive while that of δ i − 1 is negative, then this might be dangerous due to the vehicle being closer, even though in terms of magnitude δ i might be smaller than δ i − 1.

If conditions (6) and (7) are both satisfied, then \(\vert \vert \delta _{i}\vert \vert _{\infty } \leq \vert \vert \delta _{i-1}\vert \vert _{\infty }\) (Rajamani 2012).

Constant Inter-vehicle Spacing

The ACC system only utilizes on board sensors like radar and does not depend on inter-vehicle communication from other vehicles. Hence the only variables available as feedback for the upper controller are inter-vehicle spacing, relative velocity and the ACC vehicle’s own velocity.

Under the constant spacing policy, the spacing error of the ith vehicle was defined in Eq. (3).

If the acceleration of the vehicle can be instantaneously controlled, then it can be shown that a linear control system of the type
$$\ddot{x}_{i} = -k_{p}\delta _{i} - k_{v}\dot{\delta }_{i}$$
results in the following closed-loop transfer function between consecutive spacing errors
$$\hat{H}(s) = \frac{\hat{\delta }_{i}} {\hat{\delta }_{i-1}}(s) = \frac{k_{p} + k_{v}s} {s^{2} + k_{v}s + k_{p}}.$$
Equation (9) describes the propagation of spacing errors along the vehicle string.

All positive values of k p and k v guarantee individual vehicle stability. However, it can be shown that there are no positive values of k p and \(k_{v_{}}\)for which the magnitude of G(s) can be guaranteed to be less than unity at all frequencies. The details of this proof are available in Rajamani (2012).

Thus, the constant spacing policy will always be string unstable.

Constant Time-Gap Spacing

Since the constant spacing policy is unsuitable for autonomous control, a better spacing policy that can ensure both individual vehicle stability and string stability must be used. The constant time-gap (CTG) spacing policy is such a spacing policy. In the CTG spacing policy, the desired inter-vehicle spacing is not constant but varies with velocity. The spacing error is defined as
$$\delta _{i} = x_{i} - x_{i-1} + L_{\mathrm{des}} + h\dot{x}_{i}.$$
The parameter h is referred to as the time-gap.
The following controller based on the CTG spacing policy can be used to regulate the spacing error at zero (Swaroop et al. 1994):
$$\ddot{x}_{i\_\mathrm{des}} = -\frac{1} {h}(\dot{x}_{i} -\dot{ x}_{i-1} +\lambda \delta _{i})$$
With this control law, it can be shown that the spacing errors of successive vehicles δ i and δ i − 1 are independent of each other:
$$\dot{\delta }_{i} = -\lambda \delta _{i}$$
Thus, δ i is independent of δ i − 1 and is expected to converge to zero as long as λ > 0. However, this result is only true if any desired acceleration can be instantaneously obtained by the vehicle i.e., if τ = 0.
In the presence of the lower controller and actuator dynamics given by Eq. (2), it can be shown that the dynamic relation between δ i and δ i − 1 in the transfer function domain is
$$\hat{H}(s) = \frac{s+\lambda } {h\tau s^{3} + hs^{2} + (1 +\lambda h)s+\lambda }$$
The string stability of this system can be analyzed by checking if the magnitude of the above transfer function is always less than or equal to 1. It can be shown that this is the case at all frequencies if and only if (Rajamani 2012)
$$h \geq 2\tau .$$
Further, if Eq. (14) is satisfied, then it is also guaranteed that one can find a value of λ such that Eq. (7) is satisfied. Thus the condition (14) is necessary (Swaroop and Hedrick 1996) for string stability.

Since the typical value of τ is of the order of 0.5 s, Eq. (14) implies that ACC vehicles must maintain at least a 1-s time gap between vehicles for string stability.

Transitional Maneuvers

While under speed control, an ACC vehicle might suddenly encounter a new vehicle in its lane (either due to a lane change or due to a slower moving preceding vehicle). The ACC vehicle must then decide whether to continue to operate under the speed control mode or transition to the vehicle following mode or initiate hard braking. If a transition to vehicle following is required, a transitional trajectory that will bring the ACC vehicle to its steady state following distance needs to be designed. Similarly, a decision on the mode of operation and design of a transitional trajectory are required when an ACC vehicle loses its target.

The regular CTG control law cannot directly be used to follow a newly encountered vehicle, see Rajamani (2012) for illustrative examples.

When a new target vehicle is encountered by the ACC vehicle, a “range – range rate” diagram can be used (Fancher and Bareket 1994) to decide if
  1. (a)

    The vehicle should use speed control.

  2. (b)

    The vehicle should use spacing control (with a defined transition trajectory in which desired spacing varies slowly with time)

  3. (c)

    The vehicle should brake as hard as possible in order to avoid a crash.


The maximum allowable values for acceleration and deceleration need to be taken into account in making these decisions.

For the range – range rate (\(R -\dot{ R})\) diagram, define range R and range rate \(\dot{R}\) as
$$R = x_{i-1} - x_{i}$$
$$\dot{R} =\dot{ x}_{i-1} -\dot{ x}_{i} = V _{i-1} - V _{i}$$
where x i − 1, x i , V i − 1, and V i are inertial positions and velocities of the preceding vehicle and the ACC vehicle respectively.
A typical \(R -\dot{ R}\) diagram is shown in Fig. 3 (Fancher and Bareket 1994). Depending on the measured real-time values of R and \(\dot{R}_{,}\) and the \(R -\dot{ R}\) diagram in Fig. 3, the ACC system determines the mode of longitudinal control. For instance, in region 1, the vehicle continues to operate under speed control. In region 2, the vehicle operates under spacing control. In region 3, the vehicle decelerates at the maximum allowed deceleration so as to try and avoid a crash.
Fig. 3

Range vs. range-rate diagram

The switching line from speed to spacing control is given by
$$R = -T\dot{R} + R_{\mathrm{final}}$$
where T is the slope of the switching line. When a slower vehicle is encountered at a distance larger than the desired final distance R final, the switching line shown in Fig. 4 can be used to determine when and whether the vehicle should switch to spacing control. If the distance R is greater than that given by the line, speed control should be used.
Fig. 4

Switching line for spacing control

The overall strategy (shown by trajectory ABC) is to first reduce gap at constant \(\dot{R}\) and then follow the desired spacing given by the switching line of Eq. (17).

The control law during spacing control on this transitional trajectory is as follows. Depending on the value of \(\dot{R}\), determine R from Eq. (17). Then use R as the desired inter-vehicle spacing in the PD control law
$$\ddot{x}_{\mathrm{des}} = -k_{p}\left (x_{i} - R\right ) - k_{d}\left (\dot{x}_{i} -\dot{ R}\right )$$
The trajectory of the ACC vehicle during constant deceleration is a parabola on the \(R -\dot{ R}\) diagram (Rajamani 2012).

The switching line should be such that travel along the line is comfortable and does not constitute high deceleration. The deceleration during coasting (zero throttle and zero braking) can be used to determine the slope of the switching line (Rajamani 2012).

Note that string stability is not a concern during transitional maneuvers (Rajamani 2012).

Traffic Stability

In addition to individual vehicle stability and string stability, another type of stability analysis that has received significant interest in ACC literature is traffic flow stability. Traffic flow stability refers to the stable evolution of traffic velocity and traffic density on a highway section, for given inflow and outflow conditions. One well-known result in this regard in literature is that traffic flow is defined to be stable if \(\frac{\partial q} {\partial \rho }\) is positive, i.e., as the density ρ of traffic increases, traffic flow rate q must increase (Swaroop and Rajagopal 1999). If this condition is not satisfied, the highway section would be unable to accommodate any constant inflow of vehicles from an oncoming ramp. The steady state traffic flow on the highway section would come to a stop, if the ramp inflow did not stop (Swaroop and Rajagopal 1999).

It has been shown that the constant time-gap spacing policy used in ACC systems has a negative qρ slope and thus does not lead to traffic flow stability (Swaroop and Rajagopal 1999). It has also been shown that it is possible to design other spacing policies (in which the desired spacing between vehicles is a nonlinear function of speed, instead of being proportional to speed) that can provide stable traffic flow (Santhanakrishnan and Rajamani 2003).

The importance of traffic flow stability has not been fully understood by the research community. Traffic flow stability is likely to become important when the number of ACC vehicles on the highway increase and their penetration percentage into vehicles on the road becomes significant.

Recent Automotive Market Developments

The latest versions of ACC systems on the market have been enhanced with collision warning, integrated brake support, and stop-and-go operation functionality.

The collision warning feature uses the same radar as the ACC system to detect moving vehicles ahead and determine whether driver intervention is required. In this case, visual and audio warnings are provided to alert the driver and brakes are pre-charged to allow quick deceleration. On Ford’s ACC-equipped vehicles, brakes are also automatically applied when the driver lifts the foot off from the accelerator pedal in a detected collision warning scenario.

When enabled with stop-and-go functionality, the ACC system can also operate at low vehicle speeds in heavy traffic. The vehicle can be automatically brought to a complete stop when needed and restarted automatically. Stop-and-go is an expensive option and requires the use of multiple radar sensors on each car. For instance, the BMW ACC system uses two short range and one long range radar sensor for stop-and-go operation.

The 2013 versions of ACC on the Cadillac ATS and on the Mercedes Distronic systems are also being integrated with camera based lateral lane position measurement systems. On the Mercedes Distronic systems, a camera steering assist system provides automatic steering, while on the Cadillac ATS, a camera based system provides lane departure warnings.

Future Directions

Current ACC systems use only on-board sensors and do not use wireless communication with other vehicles. There is a likelihood of evolution of current systems into co-operative adaptive cruise control (CACC) systems which utilize wireless communication with other vehicles and highway infrastructure. This evolution could be facilitated by the dedicated short-range communications (DSRC) capability being developed by government agencies in the US, Europe and Japan. In the US, DSRC is being developed with a primary goal of enabling communication between vehicles and with infrastructure to reduce collisions and support other safety applications. In CACC, wireless communication could provide acceleration signals from several preceding downstream vehicles. These signals could be used in better spacing policies and control algorithms to improve safety, ensure string stability, and improve traffic flow.


  1. Fancher P, Bareket Z (1994) Evaluating headway control using range versus range-rate relationships. Veh Syst Dyn 23(8):575–596CrossRefGoogle Scholar
  2. Rajamani R (2012) Vehicle dynamics and control, 2nd edn. Springer, New York. ISBN:978-1461414322CrossRefzbMATHGoogle Scholar
  3. Santhanakrishnan K, Rajamani R (2003) On spacing policies for highway vehicle automation. IEEE Trans Intell Transp Syst 4(4):198–204CrossRefGoogle Scholar
  4. Swaroop D, Hedrick JK (1996) String stability of interconnected dynamic systems. IEEE Trans Autom Control 41(3):349–357CrossRefzbMATHMathSciNetGoogle Scholar
  5. Swaroop D, Rajagopal KR (1999) Intelligent cruise control systems and traffic flow stability. Transp Res C Emerg Technol 7(6):329–352CrossRefGoogle Scholar
  6. Swaroop D, Hedrick JK, Chien CC, Ioannou P (1994) A comparison of spacing and headway control laws for automatically controlled vehicles. Veh Syst Dyn 23(8):597–625CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MinnesotaTwin Cities, Minneapolis, MNUSA