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Active Braking Control Design for Road Vehicles focuses on two main brake system technologies: hydraulically-activated brakes with on–off dynamics and electromechanical brakes, tailored to brake-by-wire control. The physical differences of such actuators enjoin the use of different control schemes so as to be able fully to exploit their characteristics. The authors show how these different control approaches are complementary, each having specific peculiarities in terms of either performance or of the structural properties of the closed-loop system. They also consider other problems related to the design of braking control systems, namely: • longitudinal vehicle speed estimation and its relationship with braking control system design; • tire–road friction estimation; • direct estimation of tire–road contact forces via in-tire sensors, providing a treatment of active vehicle braking control from a wider perspective linked to both advanced academic research and industrial reality.



Braking Control Systems Design: Introduction and Modelling


Chapter 1. Introduction to Active Braking Control Systems

It can be certainly acknowledged that skidding has been a problem for as long as wheeled vehicles have existed. A 1952 paper by A.C. Gunsaulus of Goodyear Aircraft Corporation [28] defines skidding as simply the “unwanted sideways movement [of an automotive vehicle] not planned by the driver... Its prime cause is a combination of a lessened grip of the tyre on the road coupled with a sideways force that is greater than the tyre’s grip. Its effect is usually a partial or, it may be, a total loss of control of the vehicle by the driver.”
In road vehicles, the unwanted skidding phenomenon can be prevented by means of active braking control systems.
As a matter of fact, most modern road vehicles are equipped with electronic ABS. ABS can greatly improve the safety of a vehicle in extreme circumstances, since it can maximise the longitudinal tyre–road friction while keeping large lateral (directional) forces that ensure vehicle driveability. The use of automatic braking control systems has also been extended to electronic stability control (ESC) systems (see, e.g., [27, 39, 45, 88]).
Sergio M. Savaresi, Mara Tanelli

Chapter 2. Control-oriented Models of Braking Dynamics

This chapter is devoted to introducing the models of the braking dynamics employed for the design of the different active braking control systems developed in the next chapters, i.e., the single-corner and double-corner models.
As both single-corner and double-corner vehicle models clearly employ a tyre–road friction description, based on which the contact forces can be defined, before introducing the vehicle braking dynamics the force and friction model adopted in this book will be presented.
Note that the treatment of these topics is not intended to provide a comprehensive overview on vehicle modeling nor to present all the available contact forces and tyre–road friction models available in the literature. The aim is to provide the reader with the description of the dynamical models which are of interest for the design of braking control strategies. For a more detailed discussion on these topics, the reader may refer to, e.g., [45, 71, 84].
The chapter is structured as follows. Section 2.2 introduces tyre–road contact forces and presents the adopted friction model. Further, Section 2.3 describes the single-corner model of the braking dynamics, whereas Section 2.4 the double-corner one. In Section 2.5 the considered dynamical models of the braking dynamics are analysed and their linearised version is computed. A numerical analysis of the linearised dynamics is also proposed, to point out the sensitivity of the model dynamics to specific vehicle parameters.
Sergio M. Savaresi, Mara Tanelli

Braking Control Systems Design: Basic Solutions


Chapter 3. Braking Control Systems Design: Actuators with Continuous Dynamics

This chapter addresses the problem of braking control design based on actuators with continuous dynamics (see Section 1.3 for their dynamic description).
Of course, the actuator performance forces the engineer to design the braking control system accordingly. Here, for the case of an actuator with continuous dynamics we show how to design a wheel slip controller that can guarantee closed-loop stability and acceptable performance in all possible working conditions by solving a regulation problem. It will be also clear that, because the braking controller is a safety-oriented aid for the driver, it must be switched on and off according to the current manoeuvre. Thus, an appropriate activation and deactivation logic must be designed to accomplish this task.
Further, we investigate the wheel slip control problem starting from a double-corner model, i.e., taking into account the load transfer phenomena. The analysis highlights the effects of dynamic coupling between front and rear axles and its impact on ABS systems design. This leads to the selection of an alternative controlled variable for the braking control of the rear wheel, which arises from the idea of interlocking the rear wheels with the front wheels to achieve a more favorable dynamic behaviour while maintaining a SISO approach to wheel slip control design.
Sergio M. Savaresi, Mara Tanelli

Chapter 4. Braking Control Systems Design: Actuators with Discrete Dynamics

As discussed in the previous chapters, the design of automatic braking control systems is highly dependent on the braking system characteristics and actuator performance. This chapter addresses the problem of designing an ABS controller based on a hydraulic actuator system with on/off dynamics, which is capable of providing only three control actions: namely, one can only increase, hold and decrease the brake pressure.
Clearly, the control objectives must be traded off with the braking system capabilities. Accordingly, the aim of the control system will be that of maintaining the wheel slip around acceptable values, thus avoiding wheel locking, abandoning the goal of regulating it around a constant single value as was done in the preceding chapter for the case of a braking system with continuous dynamics.
To solve this problem, we will design a switching controller that yields closed-loop dynamics that converge to an asymptotically stable limit cycle.
For this control system, we provide necessary conditions for the limit cycle existence, which come from a detailed analysis of the state plane trajectories of the resulting braking dynamics. Further, the limit cycle stability properties are formally proved via a Poincaré map analysis (the interested reader may refer to Section A.2.2 for an introduction to the analysis tools used in this chapter).
Sergio M. Savaresi, Mara Tanelli

Chapter 5. Longitudinal Wheel Slip Estimation

As has become clear in the previous chapters of this book, the problem of estimating the longitudinal wheel slip is crucial for an effective design of ABS, TCS and ESC control systems, especially in the case when the control problem is formulated as the regulation of the wheel slip itself.
To obtain an accurate estimate of the longitudinal wheel slip one first of all needs to correctly process the wheel speed sensor measurements in order to obtain an estimate of the wheel speed. This topic is discussed in detail in Appendix B, where the two main algorithms employed for this task are discussed and their advantages and disadvantages outlined.
Further, according to the wheel slip definition, see Equation 2.3, one needs to estimate the longitudinal vehicle speed. In fact, this variable can be directly measured only by means of optical sensors, which are expensive and fragile and hence used only for prototyping purposes.
The crucial difference in setting up a speed estimation algorithm for ABS and TCS control systems is that in the former case all wheels are in general subject to braking torque (and thus to a non-zero wheel slip), whereas in the latter only the driving wheels transmit the traction torque to the ground, and thus two of the wheel speeds in fact evolve in almost free rolling (thus with no – or with negligible – wheel slip) and can hence be used estimate the vehicle speed.
Sergio M. Savaresi, Mara Tanelli

Braking Control Systems Design: Advanced Solutions


Chapter 6. Mixed Slip and Deceleration Control

As has been discussed in the previous chapters (see in particular Chapter 3), in braking control systems, two output variables are usually considered for regulation purposes: wheel deceleration and wheel slip. Deceleration control and slip control are mostly viewed as alternative strategies; when deceleration and slip are both used, the typical approach is to regulate one variable and to keep the other variable within pre-defined thresholds.
In this chapter a braking control strategy that makes use of both wheel slip and wheel deceleration is presented and analysed. It is based on the idea of designing the braking controller as a classical feedback regulation loop, where the regulated variable is a convex combination of the wheel slip and the wheel deceleration. Accordingly, this control approach is concisely named mixed slip-deceleration (MSD) control. MSD is effective and flexible; it inherits all the attractive dynamical features of slip control, while strongly alleviating the detrimental effects of poor slip measurement. Moreover, by simply changing the design parameter that governs the relative weighting between slip and deceleration in forming the convex combination it is possible to emphasise different characteristics of the controller, according to different working conditions.
Sergio M. Savaresi, Mara Tanelli

Chapter 7. Nonlinear Wheel Slip Control Design

Up to now, for the case of actuators with continuous dynamics, we have discussed how to design braking controllers based on linearised models of the braking dynamics of interest.
In this chapter, we discuss a nonlinear approach to wheel slip control design, namely a Lyapunov-based dynamic feedback control law. For an introduction to the control approach employed in this chapter, the reader is referred to Appendix A.
As a matter of fact, the braking dynamics are nonlinear due to the tyre–road interaction model; as such, the proposed approach allows us to take these nonlinearities directly into account and to consider their effects on the control algorithm. Further, the proposed control strategy is grounded on theoretical tools which allow us analyse its characteristics and to highlight its advantages with respect to the linear approaches.
Specifically, the nonlinear slip controller discussed in this chapter relies on a nonlinear dynamic feedback control law, based on wheel slip and wheel speed measurements, which guarantees bounded control action and thus copes with input constraints. The main advantage of the proposed controller with respect to a wheel slip controller based on a linearised model of the braking dynamics is that the closed-loop system properties allows one to detect if the current operating condition is such that the chosen wheel slip set-point determines a closed-loop equilibrium point which is beyond the peak of the tyre–road friction curve, thereby enabling us to adapt the set-point and yielding a significant enhancement of both performance and safety. This is a special feature of the proposed control law, which in general is not offered by other active braking control systems unless they are complemented with tyre–road friction estimators.
Sergio M. Savaresi, Mara Tanelli

Chapter 8. Identification of Tyre–road Friction Conditions

Tyre–road friction characteristics are deeply interlaced with all vehicle safety oriented control systems as road conditions strongly affect the controlled system behaviour. Thus, the capability of estimating in real-time the friction conditions may provide a valuable source of information for any active vehicle control system. In particular, friction information can be used to enhance the performance of wheel slip control systems.
In this chapter we address three different problems related with friction estimation. Specifically, Section 8.2 illustrates an approach that is capable of estimating the sign of the slope of the friction curve, thereby allowing one to detect if the system is operating in the stable or in the unstable region of the friction curve. In fact, as largely discussed in this book, the equilibrium points associated with the wheel braking dynamics are stable for values of the wheel slip before the peak and unstable for those beyond the peak.
Hence, an online detection of the slope of the friction curve can be exploited to adapt and to optimise the closed-loop performance of wheel slip control systems. The advantage of this identification method is that it can be implemented also with a very limited set of sensors.
Secondly, in Section 8.3 an approach to the problem of estimating both the slip value corresponding to the peak of the friction curve and the parameters of the Burckhardt friction model (see Section 2.1) is presented. This is done by setting up a curve fitting problem which is then solved by two different identification approaches, namely a least squares and a maximum likelihood approach, arising from different parametrisations of the friction curve. A detailed analysis of the merits and drawbacks of the two approaches is also provided, which considers both the obtained accuracy in the estimated parameters and the convergence issues which have to do with the length of the available data set.
Finally, Section 8.4 presents an approach for estimating the instantaneous vertical and longitudinal forces from in-tyre acceleration measurements. Specifically, an appropriate set of sensors and regressors is illustrated, based on the measurements provided both by standard vehicle sensors (wheel encoders) and an accelerometer mounted directly in the tyre. Such estimates are based on the idea of extracting information from the phase shift between the wheel hub and the tyre, which is due to the transmission of traction and braking forces exerted on the tyre itself.
Sergio M. Savaresi, Mara Tanelli


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