（節(jié)選）外文翻譯--增益輪滑動(dòng)控制在汽車(chē)制動(dòng)系統(tǒng)中的應(yīng)用（英文）

1、IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 6, NOVEMBER 2003 799Gain-Scheduled Wheel Slip Control in Automotive Brake SystemsTor A. Johansen, Senior Member, IEEE, Idar Petersen, Jens Kalkkuhl, and Jens

2、LüdemannAbstract—A wheel slip controller is developed and experimen- tally tested in a car equipped with electromechanical brake actu- ators and a brake-by-wire system. A gain scheduling approach is taken, where the

3、 vehicle speed is viewed as a slowly time-varying parameter and the model is linearized about the nominal wheel slip. Gain matrices for the different operating conditions are de- signed using an LQR approach. The stabili

4、ty and robustness of the controller are studied via Lyapunov theory, frequency analysis, and experiments using a test vehicle.Index Terms—Antilock braking, automotive control, gain sched- uling, nonlinear control, optima

5、l control.I. INTRODUCTION AN ANTILOCK brake system (ABS) controls the slip of each wheel of a vehicle to prevent it from locking such that a high friction is achieved and steerability is maintained. ABS brakes are charac

6、terized by robust adaptive behavior with respect to highly uncertain tire characteristics and fast changing road surface properties and they have been commercially avail- able in cars for more than 20 years [1], [2]. The

7、 introduction of advanced functionality such as electronic stability program (ESP), drive-by-wire, and more sophisticated actuators and sensors offer both new opportunities and require- ments for more accurate and flexib

8、le control in automotive brake systems. The brake system is no longer a stand-alone system whose only purpose is to generate stable and efficient braking, but is seen as a subsystem where each wheel can receive individua

9、l brake commands from higher level control systems. For example, the ESP system may achieve lateral stabilization by commanding brake torque or target slip to the ABS. The target slip may also be based on automatic monit

10、oring of the road conditions, e.g., [3]. This makes wheel slip control an interesting alternative to conventional ABSs, where the control logic usually does not include an explicit wheel slip controller [2], [4], [5]. Th

11、e contribution of this paper is a study of a model-based de- sign of wheel slip control, extending the preliminary results de- scribed in [6], [7]. We consider electromechanical actuators, [8],Manuscript received June 12

12、, 2002. Manuscript received in final form March 3, 2003. Recommended by Associate Editor A. Stefanopoulou. This work was supported by the European Commission under the ESPRIT LTR-Project 28104 H C. T. A. Johansen is with

13、 SINTEF Electronics and Cybernetics, N-7465 Trond- heim, Norway, and also with the Department of Engineering Cybernetics, Nor- wegian University of Science and Technology, N-7491 Trondheim, Norway. I. Petersen is with SI

14、NTEF Electronics and Cybernetics, N-7465 Trondheim, Norway. J. Kalkkuhl and J. Lüdemann are with DaimlerChrysler, Research and Tech- nology, D-10559 Berlin, Germany. Digital Object Identifier 10.1109/TCST.2003.81560

15、7[9], rather than conventional hydraulic actuators, which allow accurate continuous adjustment of the clamping force. Despite the fact that the wheel slip dynamics are highly nonlinear, our control design relies on local

16、 linearization and gain-scheduling. In order to analyze the effects of this simplification, we develop a somewhat idealized Lyapunov-based nonlinear stability and robustness analysis, taking into account uncertain tire f

17、riction nonlinearities. In order to also investigate the effects of sam- pling, communications delays, actuator dynamics, and the fun- damental limitations in performance, this analysis is comple- mented by a classical f

18、requency analysis. Experiments using a test vehicle are included. Other contributions to model-based wheel slip control for ABS can be found in the literature. An adaptive control-Lya- punov approach is suggested in [10]

19、, and similar ideas are pursued in [11], [12]. The use of Sontag’s formula is applied in the adaptive control Lyapunov approach in [13], which includes gain scheduling on vehicle speed and experimental testing. Feedback

20、linearization in combination with gain scheduling is suggested in [14]. In contrast, our controller contains no explicit friction model and relies on integral action rather than adaptation in order to eliminate steady-st

21、ate uncertainty. This simplifies the design and may potentially improve robustness as the friction is difficult to model accurately for a wide range of tires and surfaces. PID-type approaches to wheel slip control are co

22、nsidered in [15]–[19]. Our work is based on a gain scheduled LQ control design approach with associated analysis, and, except [13] and [19], is the only one that contains detailed experimental evaluation using a test veh

23、icle. In [20], an optimum seeking approach is taken to determine the maximum friction, using sliding modes. Sliding mode control is also considered in [21] and [22].II. MODELINGIn this section, we review a mathematical m

24、odel of the wheel slip dynamics, see also [1], [10], and [20]. The problem of wheel slip control is best explained by looking at a quarter car model as shown in Fig. 1. The model consists of a single wheel attached to a

25、mass . As the wheel rotates, driven by the inertia of the mass in the direction of the velocity , a tire reaction forceis generated by the friction between the tire surface and the road surface. The tire reaction force w

26、ill generate a torque that results in a rolling motion of the wheel causing an angular ve- locity . A brake torque applied to the wheel will act against the spinning of the wheel causing a negative angular acceleration.

27、The equations of motion of the quarter car are(1)1063-6536/03$17.00 © 2003 IEEEJOHANSEN et al.: GAIN-SCHEDULED WHEEL SLIP CONTROL 801during braking. Achieving this by manual control is difficult be- cause the slip d

28、ynamics are fast and open loop unstable when operating at wheel slip values to the right of any peak of the friction curve. We observe that a reasonable tradeoff between high longitudinal friction and lateral friction is

29、 achieved under all road conditions for longitudinal slip close to its peak value on the longitudinal slip curve. Hereafter, for simpli- fication purposes unless otherwise stated, the side slip angle will be considered t

30、o be zero with . Using (1)–(4), for and we get the wheel slip dynamics(5)(6)Notice that when , the open loop slip dynamics (5) be- comes infinitely fast with infinite high-frequency gain. Hence, the slip controller is sw

31、itched off for small . During braking it is clear that and , see also [6]. The dynamics of the wheel and car body are given by (5) and (6), respectively. Due to large differences in inertia, the wheel dynamics and car bo

32、dy dynamics will evolve on significantly different time scales. The speed will change much more slowly than the wheel slip, and is therefore a natural candidate for gain scheduling. Thus, for the control design, we consi

33、der only (5) and regard as a slow time-varying parameter. A gain scheduled control de- sign requires a set of nominal linearized models for design. Let ( ) be an equilibrium point for (5) defined by some nominal values ,

34、 andThe speed-dependent nominal linearized slip dynamics are given by(7)where h.o.t. denotes higher-order-terms, and and are lin- earization constants given by(8)(9)Notice that for nominal wheel slip values to the right

35、of any peak of the friction curve we get such that the open-loop dynamics are open-loop unstable. For nominal slip values slightly to the left of any peak, (notice that the last term in (8) is generally small) and the dy

36、namics are open-loop stable. Assuming arbitrary values of and , the wheel slip dynamics (5) can be written in the form(10)where and is the target slip (setpoint). Further- more, we have defined(11)and(12)It can be seen t

37、hat (10) has an equilibrium point given by ,since . The linearized slip model (7) with a perturbation term is written as follows:(13)where . Equation (13) will be used later on for control design and analysis.III. CONTRO

38、L DESIGN AND ANALYSISA. Control ProblemThe actual control input is the clamping force that is re- lated to the brake torque as , where the constantdepends on the friction between the brake pads and the brake disc. There

39、are limitations on the clamping force that can be applied to the brake pads by the actuator during braking. The (small) minimum force is to ensure that the brake pads are po- sitioned close to the brake disk with no air-

40、gap during braking. The maximum force is what the actuator is capable of. There is also a rate limit at how fast the torque can be changed by the actuator. The control problem is to regulate the value of the longitu- din

41、al slip to a given setpoint that is either constant or com- manded from a higher-level control system. The controller must be robust with respect to uncertainties in the tire characteristic, brake pads/discs, variations

42、in the road surface conditions, load on the vehicle etc. Integral action or adaptation must be incor- porated to remove steady-state error due to model inaccuracies, in particular the unknown maximum friction coefficient

43、 .B. Wheel Slip Control With Integral ActionLet the system dynamics (13) be augmented with a slip error integrator such that(14)where(15)The steady-state brake torque depends on road and tire prop- erties such as and mus