improving handling stability perance of four-wheel steering vehicle via μ-synthesis robust control

1、2432IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007Improving Handling Stability Performance of Four-Wheel Steering Vehicle via-Synthesis Robust ControlGuodong Yin, Nan Chen, and Pu Li (M )GMaximum singular value of M .AbstractA -synthesis robust controller of a four-wheel s

2、teer- ing (4WS) vehicle is designed with the optimized weighting func- tions to attenuate the external disturbances while the yaw rate is chosen as the only feedback signal. Numerical simulation shows that the 4WS vehicle equipped with the -synthesis robust con- troller has better maneuverability an

3、d a stronger ability to resist disturbances; it is not sensitive to the variations in vehicle parame- ters. The -synthesis controller has an advantage over the H controller; the latter is thought to be relatively conservative for normal ambient disturbances.Index TermsFour-wheel steering (4WS), robu

4、st control, yaw rate tracking, -synthesis.G = sup G(j), H norm of transferRfunction G(s), where (G) is the maximum sin- gular value of G.Set of complex numbers. Set of real numbers.R C(M )Spectral radius of a matrix M, wherei(M)is an eigenvalue of a matrix M (M ) := maxi |i(M )|.Absolute value of el

5、ement in R or C.|Subscriptsf, rFront wheel, rear wheel.NOMENCLATURETotal mass of vehicle (in kilograms).Distance from (front, rear) axle to vehicles center of gravity (CG) (in meters).Vehicle speed (velocity) at CG (in meters per second).(Front, rear) tire cornering stiffness (in Newtons per radian)

6、.Steering angle of the tire (front, rear) (in degrees). Time (in seconds).Laplace operator.Yaw moment of inertia about z-axis (in kilograms per square meter).Lateral displacement.System matrix, output matrix.Input matrices related to control and disturbance vectors.Structured singular value of a mat

7、rix M. Angular frequency.mLf, LrI. INTRODUCTIONOUR-WHEEL steering (4WS) systems for automobiles have been actively studied as one of effective vehicleFmaneuvering technologies. Properly designed systems can im- prove the maneuverability of vehicles at low speeds and en- hance their stability at high

8、 speeds. For early 4WS vehicles, a simple speed-dependent ratio between rear and front wheels has been used in an open-loop controller to achieve a zero con- stant sideslip angle during directional maneuvers 1. This con- troller is simple, but it may not perform well during the transient motion. On

9、the contrary, if rear wheels are controlled according to the input front steering wheel and the vehicle real-time situa- tion 25, the closed-loop control of the 4WS vehicle will be more robust and can improve vehicle directional stability.It is well known that the vehicle maneuvering can be highly u

10、ncertain, so it is a highly nonlinear and complex dynamical process. The parameters of a vehicle are subject to a vast range of uncertainties such as external disturbances, unmod- eled dynamics, road roughness, wind gusts, load fluctuations, braking/accelerating forces, etc. Thus, a serious robust s

11、tability problem for the 4WS vehicle control has been raised, namely, the vehicle controller has to cope with these uncertainties to keep maneuvering stability and ensure that the system perfor- mance does not deteriorate too much.Modern robust control theories have proved some useful techniques to

12、solve above problems. They give some powerful tools to deal with the uncertainties met by the 4WS vehiclecontrol. A typical robust control method is H2/H synthesis 3. Based on the linear vehicle dynamics model and the tiremodels, although those models are not very accurate underlimited conditions, H

13、2 /H synthesismayimprovetherobust- ness of the designed controller under some uncertainties 4.Kf, Krf, r ts IzyA, CB, D(M )G(s)A CB DState space realization =A) 1B + D.= C(sIManuscript received April 14, 2004; revised August 2, 2004, January 14,2005, May 6, 2005, May 11, 2006, July 19, 2006, and Sep

14、tember 5, 2006. This work was supported in part by the National Science Foundation of China Fund under Grant 50575041, by the Ford-China R&D Fund under Grant 50122153, and by the Southeast University Excellent Doctor degree Thesis Fund under Grant YBJJ0402. The review of this paper was coordinated b

15、y Dr. M. S. Ahmed.The authors are with the College of Mechanical Engineering, Southeast Uni- versity, Nanjing 210096, China (e-mail: ; ; ).Color versions of one or more of the figures in this paper are available online at .Digital Obje

16、ct Identifier 10.1109/TVT.2007.8999410018-9545/$25.00 2007 IEEEYIN et al.: IMPROVING HANDLING STABILITY PERFORMANCE OF 4WS VEHICLE2433Fig. 1. Half vehicle dynamic model.II. VEHICLE SYSTEM DYNAMIC MODELThe essential features of vehicle steering dynamics in the horizontal plane can usually be describe

17、d by a bicycle model, as shown in Fig. 1 9. The coordinate frame x ,y0 0,z0 is an Earth-fixed frame as a reference coordinate system, where the z0-axisrepresentsthedirectionnormaltothex0, y0 plane. The bodyframex, y, z isfixedatthevehicles CG. Thebodyframe is initially parallel to the Earth fixed se

18、t. Then, the vehicle is moving at a constant speed ( = 0) measured at CG and is also rotating at angular (yaw) rate r relative to the reference frame. The chassis sideslip is the angle of the CG velocity with respect to the y-axis. The body frame is rotated by yaw angle with respect to the reference

19、 set.In Fig. 1, Ff and Fr are the lateral forces due to the contact between the tire and the road surface at each of the front and rear wheels. The other definitions will be seen in the notations. The tire side forces can be projected through the steering angle into chassis coordinates x, y; thus, t

20、he longitudinal force Fx and lateral force Fy caused by front and rear tires can beexpressed asHowever, the synthesis by the standard H method is relatively conservative since, with this theory, only a boundary of theunmodeled dynamics is considered; the system perturbation has not be distinguished

21、carefully. Among an improved approach, the design performance for robust control problems can be further enhanced by -analysis 6. The recent advance in - synthesis has made it possible to analyze and to design a controller that can deal with the dynamic systems with strong uncertainties, such as the

22、 vehicle system 7.The objective of this paper is to present design issues of robust controllers for the 4WS vehicles under the yaw ratetracking architecture by using -synthesis with the DK it- eration algorithm 8. This approach is employed to improvevehicle performance such as its robustness and lat

23、eral motion stability with respect to a given class of uncertainties. The vehicle yaw rate is chosen as the only feedback signal to avoid the practical difficulty of measuring the center of gravity (CG) sideslip angle of the vehicle.In this paper, with the yaw rate tracking control architecture, the

24、 -synthesis methodology is introduced to design the 4WS vehicle robust controller in the case of the multiplicative model for the system perturbation. The weighting function matrices are selected according to the feedback tracking properties and the requirement of the considered disturbance resistan

25、ce. The low-frequency parametric perturbations and the high-frequency unmodeled dynamics are considered at the same time. More-over, because the obtained controller through DK iteration may have too-high controller orders which cause “numerical ill-conditioning,” a new approach for the -synthesis wi

26、th balance realization is proposed, by which the obtained controller is proven to be close to the optimal controller, but its orders are decreased greatly.The designed -synthesis controller provides a good ro- bustness against a rather wider system perturbation, such as fluctuating loads and change

27、of vehicle velocity. The numerical simulation shows that the 4WS vehicle equipped with the proposed controller can achieve preferable dynamic and steady- state performance. The whole control system has fine dy- namic characteristics as well as better stability and performance robustness.This paper i

28、s organized as follows. Section II is devoted to deducing a 2-DOF bicycle model for the 4WS vehicle to keep the integrality of its contents. The design of the -synthesis robust controller for the 4WS vehicle is given in Sections III and IV. Section V explains the numerical simulation results that sh

29、ow the control and robust performance of the controller for 4WS. Conclusions are provided in Section VI.Fx = Ff sin f Fr sin r Fy = Ff cos f + Fr cos r(1)(2)where f and r are the steering angle of front and rear wheels, respectively. Because f and r are generally smallcos f 1 and cos r 1.(3)From Fig

30、. 1, the following three equations are obtained to characterize the motion of the vehicle in the horizontal plane:Longitudinal motion m( + r) sin + mcos = FLateral motion(4)xm( + r) cos + msin = FYaw motionIzr = Lf Ff LrFr(5)y(6)where Iz denotes the yaw moment of inertia about its mass center z-axis

31、, and m is the vehicle mass.2434IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007The slip angles of front and rear tires are represented by fand r. If is small, and is varying slowly, thenLfLrf = f randr = r +r.(7)In general, lateral tire forceisanonlinearfunction of slipan-

32、gle. In this paper, the cornering stiffness for the front (rear) tire is denoted by Kf(Kr), and its value depends on the tireroad interaction. As long as the tire slip angle is small, a linear relationship between tire force and slip angle can be justified. Then, the lateral forces generated by the

33、front and the rear tires vary linearly with their slip angles, so Kf and Kr are constants, and we haveFig. 2. 4WS vehicle closed-loop control system.Because the front wheel steering angle is commonly pro- portional to the steering wheel angle that is controlled by the driver, the front wheel steerin

34、g angle is taken as the input signal. The -synthesis robust controller based on yaw rate tracking is designed for the rear wheel steering angle control with the consideration of some possible uncertainties.The transfer function from the front and rear steering input to yaw rate r could be expressed

35、asFf = Kff ,Fr = Krr .(8)If the vehicle is neither accelerating nor decelerating in the longitudinal direction, (4) may be omitted, and the lateral mo- tions of the vehicle are governed by (5) and (6). Therefore, com- bining (4)(8) yields the lateral equations of vehicle motion:b1s + b2s2 + a1s + a2

36、b3s + b4s2 + a1s + a2Gf (s) =and G (r s) =Lf KfIzKf Kr(Lf + Lr)mIzLrKr Izb3 = Kf +Krmb1 =,b2 =,Lf Kf LrKrm = fK + K ) m +r(r+ K ff+ K r rL2Kf+ L 2K r(9) fr+b4 = b2,a1 =L K +L K22zI r= (Lf Kf Lr Kr ) + Lf Kf f LrKrr.frrrfIzKf Kr(Lf+ L r)2 m2IzLrK r L fKf .a2 =(12)+In addition, the lateral acceleratio

37、n ay at the CG is obtained by the yaw rate r and the vehicle sideslip angle with the following relation:IzIt is obvious thatGf(s)andGr(s)are the functions ofvelocity , mass m, and cornering stiffness Kf and Kr, which may all vary under different running conditions. Therefore, a robust controller for

38、 the K, which is shown in Fig. 2, is needed in order to satisfy all robust requirements with respect to all uncertainties.The desired yaw rate is assumed to be 10, 11y = y = (r + )(10)where y is defined as the lateral displacement.The state-space describing the system dynamics is given byx = Ax + Bu

39、y = Cx + Dur = s/300 +0.8(11)(13)s/10 + 1where the state vector x = rT, control input vector u = f rT, and the out vector y is the same as the state vector x, namely, supposing y = x.The matrices A, B, C, and D in (11) will be given byKf +Krwhere r is corresponding to a yaw rate of vehicle response

40、which is of agility and without much overshoot.III. ROBUST CONTROL SYSTEM FORMULATIONOn the design of vehicle control system, it is necessary to take consideration of changes in the parameters of the vehicle due to variations in running ambient and driving velocity. In order to compensate the robust

41、ness against the changes in the parameters, the linear feedback controller K is designed by applying the -synthesis. Lf Kf +LrKr 1mm2A =L2K f+L 2K r Lf Kf +LrKr frIzIzKf m Lf KfIzKrmL KB =r r IzC =0110A. Lateral Vehicle Model With Uncertainty DescriptionTo consider the uncertainty in the vehicle run

42、ning environ- ments and driving velocity by the -synthesis, we identify a nominal plant model to design the controller. It is well known that the vehicle speed in the state-space realization matrices (A, B, C, D) is taken as a constant; therefore, the controller design is related to one speed at tha

43、t moment; in practice, theD =.0000Since the estimation of the sideslip angle is difficult but the yaw rate r is relatively easier to measure in practice, the yaw rate r is chosen as the only feedback signal to construct the control of the system (11). The 4WS vehicle control can be simplified as a y

44、aw rate-tracking problem, as shown in Fig. 2.YIN et al.: IMPROVING HANDLING STABILITY PERFORMANCE OF 4WS VEHICLE2435Generally, a frequency-dependent weighting function matrixWs = diagWsf , Wsris used for covering or representing all possible structured un- certainty distributions between each true a

45、ctual system transfer function block and its corresponding nominal value.The structured uncertainty perturbation blocks f and rare assumed to be stable but unknown, except for their stan-f 1r 1.dard conditions, namely,andThefollowing high-pass frequency weightings have been used in representing thes

46、e:Fig. 3. 4WS system model based on the framework.0.005s2 + 5s +8s2 + 20s + 45Wsf(s) =0.006s2 + 6s + 11s2 + 15s + 50Wsr(s) =.At any frequency , the magnitude of |Ws(j)| can be interpreted as the percentage of the model uncertainty at thatfrequency. Therefore, this particular weight implies that ther

47、e exist potentially about 17.8% of Wsf and 22% of Wsr model errors at low frequency, and the uncertainty in the model may be up to 100% at high frequency.Now, the performance weighting functions are used in defining design specifications. The inverse of the performance weight indicates how much the

48、external disturbances should be rejected at the output or how much steady-state tracking errorFig. 4. 4WS vehicle closed-loop control system based on synthesis.speed always varies in a range accompanying an uncertainty that comes from the set of all possible system variations. Although it is always

49、necessary to design different controller parameters with respect to different vehicle speeds and make a real-time switch to the controller parameters corresponding to the vehicle speed when running, however, the change of the controller parameters is not smooth, so it is always hoped that the contro

50、ller parameter designed for one speed could work well for a certain speed range.Based on the framework of the -analysis, which is shown in Fig. 2, it is modified into an equivalent form, as shown in Fig. 3. We treat all parameters of ai and bi as constants, except for the speed since it dominates un

51、certain situations. The blocksf and r are the plant structured uncertainty perturbations, and they are added according to the multiplicative perturbation model of the -analysis. The weighting functions Wsf and Wsr represent the uncertainty frequency response of the correspond- ing transfer function

52、blocks. Here, 0 is used to denote the nom- inal speed. Additional measuring noise, external disturbance, and uncertainty blocks could be added to the model, depending on the situation and practical requirement if a more complicated model is needed for the design.due to external input is allowed. Wpr

53、(j)for the yaw rate andWp(j)for the slid slip angle are the weights specifying sys-tem performance. To limit yaw rate, the error er of the systemtracking the yaw rate is defined as er 1/Wpr so that the weight is chosen as Wpr = (0.4s + 0.9)/(s + 0.03) for allfrequencies. The corresponding steady-sta

54、te tracking error isless than 0.03/0.9 = 3.3%. The upper bound on |1/Wp(j)|is the weight for the tolerable maximum angle ; the weightis assumed to be constant over all frequencies and is setto Wp =(0.3s + 0.5)/(s + 0.01). The corresponding steady-state control error is less than 0.01/0.5 = 2%.To pro

55、vide the standard performance-robustness balance, theexact values for all weightings are actually determined after several design iterations.C. General Description for Vehicle Control SystemFor a general controlled system, there are three basic com- ponents: the generalized system P, the controller

56、K, and the multiple uncertainties . Rearranging the feedback system in Fig. 4 leads to the general structure shown in Fig. 5.Note that the system P consists of recognizing three sets of input/output variables. The first pair (u, r) consists of control and measurement signals. Next, the external and error signals (f, er, e) constitute the performance variables. Finally, the perturbation signals (d1, d2, zf, zr) connect the system through the perturbation . Because of its general structure, this control description is suitable for synthesis, as