CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 28,aNo. 2,a2015 ·285·
DOI: 10.3901/CJME.2015.0106.006, available online at www.springerlink.com; www.cjmenet.com; www.cjme.com.cn
Generalized Internal Model Robust Control for Active Front Steering Intervention
WU Jian1, 2, ZHAO Youqun1, *, JI Xuewu3, LIU Yahui3, and ZHANG Lipeng2 1 College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 2 School of Mechanical and Automotive Engineering, Liaocheng University, Liaocheng 252059, China 3 State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China
Received January 6, 2014; revised October 9, 2014; accepted January 6, 2015
Abstract: Because of the tire nonlinearity and vehicle’s parameters’ uncertainties, robust control methods based on the worst cases, such as H∞, μ synthesis, have been widely used in active front steering control, however, in order to guarantee the stability of active front steering system (AFS) controller, the robust control is at the cost of performance so that the robust controller is a little conservative and has low performance for AFS control. In this paper, a generalized internal model robust control (GIMC) that can overcome the contradiction between performance and stability is used in the AFS control. In GIMC, the Youla parameterization is used in an improved way. And GIMC controller includes two sections: a high performance controller designed for the nominal vehicle model and a robust controller compensating the vehicle parameters’ uncertainties and some external disturbances. Simulations of double lane change (DLC) maneuver and that of braking on split-μ road are conducted to compare the performance and stability of the GIMC control, the nominal performance PID controller and the H∞ controller. Simulation results show that the high nominal performance PID controller will be unstable under some extreme situations because of large vehicle’s parameters variations, H∞ controller is conservative so that the performance is a little low, and only the GIMC controller overcomes the contradiction between performance and robustness, which can both ensure the stability of the AFS controller and guarantee the high performance of the AFS controller. Therefore, the GIMC method proposed for AFS can overcome some disadvantages of control methods used by current AFS system, that is, can solve the instability of
PID or LQP control methods and the low performance of the standard H∞ controller.
Keywords: active front steering system, generalized internal model robust control, H∞ optimization, PID, split-μ road
disturbances. And MPC system will also become unstable 1 Introduction when the constraints are violated by emergency disturbances or cornering stiffness variation, moreover Active front steering system (AFS) as an active safety MPC method requires online optimization. IMC is only system can improve vehicle’s handling and safety even in applicable to a stable controlled plant, in addition it is extreme situations and thus avoid accidents. Considering always combined with other control methods to overcome the strong nonlinearity and uncertainty in vehicle motion, the shortcomings. the control methods for AFS have been a key issue. For the reasons mentioned above, the robust control [1−2] ORABY, et al , presented an AFS control scheme based on the worst cases, such as H∞, μ-synthesis, etc, have based on optimal control theory. LI, et al[3], presented a PID been widely used in active front steering. DU, et al[8], plus fuzzy logic controller for vehicle dynamics control. A presented a robust yaw-moment controller design for model predictive control (MPC) scheme was applied to the improving vehicle handling and stability with vehicle dynamics control[4]. MEN, et al[5], proposed internal considerations of model uncertainties and control saturation. model control (IMC) based on combined brake and active ZHENG, et al[9], proposed a method to achieve the robust front steering for vehicle stability control. LI, et al[6−7], decoupling of the lateral and yaw motion. YOU, et al[10], researched parameter estimation techniques to make the put forward a two degree of freedom control system based control method adapt to different driving conditions. on H∞ loop-shaping control for automatic steering. To PID control and optimal control as high performance reduce the external uncertainties and disturbances, controller are simple and practical, but have very little nonlinear H∞ control theory with L2 gain performance was robustness against the model uncertainties and external applied in the vehicle dynamics control[11]. Linear fractional transformations (LFT) based on feedforward and feedback * Corresponding author. E-mail: [email protected] H control for vehicle handling improvement was Supported by National Natural Science Foundation of China(Grant Nos. ∞ [12] [13] 11072106, 51375009) presented . ESKI, et al , presented a neural © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015
·286· Y WU Jian, et al: Generalized Internal Model Robust Control for Active Front Steering Intervention network-based robust control system design for the active mass moment of inertia about the rotation axis, Ix is the roll steering system. moment of inertia (about vehicle x-axis), Ixz is the roll-yaw
The robust control methods used in the above situations product of inertia, Iy is the pitch moment of inertia (about have achieved relatively good results, but still have some vehicle y-axis), Iz is the yaw moment of inertia (about problems to be solved. As everyone knows, a contradiction vehicle z-axis), Qu is the general force along vehicle x-axis, between performance and stability exists in the robust Qy is the general force along vehicle y-axis, Qr is the control, therefore a trade-off between performance and general force for yaw motion, Q is the general force for stability against disturbances and parameters’ variations roll motion of roll part, ω is the wheel rotation speed. must be made. Unfortunately, the robust controller design is usually at the cost of performance, so that the robust controller is always conservative because most robust control design methods are based on the worst possible case. In this paper, a generalized internal model robust control (GIMC) that can overcome the contradiction between performance and stability is used in the AFS control. The Youla parameterization is used by GIMC in an improved way[14‒15]. GIMC controller includes two sections: a high performance controller designed for the nominal vehicle model and a robust controller. The high performance controller will solely work when there is no model uncertainties and external disturbances, while the robust controller will be active only when there are model uncertainties or external disturbances. This paper is arranged as follows. The second section introduces the nonlinear vehicle model, the tire model and driver model. The third section describes the GIMC robust method for AFS. In the fourth section, the simulation results of the several control strategies are shown, and a comparison is made among these control strategies. Finally, conclusions are drawn in the fifth section.
Fig. 1. A full vehicle model 2 Vehicle and Driver Models The equations of motion are as follows: The simulation models of the vehicle, tire and driver are presented in this section to evaluate GIMC control method. ïì ïmu()-- rv mh (2), r + r = Qu ï ïmv() +- ru mh ( - r2 ), = Q 2.1 Vehicle model ï v [16] ï It is shown in Fig. 1 that is an eight-DOF nonlinear ï íIrzzrxz+--()(), I I mh u -= rv Q r (1) vehicle model including the four wheels rotational, yaw, ï ï 2 ï()()(),I xzrxz++++-mh mh v ru I I r longitudinal, lateral, and roll motions. Meanings of the ï ï 22 variables and the parameters in Fig. 1 and in Eqs. (1) and (2) ï-+-+- = îï ()().mh Iyz I r C mgh Q are given in the latter part of this section. In Fig. 1, a is the horizontal distance between vehicle CG and front axle, b is And the general forces are given by the following horizontal distance between vehicle CG and rear axle, tw equations: is the wheel track, u is the longitudinal velocity, V is the velocity vector, v is the lateral velocity, β is the tire side slip ìQF=-+cos( ) F sin( ) F , ï uxf yf xr angle, r is the yaw rate, δ is the mean steering angle of front ï ïQF=-+sin( ) F cos( ) F , ï vxf yf yr wheels, K is the total roll damping, m is the total vehicle ï ïQaF=+-+sin( ) aF cos( ) bF mass, m is the mass of roll part, is the roll angle around ï rxfyf yr í (2) ï roll axis of vehicle, μ is the nominal tire-road friction ï tFwxflxfr[(---+ F )cos( ) ( Fyflyfr F )sin( ) ï coefficient, C is the total roll stiffness, Fx is the ï ()]/2,FF- ï xrl xrr longitudinal force in x-axis, Fy is the lateral force in y-axis, ï ïQK=- . Fz is the vertical force on tire, h is the CG height of total îï vehicle mass with regards to ground, h is the CG height of roll part (m) with regards to roll axis, Iw is the wheel total Parameters of vehicle model are shown in Table 1.
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·287· Table 1. Parameters of vehicle model driver-vehicle system block can be presented as shown in Parameter Value Fig. 3. Where, f(s) is the road input, y(s) is the lateral m/kg 1230 displacement response, V(s) is the vehicle’s transfer
m/kg 1010 function, T is the driver preview time, TaTc (, ) is the a/m 1.0 correction time, Td is neural lag time, Th is operation b/m 1.35 * lag time, C0 is the correction parameter, y is ideal tw/m 1.49 * −1 lateral acceleration, sw is the ideal steering wheel angle, Cf/(N • m • (°) ) 1580.4 −1 is actual steering wheel angle. Cr/(N • m • (°) ) 571.4 sw 2 Iz/(kg • m ) 1150 2 Ix/(kg • m ) 580 −1 Kf/(N • m • s • (°) ) 59.8 −1 Kr/(N • m • s • (°) ) 59.8 −1 Cf0(nominal)/(N • (°) ) 698.4 −1 Cr0(nominal) /(N • (°) ) 837.3 2 Ixz/(kg • m ) 0 h/m 0.54 i 16 Fig. 3. Driver-vehicle closed-loop system
2.2 Tire model Transfer function form of the driver single point preview The nonlinear tire model should be used to describe the model can be drawn through the derivation. nonlinearity of the tire forces. The Magic formula is preferred in dynamic control systems and can be expressed T [17] - d as follows : 1 s 2C =´2 0 sw ()s 2 æöTTTdhd2 T YDCBEB( )=--¢¢ sin[ arctan{ ¢¢¢ ( arctan( B ¢ ))}]. 1++çTss÷ + èøç h 22÷ (3) Ts 2 {}(1+-+++Tsccc ) f ( se ) [1 ( T Ts TTs ) y ] . (4) The variable is the tire slip angle, factors B, C, D and E can be obtained by fitting the tire experimental data in consideration of the vertical loads and the camber angle. 3 GIMC Strategy of AFS Design From Ref. [18], corresponding to a model on different adhesion coefficient roads, parameters B, C, D and E are The GIMC strategy of AFS developed in this paper is to shown in Table 2, where μ is the road adhesion coefficient. enhance vehicle stability and handling and solve the contradiction between performance and robustness. It is Table 2. Tire coefficient obvious that feedback control can be used to achieve desired performance even in the presence of external Road adhesion coefficient B C D E Front 6.765 1 1.200 0 −6426.8 −1.999 0 disturbances. Therefore, the GIMC structure of AFS =0.8 Rear 9.005 1 1.200 0 −5420.0 −1.790 8 depicted in Fig. 4 is established on the basis of feedback Front 11.275 1.560 0 −2574.7 −1.999 0 control and Youla parameterizations. =0.2 Rear 18.621 1.560 0 −1749.7 −1.790 8
The tire cornering force characteristic curves calculated with the given parameters is shown in Fig. 2.
Fig. 4. AFS control scheme based on GIMC
The GIMC structure includes a feedback Q and a -1 controller KUV0 = . The conditional compensatory Fig. 2. Cornering properties feedback controller Q included in GIMC guarantees robustness of the AFS GIMC controller, while the
2.3 Driver model controller K0 based on nominal vehicle model presents high According to optimal curvature preview control theory[18−19], performance. The Q is only active in the case where
·288· Y WU Jian, et al: Generalized Internal Model Robust Control for Active Front Steering Intervention external disturbances as lateral wind or unbalance braking forces on split-μ road and parameters variations of real vehicle model happen. The details of the GIMC strategy will be discussed in the following text of this section.
3.1 Nominal vehicle model 2-DOF vehicle model is widely used as approximate model of the vehicle dynamics. The model is of second Fig. 6. Youla parameterization T order with state vector as x = [] r and with the steering angle ( u = f ) as control input. System state 3.3 GIMC architecture space model expression is as follows: K0 is a stabilizing controller for the nominal vehicle
model G0. A new method of implementing the controller ïìxAxB =+u, KVQNUQM=-()()-1 + is presented as shown in Fig. íï (5) îïy = Cx. 7. This new method is called as the generalized internal model control. It is obvious that the scheme in Fig. 7 is not And the transfer function can be expressed as equivalent to the scheme in Fig. 6 because the input signal ri enters into the scheme from a different location. -1 Nevertheless, the internal stability of the system is not Gs0 ()=-CI ( s A ) B , (6) changed since the transfer function from y to ui is not where changed. The transfer function from y to ui in two schemes can be calculated as follows: éù-+()C C bC - aC é C ù êúfr00 r 0 f 0-u ê f 0 ú êúmu mu2 ê mu ú -1 A = êú, B = ê ú , usVUesQfsiYoula ()=+= [ () ()] êú22 êaC ú êúbCrf00--+ aC() a C f 0 b C r 0 f 0 -1 ê ú VUrsysQMesNus[ (ii ()-+ ()) ( () + ())] = êúê I ú ëûêúIUIzzë z û -1 V[ Ur (ii ( s )-+ ys ( )) QM ( { r ( s ) -+ ys ( )} Ns ( )), (8) C = [01] .
-1 After calculations, usVUesQfsiGIMC ()=+= [ () ()]
VUrsysQNusMys-1[ (()-+ ()) ( () - ())]. (9) 34.78s + 233.1 Gs()= . 0 ss2 ++11.88 54.43
3.2 Youla parameterization Fig. 5 is a standard feedback configuration. The generalized internal model robust control is the evolution from the standard feedback control. In this figure G is the real vehicle model and K is the controller. In general, the real vehicle model G is not exactly known. What one Fig. 7. Generalized internal model control structure knows is a nominal 2-DOF vehicle model G0. After some calculations, the following formulas can be obtained:
-+()()UMQ UMQ + us()=+ ys () rs (), (10) Fig. 5. Standard feedback configuration iYoula VQN-- VQN i
Let GMN= -1 , and { M-1, N } is the left coprime 0 -+()UMQ U factorization of G , and make KVU= -1 be a stabilizing us()=+ ys () rs (). 0 0 iGIMC i (11) controller. Then the set of controllers that can guarantee the VQN-- VQN feedback system internal stability is parameterized by It can be seen from Eqs. (10) and (11) that the transfer - KVQNUQM=-()(),1 + (7) functions from y to ui are equivalent, so the internal stability of the system is not changed. where QÎRH∞, where V and U can be chosen to satisfy The difference between GIMC shown in Fig. 7 and a the UN+= VM I [20−21]. The parameterized controller K of standard feedback control shown in Fig. 6 is that y instead Eq. (7) can be implemented in a standard feedback of e enters M in GIMC structure. Because of the IMC structure as shown in Fig. 6. characteristics, the Youla parameter Q is only active in the
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·289· case when there are model uncertainties or external The controller Q can guarantee the closed-loop stability disturbances. If there is no disturbance and no model by maximizing the robustness in the closed-loop system, uncertainty, Q in GIMC does not generate any control minT , i.e., wz ¥ where Twz is the closed-loop transfer signal and the control system is governed only by the Q function from signals w to z. nominal controller. Seen from Fig. 9, the following equations can be
obtained: 3.4 Nominal performance controller design
Fig. 8 is AFS control scheme based on the nominal --11 usiiiw()=--+ V [ Urs ( () Gus0 () Tz () susi ()) performance controller K0. The high performance controller -1 K0 is designed for the nominal vehicle model, and in this QNusMGus(iiw ( )--0 ( ) MTsuszi ( ) ( ))] = paper K0 is a PID controller. After repeated simulation --11 VUQMusTs[(-+ )()iwz ()+ ]. (12) analysis, K =10(1+8/s). 0
If ri = 0,
--11 usii()=- V [ UGus (0 () + Twz () susi ()) + -1 QNusMGus(()iiw--0 () MTsuszi ()())] = --11 VUQMusTs[(-+ )()iwz ()+ ]. (13)
-1 Among these formulas, GMN0 = and -1 ws()= Twz ()(), susi after some calculations, the following transfer function can be derived: Fig. 8. AFS control scheme based on K0
--11 Although PID control has little robustness against the Tswz ()=+ (1 KGV00 ) ( UQM+ ). (14) vehicle model uncertainties and external disturbances such as lateral wind or other unexpected disturbances, it is If the system is expected to have the strongest robustness, -1 -1 simple and practical as a high performance controller for min Twz Q ¥ , i.e., QUM=- , where M Î H¥ needs the nominal vehicle model. to be satisfied. In the paper, a fixed yaw rate gain is adopted to satisfy It is obviously seen from Fig. 9 that that the driver could determine the necessary steering wheel angle from visual information alone regardless of the u() s=-+- V-1 [ U ( r () s Gu ()) s QNus { () MGus ()}]. driving speed. Generally, a vehicle yaw rate gain of iiiii00 medium speed is most comfortable and drivable for driver, (15) therefore set v0=60 km/h. Considering the road friction With QUM=--1 and after simplifications, it can be restrictions, the desired yaw rate must be less than or equal obtained us()=- VUrs---111 () VUMNus () and to g /V . ii i Ts()=+ [1 KGK ]-1 , in which KVU= -1 . ruii 00 0 0 In order to solve the Q controller from the GIMC 3.5 Design of robust compensational controller Q(s) structure, the GIMC structure as seen in Fig. 10 should be In order to solve the instability of the nominal transformed into the linear fractional transformation with performance controller K0, a compensational controller Q the generalized plant P given by for robustness is designed in this section. In general, the uncertain real vehicle model can be described by a linear é(1++KG )--11 K (1 KG ) KV-1ù fractional transformation as shown in Fig. 9, where ∆ P = ê 00 0 00 0 ú, ê ú (16) includes the vehicle model uncertainties. ë -M 0 û
Fig. 10. Generalized linear fractional transformation
==min T Hence, Q is solved according to wz ¥ Fig. 9. GIMC considering model uncertainty Q
·290· Y WU Jian, et al: Generalized Internal Model Robust Control for Active Front Steering Intervention minFGQL ( , ) , Q ¥ where
--11 TFQwz==-+++- L (,)P (1 KGK00 ) 0 (1 KGQM 00 ) ( ),
< if ¥ 1/ ,internal stability is guaranteed.
4 Simulation Results
The purpose of simulation is to evaluate the effect of the
GIMC controller and compare GIMC with PID, H∞ controller on the general split-μ road and under double lane change maneuvers.
4.1 Emergency braking on split-μ road A braking manoeuvre on split-μ road was adopted for testing the stability and performance of the GIMC controllers under Matlab/Simulink environment and the results are shown in Fig. 11. It is supposed the right of split-μ road was dry (μhigh=0.8) while the left was icy
(μlow=0.2). The braking was implemented at t1=2 s and relieved at t2=3 s, and the brake master cylinder pressure was 3 MPa.
Fig. 11. Vehicle status braking on split-μ road
It can be seen from Figs. 11(a) and 11(b) that the yaw rate of the uncontrolled vehicle rises up to 30.2/s and slip angle to 10.2 at 2.84 s. Generally when slip angle exceeds 10, the vehicle will spin and go into instability. Figs. 11(e) and 11(f) illustrate that the yaw and lateral location of the uncontrolled vehicle deviates largely from the target path. In such condition, it is a difficult task for most drivers to maneuver the vehicle. Sideslip angle and yaw rate values
can be kept at relatively low values by GIMC, PID and H∞ controllers, and the active steering intervention by GIMC,
PID and H∞ controllers can maintain the stability of the vehicle and reject the disturbances without the driver action during emergency braking on split-μ road. From Figs. 11(a), 11(b), 11(e), and 11(f), it also could be concluded that the performance of GIMC, PID controllers
is higher than that of the H∞ controller, and the performance of GIMC controller is only a little bit weaker than that of
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·291· high performance PID controller based on nominal vehicle Fig. 12(e) shows the intervention steering angles under model. It can be seen from Fig. 11(c) that the intervention different controllers. The output of high performance PID steering angle of high performance PID controller is bigger controller has oscillated and can’t converge, which means than that of GIMC and H∞ controllers, which proves that the controller has been unstable. Because H∞ controller despite PID controller has the best performance in rejecting includes the feedforward, the output of H∞ controller is a the disturbances caused by emergency braking on split-μ little ahead of the output of the PID and GIMC controllers. road, high performance PID controller is at the expense of Fig. 12(f) shows the output of the Q controller. stability. Fig. 11(d) is the output of the Q controller. From what has been discussed above, the high performance PID controller is at the expense of stability,
H∞ controller guarantees the robustness at the expense of performance, and only the GIMC controller can provide both high performance and robustness, which not only rejects the disturbances quickly but also guarantees the intervention steering angle small enough.
4.2 Performance comparison during DLC maneuver The double lane change (DLC) maneuver is to make vehicle reach a certain sequence of alternate high lateral accelerations, such that the vehicle emergency obstacle avoidance ability can be evaluated. The initial speed of DLC maneuver in this simulation is 126 km/h. Fig. 12(a) shows that the uncontrolled vehicle sharply swerved in the exit of the DLC trajectory and the vehicle with GIMC, PID and H∞ controllers can pass the trajectory with a certain degree of deviation. The lateral deviation of the vehicle with H∞ controller is much bigger than those with the GIMC and PID controllers, and the lateral deviation of the vehicle with GIMC controller only a little bit smaller than that with the high performance PID controller. The yaw rate responses of the car with GIMC, PID and
H∞ controllers can be kept at relatively low values, and do not present overshot, while that of the uncontrolled car has a larger amount of overshot so that vehicle goes into instability after 6 s (Fig. 12(b)). Seen from Fig. 12(b), the yaw rate responses of the car with GIMC, PID and H∞ controllers can closely follow the respective reference, the reference yaw rate values and yaw rate values of the vehicle with H∞ controller are bigger than those with GIMC and PID controllers, and the reference yaw rate curve and yaw rate curve of the vehicle with GIMC controller are very close to those with the high performance PID controller. Figs. 12(c) and 12(g) are the slip angle and roll angle curves, and have the same trend as the Figs. 12(a) and 12(b). Seen from Figs. 12(a), 12(b), 12(c) and 12(g), the PID controller based on nominal vehicle model has the highest performance, the GIMC controller has the similar properties with the PID controller, and the H∞ controller has the lowest performance. Figs. 12(d) shows the driver steering angles under the different controllers. It can be concluded that because of the
H∞ controller conservatism, driver steering angle under H∞ controller is bigger than that under the PID and GIMC controllers.
·292· Y WU Jian, et al: Generalized Internal Model Robust Control for Active Front Steering Intervention
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CHINESE JOURNAL OF MECHANICAL ENGINEERING ·293· and Astronautics, China, and an instructor at Liaocheng JI Xuewu is a vice professor at Department of Automotive University, China. His research interests include vehicle lateral Engineering, Tsinghua University, China. His research interests dynamics control and vehicle active safety control. include vehicle steering system technology and the vehicle E-mail: [email protected]. integrated control. E-mail: [email protected]
ZHAO Youqun is a professor at Nanjing University of Aeronautics LIU Yahui is an instructor at Department of Automotive and Astronautics, China. His research interests include vehicle Engineering, Tsinghua University, China. His research interests dynamics control and automotive design theory and test methods, include vehicle steering system technology and vehicle lateral etc. dynamics control. E-mail: [email protected]. E-mail: [email protected]