A study on µ synthesis control for four wheel steering system to enhance vehicle lateral stability

中国科技论文在线 Guo-Dong Yin1 e-mail: ygdl@seu.edu.cn Nan Chen e-mail: nchen@seu.edu.cn Jin-Xiang Wang http://www.paper.edu.cn A Study on µ-Synthesis Control for Four-Wheel Steering System to Enhance Vehicle Lateral Stability e-mail: wangjx@seu.edu.cn School of Mechanical Engineering, Southeast University, Nanjing 210096, P.R China Ling-Yao Wu School of Automation, Southeast University, Nanjing 210096, P.R China e-mail: lywu@seu.edu.cn This paper presents the design of ␮-synthesis control for four-wheel steering (4WS) vehicle and an experimental study using a hardware-in-the-loop (Hil) setup First, the robust controller is designed and the selection of weighting functions is discussed in the framework of ␮-synthesis control scheme, considering the varying parameters induced by running vehicle condition Second, in order to investigate the feasibility of the four-wheel steering control system, the 4WS vehicle control system is built using dSPACE DS1005 platform The experimental tests are performed using the Hil setup which has been constructed using the devised rear steering actuating system The dynamics performance is evaluated by experiment using the Hil setup under the condition of parameter variations Finally, experimental results show that the ␮-synthesis controller can enhance good vehicle lateral maneuverability ͓DOI: 10.1115/1.4002707͔ Keywords: four-wheel steering, robust control, µ-synthesis, hardware-in-the-loop setup Introduction Four-wheel steering ͑4WS͒ technique is one of the most effective methods of vehicle active control systems, which aims to enhance handling and comfort characteristics ensuring stability in critical manoeuvring situations Several control strategies have been applied in 4WS vehicles, such as the LQG control, the fuzzy control, etc ͓1–3͔ In recent years, a great deal of attention has been paid to the Hϱ control because it not only provides a unified and general control framework for all control structures, but also yields a controller with guaranteed margins ͓4,5͔ However, the Hϱ control models all uncertainties as a single complex full block, which results in a rather conservative design ͓6,7͔ Under such circumstances, the ␮-synthesis technique, which involves the use of Hϱ optimization for synthesis and structured singular value ͑␮͒ for analysis, has been developed ͓8͔ Literature survey shows that most results related to ␮-synthesis are simulations, and there is no sufficient experimental evidence for four-wheel steering using ␮-synthesis In this paper, an active four-wheel steering controller is designed with the framework of ␮-synthesis control scheme, considering that the handling dynamic responses of 4WS vehicle can be affected by parameter variations resulting from cornering stiffness on different road conditions The control performance is evaluated by the optimal control and ␮-synthesis control simulations, based on the linear vehicle model, considering the parameter variations Finally, using the hardware-in-the-loop setup including the prototype control system, the performance of 4WS vehicle system is investigated by experiment test, which is carried out based on the dSPACE DS1005 PPC digital system Mathematical Vehicle Model 2.1 Linear Vehicle Model In developing the active controller, it is not desirable to use the complex vehicle model because of Corresponding author Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL Manuscript received May 13, 2008; final manuscript received June 13, 2010; published online November 23, 2010 Assoc Editor: Hemant M Sardar sampling time and implementation of the control system In this paper, the linear vehicle model is used for the design of a controller Figure shows a two-degree-of-freedom model including the yaw and lateral motion dynamics of 4WS vehicle, related to driver steering maneuvers, traveling on a road surface at a constant speed v In this model, the coordinate frame is fixed on the vehicle body in the center of gravity which is denoted as CG In a yaw plane representation, the sideslip angle ␤, at the center gravity of the vehicle, is assumed to be small, ͉␤͉ Ӷ The slip angles of front and rear tires ͑␣ f and ␣r͒ are, respectively, written as ͓4͔ ␣f = ␦f − ␤ − Lf r v and Lr r v ␣r = ␦r − ␤ + ͑1͒ where L f and Lr are the distances from the CG to the front and rear axles, respectively, and L = L f + Lr is the wheel base r is yaw angular velocity ␦ f and ␦r are the steering angle of front and rear wheels, respectively In general, lateral tire force is a nonlinear function of slip angle In this paper, under the assumption that the lateral tire forces F f and Fr are linear functions with slip angles ␣ f and ␣r, respectively, the following equations are used: F f = − ␮K f ␣ f and F r = − ␮ K r␣ r ͑2͒ where K f = Kcf K fn͑mgLr / L͒, Kr = KcrKrn͑mgL f / L͒, m is the total mass of vehicle, ␮ is the adhesion coefficient between road surface and the tire ranging from 0.8 ͑dry road͒ to 0.25 ͑icy road͒, and the cornering stiffness of the front ͑rear͒ tire is denoted by K f ͑Kr͒, K fn and Krn are normalized cornering stiffnesses, and Kcf and Kcr are cornering stiffness coefficients Considering the previous forces, equations of motion including lateral and yaw motions are written as ͓8͔ ͭ ͮ ␮ L f K f − ␮ L rK r mv␤˙ = − ͑K f + Kr͒␮␤ − mv + r + ␮K f ␦ f v + ␮ K r␦ r Journal of Dynamic Systems, Measurement, and Control Copyright © 2011 by ASME JANUARY 2011, Vol 133 / 011002-1 转载 中国科技论文在线 http://www.paper.edu.cn Fig A half vehicle dynamic model Izr˙ = − ͑L f K f − LrKr͒␮␤ − L2f K f + L2r Kr v Fig The 4WS closed-loop interconnection structure ␮ r + ␮ L f K f ␦ f − ␮ L rK r␦ r ͑3͒ where Iz denotes the yaw moment of inertia about its mass center z-axis In addition, the lateral acceleration ay at the CG is obtained by the yaw rate r and the sideslip angle ␤ with the following relation: ␣y = v͑r + ␤˙ ͒ ͑4͒ 2.2 State Space Representation From Eqs ͑3͒ and ͑4͒, the state space representation can be expressed as x˙ = Ax + Bu ͑5͒ y = Cx + Du where the state vector x = ͓␤ r͔T, the control input vector u = ͓␦ f ␦r͔T, and the output vector y = ͓␤ r ay͔T A= ΄ −␮ ␮ K f + Kr mv ␮ − L f K f + L rK r Iz − L f K f + L rK r −1 mv2 −␮ ΄ L2f K f + L2r Kr I zv Kf Kr ␮ ␮ mv mv B= L rK r LfKf ␮ ␮ − Iz Iz C= ΄ −␮ 0 K f + Kr L f K f − L rK r −␮ m mv ΅ ΅ ΅ ΄ ΅ , D= 0 0 ␮ Kf Kr ␮ m m The key parameters of vehicle and tires used in this paper are summarized in Table variations is considered In this system, the desired yaw rate gain G f−r is selected as ͓9͔ G f−r͑s͒ = s/300 + 3.75 s/10 + ͑6͒ where G f−r is corresponding to a yaw rate of vehicle response which is of agility See Fig Consider the steer disturbance ␦ f , controlled output z1, z2 and control input u which denotes the rear wheel angle ␦r, and y is the measured output containing only the yaw rate r n is the measurement noise The transfer function ᮀ, which represents the uncertainties between the nominal model and the actual plant, is assumed to be stable and unknown, except for the norm condition, ʈ⌬ʈϱ Ͻ ͓7,8͔ In the diagram, e is the input of the perturbation, d is its output The main performance objective is that the transfer function from ␦ f to z should be small, in the ʈ • ʈϱ sense, for all possible uncertainty transfer functions ᮀ The weighting functions W p and Wr reflect the relative importance of the different frequency domains in terms of the tracking error The weighting function Wn represents the impact of the different frequency domains in terms of the sensor noise Necessary and sufficient conditions for robust stability and robust performance can be formulated in terms of the structured singular value denoted as ␮ ͓6,7͔ Now, the design setup in Fig should be formalized as a standard design problem In order to analyze the performance and robustness requirements, the closedloop system, which is illustrated in Fig 3, is expressed by using the feedback effect u = Ky Note that the system P consists of recognizing three pairs of input/output variables The complete vehicle model for the control system is described by ΄΅ ΄ e P11 P12 P13 z = P21 P22 P23 P31 P32 P33 y ΅΄ ΅ d w u P͑s͒ Design and Evaluation of Robust Controller ͑7͒ 3.1 Synthesis of the ␮-Controller To design the active fourwheel steering controller K͑s͒, an output feedback ␮-synthesis control scheme is applied In order to obtain robust stability and performance, the additive modeling error resulting from parameter Table Parameters of the vehicle and the tires Parameter m Lf Lr Kf Kr Iz Value Unit 1740 1.035 1.655 35,000ϫ 37,500ϫ 3048 kg m m N/rad N/rad kg m2 011002-2 / Vol 133, JANUARY 2011 Fig The P − K structure with uncertainty Transactions of the ASME 中国科技论文在线 http://www.paper.edu.cn d = ⌬e ͑8͒ where z = ͓z1 z2͔ and w = ͓␦ f n͔ The system P augmented with weighting functions can be repartitioned as described in Eq ͑7͒ For the problem, the controller K can be combined with P via a lower linear fractional transformation ͑LFT͒ to yield the transfer function matrix M T T M͑s͒ = FL͑P͑s͒,K͑s͒͒ = ͫ P11 P12 P21 P22 ͬ + P13K͑I − P33K͒−1͓P31 P32 ͔ ͑9͒ which is actually obtained by substituting u = Ky into Eq ͑7͒ The LFT paradigm can be used to describe and analyze the uncertain vehicle system, where M corresponds to what is assumed as the constant in the control system and ⌬ is the block diagonal matrix Then, the matrix M is partitioned as ͫͬ ͫ e z = M 11 M 12 M 21 M 22 ͬͫ ͬ d z = FU͑M,⌬͒w = ͓M 22 + M 21⌬͑I − M 11⌬͒−1M 12͔w sup ␮⌬͓FL͑P,K͒͑j␻͔͒ ͑10͒ ͑12͒ stabilizing 3.2 Selection of Weighting Functions As is known, the weightings are included in the controller synthesis instead of in the control system implementation to yield robust performance and stability In general, in order to find a controller, they should be properly selected in advance The weighting functions W p and Wr represent the performance outputs, which are related to the components of z Now, the performance weighting function is used to define design specification The inverse of the performance weight indicates how much the external disturbances should be rejected at the output, or how much steady state tracking due to the external input is allowed W p͑j␻͒ for the sideslip angle and Wr͑j␻͒ for the yaw rate are the weights specifying system performance The upper bounds on ͉1 / W p͑j␻͉͒ and ͉1 / Wr͑j␻͉͒ are the weights for the tolerable maximum angle ␤ and the maximum tracking yaw rate; the weights are assumed to be constant over all frequencies and are set to 0.3s + 0.5 s + 0.01 Wr = s + 0.5 s + 0.01 3.3 Optimal Control Design The question of linear quadratic optimization control is to seek an optimization control signal u͑t͒ which minimizes the following performance index J with reference to the system described by Eq ͑5͒ ͵ ϱ ͑xTQx + uTRu͒dt ͑15͒ Here, Q and R are the weighting matrices, where Q Ն 0, R Ն 0,and ͑A , B͒ is assumed to be controllable and ͑A , C͒ is assumed to be observable The purpose of control is to minimize the sideslip angle; thus, Q and R take the following values: ͑11͒ Obviously, this is the standard ␮-control problem, and the design can be based on the MATLAB ␮-toolbox, in which the D − K iteration is adopted to perform the synthesis procedure D − K iteration is a two-step minimization process: the first step is a minimization of the Hϱ norm over all stabilizing controllers K, while the scaling matrix D is held fixed, and the second step is a minimization over a set of scaling D, while the controller K is held fixed ͓7,10͔ Wp = ͑14͒ where FU͑M , ⌬͒ is the upper LFT The robust performance of the closed-loop system about nominal plant perturbation is equivalent to ʈFU͑M , ⌬͒ʈϱ Ͻ The goal of the ␮-synthesis is to minimize over all stabilizing controllers K the peak value ␮⌬͑ · ͒ of the closed-loop transfer function FL͑P , K͒ The formula is as follows: ␻෈R 0.006͑s + 1͒ 500s + where the upper bound of ͉1 / Wn͑j␻͉͒ represents the maximal expected noise gain J= Moreover, the upper LFT connects w and z, which is obtained by combining Eq ͑8͒ with Eq ͑10͒ being expressed as K Wn = w M ͑s͒ out noise, all measurement signals will always be corrupted by a frequency-dependent noise The noise varying frequency should be suppressed In this study, the noise occurs at high frequency; therefore, it has to be weighted by high-pass characteristics The weighting Wn͑s͒ is given by ͑13͒ The weighting function Wn represents the impact of the different frequency domains in terms of sensor noise n In order to account for the fact that system outputs can never be sensed withJournal of Dynamic Systems, Measurement, and Control Q= ͫ ͬ ͫ ͬ 103 0 , R= 0 The control input u, which minimizes Eq ͑15͒, is u = −Kopx, where Kop is called an optimal feedback coefficient matrix given by Kop = −R−1BT P Here, P which is a positive definite matrix is the solution of the following Riccati matrix equation: − P͑t͒A − AT P͑t͒ + P͑t͒BR−1BT P͑t͒ − Q = ͑16͒ 3.4 Simulation With Full Vehicle In this section, the dynamic performance of both versions of the controller will be compared in order to validate the approximation put forward In what follows, the 4WS robust controllers are evaluated in time domain using the ␮-toolbox ͓11͔ As shown in the ␮-design procedure with the D − K iteration, the robust controller is synthesized and designed for 4WS vehicle at a velocity with 30 m/s To achieve the desired performance and cover the uncertainty for the considered vehicle, a set of frequency-dependent weightings have to be included, so the order of the generalized 4WS control system is increased, resulting in a high order controller It is difficult to implement a high order controller because the controller normally is ill-conditional A lower order model can lead to a lower order compensator By adopting the balanced model reduction via the truncated method ͓12͔, which can preserve stability and gives an explicit bound on frequency response error, the 14-order controller obtained by the above iteration could be reduced to a three-order controller The transient responses to the steering wheel angle input which changes from deg to 35 deg ͑gear ratio= 15͒ So the given front wheel steering angle ␦ f is 0.04 rad ͑step signal͒, approximately equivalent to 2.29 deg The simulation results are obtained as illustrated in Fig Results obtained from the computer simulation indicate that the vehicle with the robust controller has a superior performance compared with that with the optimal controller Figure 4͑a͒ illustrates that the steady state values of the yaw rate of the two controllers are almost equal to which of the desired yaw rate However, the yaw rate response of the robust controller is more rapid than which of the optimal controller and the peak value of the robust controller is lower than that of the optimal controller This means that the lower sensitivity of the steering system with the robust controller is achieved at high speed Furthermore, Fig 4͑b͒ indicates that the reduction in the vehicle sideslip angle is an important safety criterion, which could certainly be achieved more JANUARY 2011, Vol 133 / 011002-3 中国科技论文在线 http://www.paper.edu.cn Experimental Analyses Faced with the need to reduce the development time and cost, the hardware-in-the-loop simulation ͓13,14͔ increasingly proves to be an efficient tool in the automotive industry Hardware-in-theloop simulation is characterized by the operation of real components in connection with real-time simulated components Usually, the control-system hardware and software are the real system, as used for series production The controlled process, consisting of actuators, physical processes, and sensors, can then be either fully or partially simulated Hence, hardware-in-the-loop simulators may also contain partially simulated ͑emulated͒ control functions The complexity of practical vehicles, large on-line real-time synthesis is required for developing the control system HILS offers the possibility to investigate new chassis control systems with fewer expensive chassis dynamometer experiments and test drives (a) 4.1 Simulation Hardware The selected HILS hardware is delivered by dSPACE GmbH and consists of a DSP-processor board and I/O board connected by a fast PHS-bus The main reason to use this system is that the main parts of the system already exist and earlier experiences are good The DSP-processor offers a reasonable calculation power ͑50MFlops͒, although much more powerful solutions nowadays exist A digital waveform output board is used to simulate the incremental encoder This special board is capable to pulse widths from 250 ns to 26 s with 25 ns accuracy and so clearly satisfies the requirements of 4WS control realization (b) Fig „a… Yaw rate response for robust and optimal control laws „b… Sideslip angle response for robust and optimal control laws reduction in the robust controlled vehicle So the comparison of the robust and optimal controls for improving vehicle performance shows that the robust controller can certainly improve the vehicle handling when its performance is compared with the optimal controller Next experiment work is to further validate the superiority of the robust controller 4.2 Software Environment The MathWorks’ SIMULINK simulation tool is used in the modeling of the system SIMULINK together with the MathWorks’ real-time workshop and dSPACE’s real-time interface makes it possible to generate the whole Hilmodel from the SIMULINK model The digital control system consists of SIMULINK modeling software and a dSPACE DS2210 controller in a pentium computer The dynamics of the real system is first modeled as a SIMULINK block diagram Required I/O:s are determined by copying corresponding blocks from the real-time interface block library to the simulation model After that, a c-code is automatically generated, compiled, linked, and downloaded into the DSP board The measurements are made using the dSPACE’s trace tool 4.3 Experiment Using Hil Setup In order to evaluate the performance of the four-wheel steering control system, a series of experiments are performed using Hil setup The Hil simulation Fig 4WS vehicle HILS platform 011002-4 / Vol 133, JANUARY 2011 Transactions of the ASME 中国科技论文在线 http://www.paper.edu.cn Fig Four-wheel steering ECU development platform technique is an efficient way to realistically test the dynamic vehicle behavior in a laboratory A schematic diagram of the experiment is shown in Fig and a Hil simulation platform based on MATLAB/SIMULINK/dSPACE is developed in Figs and The dynamic behavior in a vehicle induced by a driver steering maneuver is simulated in the computer The real rear wheel steering angle is calculated using the feedback signal of the yaw angular velocity The control signal corresponding to the desired yaw rate is transmitted to the servo motor through an interfacing board with a D/A converter of dSPACE The real rear wheel steering angle is measured by an absolute encoder and delivered to the computer through an interfacing board with an A/D converter The actual yaw rate and real steering angle are exerted on the vehicle handling dynamic model The most interesting results are yaw rate and sideslip angle response through the HILS platform system during handling operation The results in Figs 8–10 are obtained at a constant forward speed of 30 m/s, with driver steering input as a lane change maneuver in Fig The four-wheel steering vehicle sideslip angle and yaw rate are shown in Figs and 10 for comparison with the HILS and robust control simulation, to illustrate the scale of the change, which is brought about by the HILS platform using the ␮-synthesis robust controller The results of the Hil setup have some deviations from the simulation results, and Fig is shown for comparison with trajectories in yaw rate, to represent vehicle lateral stability The resulting maximum value of sideslip angle is 0.04 rad ͑2.29 deg͒ in Fig 10, which explains that the four-wheel steering vehicle has better steering and active safety performance Furthermore, the change current of which shows that it follows the desired yaw rate better Fig The front wheel steering angle under lane change maneuver variation by cornering stiffness to the 4WS is studied Meanwhile, an optimal controller is also implemented for comparison From the Hil experimental tests, the following conclusions can be drawn that the ␮-synthesis method is proved to be effective to cope with the possible vehicle system perturbation and disturbance A fourwheel steering prototype vehicle comprising electric motor, rear steering mechanism and sensors, etc., is constructed Finally, by Conclusions In this paper, the robust ␮-method has been applied in designing the four-wheel steering system, and proper selection of the weightings is discussed In order to investigate the robustness of the synthesized controller for active control, the case of parameter Fig Yaw rate response Fig dSPACE and control platform Fig 10 Sideslip angle response Journal of Dynamic Systems, Measurement, and Control JANUARY 2011, Vol 133 / 011002-5 中国科技论文在线 experimental work using the Hil setup, a useful method to investigate the characteristics of a prototype hardware system, it is shown to produce that the ␮-synthesis robust controller can enhance the vehicle lateral stability Acknowledgment This research is sponsored by the NSFC Fund ͑Contract Nos 50975047, 60904026, and 50575041͒ and the Southeast University Technology Foundation ͑Contract No KJ2009346͒ References ͓1͔ Cho, Y H., and Kim, J., 1995, “Design of Optimal Four-Wheel Steering System,” Veh Syst Dyn., 24͑9͒, pp 661–682 ͓2͔ Szosland, A., 2000, “Fuzzy Logic Approach to Four-Wheel Steering of Motor Vehicle,” Int J Veh Des., 24͑4͒, pp 350–359 ͓3͔ El Hajjaji, A., Ciocan, A., and Hamad, D., 2005, “Four Wheel Steering Control by Fuzzy Approach,” J Intell Robotic Syst., 41͑2–3͒, pp 141–156 ͓4͔ You, 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LFT paradigm can be used to describe and analyze the uncertain vehicle system, where M corresponds to what is assumed as the constant in the control system and ⌬ is the block diagonal matrix... behavior in a laboratory A schematic diagram of the experiment is shown in Fig and a Hil simulation platform based on MATLAB/SIMULINK/dSPACE is developed in Figs and The dynamic behavior in a vehicle. .. input as a lane change maneuver in Fig The four- wheel steering vehicle sideslip angle and yaw rate are shown in Figs and 10 for comparison with the HILS and robust control simulation, to illustrate