外文翻译--不同主动悬架系统的乘坐舒适度表现 英文版.pdf

Ride comfort performance of different active suspension systems Daniel A Ma ntaras and Pablo Luque A rea de Ingenier a e Infraestructura de los Transportes Universidad de Oviedo Campus de Viesques s n 33203 Gijo n Spain E mail mantaras uniovi esE mail luque uniovi es Corresponding author Abstract This paper presents a comparative of different active and semi active suspension systems in order to research the improvement in ride comfort and handling stability compared with the equivalent passive systems The two degrees of freedom model quarter car model is employed in the analysis of seven different active suspension control strategies LQR LQG Robust design Kalman filter Skyhook damper Pole assignment Neural network and Fuzzy logic Computer simulations of the different active models and the equivalent passive systems are performed to obtain the vertical acceleration of the sprung mass and the vertical wheel load variation The vertical acceleration of the sprung mass is processed to evaluate the ride comfort behaviour and the vertical wheel load variation is used to evaluate the handling stability of the different active suspension control systems Keywords active suspension Fuzzy logic handling stability Kalman filter LQR LQG Neural network passive suspension Pole assignment ride comfort robust design Skyhook damper Reference to this paper should be made as follows Ma ntaras D A and Luque P 2006 Ride comfort performance of different active suspension systems Int J Vehicle Design Vol 40 Nos 1 2 3 pp 106 125 Biographical notes Daniel A Ma ntaras is an Engineer and Professor at Oviedo University Spain in the division of Mechanical and Transportation Engineering where he leads the testing area in the Vehicle Laboratory He is working with local and national Spanish Administrators to evaluate vehicles and technical conditions of working In collaboration with diverse manufacturers and race teams Dr Ma ntaras is working in the design test and optimisation of vehicles and components Pablo Luque is an Engineer and Professor at Oviedo University Spain in the division of Mechanical and Transportation Engineering where he leads the Vehicle Laboratory Dr Luque is working with local and national Spanish Administrators to evaluate vehicles and technical conditions of working In collaboration with diverse manufacturers he is developing optimised heavy vehicles and components Int J Vehicle Design Vol 40 Nos 1 2 3 2006106 Copyright 2006 Inderscience Enterprises Ltd 1Introduction The design of a vehicle suspension system must perform two opposite tasks on the one hand it should isolate the vehicle body from road disturbances and therefore improve the passenger s comfort on the other it should maintain the contact between the tyre and the road in order to maximise the handling stability Active and semi active suspension systems were developed in order to improve vehicle performance given the difficulty in obtaining an acceptable balance between ride comfort and handling stability with passive suspension systems It is well documented Hrovat 1997 Sharp and Crolla 1987 Sharp and Hassan 1986 that computer controlled vehicle suspensions offer substantial benefits in ride comfort handling and attitude control over traditional passive systems A variety of control techniques such as PID or lead lag compensation LQG H2 H1 synthesis and feedforward control have been used in active systems e g Hrovat 1997 Karnopp 1995 Serrand and Elliott 2000 Ulsoy et al 1994 Yang et al 2001 Zuo and Nayfeh 2003 A closer examination of these papers however shows that most of these published results were focused on the main loop design i e on figuring out the active force to be applied as a function of vehicle states and road disturbance input A comparative review of different active suspension control systems is presented in this paper The models contrasted are LQR LQG robust design Kalman filter Skyhook damper Pole assignment neural network fuzzy logic In the active suspension system the spring and damper of the passive system are replaced or supplemented by active force actuators that are regulated using the different control laws described This control system can be optimised in order to improve the handling stability or the ride comfort In the present paper all the active models simulated were optimised to improve the ride comfort of the passive system 2Vehicle and road models The two degrees of freedom quarter car model Figure 1 is employed in the analysis of the different active suspension control strategies Table 1 The equation of motion of the two degrees of freedom quarter car model in state space form is x t A0 x t B1z0 t B2u t 1 where x t is the state vector xT z1z2 z1 z2 2 Ride comfort performance of different active suspension systems107 Figure 1The two degrees of freedom quarter car model Table 1Parameters of the quarter car model ParameterSymbolUnitValue Mass of tyremtkg40 Mass of bodymbkg275 Spring stiffnesskbN m16 000 Tyre stiffnessktN m150 000 The four state variables are z1is the vertical movement of the unsprung mass z2is the vertical movement of the sprung mass of the vehicle body dz1 dt is the derivative of the displacement of the unsprung mass dz2 dt is the derivative of the displacement of the sprung mass The road roughness is typically specified as a random process of a given displacement power spectral density Hrovat 1997 In this paper the road profile represented by z0 is modelled as a first order linear process derived from integrated white noise Michelberg et al 1993 with the coefficients proposed by this author for an asphalt road The equation is z0 t avz0 t w t 3 Thus the equation of motion of the two degrees of freedom quarter car model in state space form can be written as x t A0 x t B1w t B2u t 4 D A Ma ntaras and P Luque108 Now x t is the state vector with five variables xT z0z1z2 z1 z2 5 where z0is the vertical displacement of the road road excitation The vector u t is the actuator force In a passive suspension system this is the damper force calculated by multiplying the damper constant by the relative speed between the sprung and unsprung mass The state and input matrix are A0 av0000 00010 00001 kt mt kb kt mt kb mt 00 0 kb mb kb mb 00 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 B1 1 0 0 0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 B2 0 0 0 1 mt 1 mb 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 6 To get the necessary data to compare the different suspension systems the state vector x t must be multiplied by a matrix as can be seen in the next expression y t Cx t C kt kt000 00001 01 100 1 1000 2 6 6 6 6 4 3 7 7 7 7 5 7 Consequently the model outputs are the vertical acceleration of the sprung mass the vertical wheel force kt z0 z1 the suspension displacement z1 z2 the relative displacement between the unsprung mass and the road z0 z1 3Active suspension control system models 3 1Standard LQR LQG methodology The standard LQR LQG methodology is one of the first strategies used in the active control of the vehicle suspension Zuo and Nayfeh 2003 This model is considered to be linear and it is assumed that the parameters of the system are perfectly well known The LQR LQG methodology used in this paper is described by Chen et al 1995 Michelberg et al 1993 and Palkovics and Venhovens 1992 Figure 2 Ride comfort performance of different active suspension systems109 Figure 2The block diagram of LQR LQG model D A Ma ntaras and P Luque110 The vector u t is the force in the actuator that should minimise the following index J lim T 1 1 T E T 0 qw1ktz0 z1 2 qw2z2 2 qw3z1 z2 2 dt 8 qwiare the weighting factors tyre load ride comfort and suspension working space respectively To minimise the index it is necessary to solve the algebraic Riccati equation to obtain the optimal state feedback vector KLQR XLQRA0 B2R 1 0 NT 0 A0 B2R 1 0 NT 0 X LQR XLQRB2R 1 0 BT 2XLQR Q0 N0R 1 0 NT 0 0 9 The actuator force will be u KLQRx R 1 0 NT 0 BT 2XLQR x 10 where the XLQRis the solution of the Riccati equation After the analysis of several pre simulations the proper weighting factors that have a significant improvement in ride comfort of the vehicle with a low variation of the wheel load are qw1 0 01 qw2 5000 qw3 0 01 Beside that the actuator force values are not excessive so the suspension system has a low energy cost 3 2Kalman filter methodology A Kalman filter is applied to the measurable variables of a real suspension system In this case the standard LQR LQG system developed is used to apply the Kalman filter to the measurable output the suspension travel The outputs of the Kalman filter methodology are yKl Hx Du v 11 where v is the white noise of the measured variable and H and D are the following matrices D 1 mb H 0 kb mb kb mb 0 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 12 The optimal value of the actuator force is calculated using the following expression u KLQR x R 1 0 NT 0 BT 2XLQR x 13 where x is calculated using the following expression x A x Bu L yv H x Du 14 Ride comfort performance of different active suspension systems111 A and B are the matrix A av0000 00010 00001 kt mt kb kt mt kb mt 00 0 kb mb kb mb 00 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 B 0 0 0 1 mt 1 mb 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 15 To obtain the L matrix it is necessary to solve the algebraic Riccati equation applied to the Kalman algorithm The Kalman filter block Figure 3 represents the solution of the differential equation of the Kalman filter 3 3Skyhook Damper control strategy To develop the Skyhook Damper control system design the quarter car model shown in Figure 4 is used As can be seen in the block diagram the damper force in the Skyhook subsystem is calculated using the following expression U t Ch z2 t 16 The value of the parameter Chwas obtained to improve the ride comfort after iterative simulations checking values from 700 to 10 000Ns m 3 4Pole assignment control To develop the Pole assignment control the one degree of freedom quarter car model was used It reduces the complexity of the control laws based upon a full model without a significant variation of the simulation results following Dukkipati et al 1993 and Ramsbottom and Crolla 1999 The equation of motion of the one degree of freedom quarter car model can be written as mb z2 t kbz2 F2 t 17 Where F2is calculated using the following expression F2 t u t kbz0 t 18 The Laplace transfer function of the model is z2 s F2 s 1 mbs2 kb B A 19 Applying the Laplace transfer function to the road equation z0 s w s 1 s av 20 D A Ma ntaras and P Luque112 Figure 3The block diagram of Kalman filter Ride comfort performance of different active suspension systems113 Figure 4Car model and block diagram of skyhook damper control In order to design the controller classic pole assignment control theory is applied This is shown in the block diagram representation of the full system Figure 5 The expressions of the transfer functions used in the controller as can be seen in the block diagram are T s R s kbs a0kb s a0 Cop mb 21 S s R s a0Chs kb Cop mb s a0 Cop mb 22 The value of the parameter Copwas obtained to improve the ride comfort after iterative simulations checking values from 700 to 2500Ns m D A Ma ntaras and P Luque114 Figure 5The block diagram of pole assignment control system Ride comfort performance of different active suspension systems115 3 5Robust design The robust design of active suspension controls considers the non linearities of the system given that its parameters are not perfectly known Many uncertainties can be found in the one quarter car model but in this study only the stiffness of the tyre has been chosen A complete description of the robust design can be found in Michelberg et al 1993 The uncertainties must be introduced in the state space equation of the system To do this the state matrix of the uncertain system is considered as A A0 L NT 23 So the state space equation can be written as x t A0 x t L NTx t B1z0 t B2u t 24 Where the matrixes can be written as A0 0010 0001 kb kt mt kb mt 00 kb mb kb mb 00 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 L 0 0 kt mt s 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 NT kt mt s 000 25 B1 0 0 kt mt 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 B2 0 0 1 mt 1 mb 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 26 The matrix is the scale factor of the uncertainties 1 Figure 6 The feedback force should minimise the following performance index JR lim T 1 1 T E T 0 xTQ0 NNT x uTR 0u 2zT0z0 dTd dt 27 To minimise the index it is necessary to solve the algebraic Riccati equation to obtain the optimal state feedback vector KR XRA0 A0XR XR 1 2 B1BT 1 1 LLT B2R 1 0 BT 2 XR Q0 NNT 0 28 The expression of the feedback force will be the following u KRx R 1 0 BT 2XRx 29 where the XRis the solution of the Riccati equation D A Ma ntaras and P Luque116 Figure 6The block diagram of robust design control system Ride comfort performance of different active suspension systems117 The control system will be solved as a SISO problem to simplify it The measurable output of the system will be the following y1 C1x D12u 30 C1 kb mb kb mb 00 D12 1 mb 31 The matrix Q0and R0are the following R0 1 10 6Q0 CT 1C1 32 The value of the parameters and were obtained after a deep analysis of their influence on the acceleration of the sprung mass This was done to compare the bode diagram for different values of both parameters 3 6Fuzzy control As can be seen in the block diagram Figure 7 the active control of the suspension is constructed by using fuzzy reasoning A detailed analysis and description of the fuzzy logic control of suspension systems can be found in Chou et al 1998 Li et al 1998 Lin et al 1993 and Yoshimura et al 2000 Figure 7The block diagram of fuzzy logic control system The membership functions used for the fuzzy control are triangular and singleton types for the input and output variables respectively see Figure 8 The input variables to the fuzzy control are 1 z2 dz2max 2 z2 z2max 33 D A Ma ntaras and P Luque118 Figure 8Membership functions used for the fuzzy control They are the scaled values of the speed and displacement of the sprung mass The active control is obtained as a weighted sum of the defuzzificated output variables 1 and 2 in the fuzzy control rules where the product sum gravity method Yoshimura et al 2000 is applied to obtain the defuzzificated value of the output u X 2 i 1 gi i 34 where g1and g2are constants Figure 9 The type of fuzzy control rules are If iis Aijthen iis Biji 1 2j 1 2 3 4 5 Figure 9Fuzzy control subsystem and fuzzy control rules 3 7Neural network control The inputs of the neural network are the following Figure 10 y1 z1 z2y2 z1 z2 35 The neural network used is a feed forward type with two layers the first layer has six neurons with tansig transfer functions the second layer has one neuron with a purelin transfer function Batch training with the Levenberg Marquardt algorithm was used to train the neural network This allowed us to obtain the weights and biases necessary either to improve ride comfort or to improve handling stability In this case the neural network was trained to improve ride comfort with a 4000 point target vector Ride comfort performance of different active suspension systems119 Figure 10 The block diagram of neural network control system 4Simulation results 4 1Comfort analysis results The vertical acceleration of the sprung mass one of the simulation outputs of the described models was used to evaluate the ride comfort of the different active suspension control systems The procedure is described below All the suspension models described were simulated for 300 seconds To take into account the highly frequency dependent nature of human sensitivity to vibration the following transfer functions were applied to the vertical acceleration of the sprung mass to obtain the frequency weighted acceleration H s 39 48 s2 f 2 6 s2 8 85 f1 s 39 48 f 2 1 s2 8 85 f6 s 39 48 f 2 6 s 6 28 f4 s2 7 85 f2 s 39 48 f 2 2 s2 9 97 f5 s 39 48 f 2 5 s2 7 85 f3 s 39 48 f 2 3 2 51 f 2 5 f 2 3 f4 f 2 2 36 where f1 0 4 f2 2 5 f3 4 f4 16 f5 16 f6 100 37 The next step was to calculate weighted root mean square acceleration in five second intervals and elaborate its histogram Figures 11 and 12 The parameter used to evaluate the different ride comfort behaviour is the 95 percentile of the histograms Figure 13 The acceleration improvement of the different active suspension systems with respect to the passive system is represented in Figures 14 and 15 This represents the following expression impr 95 percentile of active sys 95 percentile of passive sys 95 percentile of passive system 100 38 D A Ma ntaras and P Luque120 Figure 11 Comparative comfort percentage of events for weighted root mean square acceleration in five second intervals Figure 12 Comparative comfort percentage of events for root mean square acceleration in five second intervals Figure 13 95 percentile of the percentage events for weighted root mean square acceleration for each active system Ride comfort performance of different active suspension systems121 Figure 14 Improvement of the different active suspension systems respect to the passive system Figure 15 PSD of vertical acceleration for each suspension system 4 2Handling stability The vertical wheel load variation one of the simulation outputs of the described models was used to evaluate the handling stability of the different active suspension control systems All the suspension models described were simulated for 300 seconds in order to calculate the root mean square force in one second intervals The results can be seen in the histogram of Figures 16 and 17 D A Ma ntaras and P Luque122 Figure 16 Percentage of events for wheel force in one second intervals Figure 17 PSD of vertical wheel load for each suspension system 5Conclusions Seven different active suspension systems have been analysed in order to research the improvement in ride comfort and handling stability compared with the equivalen