Active Suspension Control with Direct-Drive Tubular Linear Brushless Permanent-Magnet Motor

Transcription

1 29 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 1-12, 29 FrC1.4 Active Susension Control ith Direct-Drive ubular Linear Brushless Permanent-Magnet Motor Seungho Lee and Won-jong Kim Deartment of Mechanical Engineering exas A&M University College Station, exas 77843, USA Abstract Recently, active susension has been alied to many commercial automobiles. o develo the control algorithm for active susension, a quarter-car test bed as built by using a direct-drive tubular linear brushless ermanent-magnet motor (LBPMM) as a force-generating comonent. Modified lead-lag control, linear-quadratic (LQ) servo control ith a Kalman filter, the fuzzy control methodologies ere imlemented for active-susension control. In the case of fuzzy control, an asymmetric membershi function as introduced. his controller could attenuate road disturbance by u to 78%. Both simulation and exerimental results are resented to demonstrate the effectiveness of these control methodologies. Keyords tubular linear actuator, quarter-car, lead-lag control, LQ servo, fuzzy control. I. INRODUCION Active susension suorts a vehicle and isolates assengers from road disturbances for ride quality and vehicle handling using force-generating comonents under feedback control. Develoment of an active-susension system should be accomanied by the methodologies to control it. Considering costly commercial vehicles ith active susension, Allen constructed a quarter-car test bed to develo the control algorithms [1]. Many researchers develoed active-susension control techniques [2] [7]. hese researches could be categorized according to the alied control theories. When it comes to the LQ control, Peng, et al. resented the virtual inut signal determined by the LQ otimal theory for active-susension control [2]. ang and Zhang alied linear-quadratic-gaussian (LQG) control, neural netorks, and genetic algorithms in an active susension and resented simulation results [3]. Sam, et al. alied LQ control to simulate an active-susension system [4]. As for the robust control, Lauerys, et al. develoed a linear robust controller based on the µ-synthesis for the active susension of a quarter car [5]. Wang, et al. resented the algorithm to reduce the order of the H controller in the alication of active susension [6]. hey ere able to reduce the controller s order by nearly one third hile the erformance as only slightly degraded. Abbas, et al. alied sliding-mode control for nonlinear full-vehicle active susension [7]. hey considered not only the dynamics of the nonlinear full-vehicle active-susension system but also the dynamics of the four actuators. he LBPMM is directly alicable to active susension ithout converting rotary motion to linear motion [8]. Besides its smooth, recise translational motion ithout cogging, the fact that the length of the mover can be conveniently adjusted makes it aroriate for the force-generating comonent in an active-susension system. Since a human body is most suscetible to vibration at around 3 Hz (2 rad/s) [1], disturbance from the road is modeled as a sinusoidal inut ith a frequency of 3.5 Hz (22 rad/s) and a magnitude of ±.3 m in this research. he LBPMM as designed to be able to generate the force u to 29.6 N ith a ±6-A hase current [8]. Since NdFeB magnet ould lose magnetization around 15 C, control erformance is comromised ith the maximum current sing that yields temerature rise. As a result, controllers are designed to have the current limit of around ±4 A. he iezoelectric accelerometers (Piezotronics model 336B18) used in the quarter-car test bed also limit the erformance. hese accelerometers can be used only in the frequency range of.5 to 3 Hz (3 to 2 rad/s). Particularly, this imlies that our active-susension system is not able to attenuate the disturbance ith a frequency comonent loer than.5 Hz. he fact that the LBPMM is used for active susension and three different classes of control methodologies are develoed and successfully imlemented is the key contribution of this research and differentiates this aer from others. Esecially, in the case of fuzzy control, an asymmetric fuzzy controller as imlemented to comensate for dc offset in sensor data. his aer is organized as follo. In Section II.A, orking rinciles of the LBPMM are summarized. Section II.B resents the modeling of the quarter-car test bed. In Section III.A, imlementation of a modified lead-lag controller and its disturbance attenuation are resented. Section III.B describes the design and erformance of an LQ servo controller and the state estimation by a Kalman filter. Section III.C resents a fuzzy controller ith asymmetric membershi functions and its erformance. he conclusions follo in Section IV /9/$ AACC 5498

2 II. ES BED FOR ACIVE SUSPENSION CONROL A. Linear Brushless Permanent-Magnet Motor Fig.1 shos a concetual configuration of the LBPMM. he mover of the LBPMM consists of a series of cylindrical ermanent magnets. he magnets are fixed in a brass tube and connected each other in an NS NS SN SN fashion ith sacers beteen the airs. he stator consists of 9 coils (3 er each hase). Fig. 1. Schematic diagram of the LBPMM. he direction of the generated force on the mover is in the negative z-direction in this articular current distribution. By the Lorentz force equation, the generated force is the vector cross roduct of the current density J in the coils and the magnetic flux density B generated by the magnets, F J B [8]. he inverse Blondel-Park transformation in the LBPMM that governs the relationshi beteen the three-hase currents and the desired force is defined as follos [8]. LBPMM force can act on this quarter-car test bed. he rotating cam shon at the bottom of Fig. 2 simulates sinusoidal road disturbance at various frequencies. As in [9], the states of the quarter-car test bed are defined as xs ( t) xus ( t) xs( t) xus ( t) xus ( t) xr ( t), and its dynamics is exressed as the folloing state-sace matrix form. k 1 xs( t) M s xs( t) M s xus ( t) c k k x ( ) us t 1 Fact () t x r () t x s( t) x us ( t) Mus Mus Mus xs( t) xus ( t) Mus xus ( t) x ( ) 1 1 ( ) ( ) r t xus t xr t 1 1 (2) here x s t and x us t are the velocities of the srung and unsrung masses, resectively, xr () t is the sinusoidal disturbance generated by the rotating cam, and F () t is the force generated by the LBPMM. Additionally, the heel is modeled by the sring constant k and the viscous daming coefficient c. he arameter values are given in able I. he tire is assumed to be made of natural isorene hich has modulus of elasticity of.1 GPa. LVD LBPMM accelerometer srung mass accelerometer unsrung mass heel act i 2 a() t cos1zo ib ( t) C 1 3 fzd t sin1z ic () t 1 3 (1) cam here ia( t), ib( t ), and ic () t are the currents floing in hases ABand,, C, resectively. fzd is the desired force in the axial direction. 1 2 /l, here l is the itch of the motor (63.3 mm). z is relative dislacement beteen the mover and the stator. In active susension, it is the distance beteen the srung and unsrung masses. he inverse force constant C as determined as.1383 A/N by exeriments [8]. B. Quarter-Car est Bed Fig. 2 shos a hotograh of the quarter-car test bed. he srung mass ( M s ) is considered to be the body of a car, and the unsrung mass ( M us ) reresents the mass beteen its susension and a heel. As shon in Fig. 2, to masses are connected ith a mechanical sring and the LBPMM. he stator of the LBPMM is fixed to the srung mass and one end of the mover is fixed to the unsrung mass so that the Fig. 2. Photograh of the quarter-car test bed ith active susension. ABLE I PARAMEERS AND CORRESPONDING VALUES OF QUARER-CAR Parameters Values M s kg M us kg k 1521 N/m c 5 N-s/m k 156 N/m 5499

3 Srung Mass Velocity (m/s) III. CONROL MEHODOLOGIES AND EXPERIMENAL RESULS In this Section, three classes of controllers are designed and imlemented in the quarter-car test bed and their exerimental results are resented. A. Modified Lead-Lag Control he outut force controls the velocity of the srung mass rather than its osition. From (2) and able I, the transfer function from Fact t to x s() t is determined as follos s 9.547s 29.79s Gs ( ) (3) s 21.95s 1398s s he control objectives are as follos. First, high loo gain is desirable around the oerating frequency at 22 rad/s for good disturbance attenuation and command folloing. Hoever, this high gain ould yield large current flo in the LBPMM, hich ould raise its temerature and demagnetize the magnets. herefore, the gain as limited by examining the simulation result of the maximum current flo (±4 A) in the LBPMM. Finally, the loo gain of the controller at around the oerating frequency as determined as 56 db. Second, the control bandidth as set to be [1 rad/s, 8 rad/s]. Since the oen-loo frequency resonse of this quarter car has lo gain in lo and high frequencies and high gain in the middle frequency ith to cross-over frequencies, the bandidth could be adjusted by changing either the loer cross-over frequency or the higher cross-over frequency. In this aer, a lag comensation.2252s1.15 s 1.5 as alied in the lo-frequency range to achieve this goal. hird, since the gain should be lo in the high frequency range to attenuate noise, another lag comensation.4681 s1.5 s1.54 as alied. Finally, sufficient gain and hase margins should be obtained due to modeling uncertainties. o achieve this objective, a lead comensation s1 s 1.2 as introduced beteen the to lag controllers. he loer-frequency lag controller yields a loer loo gain. he lead controller around the oerating frequency broadens the bandidth. herefore, each lead or lag controller should be fine-tuned by examining the overall loo transfer function. o decide the exact corner frequencies in each of the lead or lag controllers, the Matlab SISO (single-inut-single-outut) tool as used. he modified lead-lag controller ith one lead and to lag controllers as finalized in the s domain as (4). Fig. 3 shos the frequency resonse of the loo transfer function. As seen from Fig. 3, the loo-transfer-function gain is higher than that of the quarter-car transfer function around the oerating frequency (22 rad/s). he bandidth is accetable since it is close to the frequency range of [1 rad/s, 8 rad/s]. he simulation and exerimental results of disturbance rejection are resented in Fig. 4. Fig. 3. Quarter-car and loo-transfer-function frequency resonses of the quarter-car dynamics (3) and the modified lead-lag controller (4). Gain and hase margins are 28.2 db and 66.4, resectively ( s 2147)( s51.31)( s 5.17) C() s 12 (4) ( s 1.54)( s1.2)( s1.5) ime (s) Fig. 4. Exeriment and simulation results of the modified lead-lag control for the 3.5 Hz disturbance. B. Linear-Quadratic Servo Control LQ servo control is develoed by introducing the command inut and the outut disturbance. From (2), quarter-car model can be exressed as follos. x t Ax t BPu t x y t C t here x t x t x t x t x t x t x t ( ) ( ) ( ) ( ) ( ) ( ) ( ), s us s us us r k M 1 s M 1 s c k k 1 A, B, C Mus Mus Mus M as in (2) us y t x t x t is artitioned as follos. hus, (), and s (5) 55

4 x t y t x t R xs t xus t xs t xus t xus t xr t he vertical line indicates that y t x t s. As shon in Fig. 5, the control gain matrices are alied to y s and x R s (6) G y and G r, resectively. o eliminate a non-zero steady-state error for the ste command inut or the outut disturbance, this LQ servo controller is imlemented ith an integrator. In this alication, the LQ servo model is determined by considering the frequency resonses of the loo transfer function of model given in Fig. 5. rs 1 s Gy z s G i us G r B x s R s y s Fig. 5. Block diagrams of LQ servo control in this alication. he control objectives are similar to those of the modified lead-lag control. As shon in Fig. 5, the control gains for the integrator, outut state, and rest states are Gi, Gy, Gr, resectively [11]. his LQ servo system consisted of the standard LQ servo dynamics (5) and the integrator dynamics. With rs in a regulation roblem, I I z s y s x s s. (7) s s he augmented system is defined as follos. t A t Bu t, here t z t y t t x x R x x (8), A CP AP and B B. he control la is defined as follos. u( t) Gx t, (9) here G Gi Gy Gr. o obtain G, a control algebric Riccati equation (CARE) should be solved. o construct this CARE, a symmetric ositive definite matrix R and a symmetric ositive semi-definite matrix Q should be determined. he R matrix affects the loo gain that determines the system bandidth. After several design iterations, R as set to be.5. he diagonal elements of the Q matrix are the eights of each state, after several design iterations, the Q matrix as determined as follos diag Q (1) Qij for i j A unique ositive semi-definite symmetric matrix K is determined by the folloing CARE. 1 KA A K Q KBR B K (11) K is solved ith Matlab as follos K he feedback gain G is determined as follos. 1 (12) G R BK (13) 1) Kalman Filter Design LQ servo requires full state feedback. he last state of the system is defined as the tire deflection ( xus ( t) xr ( t) ) hich is difficult to be measured because locating any tye of the sensor in the rotating heel is inconvenient. herefore, it is estimated by a Kalman filter. o solve the filter algebric riccati equation (FARE) and obtain the Kalman filter gain, a ositive value and a non-negative value should be determined in (15). hey are adjusted and determined as.1 and.1after several design iterations. he disturbance matrix L defined as follos. L 1 (14) hen the unique ositive semi-definite symmetric matrix P is determined by solving the folloing FARE. 1 AP PA L L PC CP (15) he solution is obtained as follos by Matlab CARE function P he Kalman filter gain H is determined as follo. H PC (16) (17) Fig. 6 shos the estimated tire deflection ( xus ( t) xr ( t) ) by the Kalman filter algorithm in closed-loo control. he solid line reresents the estimated state generated from Control Desk during the exeriment. he dashed line reresents the 551

5 Srung Mass Velocity (m/s) Fuzzified inut ire deflection (mm) ire deflection [mm] Fuzzy sets estimated state generated from the Simulink block ithout the exeriment setu. here is some discreancy beteen the simulation and exerimental results of state estimation. In the Kalman filter algorithm, the measured outut and the disturbance are assumed as zero-mean hite Gaussian noises. In the quarter-car model, there is some discreancy beteen the measured outut ( x s() t ) and the zero-mean hite Gaussian noise (Figs. 4 and 7), hich limits the erformance of the state estimator. Noises from three sensors also degrade the erformance of the state estimator Simulation Exeriment ime 1.5 [s] ime (s) Fig. 6. Estimated state comarison beteen simulation and exeriment results. he erformance of the disturbance attenuation is resented in Fig. 7. Due to the error from the state estimator, disturbance attenuation contains some discreancy beteen the exeriment and simulation results ime (s) Fig. 7. Exeriment and simulation results of the LQ servo control hen the controller is turned on. C. Fuzzy Control A Mamdani-tye fuzzy controller is imlemented in this section [12]. he inut to this fuzzy controller is the system error ( et ()), and the outut is the control inut ( Fact t ). o determine Fact t, et () is fuzzified by the membershi functions shon in Fig. 8 (a) and defuzzified by the membershi functions shon in Fig. 8 (b). he area under the membershi functions are defined by i ( i 1, 2,,7 ). he range of error in Fig. 8 (a) as set as [.8,.8] because the magnitude of the largest measured error ( x () t ) s as.8 m/s. he range of oututs in Fig. 8 (b) as set as [ 3, 3] because the LBPMM could generate force u to near ±3 N. Since this active-susension test bed is a single-inut, single-outut system, the inut and the outut from single-dimension arrays. Each fuzzified value is one-to-one matched for the defuzzification. For examle, if the error is NLE, the outut is NLF. Each rule has the same eight. NLE NME NSE ESE PSE PME PLE (a) NLF NMF NSF ESF PSF PMF PLF (b) Fig. 8. (a) Membershi functions for fuzzification. (b) Membershi functions for defuzzification. he control inut as the result of this fuzzy controller is determined by the center of gravity (COG) method. he COG method comutes F as follos [12]. act Error (m/s) Oututs (N) F g, (18) act i i i i1 i1 here g i is defined as the COG of the each membershi function. Fig. 9 shos the relation beteen the error and the generated control force. his inut-outut curve as designed not to be symmetric ith resect to the origin. When the active-susension system is under closed-loo control, the srung mass s maximum absolute velocity is larger hen its velocity is ositive comared to that ith a negative velocity. his henomenon as examined through the modified lead-lag control and the LQ servo control (Figs. 4 and 7). 552

6 Srung Mass Velocity (m/s) Force (N) Inut-outut curve Reference line Error (m/s) Fig. 9. Inut-outut relation of the asymmetric fuzzy controller. his henomenon indicates that the osition of the srung mass is higher than the desired osition and its insufficient control. herefore, additional control inut should be generated. o solve the roblem, a membershi function PSF in Fig. 8 (b) as idened. In Fig. 1, this henomenon is reduced in comarison ith Figs. 4 and 7 due to the additional control inut generated in the hum here.1 < Error (m/s) <.4 in Fig ime (s) Fig. 1. Fuzzy control result of exeriment and simulation hen controller is turned on. IV. CONCLUSIONS An active-susension system ith a quarter-car test bed as constructed ith an LBPMM in this research. Modified lead-lag, LQ servo, fuzzy controllers ere designed and imlemented to attenuate modeled road disturbance. he fuzzy controller as able to reject the disturbance by u to 78%. he LQ servo s erformance in disturbance rejection as slightly inferior to the to other controllers. he reason is that the estimator could not erfectly generate the estimated state because the noise and the disturbance ere not zero-mean hite Gaussian. Moreover, an additional sensor (the LVD) as used in this control method. he erformance of the modified lead-lag control as fairly accetable. he fuzzy controller turned out to be the most suitable control methodology for this active-susension alication. It is because its asymmetric membershi functions alloed the LBPMM to generate the most suitable control force. Due to the asymmetric membershi functions, the discreancy beteen the ideal and ractical test beds as reduced. Hoever, a fuzzy controller is difficult to design since it has infinitely many design arameters such as selecting the domain for the fuzzification and defuzzyfication. In this research, these design arameters ere finalized ith the results from the modified lead-lag and LQ servo controllers. REFERENCES [1] J. Allen, Design of Active Susension Control Based Uon Use of ubular Linear Motor and Quarter-Car Model, Master s hesis, exas A&M University, August 28. [2] H. Peng, R. Stratharn, and A. Ulsoy, A Novel Active Susension Design echnique Simulation and Exerimental Results, Proceedings of 1997 American Control Conference, , June [3] C. ang and. Zhang, he Research on Control Algorithms of Vehicle Active Susension System, Proceedings of IEEE International Conference on Vehicular Electronics and Safety, , October 25. [4] Y. M. Sam, M. R. H. A. Ghani, and N. Ahmad, LQR Controller for Active Car Susension, Proceedings of ENCON 2, , Setember 2. [5] C. Lauerys, J. Severs, and P. Sas, Design and Exerimental Validation of a Linear Robust Controller for an Active Susension of a Quarter-car, Proceedings of 24 American Control Conference, , July 24. [6] J. Wang, A. C. Zolas, and D. A. Wilson, Active Susension: A reduced-order H Control Design Study, Proceedings of Mediterranean Conference on Control and Automation, , July 27. [7] C. Abbas,. Rahaijaona, and H. Noura, Sliding Mode Control Alied to Active Susension Using Nonlinear Full Vehicle and Actuator Dynamics, Proceedings of 26 IEEE conference on Decision and Control, , December 26. [8] W.-J. Kim and B. Murhy, Develoment of a Novel Direct-Drive ubular Linear Brushless Permanent-Magnet Motor, International Journal of Control, Automation, and System, vol. 2, no. 3, , Setember 24. [9] R. Rajamani, Adative Observers for Active Automotive Susensions: heory and Exeriment, IEEE ransactions on Control Systems echnology, vol. 3, no. 1, , March [1] G. Stein, Resect the Unstable, IEEE Control Systems Magazine, vol. 23, no. 4, , August 23. [11] B. Anderson and J. Moore, Otimal Control: Linear Quadratic Methods, Prentice-Hall,. 74, [12] K. Passino and S. Yurkovich, Fuzzy Control, Addison-Weslsy,. 42,