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1、Analysis and experiment of time-delayed optimal control for vehicle suspension system Gai Yan, Mingxia Fang*, Jian Xu School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China a r t i c l ei n f o Article history: Received 5 July 2018 Received in revised form 5 Januar
2、y 2019 Accepted 10 January 2019 Available online 11 January 2019 Handling Editor: Ivana Kovacic Keywords: Vehicle active suspension Time-delayed optimal control Active time delay Stability Experiment a b s t r a c t In this paper, the performance of vehicle suspension system under time-delayed optim
3、al control is investigated. The effect of time delay on control stability of the active suspension system is discussed. The mathematical simulation is used to verify the correctness of the stable interval obtained by differential equation theory for linear systems with constant coeffi cients and tim
4、e delay. In order to keep the stability of the system, time-delayed optimal control is designed through the method of state transformation and optimal control theory. The results show that the control strategy could not only guarantee the stability of the system regardless of the variation on contro
5、l time delay, but also improve the performance of the suspension system. Additionally, infl uence of the designed active time delay on the amplitude of sprung mass acceleration, suspension defl ection, rode holding are analyzed to provide guidance for its further practical engineering application. V
6、ibration control experiments of the active suspension system under harmonic excitation with time-delayed optimal control are fi gured to compare with simulations. It is seen that the designed time-delayed optimal control strategy has the effectiveness and advantage. 2019 Elsevier Ltd. All rights res
7、erved. 1. Introduction With the progress of the society and the development of science and technology, the requirements for the performance of the vehicle are becoming higher and higher. As an important part of the vehicle driving system, the performance of the suspension system has a direct impact
8、on the comfort and safety of the occupants 1e4. At present, there are three types of vehicle suspension system,including passive,semi-active andactive suspension. Amongthem, the passivesuspension system isused in most cars. Although it can improvethe comfort andhandling stabilityof the vehicle toace
9、rtain degree,its structural parameters cannot be changed in real time with the running state of the vehicle. Active suspension system is more elastic and effi cient than the other two kinds in getting better driving comfort and handling stability 5,6. Therefore, studying the active suspension is ver
10、y necessary for the development of the automobile industry and comfortable requirement of the people. An active suspension system is generally including an actuator, control unit and sensors. The control is a kennel unit for any active suspensions. To this end, a lot of scholars have been researchin
11、g the active control to achieve good performances. Aimed on different performance objectives, various active suspension control strategy have been proposed such as optimal control 7e10, H control 11,12, sliding mode control 13,14, adaptive control 15,16, fuzzy control 17, and so on. Although some go
12、od results have been displayed, the active suspension has not been widely used due to its complex * Corresponding author. School of Aerospace Engineering and Applied Mechanics, Tongji University, No.1239, Siping Road, Shanghai, 200092, China. E-mail address: mxfang_ (M. Fang). Contents lists availab
13、le at ScienceDirect Journal of Sound and Vibration journal homepage: https:/doi.org/10.1016/j.jsv.2019.01.015 0022-460X/ 2019 Elsevier Ltd. All rights reserved. Journal of Sound and Vibration 446 (2019) 144e158 structure. It implies that the active suspension requires to be investigated further in o
14、rder to simplify its structure. One of challenging problems is how to consider time delay occurring in the control loop 18. Furthermore, it often degrades the control performance and even leads tothe system losing its stability such that the control is invalid 19e21. Thus, a great deal of time delay
15、 elimination and compensation technology has emerged such as methods of Smith predictor, phase-shift and recursive response 22e25. These methods can deal with time delay and improve the control effect to a certain extent. It is worth mentioning that in Ref. 22 the time delay caused by the response o
16、f Magneto-rheological (MR) damper actuator is measured and the authors apply the Smith predictive control to design time delay compensation controller and to improve the performance of the vehicle semi-active suspension effectively. On the other hand, utilizing active delays can also improve the per
17、formance of a controlled system 26. Following such view, studies have showed that the reasonable time delay used in control can also improve the stability and damping effect of the system 27,28. And the time-delayed feedback forms are diversity. Such as displacement, acceleration and velocity combin
18、ed time-delayed feedback respectively 28e30. These could improve the performance of the system effectively when the feedback gain and time delay are properly selected. For example, Zhao 31 studied the damping mechanism of nonlinear dynamic vibration absorber with time delay. The response curve of th
19、e main system is gained through the method of multiple scales. It shows that the damping effect of the nonlinear dynamic vibration absorber with time delay is obviously better than that with no time delay dynamic vibration absorber. Xu 32 found that the time delay can change the effective frequency
20、range of saturation control, which can be used as a control parameter to suppress the systemvibration effectively. Sun 33,34 developed a time-delayed absorber considering the inherent and active time delay. The theoretical predictions, numerical simulations and experimental results show that it supp
21、resses the vibration of the traditional system when the passive absorber fails. Moreover, choosing the proper control parameters may broaden the effective frequency band of vibration absorption. Sun 35 studied the stability and response of a nonlinear isolator with time-delayed active control to obt
22、ain the standards for time delay and control strengths, and the optimal value of time delay is given on the base of the vibration dissipation time through eigenvalues analysis. Zhang 36 studied the modelling and tuning of the TDVA (time-delayed vi- bration absorber) with friction through integrated
23、analysis and experiment. The simulation and experimental results both show that it provides a direct method to tune the vibration absorption ratio via tunable parameters. And the time-delayed control can be achieved accurately by using the modelling and tuning technique. However, there are still som
24、e shortcom- ings in researches mentioned above such as there is no fundamental solution to the problem of time delay control stability, almost no research measure the time delay in active suspension systems and verify the effectiveness of control strategy by experiment. Driven by these problems, thi
25、s paper tries to propose a new method for designing control strategy considering time delay to active suspension, aiming at solving the stability of suspension control and enhancing the practical engineering value of application. In this paper, to guarantee the stability of the system, the time-dela
26、yed optimal control of active suspension system is designed by the method of state transformation and optimal control theory. The state transformation can be used to consider the time delay of the system. The optimal control can get a better control law. In order to accurately compensate the time de
27、lay and utilize it, we fi rst measure the time delay in active suspension system on time domain. Meanwhile, infl uence of the inherent time delayand the active time delayon the control characteristicsof the systemis analyzed. Atlast, the experimental platform of the time-delay optimal control for th
28、e suspension system is confi gure to verify the effectiveness of the control strategy designed and to provide fundamental for engineering practice in this paper. The present paper is organized as follows. After an introduction, the active suspension system and mathematic formu- lation is modeled in
29、section 2. The unstable state of the control system caused bytime delay is analyzed in section 3. In section 4, the control equation of active suspension system is updated and the stability of all time delay is proved. The infl uence of active time delay on suspension system is shown in section 5. I
30、n section 6, the effi ciency of the time-delayed optimal control strategy is illustrated experimentally and numerically. The dynamic characteristics of system under random excitation is analyzed by simulation in section 7. The conclusion is given in the fi nal section. 2. Modelling and formulation I
31、n order to illustrate the effectiveness of the time-delayed optimal control for suspension system, a suspension system control model is established. According to the vertical vibration of the vehicle body is the main factor affecting the ride comfort of the vehicle. And the structure of the vehicle
32、is quite complex. Now neglecting the pitch motion and roll motion of the vehicle body, a quarter-vehicle suspension model is set up shown in Fig. 1 by using passive suspension components and Magneto rheological (MR) damper as actuator. It is a 2-DOF simplifi ed model widely used in the literature 37
33、,38. In the model, msis sprung mass, which denotes the vehicle chassis; mwis unsprung mass, which denotes the mass of wheel as- sembly; ksand csare the stiffness and damping of the suspension system, respectively, ktand ctare compressibility and damping of pneumatic tire, respectively, xsand xware t
34、he displacement of the sprung mass and the unsprung mass, respectively, ut ?t is active control force,tis the time delay in the suspension control system, xgis the road displacement input. The dynamic equation of the quarter-vehicle suspension model is established as follows: G. Yan et al. / Journal
35、 of Sound and Vibration 446 (2019) 144e158145 ? msxs ksxs? xw cs_xs?_xw ? ut ?t 0 mwxw? ksxs? xw ? cs_xs?_xw kt?xw? xg? ct?_xw?_xg? ut ?t 0 (1) The main control objectives of the active suspension system are ride comfort, suspension defl ection and road holding. In fact, the ride comfort can be quan
36、titatively analyzed by the sprung mass acceleration. That is why to choosexsas a most important control output y1. Consequently, we want it to be controlled as small as possible. Due to the suspension structure, the relative displacement between the sprung mass and the unsprung mass, xs? xw, must be
37、 within a certain range, that is the reason for making it as the second control output y2in this paper. Moreover, to maintain the contact of wheel with road, the tire dynamic load ktxw? xg ct_xw?_xg which is another control output y3considered must be less than static tire load ms mw g. Now defi ne
38、thestatevariables: x1 xsdenotes the displacementof the sprung mass, x2_xsdenotes the speedof the sprung mass, x3 xwdenotes the displacement of the unsprung mass, and x4_xwdenotes the speed of the sprung mass. Meanwhile defi ne the state vector x x1;x2;x3;x4?Tand control output vector y y1;y2;y3?T, s
39、o the state-space equations of the suspension system can be written as following form: _x Ax But ?t EW y Cx Dut ?t GW (2) Where A 2 6 6 6 6 6 6 6 6 4 0100 ? ks ms ? cs ms ks ms cs ms 0001 ks mw cs mw ?ks? kt mw ?cs? ct mw 3 7 7 7 7 7 7 7 7 5 , B 2 6 6 6 6 6 6 6 4 0 1 ms 0 ? 1 mw 3 7 7 7 7 7 7 7 5 ,
40、E 2 6 6 6 6 6 6 4 00 00 00 kt mw ct mw 3 7 7 7 7 7 7 5 , W ? xg _xg ? . C 2 6 6 6 6 4 ? ks ms ? cs ms ks ms cs ms 10?10 00ktct 3 7 7 7 7 5 , D 2 6 6 6 6 4 1 ms 0 0 3 7 7 7 7 5 , G 2 4 00 00 ?kt?ct 3 5. This paper proposed the control force model ut ?t Kxt ?t(3) Where K is the control gain matrix. In
41、 order to distinguish the time delay which is intentionally added in the system, hence defi ne the time delay caused by signal acquisition, transmission, controlling calculation and actuator actuation as inherent Fig. 1. Quarter-vehicle suspension model. G. Yan et al. / Journal of Sound and Vibratio
42、n 446 (2019) 144e158146 time delay,t1 . And defi ne the time delay which intentionally added in the system as active time delay,t2. Therefore, the time delay in whole control system can be described astt1t2. Aim to use as small energyas possible to improve the performance of the suspension system, t
43、he objective function can be written as follows according to the method of classical quadratic optimal control. J lim T/ 1 T Z T 0 h q1y 2 1 q2y 2 2 q3y 2 3 ru2t ?t i dt(4) Where q1, q2, q3 and r are the weighting coeffi cient of the vehicle body acceleration, suspension defl ection, road holding an
44、d the control force, respectively. The size of the weighting coeffi cient indicates the importance of the performance index in designing of the suspension. So the safety and comfort of the suspension must be considered synthetically when choose the weighting coeffi cient. Now write Eq. (4) into matr
45、ix form as follows: J lim T/ 1 T Z T 0 yTtQyyt ut ?tTRyut ?tdt(5) Where Qy diagq1;q2;q3?, Ry r?. Substituting the control output equation into Eq. (5), If the time delayt 0, the classical quadratic optimal control standard form could be obtained. J lim T/ 1 T Z T 0 h xtTQxt utTRut 2xtTNut i dt(6) Wh
46、ere Q CTQy C is non-negative defi nite symmetric matrix, R DTQyD Ry is positive defi nite gain matrix and N CTQyD. Then the control strategy is easy to get the control gain matrix K based on optimal control. It is very extensively accepted in control applications. However, the suspension control loo
47、p has time delay, so the control law is different with the classical quadratic optimal control. The state equation of suspension system becomes a functional equation because of the time delay, so the stability of the system must be analyzed. It needs to be explained that when the control force is ze
48、ro, the active suspension actuator MR damper will be used as a passive suspension device, and the stiffness and damping of it are added into ksand csdirectly. 3. Time delay unstable state Due to the time delay may lead to instability of the control system 21. Now the stability of the system with tim
49、e delay is analyzed in this section. According to differential equation theory for linear systems with constant coeffi cients and time delays 39,40, it is known that the solution of the Eq. (1) is in the form of formula (7). xr Xreltr 1;2(7) Take K g1;g2;g3;g4?, available from Eq. (3) ut ?t Kxt ?t g1xst ?t g2_xst ?t g3xwt ?t g4_xwt ?t(8) Substituting Eqs. (7) and (8) into Eq. (1), and in Laplace domain, according to the nonzero solution, the characteristic equation of the system is ? ? ? ? ? msl2 csl ks? g1 g2le?lt?csl? ks? g3 g4le?lt ?csl? ks g1