PID Control Analysis of Brake Test Bench

Transcription

1 PID Control Analysis of Brake Test Bench Rui Zhang, Haiyin Li, and Huimin Xiao, Department of Mathematics and Information, Henan University of Finance and Economics, Zhengzhou , Henan, China Abstract. In this paper, the control methods of brake test bench are studied. According to as much as possible consistent principle of braking process between on the test bench and road test vehicle, by controlling motor to work under certain regular current, compensate lack energy owing to the shortage of electricity, establish control model of equivalent inertia and vehicle load. At the same time, we make the model of current with speed, torque, inertia and other observable, also introduce PID control technology, and obtain PID control model which compensate for deficiency in control current model. Through stochastic simulation and disturbance simulation, we verify the accuracy and reliability of PID model. Keywords: brake; PID control; stochastic simulation; disturbance simulation; stability. 1 Introduction Drive brake (denoted brake in the paper)of car connects with wheel, its role is making the vehicle to slow or stop. The design of brake is one of the most important parts of designing vehicle, because it has a direct impact on personal and vehicle safety. To test the merits of the design we must carry out appropriate tests. In order to check the overall performance of brake, it is needed a lot of road tests in a variety of different situations. But vehicle can not been tested on the road in the design phase. What we can do is simulating road test designed on specialized brake test bench. The principle of simulation is as much as possible consistent principle of braking process between on the test bench and road test vehicle. Brake test bench is generally composed spindle installed flywheel group, motor of spindle-driven, base, auxiliary device of braking, measurement and control system and so on. The brake installed at the end of spindle, which makes spindle slow down if brake works. When test bench works, motor will make spindle and flywheel whirl, and when motor reaches the designed speed (in the simulation experiment, we can assume that angular velocity of spindle is always the same as angular velocity of wheel), motor will be powered off and at the This work is supported by the National Natural Science Foundation of China (No ). Corresponding author. Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part II, LNCS 6146, pp , c Springer-Verlag Berlin Heidelberg 2010

2 702 R. Zhang, H. Li, and H. Xiao same time brake will be applied. When it meets the designed ending condition, we say that one brake is finished. Because of the complexity of brake performance, it is difficult to obtain the relationship of motor drive current and time. Computer-controlled method commonly used in engineering is letting the whole braking time scatter into a lot of discrete time, for example, 10ms of every period, then according to instantaneous speed (or instantaneous torque) observed in the previous period, design the value of drive current in this period, this process is carried out successively until the brake is finished. An important indicator of evaluation of advantages and disadvantages of control methods is the size of energy error, the energy error in this article is brake energy of designed road test minus the consumed energy during braking on the corresponding test bench. In general, we don t consider observation error, random error and error arising from discrete to continuous problem. Automobile brake has been payed attention by scholars at home and abroad, there are a lot of papers discussing this subject [1 5]. In this article we not only establish the control model of brake test bench, but also carry on the stochastic simulation and disturbance simulation to the model, and verify the accuracy and reliability of the model, it is a breakthrough in the PID control of brake. 2 Mathematic Model of Control System for Brake Test Bench The designated wheels of road test vehicle bear load during braking, energy which vehicle has at the time of moving because of the load (ignore energy for vehicle rotating itself) equivalently changes into energy which flywheel and spindle produce during rotating on the best bench. Rotating inertia (referred to as inertia in the following) corresponding to the energy is called as equivalent rotating inertia in this article. The inertia of spindle etc non-detachable agencies on the best bench is called as basic inertia. Flywheel group was assembled with several flywheel, we can select a few required flywheel to fix on spindle, the total of these flywheel inertia plus basic inertia is called as mechanical inertia. In general, we assume that motor drive current of test bench is proportional to torque generated by motor (in this article, we assume the ratio is equal to 1.5A/N.m), and instantaneous speed and instantaneous torque of spindle are observable discrete during work of test bench. It is specially noted that units of measurement in this article are legal units, and don t indicate in detail. 2.1 Model of Equivalent Inertia and Vehicle Load Energy which vehicle has at the time of moving because of the load E 2 equivalently converts into the energy of flywheel and spindle rotating E 1,thatisto say, 1 2 kω2 = E 1 = E 2 = 1 2 mv2, (1)

3 PID Control Analysis of Brake Test Bench 703 where m is load mass, v is moving velocity of vehicle, k is equivalent rotating inertia, ω is rotating angular velocity of spindle and flywheel. When test bench works, motor will make spindle and flywheel whirl. When motor reaches the designed speed, motor will be powered off and at the same time be imposed on brake, therefore v = rω, wherer is radius of wheel, from (1), we can get k = mr 2 = G g r2, (2) where G is load of wheel, equation (2) describes relationship between equivalent rotating inertia and load of wheel. 2.2 Model of Motor Drive Current From the inertia of homogeneous hollow cylinder [6] J z = 1 2 M(R2 2 + R 2 1), we can obtain model of flywheel inertia J z = 1 2 πρh(r2 2 R2 1 )(R2 2 + R2 1 ), (3) where M is rigid-body mass, ρ is rigid-body density, h is rigid-body thickness, R 1, R 2 are inner and outer diameter of ring steel flywheel, respectively. Depending on instantaneous speed and instantaneous torque and so on observable discrete during test bench working, we can establish mathematical model of motor drive current. In the process of braking, due to short mechanical inertia, the lack of energy which is compensated by motor drive current ΔE is equal to energy of brake E 4 minus energy of flywheel and spindle rotating E 3, ΔE = 1 2 k 1(Δω) k 2(Δω) 2, (4) where k 1 is equivalent rotating inertia when brake works, k 2 is mechanical inertia when flywheel and spindle rotate. It is not difficult to obtain the following by the relationship of variables, ΔE = M ωδt, Δω = ω 1 ω 2, I =1.5M, (5) where M is torque braking, ω = ω1+ω2 2, ω 1 and ω 2 are initial and final angular of flywheel and spindle, respectively. Put (4) together with (5), we can get IωΔt 1.5 = k 1 2 (ω2 1 ω2) 2 k 2 2 (ω2 1 ω2). 2 (6) That is to say, we can get the function about current, variation of speed and inertia: I = 3 2 (k 1 k 2 ) Δω Δt. (7)

4 704 R. Zhang, H. Li, and H. Xiao 2.3 Model of Control Current Bythemodelofdrivecurrent,ifwewant to design computer control method of calculating current value in this period, then instantaneous speed and instantaneous torque observed in the previous period are needed. By the principle that drive current will compensate lack of energy due to short mechanical inertia, and because of the functional relationship of the system, we can have the energy equation, 1 2 J aω J aω4 2 = 1 2 J bω J bω4 2 + ω 3 + ω 4 M e Δt, (8) 2 where J a is equivalent rotating inertia; J b is mechanical inertia produced by flywheel and spindle; ω 3 and ω 4 are observable speed in previous period and forecasting speed in present period, respectively; M e is torque caused by drive current. From the relationship of observable speed in previous period and in present period, we can get ω 4 = ω 3 M s J a Δt, M e = I 1.5, (9) where M s is observable torque. From (8) and (9), we can obtain equation of control current, I =1.5M s J a J b J a. (10) By instantaneous torque of previous period, we can calculate current value of present period and control it, but because in this method observed value of previous period directly is used in present period control, doesn t have mathematical treatment to deviation of present period. Therefore, this control method has a longer response and not high accuracy on the system regulating, then output is easy to oscillate. 3 PID Control System 3.1 Model of PID Control To defect of control current model, we improve the model. PID control technology just can make up for lack of above model. On the basis of control current model, we can build state-space discrete model of this system, and let observed speed before a moment and output current become state variable, instantaneous torque before a moment become input, output current of present period become output, establish the following state equation, x t+1 = Ax t + Bu t, y t = Cx t, (11)

5 the model of first equation of (11) is PID Control Analysis of Brake Test Bench 705 ω t+1 = ω t M t, I t+1 = 3M t(j a J b ), (12) J a 2J a the model of second equation of (11) is I t+1 = I t, (13) the coefficients of state equation are ( ) ( ) 1 10 J A =, B = a 3(J 00 a J b ), C = ( 01 ), 2J a then transfer equation of the system T = s s 2. (14) s 3.2 PID Simulation By transfer equation (14), we can have PID simulation using Simulink toolbox of MATLAB [8], as figure 1 In figure 1, part of Step provides a step-wave, part of Discrete PID Controller is PID control provided by Simulink, Transfer Fcn refers transfer equation of the system, Scope is oscilloscope which is used to observe test data, Clock is clock which is used to output time. In MATLAB, if we have three parameters, then PID simulation will can go on. After a large number of simulation experiments, we have a set of good parameters K p =80,K i =30,K d = 4. At the same time, we have figure 2 on oscilloscope, Trend of the graphic can reflect relationship between time and current, computer can use this method to control current, accordingly achieves accurate, rapid and stable effect. Fig. 1. PID Simulation using MATLAB

6 706 R. Zhang, H. Li, and H. Xiao Fig. 2. Simulation diagram of time and current on oscilloscope 4 Test and Analysis Good or bad performance of a control system depends on if it could be quickly stabilized when it is disturbed, that is to say, if it could overcome deviation because of disturbance and return to designed value accurately and fast [10]. Therefore, we improve model of figure 1, add to 10 4 length disturbance at the time 3 second (generated from Pulse Generator of the following figure 3) in order to test model of figure 1. As shown in Figure 3, After simulation, we can observe the following graphic on the oscilloscope, as figure 4: Although the above graphic is obtained by stochastic simulation, and the values of figure come from assume, from figure 4 we see the model could quickly restore stability when it is disturbed. Namely, there is higher accuracy and quickness of overcoming deviation for disturbance and returning to designed values, therefore, the control system performs well. Fig. 3. Simulation joined disturbance

7 PID Control Analysis of Brake Test Bench 707 Fig. 4. Simulation diagram of time and current subjected to disturbance 5 Conclusion This model not only has a faster response time, accurate adjustment, higher stability, better robustness, strong anti-interference ability and so on advantages, but also facilitate to simulate and control for computer, thus contribute to engineering automation. References 1. Harifi, A., et al.: Designing a sliding mode controller for slip control of antilock brake systems. Transportation Research Part C: Emerging Technologies 6(16), (2008) 2. Liu, G.P., Deley, S.: Optimal-tunning PID control for industrial systems. Control Engineering Practice 11(9), (2001) 3. Luo, J., et al.: Numerical simulation of jet breakup due to amplitude-modulated (A-M) disturbance. Transactions of Nonferrous Metals Society of China 3(18), (2008) 4. Blau, P.J., Jolly, B.C.: Wear of truck brake lining materials using three different test methods. Wear 7-12(259), (2005) 5. Harifi, A., Aghagolzadeh, A., Alizadeh, G., Sadeghi, M.: Designing a sliding mode controller for slip control of antilock brake systems. Transportation Research Part C 16, (2008) 6. Baishi, W.: Base of University Physics. Science Press, Beijing (2007) (in Chinese) 7. Yuantao, Z., Zhaoli, X., Yinan, F.: Modeling and simulation for brake-pipe-pressure servo system in automobile brake test. Modern Manufacturing Engineering 25, (2006) (in Chinese) 8. Shujun, L., et al.: MATLAB7.0 Control system Applications and Examples. Machinery Industry Press, Beijing (2006) (in Chinese) 9. Suqing, W., Weifu, J.: PID tunning based on MATLAB/Simulink. Techniques of automation and applications 328, (2009) (in Chinese) 10. Yanshuo, R., Yiding, Z.: Automatic Control System. Beijing University of Posts and Telecommunications Press, Beijing (2006) (in Chinese)