二轴陀螺仪动态平台控制及应用於自动测试系统 - 中华科技大学

1、Journal of China University of Science and Technology Vol.49-2011.10The Implementation of PC-Based Real-Time Control Systems Using Pole-Zero Cancellation Method PC-Based即時控制系統之極點零點消去法實現 中華科技大學 11581 台北市南港區研究院路三段245號Tel: E-mail:ABSTRACTThe goal of this paper is to implement the pole-zero cancellation

2、 approach in PC-based real time control systems by using mathematic model blocks in VisSim software package environment. We use PC as a controller to apply pole-zero cancellation approach in PC-based real time control systems. A design example using the real control system, FB-33 servo control syste

3、m, based on the pole-zero cancellation approach is given. The pole-zero cancellation controller can be easily obtained according to the desired dynamic performance specifications of the closed-loop system. Finally, the PC controller uses the pole-zero cancellation controller and data acquisition sys

4、tem to control the FB-33 servo control system. The satisfied results are shown in this paper.Keywords：PC-Based Real-Time Control System, Pole- Zero Cancellation Method, Data Acquisition摘要本文以VisSim為系統發展環境，以數學方塊圖模式設計開發極點零點消去法，以PC為主控制器，實現極點零點消去法於PC-Based 即時控制系統。首先以實際FB-33伺服控制系統為例，應用極點零點消去法，以滿足性能規格需求，得到

5、控制器適當之極點與零點；另外以VisSim為系統發展環境，模擬加入控制器後系統動態響應，並與性能規格需求作比對，驗證其正確性。然後再配合資料擷取模組與動態連結函數庫整合FB-33控制系統，完成PC-Based 即時控制系統，實現極點安置演算法於伺服控制系統。 關鍵詞：PC-Based即時控制系統、極點零點消去法、資料擷取I. IntroductionThe root-locus approach to design is very powerful when the performance specifications are given in terms of time-domain qua

6、ntities. In designing a control system, if other than a gain adjustment is need, we have to reshape the original root loci by adding a suitable compensator 1. The transfer functions of many systems contain one or more poles that are very close to the imaginer axis of the s-plane. These poles may cau

7、se the system to be slowly response or lightly damped. We may insert a pole-zero cancellation compensator that has a transfer function with zero selected, which would cancel the undesired pole of the uncompensated open-loop transfer function, and to place the pole of the compensator at more desirabl

8、e location in the s-plane to meet the desired performance specifications23. This paper uses VisSim as a system developing environment to design and develop advance modern control algorithms by mathematic model blocks. VisSim is a Windows-based program for the modeling, design and simulation of compl

9、ex control systems without writing a line of code 45. It combines an intuitive drag-and-drop block diagram interface with a powerful mathematical engine. The visual block diagram interface offers a simple method for constructing, modifying and maintaining complex control system models. Furthermore,

10、VisSim offers unprecedented ease-of-use and consequently a shorter learning curve than competitive systems. Setting up a simulation in VisSim is simple. Connect the controller with pole-zero cancellation to the mathematical model and to a plot block. In practical, the pole of the transfer function o

11、f control systems may vary due to external disturbance or noise during the operation of the system 6-10. However, we choose the varied range of the pole of the plant to be less than 10%. The resulting responses can be easily analyzed in VisSim. Once a satisfactory mathematical model has been obtaine

12、d, the pole-zero cancellation compensator using operational amplifiers are applied in the FB-33 control system. Notice that the resulting responses show that exact cancellation is not necessary to precisely negative the influence of the undesirable poles 2 Once the pole-zero cancellation design is c

13、omplete, the PC using VisSim/Real-Time can be used as on-line servo controller through a high-speed data acquisition card. PC-based real time control systems can be configured and executed by interfacing VisSim controller mode with the FB-33 control system. II. Mathematical Modeling of DC Motor Cont

14、rol SystemTo establish a mathematical model of PM dc motor 26, we have the equivalent circuit diagram in Figure 1. Figure 1 Model of a separated excited dc motor.where is the angular displacement of the motor shaft, is the angular velocity of the motor shaft, is the rotor inertial, is the viscous-fr

15、iction coefficient, is the torque of the motor, is the applied voltage, is the armature resistance, is the armature current, is the armature inductance, is the back emf, and is the magnetic flux in the air gap.When the armature is rotating, the back-emf votage is proportional to the angular velocity

16、, we obtain (1)where is the back-emf constant of the motor.Applying Kirchhoffs voltage law to the system, the differential equation for the armature circuit is (2)The torque equations of the system is (3)Taking the Laplace transform of both sides of Equation (1),(2)and (3), we obtain (4) (5) (6)Subs

17、tituting Equation (5)sand (6)sto Equation (4)，we have (7)If the armature inductance is really very small (), it can be neglected, and the transfer function relating and is given by (8)where is DC motor gain and is DC motor time constant. Consider a DC motor position system with proportional controll

18、er shown in Figure 2, the open-loop transfer is (9)where =，is the amplifier gain,P1 is the attenuator gain, =， is the motor gain, and is the potentiometer gain.If the DC motor position control system with P controller shown in Figure 2 is a unit-feedback, the closed-loop transfer function is (10)whe

19、re is the open-loop gain and.Figure 2 Block diagram of the DC motor position control system with P controller.III. Controller Design by Pole-Zero Cancellation The open-loop transfer function of the DC motor control system shown in Figure 2 contains one pole that is close to the imaginer axis of the

20、s-plane in the root locus plot. This pole may cause the closed-loop system to be slowly response or lightly damped. We can insert a pole-zero cancellation compensator that has a transfer function with zero selected, which would cancel the undesired pole of the open-loop transfer function, and to pla

21、ce the pole of the compensator at more desirable location in the s-plane to meet the desired performance specifications12.Because the transfer function of elements in cascade is the product of their individual transfer functions 1, some undesirable poles and zeros can be cancelled by inserting a com

22、pensating element in cascade. For example, the large time constant in Equation (9) may be cancelled by use of the compensator as follows:If is much smaller than, we can effectively obtain the small motor time constant by canceling the large time constantIn practical, the transfer function of the pla

23、nt is usually obtained through testing and physical modeling; linearization of a nonlinear process and approximation of a complex process are needed 2. Thus, the true poles and zeros of the transfer function of the plant may not be accurately modeled. In fact, the true order of the system may even b

24、e higher than that presented by the transfer function used for modeling purpose. Another difficulty is the dynamic properties of the plant may vary due to external disturbance or noise, so the poles and zeros of the transfer function may vary during the operation of the system. It is obvious that ex

25、act cancellation is physically impossible because of inaccuracies involved in the location of the poles and zeros of control systems. This problem can be solved in the following by showing that exact cancellation is not necessary to precisely negative the influence of the undesirable poles. Let us a

26、ssume that the plant of the system is represented by (11)where is the pole that is to be cancelled and is a very small value due to external disturbances of the plant. .Let the transfer function of the compensator be (12)where is the zero and c is the pole of the controller. The open-loop transfer f

27、unction of the compensated system is (13)Because of inexact cancellation, it is obvious that the terms can not be cancelled in the denominator of Equation (13). The closed-loop transfer function is (14)Notice that one closed-loop pole due to a result of inexact cancellation is very close to the open

28、-loop pole at. Thus, Equation (14) can be approximated as (15)The partial-fraction expansion of Equation (10) isterms due to the remaining poles (16)We can show that is proportional to, which is a very small value. It can be seen that although the pole at cannot be cancelled accurately, the resultin

29、g transient-response term due to inexact cancellation will have a very small magnitude. The effect causing by inexact cancellation can be neglected for all practical purpose. IV. Verifying Pole-Zero Cancellation Method with VisSimTo verify the Pole-Zero Cancellation method for FB-33 control system 6

30、 described above, it can be easily done using VisSim. It is convenient to use VisSim for the modeling, design and simulation of FB-33 control system without writing a line of code 4-5. It combines an intuitive drag-and-drop block diagram interface with a powerful mathematical engine. The visual bloc

31、k diagram interface offers a simple method for constructing, modifying and maintaining complex control system models.The FB-33 Feedback control system shown in Figure 3, the apparatus of automatic control laboratory is a typical second-order system 7. The objective of this system is to control the p

32、osition of the mechanical load in accordance with the reference position. The open-loop transfer function of the FB-33 control system obtained by the author 7 can be written as (17)where , , , and . The desired performance specifications areMaximum overshoot is less than 32% (i.e.)Natural frequency

33、To meet the performance specifications, let us modify the closed loop poles so that an is obtained, without changing the value of the damping ratio, . The desired poles will be at to have the same maximum overshoot and two times the undamped natural frequency of the uncompensated FB-33 control syste

34、m. We can insert a compensator that has a transfer function with zero selected, which would cancel the undesired pole of the uncompensated open-loop transfer function, and place the pole of the compensator at more desirable location in the s-plane to achieve the desired dynamic performances. The pol

35、e of the compensator can be selected at 4 according to the desired specifications of the closed-loop system by using the root-locus approach shown in Figure 2.The suggested compensator can be of the formReferring to the root-locus plot for the compensated system shown in Figure 3, the gain k is eval

36、uated from the magnitude condition asHencek=4 at point Figure 3 Root-locus plot for the compensated FB-33 control system.VisSim diagram in Figure 4 produces the unit-step responses of the compensated and uncompensated systems. From the unit-step responses, we find that the requirement on the respons

37、e to the step reference input is met by designed system. In practical, the pole of the transfer function of FB-33 control system may vary due to external disturbance or noise during the operation of the system. However, we choose the varied range of the pole of the motor to be less than 10%. Unit-st

38、ep response curves for three cases are shown in Figure 5. Notice that the resulting responses show that exact cancellation is not necessary to precisely negative the influence of the undesirable poles. Figure 4 Unit-step responses of the compensated and uncompensated systems.Figure 5 Unit-step respo

39、nse curves for three compensated cases.V. Pole-Zero Cancellation Method in Real Systems Once the pole-zero cancellation design is complete, we can test the controller performance with the FB-33 control system. VisSim/Real-Time 711 is a VisSim option that provides the capability to couple a VisSim sy

40、stem model directly with a real-world hardware. Hardware-in-the-loop (HIL) systems can be configured and executed by interfacing VisSim controller mode with the FB-33 control system. With VisSim/Real-Time, the PC can be used as on-line servo controller for the FB-33 control system through a high-spe

41、ed data acquisition card (NI-DAQ-6024E) 12. The NI-DAQ-6024E interface card plugged in the PC offers a 200 kHz sampling rate for A/D and D/A channels larger than the bandwidth of the FB-33 servomotor. The control diagram and the actual step response of the PC-based real-time control system without c

42、ompensator are shown in Figure 6. The actual step response of the PC-based FB-33 control system is close to the simulation response of the theoretical model in Figure 4. Figure 6 Step response of PC-based real-time control system without compensator.The control diagram and the actual step response o

43、f the PC-based real-time control system with compensator are shown in Figure 7. It is obvious that the PC-based control system with the compensator satisfies the given specifications and is satisfactory.Figure 7 Step response of PC-based real-time control system with compensator.VI. Concluding Remar

44、ks1. VisSim is a Windows-based software package for the modeling, design and simulation of complex control systems without writing a line of code. It offers unprecedented ease-of-use and consequently a short learning curve.2. This paper has presented the pole-zero cancellation approach to the real-t

45、ime control systems design by using mathematic model blocks in VisSim software package environment. 3. The resulting transient-response term due to inexact cancellation has a very small magnitude. The effect causing by inexact cancellation can be neglected for all practical purpose4. Noted that the small difference of the resulting responses between the theoretical model and the real-time system may be caused by the linearization of a nonlinear process and