1、英文資料The con tral tech niq ues drives and con trals han dbookChapter A4Torque, speed and positi on con trolA4.1 Gen eral prin ciplesMany applicati ons exist where somethi ng has to be con trolled to follow a referenee quantity. For example, the speed of a large motor may be set from a low-power contr

2、ol signal. This can be achieved using a variable-speed drive as described in the follow ing.Ideally, the relati on ship betwee n the refere nce and the motor speed should be linear, and the speed should change instantly with changes in the reference. Any con trol system can be represe ntedas in Figu

3、re A4.1b, with an in put refere nce signal, a transfer function F and an output. For the system to be ideal, the transfer fun cti on F would be a simple con sta nt, so that the output would be proporti onal to the refere nce with no delay.Figure A4.1 Variable-speed drive and motorUnfortunately, the

4、transfer function of many practical systems is not a con sta nt, and so without any form of feedback from the output to correct for the non-ideal n ature of the tran sfer fun cti on, the output does not follow the dema nd as required. Using an in ducti on motor supplied bya simple ope n-loop variabl

5、e-speed drive as an example, the followi ng illustrates some unwan ted effects that can occur in practical systems:Speed regulation. The output of a simple open-loop drive is a fixed frequency that is proporti onal to the speed refere nce, and so the freque ncy applied to the motor remains constant

6、for a constant speed reference. The speed of the motor drops as load is applied because of the slip characteristic of the motor, and so the speed does not rema in at the required level.In stability. It is possible un der certa in load con diti ons and at certa in frequencies for the motor speed to o

7、scillate around the required speed, even though the applied freque ncy is con sta nt. Ano ther major source of in stability in rotating mechanical systems is low-loss elastic couplings and shafts.Non-linearity. There are many possible sources of non-linearity. If, for example, the motor is connected

8、 to a gearbox, the speed at the output of the gearbox could be affected by backlash between the gears.Variations with temperature. Some aspects of the system transfer function may vary with temperature. For example, the slip of an induction motor increases as the motor heats up, and so for a given l

9、oad the motor speed may reduce from the starting speed when the motor was cold.Delay. With a simple open-loop inverter and induction motor there can be a delay before the motor speed reaches the demanded level after a change in the speed reference. In very simple applications such as controlling the

10、 speed of a conveyor belt, this type of delay may not be a problem. In more complex systems, such as on a machine tool axis, delays have a significant effect on the quality of the system.These are just some of the unwanted effects that can be produced if an open-loop control system is used. One meth

11、od that improves the quality of the controller is to use a measure of the output quantity to apply some feedback to give closed-loop control.Although a modern variable-speed drive includes many features, the basic function of the drive is to control torque (or force), speed or position. Before proce

12、eding to the specific details of how different types of variable-speed drive function, the theory of control for each of these quantities is discussed. A position control system is shown in Figure A4.5. This includes an inner speed controller, and within the speed controller there is an inner torque

13、 controller. It is possible to create a system where the position controller determines the mechanical torque that is applied to the load directly without the inner speed and torque loops. However, the position controller would need to be able to control the complex combined transfer function of the

14、 motor windings, the mechanical load and the conversion from speed to position.Therefore it is more usual to use the format shown in Figure A4.5. The other advantage of this approach is that limits can be applied to the range or rate of change of speed and torque between each of the controllers. Whe

15、n a system is required to control speed only, the position controller is omitted, and when a system is required to control torque only, the position and speed controllers are omitted.A position sensor is shown providing feedback for the system, but this may be replaced by a speed sensor or it may be

16、 omitted altogether as follows. Position information is required by the torque controller to function in an a.c. motor drive (see the dotted line). If position feedback is provided the speed feed-back is derived as the change of speed over a fixed sample period. Sensorless schemes are possible for s

17、peed and torque control of a.c. motors, in which case the sensor is not required.Position feedback is not necessaryfor the torque controller in a d.c. motor drive, so a speed feedback device such as a tacho-generator can be used to provide the feed-back for the speed controller. Again, sensorlesssch

18、emes are possible where a speed feedback device is not required.A torque controller for a rotary motor, or a force controller for a linear motor, is the basic inner loop of most variable-speed drives. Only torque control is discussed here, but the principles also apply to force control for a linear

19、application. In order to explain the principles of torque control, the simple d.c. motor system in Figure A4.6 is used as an example. The analysis of torque control in an a.c. motor can be done in exactly the same way, provided suitable transformations are carried out in the drive. These transformat

20、ions will be discussed later.The torque demand or reference (Te*) is converted by the torque controller into a current in the motor armature, and the motor itself converts the current into torqueFigure A4.6 Torque and current controllers in a d.c. motor drive: (a) torque control;(b) current control

21、to drive the mechanical load. Figure A4.6b shows the system required to convert the torque reference into motor current. The torque reference (Te*) is first transformed into a current reference (ia*) by including the scaling effect of the motor flux. The motor flux, controlled by the motor field cur

22、rent (if ), is normally reduced from its rated level at higher speeds when the terminal voltage would exceed the maximum possible output voltage of the power circuit without this adjustment. Current limits are then applied to the current reference so that the required current does not exceed the cap

23、a-bilities of the drive. The current reference (limited to a maximum level) becomes the input for the PI controller. The electrical equivalent circuit of the motor consists of a resistance (Ra), an inductance (La) and a back emf that is proportional to flux and speed (Kevc/crated).The PI controller

24、alone could successfully control the current in this circuit becauseas the speed increases,the voltage required to overcome the back emf would be pro-vided by the integral term. The integral control is likely to be relatively slow, so to improve the performance during transient speed changes a volta

25、ge feed-forward term equivalent to Kevc/crated is included. The combined output of the PI controller and the voltage feed-forward term form the voltage reference (va*), and in response to this the power circuit applies a voltage (va) to the motor ' s electrical circuit to give a current (ia). Th

26、e current is measured by a sensor and used as feedback for the current controller.As well as the linear components shown in Figure A4.6, the current control loop in a digital drive includes sample delays as well as delays caused by the power circuit. In practice, the response of the controller is do

27、minated by the proportional gain. In particular, if a voltage feed-forward term is used, the integral term has very little effect on the transient response.Setting of the control loop gains is clearly very important in optimising the per-formance of the control loop. One of the simplest methods to d

28、etermine a suitable proportional gain is to use the following equation: where La is the motor inductance and Ts the current controller sample time. K is a con-stant that is related to the current and voltage scaling, and the delays present in the control system and power circuit. Most modern variabl

29、e-speed drives include auto-tuning algorithms based on measurement of the electrical parameters of the motor taken by the drive itself, and so the user does not normally need to adjust the current controller gains.It is useful to know the closed-loop transfer function of the torque controller (i.e.

30、Te/Te*) so that the response of a stand-alone torque controller, or the effect of an inner torque controller on outer loops such as a speed controller, can be predicted. As the response is dominated by the system delays it is appropriate to represent the closed-loop response as simple gains and a un

31、ity gain transport delay as shown in Figure A4.7.The torque reference could be in N m, but it is more conventional to use a value that is a percentage of the rated motor torque. Figure A4.7a gives the transfer function when the torque controller is used alone. Kt is the torque constant of the motor

32、in N m A21. If the torque controller is used with an outer speed controller a slightly different representation must be used, as in Figure A4.7b. The speed controller pro-duces a torque reference where a value of unity corresponds to a current level that is specified for the size or rating of the dr

33、ive. From a control perspective it is unimportant whether this is the maximum current capability of the drive, the rated current or some other level. The actual levelClosed-loop speed control can be achieved by applying a simple PI controller around the torque controller described previously. For th

34、e purposes of this analysis it is assumed that the load is an inertia J, with a torque Td that is not related to speed (friction is neglected). The resulting system is shown in Figure A4.10 Figure A4.10 Speed controllerIf the PI controller is represented as Kp t Ki/s, the torque controller is assume

35、d to be ideal with no delays so that the unity transport delay can be neglected, and the inertia load is represented as 1/Js then the forward loop gain in the s domain is given byThe closed-loop transfer function in the s domain v(s)/v*(s) is given by G(s)/ 1 t G(s). Substituting for G(s) and rearra

36、nging givesIf the natural frequency of the system is defined as vn ? (KcKtKi=J ) and the damping factor is defined as j ? vnKp/(2Ki) thenAs with the torque controller, it is useful to know the closed-loop response so that the response of a stand-alone speed controller, or the effect of an inner spee

37、d controller on an outer position loop, can be predicted. If a moderate response is required from the speed controller it is not significantly affected by system delays, and a linear transfer function such as equation (A4.4) can be used. All the constants in these equations and the delays associated

38、 with the current controllers are normally provided to users so that calculations and/or simulations can be carried out to predict the performance of the speed controller.In addition to providing the required closed-loop step response, it is important for the system to be able to prevent unwanted mo

39、vement as the result of an applied torque transient. This could be because a load is suddenly applied or because of an uneven load. The ability to prevent unwanted movement is referred to as stiffness. The com-pliance angle of the system is a measure of Figure A4.11 Responses of an ideal speed contr

40、oller: (a) closed-loop step response;Figure A4.12 Unwanted delays in a practical digital driveDynamics 115UMC 3 000 rpm servo motor (Kt ? 1.6 N m A21, J ? 0.00078 kg m2) with the speed controller gains set to Kp ? 0.0693j and Ki ? 14.32.As the damping factor is increased, the closed-loop response ov

41、ershoot is reduced and the speed of response improves. The closed-loop response includes 10 per cent overshoot with a damping factor of unity becauseof the s term in its numerator.As the damping factor is increased, the overshoot of the response to a torque tran-sient is reduced and the response bec

42、omes slower.In this case there is no s term in the numerator and the responseincludes no overshoot with a damping factor of unity.It would appear from these results that the higher the proportional gain, and hence the higher the damping factor the better the responses; however, the results so far as

43、sume an ideal torque controller and no additional unwanted delays. In a real digital drive system the delays given in Figure A4.12 are likely to be present. A delay is included to represent the sample period for speed measurement, but this is only relevant if the speed feedback is derived from a pos

44、ition feedback device such as an encoder and is measured as a change of position over a fixed sample period.The effect of the unwanted delays can be seen in the closed-loop step response for a real system as shown in Figure A4.13. In each case the response of the real system has more overshoot than

45、the ideal system. If the damping factor is set to unity then the overshoot may be acceptable, but with a damping factor of 1.25 the response is quite oscillatory and is likely to be unacceptable. The effect of the unwanted delays is more pronounced the longer the delay and also as the set bandwidth

46、of the speed controller is increased.The effect of the additional delays can be seen in the Bode plot of the closed-loop response of the speed controller set up to give unity damping factor (Figure A4.14). The frequency at the 23 dB point of the gain characteristic has increasedsignificantly from th

47、e ideal speed controller, whereas the frequency at the 608 point of the phase characteristic is almost unchanged. If this is to be used as a stand-alone controller the gain characteristic could be used to predict the bandwidth, although it should be noted that the gain is greater than unity at some

48、frequencies. Often the bandwidth based on the gain characteristic is the only bandwidth that is quoted, because this makes the performance appear to be better, in this case 2 000 rad s21.Figure A4.13 Effect of delays on a closed-loop step response:(a) damping factor ? 1; (b) damping factor ? 1.25 ba

49、sed on the phase delay(672 rad s21 for this example) must be used, as this affects the performance of the outer loop.Unwanted delays limit the performance of the speed controller. The quantised nature of speed feedback when it is derived from a position sensor as the change of position over a fixed

50、sample period can also limit this. A high proportional gain in the speed controller, and hence high bandwidth, generate high-frequency torque ripple and acoustic noise from the quantised speed feedback.Figure A4.14 Bode plot of closed-loop response of a speed controllerThe characteristic defined by

51、equation (4.11) is shown in Figure A4.15. The required damping factor must first be selected, and from this the ratio vbw/vn is taken from the graph. For example, if a damping factor of unity is required, the value of vbw/vn is 2.5.Figure A4.15 Effect of damping factor on bandwidthThe definition of

52、damping factor is j ? vnKp/(2Ki). By rearranging this and substitut-ing for natural frequency, a suitable value for the proportional gain can be derived:Selection based on compliance angleFrom equation (A4.9) the steady-state response to a torque transient can be derived by setting s ? 0. The result

53、ing change of output angle for a given steady-state torque Td isIf the compliance angle is defined as the change of output angle with a steady-statetorque equal to KcKt (i.e. the torque produced by the motor with a torque producing current equal to Kc) then a suitable value of integral gain can be s

54、elected for a given compliance angle uc asThe proportional gain can then be determined in the same way as for the selection based on bandwidth using equation (A4.14).If the load (including motor) inertia is known, or it is possible for the drive to measure the load inertia using an auto-tuning algor

55、ithm, then the user need only enter the required damping factor and the speed loop bandwidth or compliance angle into the drive parameters. The drive can then automatically set suitable values for the speed controller gains.There are situations where it is desirable to have low-speed controller gain

56、s, for example in a winder application where the inertia of the load may change in time,Figure A4.16 Speed controller with torque feed-forward or an elevator where high gains would excite resonances in the system, making it unstable. The disadvantageof using low gains is that the responseof the spee

57、d con-troller is degraded. If the load torque characteristic is known, it is possible to use a torque feed-forward term to significantly improve the response and still use low-speed controller gains. Figure A4.16 shows a system with a known inertia load using torque feed-forward.The profile generato

58、r produces the required speed reference v*, which may be a simple linear ramp that accelerates and decelerates the load. It also produces the rate of change of speed or acceleration sv* (dv*/dt in the time domain). When this is scaled to include the inertia and the system constants, it gives a curre

59、nt reference that should accelerate and decelerate the load as required. Now the speed controller is only required as a trim to compensate for inaccuracies in the torque feed-forward. The response of the system is only limited by the response of the current controllers and the sample rate of the profile generator, and not by the response of the speed controller.If position control is requi