Adjusting Servo Drive Compensation

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A drive sizing makes sure that the drive has enough torque to satisfy the load requirements to overcome friction losses, provide sufﬁcient load thrust, and provide enough acceleration torque for the type of acceleration required. Once the drive-sizing requirement is satisﬁed, the servo drive stability must be addressed. Stabilizing the servo drive is a matter of adjusting the servo compensation. All industrial servo drives require some form of compensation often referred to as proportional, integral, and differential (PID). The process of applying this compensation is known as servo equalization or servo synthesis. In general, commercial industrial servo drives use proportional...

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Nội dung Text: Adjusting Servo Drive Compensation

14 Adjusting Servo Drive Compensation 14.1 MOTOR AND CURRENT LOOP A drive sizing makes sure that the drive has enough torque to satisfy the load requirements to overcome friction losses, provide sufﬁcient load thrust, and provide enough acceleration torque for the type of acceleration required. Once the drive-sizing requirement is satisﬁed, the servo drive stability must be addressed. Stabilizing the servo drive is a matter of adjusting the servo compensation. All industrial servo drives require some form of compensation often referred to as proportional, integral, and differential (PID). The process of applying this compensation is known as servo equalization or servo synthesis. In general, commercial industrial servo drives use proportional plus integral compensation (PI). It is the purpose of this discussion to analyze and describe the procedure for implementing PI servo compensa- tion. The block diagram of Figure 1 represents DC and brushless DC motors. All commercial industrial servo drives make use of a current loop for torque regulation requirements. Figure 1 includes the current loop for the servo drive with PI compensation. Since the block diagram of Figure 1 is Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 1 Motor and current loop block diagram. not solvable, block diagram algebra separates the servo loops to an inner and outer servo loop of Figure 2. For this discussion a worst-case condition for a large industrial servo axis will be used. The following parameters are assumed from this industrial machine servo application: Motor: Kollmorgen motor—M607B Machine slide weight: 50,000 lbs Ball screw: Length—70 in. Diameter—3 in. Lead—0.375 in. revolution Pulley ratio: 3.333 JT ¼ total inertia at the motor ¼ 0.3511 lb-in.-sec2 te ¼ electrical time constant ¼ 0.02 second ¼ 50 rad/sec t1 ¼ te Ke ¼ motor voltage constant ¼ 0.646 volt-sec/radian KT ¼ motor torque constant ¼ 9.9 lb-in./amp KG ¼ ampliﬁer gain ¼ 20 volts/volt Kie ¼ current loop feedback constant ¼ 3 volts/40A ¼ 0.075 volt/amp Fig. 2 Rearranged motor and current loop block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Ra ¼ motor armature circuit resistance ¼ 0.189 ohm Ki ¼ integral current gain ¼ 735 amp/sec/radian/sec The ﬁrst step in the analysis is to solve the inner loop of Figure 2. The closed-loop response I =e1 ¼ G=1 þ GH where: G ¼ 1=Ra ðte S þ 1Þ ¼ 5:29=½te S þ 1 ð5:29 ¼ 14:4 dBÞ GH ¼ 0:64669:9=½0:18960:3511½ðte S þ 1ÞS GH ¼ 96=S½te S þ 1 96 ¼ 39 dB 1=H ¼ J T S=K e K T ¼ 0:3511 S=0:64669:9 1=H ¼ 0:054 S ð0:054 ¼ À25 dBÞ Using the rules of Bode, the resulting closed loop Bode plot for I =e1 is shown in Figure 3. Solving the closed loop mathematically: 1 I G ¼ ¼ e1 1 þ GH Ra ðte S þ 1Þ þ Ke KT =JT S JT S ¼ (14.1-1) JT Ra S ðte S þ 1Þ þ Ke KT I JT S ¼ 2 e1 J T Ra te S þ J T Ra S þ K e K T J T =K e K T S ¼ (14.1-2) ½ðJ T Ra =K e K T Þte S 2 þ ðJ T Ra =K e K T ÞS þ 1 ð0:3511=0:64669:9ÞS I ¼ tm te S 2 þ tm S þ 1 e1 0:054 S ¼ (14.1-3) 0:0160:02 S 2 þ 0:01 S þ 1 0:351160:189 J T Ra where : tm ¼ ¼ ¼ 0:01 sec; 0:64669:9 K eK T om ¼ 1=tm ¼ 100 rad=sec (14.1-4) te ¼ 0:02 sec oe ¼ 1=te ¼ 50 rad=sec For a general quadratic: S 2 2 delta þ Sþ1 o2 or r or ¼ ½om oe 1=2 ¼ ½1006501=2 ¼ 70 rad=sec (14.1-5) 0:054S I ¼2 (14.1-6) 2 þ ð2 delta=70ÞS þ 1 e1 S =70 Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 3 Current inner loop frequency response. Fig. 4 Current loop block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Having solved the inner servo loop it is now required to solve the outer current loop. The inner servo loop is shown as part of the current loop in Figure 4. In solving the current loop, the forward loop, open loop, and feedback loop must be identiﬁed as follows: The forward servo loop: K i K G 60:054 ð0:02 S þ 1Þ 73562060:054 ð0:02 S þ 1Þ G¼ ¼ 0:0002 S 2 þ 0:01 S þ 1 0:0002 S 2 þ 0:01 S þ 1 (14.1-7) 794ð0:02 S þ 1Þ 15:88 S þ 794 G¼ ¼ 0:0002 S þ 0:01 S þ 1 0:0002 S 2 þ 0:01 S þ 1 2 (14.1-8) Where : K G ¼ 20 volt=volt K ie ¼ 3=40 ¼ 0:075 volt=amp K i K G 60:054794 ð58 dBÞ K i ¼ 794=ð2060:054Þ ¼ 735 79,400 S þ 3,970,000 G¼ (14.1-9) S2 þ 50 S þ 5000 The open loop: 79,400 S þ 3,970,000 GH ¼ 0:0756 (14.1-10) S 2 þ 50 S þ 5000 5955 S þ 297,750 GH ¼ 2 (14.1-11) S þ 50 S þ 5000 The feedback current scaling is the following: H ¼ 3 volts=40 amps ¼ 0:075 volts=amp 1=H ¼ 13:33 ¼ 22:4 dB The Bode plot frequency response is shown in Figure 5. The current loop bandwidth is 6000 rad/sec or about 1000 Hz, which is realistic for commercial industrial servo drives. The current loop as shown in Figure 5 can now be included in the motor servo loop with reference to Figure 2 and reduces to the motor servo loop block diagram of Figure 6. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 5 Current loop frequency response. Fig. 6 Motor servo loop block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
The completed motor servo loop has a forward loop only (as shown in Figure 6) where: J T ¼ total inertia at the motor ¼ 0:3511 lb-in.-sec2 K T ¼ motor torque constant ¼ 9:9 lb-in./amp 13:369:9 375 G¼ ¼ ð51:5 dBÞ 0:3511 S ðð jw=6000Þ þ 1Þ S ð0:000166 S þ 1Þ (14.1-12) 375 2,250,090 G¼ ¼2 (14.1-13) 2 0:000166 S þ S þ 0 S þ 6000 S þ 0 jI j 13:1ð0:02 S þ 1Þ 9: 9 vo KT ¼ ¼ (14.1-14) 6 6 ei J T S jvi j 0:3511 S 0:00000331 S 2 þ 0:02 S þ 1 375ð0:02 S þ 1Þ vo ¼ (14.1-15) ei S 0:00000331 ðS þ 50ÞðS þ 5991Þ 375ð0:02 S þ 1Þ vo ¼ (14.1-16) ei S 0:0000033165065991ððS=50Þ þ 1Þððs=5991Þ þ 1Þ 375ð0:02 S þ 1Þ vo ¼ (14.1-17) ei S ð0:02 S þ 1Þð0:000166 S þ 1Þ 375 vo ¼ (14.1-18) ei S ðð jw=5991Þ þ 1Þ The Bode frequency response for the motor and current loop is shown in Figure 7. The motor and current closed-loop frequency response indicate that the response is an integration that includes the 6000 rad/sec bandwidth of the current loop. This is a realistic bandwidth for commercial industrial servo drives. 14.2 MOTOR/CURRENT LOOP AND POSITION LOOP Usually this response is enclosed in a velocity loop and further enclosed in a position loop. However, there are some applications where the motor and current loop are enclosed in a position loop. Such an arrangement is shown in Figure 8. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 7 Motor and current loop frequency response. The forward loop transfer function is as follows: Kv G¼ (14.2-1) S2 ðð jw=6000Þ þ 1Þ where: K v ¼ K 2 6375. For most large industrial machine servo drives a position loop kv ¼ 1 ipm/mil or 16.6 rad/sec can be assumed. The frequency response for the position loop is shown in Figure 9. This response is obviously unstable Fig. 8 Position loop block diagram using motor and current loop only. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 9 Position loop frequency response without an inner velocity servo. with a minus two slope at the zero gain point. It is also obvious that there are two integrators in series, resulting in an oscillator. If a velocity loop is not used around the motor and current loop, some form of differential function is required to obtain stability. By adding a differential term at about 10 rad/sec in Figure 9, the response can be modiﬁed to that of a type 2 control, which could have performance advantages. The absence of the minor velocity servo loop bandwidth could make it possible to increase the position-loop velocity constant (position-loop gain) for an increase in the position loop response. 14.3 MOTOR/CURRENT LOOP WITH A VELOCITY LOOP For the purposes of this discussion it will be assumed that the motor and current loop are enclosed in a velocity servo loop. Such an arrangement is shown in Figure 10. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 10 Velocity loop block diagram. 14.4 PI COMPENSATION The servo compensation and ampliﬁer gain are part of the block identiﬁed as K 2 . Most industrial servo drives use PI compensation. The ampliﬁer and PI compensation can be represented as in Figure 11. h i K Ki Kpi s þ 1 Kp s þ Ki K2 ½t2 s þ 1 I Ki ¼ Kp þ ¼ ¼ ¼ (14.3-1) V2 s s s s Kp t2 ¼ o2 (corner frequency) Ki The adjustment of the PI compensation is suggested as follows: 1. For the uncompensated servo Bode plot, set the ampliﬁer gain to a value just below the level of instability. 2. Note the forward loop frequency ðog Þ at À1358 phase shift (458 phase margin) of Figure 14. Fig. 11 PI compensation block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
3. From the Bode plot for PI compensation of Figure 12, the corner frequency o2 K i =K p should be approximately og =10 or smaller as a ﬁgure of merit. The reason for this is that the attenuation characteristic of the PI controller has a phase lag that is detrimental to the servo phase margin. Thus the corner frequency of the PI compensation should be lowered about one decade or more from the À1358 phase shift point ðog Þ of the open loop Bode plot for the servo drive being compensated. For the servo drive being considered, og occurs at 6000 rad/sec. Fig. 12 PI frequency response. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 13 Velocity servo block diagram with PI compensation. Applying the PI compensation of Figure 11, to the velocity servo drive is shown in Figure 13. In general the accepted rule for setting the servo compensation begins by removing the integral and/or differential compensation. The propor- tional gain is then adjusted to a level where the velocity servo response is just Velocity loop frequency response without PI compensation—K 2 ¼ 1000. Fig. 14 Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
stable. The proportional gain is then reduced slightly further for a margin of safety. For a gain K 2 ¼ 1000 of the uncompensated servo, the Bode plot is shown in Figure 14. It should be noted that the motor and current loop have a bandwidth of 6000 rad/sec as shown in Figure 7. This is a normal response for industrial servo-drive current loops. The transient response for this servo is shown in Figure 15 as a damped oscillatory response. If the gain K 2 is reduced to a value of 266 for a forward loop gain of 100,000, the Bode plot is shown in Figure 16 with a stable transient response shown in Figure 17. At this point the PI compensation is added as shown in Figure 13. The index of performance (I.P.) for the PI compensation is that the corner frequency o2 ¼ K i =K p , should be a decade or more lower than the À135 degree phase shift (458 phase margin) frequency ðog Þ of the forward loop Bode plot (Figure 16) for the industrial servo drive being considered. Fig. 15 Velocity servo transient response for frequency response of Figure 14. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
With reference to Figure 16 and Figure 17 of the stable uncompen- sated servo, the À135 degree phase shift (458 phase margin) occurs at 6000 rad/sec frequency, which is also the bandwidth of the motor/current loop. Using the I.P. of setting the PI compensation corner frequency at one decade or more lower in frequency that the À1358 phase shift frequency point, the corner frequency should be set at 600 rad/sec or lower. With the corner frequency of the PI compensation set at 600 rad/sec (0.001666 sec) the compensated servo is shown in the Bode plot of Figure 18. The transient response is shown in Figure 19 as a highly oscillatory velocity servo drive. Obviously this servo drive needs to have the PI compensation corner frequency much lower than one decade (600 rad/sec) I.P. For a two decade, 60 rad/sec (0.0166 sec) lower setting for the PI corner frequency the Bode response is shown in Figure 20 with a transient response shown in Figure 21 having a single overshoot in the output of the velocity servo drive. K By lowering the PI compensation corner frequency ðo2 ¼ Kpi Þ to 20 rad/sec (0.05 sec), a stable velocity servo drive results. The forward loop Velocity loop frequency response without PI compensation—K 2 ¼ 266. Fig. 16 Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
and open loop are deﬁned as follows: H ¼ 0:0286 v=rad=sec 1=H ¼ 34:9 ð30:8 dBÞ Gain @ o ¼ 1 rad=sec ¼ 100 dB ¼ 100,000 K 2 ¼ 100,000=375 ¼ 266 (14.3-2) K 2 6375ðð j o=20Þ þ 1Þ 100,000ðð j o=20Þ þ 1Þ G¼ ¼ ð100 dBÞ S 2 ðð j o=6000Þ þ 1Þ S 2 ðð j o=6000Þ þ 1Þ (14.3-3) Fig. 17 Velocity servo transient response for frequency response of Figure 16. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
100,000ð0:05 S þ 1Þ 5000 S þ 100,000 G¼ ¼2 (14.3-4) 2 S ð0:000166 S þ 1Þ S ð0:000166 S þ 1Þ 30,120,481 S þ 602,409,638 G¼ (14.3-5) S3 þ 6024 S 2 þ 0 S þ 0 2860ðð j o=20Þ þ 1Þ GH ¼ 0:02866G ¼ ð69 dBÞ (14.3-6) S 2 ðð j o=6000Þ þ 1Þ The Bode plot for the velocity loop with PI compensation is shown in Figure 22, having a typical industrial velocity servo bandwidth of 30 Hz (188 rad/ sec). The transient response is stable with a slight overshoot in velocity as shown in Figure 23. Fig. 18 Velocity loop frequency response with PI compensation—K 2 ¼ 266, T 2 ¼ 0:001666 sec. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 19 Velocity servo transient response of Figure 18. 14.5 POSITION SERVO LOOP COMPENSATION Having compensated the velocity servo, it remains to close the position servo around the velocity servo. Commercial industrial positioning servos do not normally use any form of integral compensation in the position loop. This is referred to as a ‘‘naked’’ position servo loop. However, for type 2 positioning drives, PI compensation would be used in the forward position loop. There are also some I.P. rules for the separation of inner servo loops by their respective bandwidths (see Chapter 9). The ﬁrst I.P. is known as the three to one rule for the separation of a machine resonance from the inner velocity servo. All industrial machines have some dynamic characteristics, which include a multiplicity of machine resonances. It is usually the lowest mechanical resonance that is considered; and the I.P. is that the inner velocity servo bandwidth should be one-third lower than the predominant machine structural resonance. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 20 Velocity loop frequency response with PI compensation—K 2 ¼ 266, T 2 ¼ 0:01666 sec. A second I.P. is that the position servo velocity constant ðK v Þ or position loop gain, should be one-half the velocity servo bandwidth. These I.P.s are guides for separating servo loop bandwidths to maintain some phase margin and overall system stability. Industrial machine servo drives usually require low position-loop gains to minimize the possibility of exciting machine resonances. In general for large industrial machines the position-loop gain ðK v Þ is set about 1 ipm/mil (16.66/sec). The example being studied in this discussion has a machine slide weight of 50,000 lbs, which can be considered a large machine that could have detrimental machine dynamics. There are numerous small machine applications where the position-loop gain can be increased several orders of magnitude. The technique of using a low position-loop gain is referred to as the ‘‘soft servo.’’ A low position-loop gain can be detrimental to such things as servo drive stiffness and accuracy. The soft servo technique also requires a high- performance inner velocity servo loop. This inner velocity servo loop with its high-gain forward loop, overcomes the problem of low stiffness. For example, as the machine servo drive encounters a load disturbance the Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 21 Velocity servo transient response of Figure 20. velocity will instantaneously try to reduce, increasing the velocity servo error. However the high velocity servo forward loop gain will cause the machine axis to drive right through the load disturbance. This action is an inherent part of the drive stiffness. For this discussion it will be assumed that the industrial machine servo drive being considered has a structural mechanical spring/mass resonance inside the position loop. The machine as connected to the velocity servo drive is often referred to as the ‘‘servo plant.’’ The total machine/servo system can be simulated quite accurately to include the various force or torque feedback loops for the total system. For expediency in this discussion, a predominant spring/mass resonance will be added to the output of the velocity servo drive. Thus the total servo system is shown in the block diagram of Figure 24. Position feedback is measured at the machine slide to attain the best position accuracy. As stated previously, the I.P. for the separation of the velocity servo bandwidth and the predominant machine resonance should be three to one; Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
where the velocity servo bandwidth is lower than the resonance. The machine resonance is shown if Figure 24 as or . Since the velocity servo bandwidth of this example is 30 Hz (188 rad/sec), as in Figure 22, the lowest machine resonance should be three times higher or 90 Hz (565 rad/sec). It is further assumed that this large machine slide has roller bearing ways with a coefﬁcient of friction ¼ 0.01 lbf/lb, and a damping factor ðd ¼ 0:1Þ. Additionally, this industrial servo-driven machine slide (50,000 lbs) will have a characteristic velocity constant ðK v Þ of 1 ipm/mil (16.66/sec). This large machine was used for this discussion as a worst-case scenario since the large weight aggravates the reﬂected inertia and machine dynamics problems. Most industrial machines are not of this size. The closed-loop frequency response with a mechanical resonance ðor Þ of 90 Hz (565 rad/sec) is shown in Figure 25. A unity step in position transient response is shown in Figure 26. These are acceptable servo responses for the machine axis being analyzed. Fig. 22 Velocity loop frequency response with PI compensation—K 2 ¼ 266, T 2 ¼ 0:05 sec. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved