ADJUSTING SERVO DRIVE COMPENSATION George W. Younkin, P.E. Life Fello IEEE Industrial Controls Research, Inc. Fond du Lac, Wisconsin 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 knon as servo equalization or servo synthesis. In general, commercial industrial servo drives use proportional, and integral compensation (PI). It is the purpose of this discussion to analyze and describe the procedure for implementing PI servo compensation. 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 ith PI compensation. Since the block diagram of figure 1 is not solvable, block diagram algebra separates the servo loops to an inner and outer servo loop of figure 2. For this discussion a orst case condition for a large industrial servo axis ill be used. The folloing parameters are assumed from this industrial machine servo application: Motor - Kollmorgen motor - M67B 1
Machine slide eight - 5, lbs Ball scre: Length - 7 inches Diameter - 3 inches Lead -.375 inches/revolution Pulley ratio - 3.333 J T Total inertia at the motor.3511 lb-in-sec 2 t e Electrical time constant.2 second 5 t 1 t e K e Motor voltage constant.646 volt-sec/radian K T Motor torque constant 9.9 lb-in/amp K G Amplifier gain 2 volts/volt K ie Current loop feedback constant 3 volts/4a.75 volt/amp R a Motor armature circuit resistance.189 ohm K i Integral current gain 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/e 1 G/1+GH here: G 1/R a (t e S+1) 5.29/[t e S+1] (5.29 14.4 db) GH.646 x 9.9/[.189x.3511[(t e S+1)S] GH 96/S[t e S+1] 96 39dB 1/H J T S/K e K T.3511S/.646 x 9.9 1/H.54 S (.54-25 db) Using the rules of Bode, the resulting closed loop Bode plot for I/e 1 is shon in figure 3. Solving the closed loop mathematically : I e 1 G 1 + GH 1 R ( t S + 1) + K K J S a e e T / T JT S J R S( t S + 1) + K K T a e e T I J T S J T / K e K T S e 1 J T R a t e S 2 + J T R a S+K e K T [(J T R a /K e K T )t e S 2 +(J T R a /K e K T ) S + 1] I (.3511/.646x9.9) S.54 S e 1 t m t e S 2 + t m S +1.1x.2 S 2 +.1 S + 1 here: t m J T R a.3511 x.189.1 sec, m 1/t m 1 ra/sec K e K T.646 x 9.9 t e.2 sec e 1/t e 5 For a general quadratic- S 2 + 2 delta S + 1 Wr 2 r r [ m e ] 1/2 [1 x 5] 1/2 7 I.54S 2 2 e1 S / 7 + (2delta / 7) S + 1 2
1 75 5 Attenuation db 2 log g( j ) 2. log b( j ) 2 log gh( j ) 2 log c( j ) 25 25 5 75 1 125 15.1 1 1 1 1.1 3 1.1 4 g(s) 1/h(s) gh(s) I/e1 Fig 3 Current inner loop Having solved the inner servo loop it is no required to solve the outer current loop. The inner servo loop is shon as part of the current loop in figure 4. In solving the current loop, the forard loop, open loop, and feedback loop must be identified as follos: The forard servo loop- 3
G K i K G x.54 (.2S+1) 735 x 2 x.54 (.2S+1).2 S 2 +.1 S + 1.2S 2 +.1 S + 1 G 794(.2 S+1) 15.88S + 794.2S 2 +.1S + 1.2S 2 +.1S + 1 Where: K G 2 volt/volt K ie 3/4.75 volt/amp K i K G x.54 794 (58 db) K i 794/(2 x.54)735 G 79,4S + 3,97, S 2 + 5 S + 5 The open loop- GH.75 x 79,4S +3,97, S 2 + 5S + 5 GH 5955S + 297,75 S 2 + 5S + 5 1 Magnitude db,phase- Degrees 2 log g( j ) 2 log b( j ) 2 log gh( j ) 2 log c( j ) 18 arg gh j π ( ( )) 5 5 1 1 1 1 1. 1 3 1. 1 4 1. 1 5 g(s) 1/h(s) gh(s) I/ei Phase Fig 5 Current loop response 4
The feedback current scaling is- H 3 volts/4 amps.75 volts/amp 1/H 13.33 22.4 db The Bode plot frequency response is shon in figure 5. The current loop bandidth is 6 radians/second or about 1 Hz, hich is realistic for commercial industrial servo drives. The current loop as shon in figure 5 can no be included in the motor servo loop ith reference to figure 2 and reduces to the motor servo loop block diagram of figure 6. The completed motor servo loop has a forard loop only (as shon in figure 6) here: J T Total inertia at the motor.3511 lb-in-sec 2 K T Motor torque constant 9.9 lb-in/amp G 13.3 x 9.9 375 (51.5 db).3511s ((j/6) +1) S(.166S + 1) G 375 2,25,9.166S 2 + S + S 2 + 6S + v o K T x I 9.9 x 13.1(.2S + 1) e i J T S v i.3511s.331s 2 +.2S + 1 v o 375 (.2S + 1) e i S.331 (S + 5)(S + 5991) v 375 (.2S + 1) e i S.331 x 5 x 5991 ((S/5) + 1)(((s/5991) + 1) v o 375 (.2S + 1) e i S (.2S + 1)(.166S + 1) v o 375 e i S ((j/5991) + 1) 5
22.98 5.53 Vel. (), Phase-degrees 2 log c( j ) 18 π arg( c( j ) ) 34.3 62.54 91.4 119.54 148.5 176.55 1 1 1 1. 1 3 1. 1 4 1. 1 5 Vo/ei Phase Fig 7 Mot.&I loop freq. response The Bode frequency response for the motor and current loop is shon in figure 7. The motor and current closed loop frequency response, indicate that the response is an integration hich includes the 6 bandidth of the current loop. This is a realistic bandidth for commercial industrial servo drives. Usually this response is enclosed in a velocity loop and further enclosed in a position loop. Hoever, there are some applications here the motor and current loop are enclosed in a position loop. Such an arrangement is shon in figure 7a. 6
The forard loop transfer function is Kv G 2 S (( j / 6) + 1) Where: K v K 2 x 375 For most large industrial machine servo drives a position loop k v 1 ipm/mil or 16.6 can be assumed. The frequency response for the position loop is shon in figure 7b. This response is obviously unstable ith a minus 2 slope at the zero gain point. It is also obvious that there are to 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 1g in figure 7b, the response can be modified to that of a type 2 control hich could have performance advantages. The absence of the minor velocity servo loop bandidth could make it possible to increase the position loop velocity constant (position loop gain) for an increase in the position loop response. Magnitude db 2 log g( j ) 2 log g2( j ) 1 75 5 25 25 5 75 1 125 15.1 1 1 1 1.1 3 1.1 4 Fig. 7b Position loop response For the purposes of this discussion it ill be assumed that the motor and current loop are enclosed in a velocity servo loop. Such an arrangement is shon in figure 8. 7
The servo compensation and amplifier gain are part of the block identified as K 2. Most industrial servo drives use proportional plus integral (PI) compensation. The amplifier and PI compensation can be represented as in figure 9 1. Figure 9 I V 2 K p + K s i K s + K p s i K p Ki s + 1 Ki s [ t s ] K 1 2 2 + s t 2 K K p i K i 2 (Corner frequency) K p The adjustment of the PI compensation is suggested as- 1.For the uncompensated servo Bode plot, set the amplifier gain to a value just belo the level of instability. 2. Note the forard loop frequency ( g ) at 135 degree phase shift (45 degrees phase margin) of figure 12. 3. From the Bode plot for PI compensation of figure 1, the corner frequency 2 K i /K p should be approximately g /1 or smaller as a figure of merit 1. 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 loered about one decade or more from the 135 degree phase shift point ( g ) of the open loop Bode plot for the servo drive being compensated. For the servo drive being considered, g occurs at 6. 8
6 4 2 Magnitude db, Phase-deg 2 log c( j ) 18 π arg( c( j ) ) 2 4 6 8 1 1 1 1 1. 1 3 1. 1 4 1. 1 5 Magnitude Phase Fig 1 PI Compensation Applying the PI compensation of figure 9, to the velocity servo drive is shon if figure 11. In general the accepted rule for setting the servo compensation begins by removing the integral and/or differential compensation. The proportional gain is then adjusted to a level here the velocity servo response is just stable. The proportional gain is then reduced slightly further for a margin of safety. For a gain K 2 1 of the uncompensated servo, the Bode plot is shon in figure 12. It should be noted that the motor and current loop have a bandidth of 6 as shon in figure 7. This is a normal response for industrial servo drive current loops. The transient response for this servo is shon in figure 9
13 as a damped oscillatory response. If the gain K 2 is reduced to a value of 266 for a forard loop gain of 1,, the Bode plot is shon in figure 14 ith a stable transient response shon in figure 15. Magnitude db,phase Degrees 2 log c( j ) 2 log g( j ) 18 arg gh j π ( ( )) 2 16 12 8 4 4 8 12 16 2.1 1 1 1 1.1 3 1.1 4 Vel/ei g(s) Phase Fig 12 Velocity loop response 5 45 4 35 c( t) 3 25 2 15 1 5 8.33333. 1 4.167.25.333.417.5 t Time (sec) c(t) Fig 13 Transient response 1
Magnitude db,phase degrees 2 log g( j ) 2 log b ( j ) 2 log c( j ) 18 π arg ( c( j )) 2 16 12 8 4 4 8 12 16 2.1 1 1 1 1.1 3 1.1 4 Fig 14 Velocity loop response ct () 4 36 32 28 24 2 16 12 8 4.1.2.3.4.5.6.7.8.9.1 t Time-sec Fig 15 Transient response 11
At this point the PI compensation is added as shon in figure 11. The index of performance for the PI compensation is that the corner frequency 2 K i /K p, should be a decade or more loer than the 135 degree phase shift (45 degree phase margin) frequency ( g ) of the forard loop Bode plot (figure 14) for the industrial servo drive being considered 1. With reference to figure 14 of the stable uncompensated servo, the 135 degree phase shift (45 degree phase margin) occurs at 6 frequency hich is also the bandidth of the motor/current loop. Using the index of performance of setting the PI compensation corner frequency at one decade or more loer in frequency that the 135 degree phase shift frequency point, the corner frequency should be set at 6 or loer. With the corner frequency of the PI compensation set at 6 (.1666 sec) the compensated servo is shon in the Bode plot of figure 16. The transient response is shon in figure 17 as a highly oscillatory velocity servo drive. Obviously this servo drive needs to have the PI compensation corner frequency much loer than one decade (6 ) index of performance. For a to decade, 6 (.166 sec) loer setting for the PI corner frequency the Bode response is shon in figure 18 ith a transient response shon in figure 19 having a single overshoot in the output of the velocity servo drive. By loering the PI compensation corner fequency ( 2 K p servo drive results. The forard loop and open loop are defined as follos: H.286 v/ 1/H 34.9 (3.8 db) Gain @ 1 1 db 1, K 2 1,/375 266 G K 2 x 375 ((j/2)+1) 1, ((j/2)+1) (1 db) S 2 ((j/6)+1) S 2 ((j/6)+1) G 1, (.5S+1) 5S + 1, S 2 (.166S +1) S 2 (.166S +1) G 3,12,481S + 62,49,638 S 3 + 624S 2 + S+ K i ) to 2 (.5 sec), a stable velocity GH.286 x G 286 ((j/2)+1) S 2 ((j/6)+1) (69 db) The Bode plot for the velocity loop ith PI compensation is shon in figure 2, having a typical industrial velocity servo bandidth of 3 Hz (188 ). The tansient response is stable ith a slight overshoot in velocity as shon in figure 21. 12
Magnitude db 2 log c ( j ) 18 arg ( c ( j ) ) π 4 8 12 16 2 1 1 1 1. 1 3 1. 1 4 g(s) 1/h(s) gh(s) Vel/ei Phase Fig 16 Velocity loop response 8 72 64 56 vel.- ct () 48 4 32 24 16 8.2.4.6.8 1 1.2 1.4 1.6 1.8 2 t Time - sec Vel. Fig 17 Vel. servo trans. resp. 13
15 115 8 Magnitude-dB,Phase-degrees 2 log g( j ) 2 log b( j ) 2 log gh( j ) 2 log c( j ) 18 arg gh j π ( ( )) 45 1 25 6 95 13 165 2 1 1 1 1.1 3 1.1 4 Fig 18 Vel. loop response 14
5 45 4 35 c( t) 3 25 2 15 1 5.3.6.9.12.15.18.21.24.27.3 t Time-sec c(t) Fig 19 Vel. servo trans. resp. Magnitude-dB,Phase-degrees 2 log g( j ) 2 log b( j ) 2 log gh( j ) 2 log c( j ) 18 π arg( c( j )) 2 16 12 8 4 4 8 12 16 2 1 1 1 1.1 3 1.1 4 g(s) 1/H(s) gh(s) Vel./Vr Phase Fig. 2 Velocity servo response 15
Vel.- c( t) 4 36 32 28 24 2 16 12 8 4.2.4.6.8.1.12.14.16.18.2 t Time-sec Vel. Fig. 21 Vel. servo trans. resp. 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. Hoever, for type 2 positioning drives, PI compensation ould be used in the forard position loop. There are also some indexes of performance rules for the separation of inner servo loops by their respective bandidths 3. The first index of performance is knon as the 3 to 1 rule for the separation of a machine resonance from the inner velocity servo. All industrial machines have some dynamic characteristics, hich include a multiplicity of machine resonances. It is usually the loest mechanical resonance that is considered; and the index of performance is that the inner velocity servo bandidth should be 1/3 loer than the predominant machine structural resonance. A second index of performance is that the position servo velocity constant (K v ) or position loop gain, should be ½ the velocity servo bandidth 3. These indexes of performance are guides for separating servo loop bandidths to maintain some phase margin and overall system stability. Industrial machine servo drives usually require lo position loop gains to minimize the possibility of exciting machine resonances. In general for large industrial machines the position loop gain (K v ) is set about 1 ipm/mil (16.66/sec). The example being studied in this discussion has a machine slide eight of 5, lbs., hich can be considered a large machine that could have detrimental machine dynamics. There are numerous small machine applications here the position loop gain can be increased several orders of magnitude. The technique of using a lo position loop gain is referred to as the soft servo. A lo 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 ith its high-gain forard loop, overcomes the problem of lo stiffness. For example, as the 16
machine servo drive encounters a load disturbance the velocity ill instantaneously try to reduce, increasing the velocity servo error. Hoever the high velocity servo forard loop gain ill cause the machine axis to drive right through the load disturbance. This action is an inherent part of the drive stiffness 3. For this discussion it ill 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 4. For expediency in this discussion, a predominant spring/mass resonance ill be added to the output of the velocity servo drive. Thus the total servo system is shon in the block diagram of figure 22. Position feedback is measured at the machine slide to attain the best position accuracy. As stated previously, the index of performance for the separation of the velocity servo bandidth and the predominant machine resonance should be 3 to 1; here the velocity servo bandidth is loer than the resonance. The machine resonance is shon if figure 22 as r. Since the velocity servo bandidth of this example is 3 Hz (188 ) as in figure 2, the loest machine resonance should be three times higher or 9 Hz (565 ). It is further assumed that this large machine slide has roller bearing ays ith a coefficient of friction.1 lbf/lb, and a damping factor (δ.1). Additionally, this industrial servo driven machine slide (5, lbs) ill have a characteristic velocity constant (K v ) of 1 ipm/mil (16.66/sec). This large machine as used for this discussion as a orst case scenario since the large eight aggravates the reflected inertia and machine dynamics problems. Most industrial machines are not of this size. The closed loop frequency response ith a mechanical resonance ( r ) of 9 Hz (565 ) is shon in figure 23. A unity step in position transient response is shon in figure 24. These are acceptable servo responses for the machine axis being analyzed. In reality a machine axis eighing 25 tons ill have structural resonances much loer than 9Hz. Machine axes of this magnitude in size ill characteristically have structural resonances of about 1Hz to 2Hz. Using the same position servo block diagram of figure 22 ith the same position loop gain of 1 ipm/mil (16.66/sec), and a machine resonance of 1 Hz; the servo frequency response is shon in figure 25 ith the transient response shon in figure 26. The position servo frequency response shos a 8 db resonant (62.8 ) peak over zero db, hich ill certainly be unstable as observed in the transient response. Repeating the position servo analysis ith a 2 Hz resonance in the machine structure, results in an oscillatory response as shon in figure 27 and 28. One of the most significant problems ith industrial machines is in the area of machine dynamics. Servomotors and their associated amplifiers have very long mean time before failure characteristics. It is 17
quite common to have an industrial velocity servo drive ith 2 Hz to 3 Hz bandidths mounted on a machine axis having structural dynamics (resonances) near or much loer than the internal velocity servo bandidth. There must be some control concept to compensate for these situations. There are a number of control techniques that can be applied to compensate for machine structural resonances that are both lo in frequency and inside the position servo loop. The first control technique is to loer the position loop gain (velocity constant). Depending on ho lo the machine resonance is, the position loop gain may have to be loered to about.5 ipm/mil (8.33/sec.). This solution has been used in numerous industrial positioning servo drives. Hoever, such a solution also degrades servo performance. For very large machines this may not be acceptable. The index of performance that the position loop gain (velocity constant) should be loer than the velocity servo bandidth by a factor of to, ill be compromised in these circumstances. A very useful control technique to compensate for a machine resonance is the use of ien bridge notch filters 3. These notch filters are most effective hen placed in cascade ith the position forard servo loop, such as at the input to the velocity servo drive. These notch filters should have a tunable range from approximately 5 Hz to a couple of decades higher in frequency. The notch filters are effective to compensate for fixed machine structural resonances. If the resonance varies due to such things as load changes, the notch filter ill not be effective. There are commercial control suppliers that incorporate digital versions of a notch filter in the control; ith a future goal to sense a resonant frequency and tune the notch filter to compensate for it. This control technique can be described as an adaptive process. Another technique that has been very successful ith industrial machines having lo machine resonances, is knon as frequency selective feedback 5,6,7. This control technique is the subject of another discussion. In abbreviated form it requires that the position feedback be located at the servo motor eliminating the mechanical resonances from the position servo loop, resulting in a stable servo drive but ith significant position errors. These position errors are compensated for by measuring the machine slide position through a lo pass filter; taking the position difference beteen the servo motor position and the machine slide position; and making a correction to the position loop; hich is primarily closed at the servo motor. Conclusions Commercial industrial electric brushless DC servo drives use an inner current/torque loop to provide adequate servo stiffness. The servo loop bandidth for the current loop is usually about 1 Hz. In analysis this servo loop is often neglected because of its ide bandidth. Including the current loop as in figure 2, results in a motor and current loop response (figure 7) that is an integration ith the current loop response. A classical servo technique is to enclose the motor/current loop in a velocity servo loop. Since most commercial industrial brushless DC servo drives have position feedback from the motor armatrure for the purpose of current commutation; this signal is differentiated to produce a synthetic velocity loop. Additionally, commercial industrial servo drives use proportional plus integral (PI) servo compensation to stabalize the synthetic velocity loop. The PI type of compensation has a corner frequency that must be a decade or more loer in frequency than the 45 degree phase shift frequency of the uncompensated open loop Bode plot. This requirement is needed to avoid excessive phase lag from the PI compensation here the open loop 45-degree phase shift frequency occurs. Commercial industrial electric servo drives have a very long mean time beteen failure, and 18
therefore are very reliable. The servo plant (the machine that the servo drive is connected to) has consistent problems ith structural dynamics. When these mechanical resonances have lo frequencies that occur ithin the servo bandidths, unstable servo drives can result. The solution to such a problem can be a degradation in performance by loering the position loop gain; using notch filters to tune out the resonance; or using a control technique referred to as frequency selective feedback 5,6,7. References 1. Kuo, B. C., AUTOMATIC CONTROL SYSTEMS, Prentice Hall, 7th edition, 1993 2. Younkin, G.W., Brushless DC Motor and Current Servo Loop Analysis Using PI Compensation. 3. Younkin, G.W., INDUSTRIAL CONTROL SYSTEMS, Marcel Dekker, Inc.,1996, ISBN -8247-9686-1 4. Younkin, G.W., Modeling Machine Tool Feed Servo Drives Using Simulation Techniques to Predict Performance, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL 27, NO. 2, 1991. 5. Shinners,S.M., Minimizing Servo Load Resonance Error, CONTROL ENGINEERING, January, 1962, P.51. 6. Jones, G.H., U.S. Patent 3,358,21, Apparatus for Compensating Machine Feed Drive Servomechanisms, December 12, 1967. 7. Young,G. and Bollinger, J.G., A Research Report on the Principles and Applications of Frequency Selective Feedback, University of Wisconsin, Department of Mechanical Engineering, Engineering Experimental Station, March 1969. 8. Younkin, G.W., INDUSTRIAL CONTROL SYSTEMS, 2 nd Ed, Marcel Dekker, Inc.,23, ISBN -8347-836-9 Figure 23 and 24 ω 9 Hz δ.1 r 16.66 g( s ) s(.53s + 1)(.31s h ( s) 1 Figure 25 and 26 ω 1 Hz δ.1 r g( s ) h ( s) 16.66 s(.53s + 1)(.253s 1 Figure 27 and 28 ω 2 Hz δ.1 r 2 2 +.354s + 1) +.3128s + 1) g( s ) 16.66 s(.53s + 1)(.64s 2 +.16s + 1) 19
h ( s) 1 Magnitude-dB, Phase-degrees 2 log g ( j ) 2 log h ( j ) 2 log c ( j ) 18 arg ( c ( j ) ) π 179.82 143.84 17.87 71.89 35.92.55224 36.3 72 17.98 143.95 179.93 1 1 1 1. 1 3 1. 1 4 g(s) h(s) Pos. Phase Fig 23 Position loop response-wr9hz 2
Position-in c( t) 1.9.8.7.6.5.4.3.2.1.3.6.9.12.15.18.21.24.27.3 t Time-sec Pos. Fig 24 Pos. transient response 2 16 12 Magnitude-dB 2 log g( j ) 2. log h( j ) 2 log c( j ) 18 π arg( g( j ) ) 8 4 4 8 12 16 2 1 1 1 1. 1 3 1. 1 4 g(s) h(s) Pos. Phase Fig 25 Pos. loop response-wr1hz 21
Pos-in c( t) 2 1.8 1.6 1.4 1.2 1.8.6.4.2.6.12.18.24.3.36.42.48.54.6 t Time-sec c(t) Fig 26 Pos. transient response Magnitude-dB,Phase-Degrees 2 log g( j ) 2 log h ( j ) 2 log c ( j ) 18 arg ( c ( j ) ) π 2 16 12 8 4 4 8 12 16 2.1 1 1 1 1. 1 3 1. 1 4 Fig. 27 Pos. loop response-wr2hz 22
Position (inches) ct () 1.5 1.35 1.2 1.5.9.75.6.45.3.15.3.6.9.12.15.18.21.24.27.3 t Time-sec Fig 28 Transient resp.-wr2hz 23