Bode Plot based Auto-Tuning Enhanced Solution for High Performance Servo Drives

Transcription

1 Bode lot based Auto-Tuning Enhanced Solution or High erormance Servo Drives. O. Krah Danaher otion GmbH Wachholder Str Düsseldor Germany Tel Fax Abstract This paper describes a new sel-tuning method using an automatic bode plot based ilter-coniguration algorithm. ost existing auto-tuning algorithms are based on simple load inertia estimation. They work ine, i the user more or less does not need them, but they struggle in more complex systems. The described approach is also targeting complex and diicult problems, which usually appear in low damped direct drive machines with signiicant motor inertia to load inertia mismatches [1]. Introduction High perormance servo drives are still a ast growing market segment. The brush-less technology oers signiicant advantages in terms o reliability and motor size. The higher drive complexity is covered with additional logic and new power electronic components. Due to the innovation cycles o the semiconductor suppliers the size and the cost o the more and more complex drives did not increase. Each new drive generation oers more perormance, more lexibility and higher integration density [,3]. The servo drive component cost will continually decrease in the near uture. On the other side the mechanical design is more and more cost and weight optimized, which results in less sti constructions. Thereore the installation and set-up time (= cost is a steady growing issue. A reliable sel-tuning algorithm can help to decrease the set-up time. Control structure A very common control architecture in the motion control is the cascaded control structure. The most inner loop is also the astest loop, the current control. The next outer loop is the velocity control. The slowest and most outer loop is the position control. Due to eed orward and other extensions like observers the perormance o the system can be signiicant improved [4]. At least the eed orward does not change control loop stability. Thereore the cascaded control structure is the base or the developed sel-tuning algorithms. The current control loop depends mainly on the drive internal delay times and on the motor winding inductance and resistance. It is very common that motor and drive are selected rom one supplier. This vendor should also provide the current control parameter set [5]. Due to the act that the position is just the integration o the speed the position loop gain depends mainly on the achieved velocity loop bandwidth. Tuning the velocity loop is the real task. Here we ind the motor inertia and the load inertia connected via compliant mechanics. The system contains also riction and backlash. This can result in complex resonance issues. Friction or example can help to damp the behavior. In systems with low mechanical damping the drive has to provide electronic damping by using ilters in the current command path. Compliantly coupled load The load and the motor are two independent inertias connected by non-rigid components. The equivalent spring constant o the entire transmission is c, illustrated in Figure 1. It describes the torque produced by a position dierence between motor and load. The viscous damping term d describes the torque, which is produced by the velocity dierence o motor and load. An ideal servo system would ollow the position/velocity command without delay and ollowing error. This would require an extreme high loop gain. In real systems the gain is limited. A too high gain produces signal overshoot or excessive ringing. A very easy and common way to set-up and t c, d Figure 1: Compliantly coupled motor and load 38 CI EUROE 4 ROCEEDINGS

2 check the gain is reviewing a velocity loop step response scope plot, igure. This is perect to check the result, but it is more or less tuning by trial and error. I more then one parameter needs to be determined only a real servo expert has a chance to get an optimal result. There is no direct way to see the actual valid stability margin to system rigging. Reviewing the velocity command to measured velocity Bode plot gives easier to use inormation. This is called closed loop Bode plot and is shown in igure 3. We can read directly the peaking (output amplitude is higher than input amplitude and the bandwidth (9 or 3dB o the velocity loop. These are key perormance indices o the velocity control loop. The closed loop Bode plot allows an easier interpretation o the inormation than the pure step response, but a straightorward gain setting is still not possible. The tuning process is still more or less trial and error. Using the velocity error to measured velocity Bode plot oers more inormation, igure 4. This is called open loop bode plot. Using this graph the gain margin and the phase margin are directly visible. Knowing the selected margins we can directly calculate how much the gain should be changed (x db to get the desired tuning. This is very practical with simple single inertia systems. ost real systems are not that simple. Oten we ind compliant coupled two or more mass systems, sometimes with signiicant riction and/or some backslash. Filters in motion control systems There are two main reasons why current set-point ilters are used in motion control systems: Reduce noise rom a low resolution eedback system Increase gain and stiness o compliant load ow-resolution encoder and resolver-based systems with high velocity loop gains generates audible noise in the motor due to the noise on top o the current command. Simple low pass ilters can reduce this noise. A more advanced solution is a uenberger observer to estimate a velocity signal with less noise [4]. To suppress oscillations o compliant coupled load ilters are used to increase gain in the lower requency area or more stiness without decreasing the gain margin at the irst phase crossover. To control a motor without load an additional ilter is usually not necessary. The behavior o a good servomotor is like a simple integration (torque (~current -> speed. Due to the compliant coupling with an additional inertia we can get a system with torsional oscillations. The ideal ilter would be the inverse transer unction o the compliant coupling [6]. This is easy to see in the open loop bode plot. The requency related gain reduction is requested, some phase lag comes with it. It is rather oten that in a real system placing a ilter in ront o the current controller can help to increase the gain margin. Sizing that ilter is not an easy job. In general there are three common ways to get the current set-point ilter parameters: Figure : Velocity loop: step response scope plot AK 4 with ServoStar 3 no load Figure 3: Velocity loop: closed loop bode plot AK 4 with ServoStar 3 no load Figure 4: Velocity loop: open loop bode plot AK 4 with ServoStar 3 no load (controller integral part is switched o CI EUROE 4 ROCEEDINGS 383

3 1. Set the ilter parameters by experience o a servo expert. Set the ilter parameters by more or less trial and error 3. Use the measured or calculated inverse transer unction to set the ilter ost applications are not just compliant two mass systems. We ind oten compliant systems with distributed mass or multiple mass conigurations. Here is the calculation or the estimation o the inverse transer unction with an acceptable order very diicult or not possible. For these applications setting the ilters is the real task. To get an understanding o the behavior o these systems we can look at an ideal two mass system. It is always assumed that the eedback system is mounted on the motor side as usual in the industry. Detailed analyses o a two mass system in the requency domain In the ollowing chapter the motor inertia is always constant. The load inertia and the spring constant are the characterizing variables. In the industry it is well known that a good approach is matching motor inertia with load inertia (1:1. The coupling should be as sti as possible. Figure 5 shows a Bode plot o a low damped non inertia-matched system with ~ 9. Three dierent sections characterize the plot: In the requency range well below the antiresonant requency the plant acts like a scaled (low requency inertia with =. The + requency o the anti-resonance (zero is deined by: 1 c AR = Hz (1 π In the requency range well above the low damped resonant requency the plat acts like a scaled inertia with only the value o the motor (high requency inertia =. The requency o the resonance (pole is deined by: 1 c R = ( π + The inertia ratio, the spring constant and the system damping are deining the plot in the transition area o the resonant requencies. The transer unction can be ormed in a two-part notation [6]: ( V s 1 ( s + ds + c = (3 T S s + + s + ds + c The term on the let is just the motor behavior and the term on the right is the eect o the compliant Figure 5: Bode plot o motor / load plant with coupled load [6] black: motor only red: blue: motor with ideal coupled load motor with compliantly coupled load (low damped coupled load. An ideal ilter would be similar or equal to the inverse unction o the right part. The high requency gain ( s = o the right part is one and the low requency gain (s = is G( s = = (4 + Figure 6 shows the low requency gain drop as a unction o the load to inertia ratio. This gain drop results also in a bandwidth drop. Figure 7 shows the resonance requencies as a unction o the load to motor inertia ratio. The anti-resonance is strictly moving with the square root o the inertia ratio. The resonance requency is only slightly moving. + R = (5 AR In a servo system with matching inertia ( = the resonance requency is higher than the antiresonance requency. I the load inertia is signiicant higher than the motor inertia, the anti resonance requency will drop down. To achieve an appropriate bandwidth a stier coupling c is required (or example a direct drive solution. By comparing the anti-resonance (nominator rom the second part o equation 3 with the standard oscillation equation: 384 CI EUROE 4 ROCEEDINGS

4 db ( F ω << ω / 1 1 Figure 6: ow requency gain drop as unction o the load to inertia ratio 1 1 / anti-resonance 1 resonance ϕ +1 F - -3 db Hz 65 Hz - 17 db - 34 db k Hz 1 k Figure 9: Bode plot o a irst order and second order lag ilter z = 38 Hz, p = 65 Hz and D AR =D R =.5 ω + Dω s + s = (6 we get or the anti-resonance damping: D = d ( c (7 and the resonance damping: = D d + c (8 Interesting is the relation between these equations: D + D = ( / 3 1 Figure 7: resonance / anti-resonance requency as unction o the load to motor inertia ratio The equation shows that in an ideal two mass system the resonance requency (pole is always better damped than the anti-resonance (zero. Figure 8 shows several idealized Bode plots (damping not shown with dierent load inertia conigurations. arameter or the plots is the inertia ratio. Wrapped up the key behaviors o the compliantly F db -3-4 = Figure 8: several idealized bode plots with dierent load inertia conigurations damping is not shown see igure 5 coupled two mass system are: ow gain at low requencies < AR High gain at high requencies > R hase lead between AR and R (igure 5 Signiicantly inluenced by the inertia ratio The desired ilter to compensate a two inertia compliantly coupled system should own high gain at low requencies and low gain at high requencies. The phase lag between the corner requencies should be less than the phase lead o the system in this area. I the gain drop at low requencies is low (< 15 db and the system is well damped a two parameter, ω irst order lag ilter is a good approach: s + ω ω F1 ( s = ; ω = π (1 s + ω ω In case o higher inertia mismatches which results in a higher gain drop or a low damped system a our parameter,ω, D, D second order lag ilter (bi quad could be better itting: s + Dω s + ω ω p F ( s = (11 s + Dω s + ω ω Figure 9 shows two Bode plots o these ilters. Setting these our parameters without experience or guideline just by trial and error is nearly impossible. An optimization criterion is to maximize the closed loop bandwidth with a stable well-damped behavior. The Nyquist stability criteria can help to determine that parameter set. Bode plot based velocity loop autotuning The task is to determine the two parameters o the velocity loop Icontroller and - i necessary - to select and parameterize the irst or second order lag ilter to compensate the CI EUROE 4 ROCEEDINGS 385

5 the velocity observer is well adapted [4]. Figure 11 shows such a measured open loop Bode plot (green o an industrial linear motion system, igure 1. The red marked plot shows the gain increasing eect o the compliant coupled load. Figure 1: Belt driven linear motion system produced by ontech described compliant coupled load behavior. This is executed in three major steps: 1. easure system transer unction. Select and parameterize anti resonance lag ilter 3. Set I-gains according to Nyquist stability criteria easure system transer unction In the shown plots we can see that the system has mainly the behavior o a I-controlled single integrator with delay. Due to the integral part o the velocity loop controller the phase shit in the low requency area is close to 18. The behavior in the high requency area is signiicantly inluenced by the current loop perormance and the velocity estimation method. By using an observer to eliminate the digital dierentiation delay this observer has to be adapted to the system. The digital drive ServoStar 3 can run a simple irst order - inertia estimation at a deined requency (usually 3-5Hz. This requency should be set above o the system resonance requency to adapt the observer to the high requency (motor only inertia. According to this constrains the system transer unction can be best measured when: all internal ilters are switched o, the current loop is well tuned [5], the velocity I-integral part is o or very low and Select and parameterize anti resonance lag ilter Figure 1 shows more detailed the gain increase eect o the compliant coupled load (red. The blue plot shows the gain approximation o a nd order lead ilter. The parameters ( and = 74 Hz, = 146 Hz dc-gain = -1.5dB are itted by using a simple least square adoption to the measured plot. The itting works considerably good as shown in the Bode plot. The highlight is that here a compliant coupled threeinertia system (belt gear box + linear motion belt is compensated quite well. The requested current set-point ilter is the inverse transer unction o the approximated nd order ilter ( = 74 Hz, = 146 Hz. The dc-gain can be ignored and the irst approximation o the damping is set to: D D = 1 = The requency with the highest phase lag is here: = = Hz lag 15 This is itting perectly with the phase lead (at 15 Hz o the measured open loop bode plot, igure 13. According to the Nichols diagram here is a minimum gain requested. Due to digital delays and system time constants the phase shit crosses 18 in an area o several 1 Hz. Here is a maximum gain requested. The requency dependent minimum or maximum value is deined by the corresponding open loop phase shit. Nyquist stability criteria and the Bode plot The easiest interpretation o the Nyquist stability criteria is to check o the gain margin (at 18 phase and the phase margin (at db gain in the open loop Bode plot, Figure 13. But this is a veriication at only two points. What is beside these two points? We can see the detailed behavior much better in the Nichols diagram, igure 14. Here we see the phase-gain relation or each requency. db Hz Figure 11: Open loop Bode plot (green and the gain increase eect o the compliant coupled load (red ontech linear motion system 386 CI EUROE 4 ROCEEDINGS Figure 1: easured gain increase eect o the compliant coupled load (red nd order ilter approximation (blue ontech linear motion system

6 F ϕ db db Gain Hz 1 k 5 phase margin 1 db gain margin 18 phase Figure 13: Open loop Bode plot with 5 phase margin and 1 db gain margin Accepting 1dB peaking (red oval we can see that the gain is limited i the phase shit is between 115 to 45. For low requencies due to the phase shit - a minimum gain is requested, or high requencies the maximum is shown. The gain margin would be 5 db; phase margin would be 65. Set I-gains according to Nyquist stability criteria By switching the integral part o we get the doted line in igure 14. Now the gain can be increased until the plot touches the lower side o choused peaking oval (the plot is moving up. In the next step the integral part can be increased until the plot touches the upper side o the oval (the plot is moving let. Figure 15 shows the Bode plot o the tuned velocity loop. Summary Instead o tuning a servo drive system by highly educated servo experts the Bode plot based auto-tune can provide reliable results in a short time rame. The result is documented with an open and closed loop Bode plot that shows key perormance indices like bandwidth and stability margin. The key advantages o using a Bode plot based autotuning are: Ease o use Very good results in a short time rame ess set up cost This results in advantages or the designed product: Higher reliability and lower ault risk Higher productivity Fast development time Reerences [1] V. Wesselak,. Köhler and G. Schäer, Robust Speed Control Based on the Identiication o echanical arameters, CI, Germany 3. []. O. Krah and K. Neumayer, otorsteuerung kompakt und lexibel, Elektronik Het 3/4, Weka Verlag. F db db 6 db 36 1 db velocity observer & digital delay.5 db 1 db I-art ϕ Figure 14: Nichols plot - always open loop here with.5 db peaking doted line: without integral part Figure 15: Open loop (green and closed loop (red bode plot o the ontech linear motion system with nd order lag ilter and switched on integral [3]. O. Krah, S. Geiger and G. askowski, Free rogrammable Signal rocessing inside a High erormance Servo Ampliier, CI, Germany, [4]. O. Krah, Sotware Resolver-to-Digital Converter or High erormance Servo Drives, CI, Germany [5]. O. Krah,. Holtz, High erormance Current Regulation or ow Inductance Servo otors, IEEE Industry Appl. Soc. Annual eeting, St ouis, Oct [6] G. Ellis, Cures or echanical Resonance in Industrial Servo Systems, CI Germany 1. CI EUROE 4 ROCEEDINGS 387