Bode Plot based Auto-Tuning –

Enhanced Solution for High Performance Servo Drives

J. O. Krah

Danaher Motion GmbH

Wachholder Str. 40-42 40489 Düsseldorf Germany

Email:

j.krah@danaher-motion.de

Tel. +49 203 9979 133 Fax. +49 203 9979 200

Abstract

This paper describes a new self-tuning method using

an automatic bode plot based filter-configuration

algorithm. Most existing auto-tuning algorithms are

based on simple load inertia estimation. They work

fine, if the user more or less does not need them, but

they struggle in more complex systems. The described

approach is also targeting complex and difficult

problems, which usually appear in low damped direct

drive machines with significant motor inertia to load

inertia mismatches [1].

Introduction

High performance servo drives are still a fast growing

market segment. The brush-less technology offers

significant advantages in terms of reliability and

motor size. The higher drive complexity is covered

with additional logic and new power electronic

components. Due to the innovation cycles of the

semiconductor suppliers the size and the cost of the

more and more complex drives did not increase. Each

new drive generation offers more performance, more

flexibility and higher integration density [2,3].

The servo drive component cost will continually

decrease in the near future. On the other side the

mechanical design is more and more cost and weight

optimized, which results in less stiff constructions.

Therefore the installation and set-up time (= cost) is a

steady growing issue. A reliable self-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 fastest loop, the current control.

The next outer loop is the velocity control. The

slowest and most outer loop is the position control.

Due to feed forward and other extensions like

observers the performance of the system can be

significant improved [4]. At least the feed forward

does not change control loop stability. Therefore the

cascaded control structure is the base for the

developed self-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 from one supplier. This

vendor should also provide the current control

parameter set [5].

Due to the fact that the position is just the integration

of the speed the position loop gain depends mainly on

the achieved velocity loop bandwidth.

Tuning the velocity loop is the real task. Here we find

the motor inertia and the load inertia connected via

compliant mechanics. The system contains also

friction and backlash. This can result in complex

resonance issues. Friction for example can help to

damp the behavior. In systems with low mechanical

damping the drive has to provide electronic damping

by using filters 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 of the entire transmission is c,

illustrated in Figure 1. It describes the torque

produced by a position difference between motor and

load. The viscous damping term d describes the

torque, which is produced by the velocity difference

of motor and load.

An ideal servo system would follow the

position/velocity command without delay and

following 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

M

t

dc,

M

J

L

J

M

t

dc,

M

J

L

J

Figure 1: Compliantly coupled motor and load

382 • PCIM EUROPE 2004 • PROCEEDINGS

check the gain is reviewing a velocity loop step

response scope plot, figure 2. This is perfect to check

the result, but it is more or less tuning by trial and

error. If 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 information.

This is called closed loop Bode plot and is shown in

figure 3. We can read directly the peaking (output

amplitude is higher than input amplitude) and the

bandwidth (90° or –3dB) of the velocity loop. These

are key performance indices of the velocity control

loop. The closed loop Bode plot allows an easier

interpretation of the information than the pure step

response, but a straightforward 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 offers more information, figure 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. Most real systems are not that

simple. Often we find compliant coupled two or more

mass systems, sometimes with significant friction

and/or some backslash.

Filters in motion control systems

There are two main reasons why current set-point

filters are used in motion control systems:

• Reduce noise from a low resolution feedback

system

• Increase gain and stiffness of compliant load

Low-resolution encoder and resolver-based systems

with high velocity loop gains generates audible noise

in the motor due to the noise on top of the current

command. Simple low pass filters can reduce this

noise. A more advanced solution is a Luenberger

observer to estimate a velocity signal with less noise

[4].

To suppress oscillations of compliant coupled load

filters are used to increase gain in the lower

frequency area for more stiffness without decreasing

the gain margin at the first phase crossover. To

control a motor without load an additional filter is

usually not necessary. The behavior of 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 filter would be the

inverse transfer function of the compliant coupling

[6].

This is easy to see in the open loop bode plot. The

frequency related gain reduction is requested, some

phase lag comes with it. It is rather often that in a real

system placing a filter in front of the current

controller can help to increase the gain margin. Sizing

that filter is not an easy job. In general there are three

common ways to get the current set-point filter

parameters:

Figure 2: Velocity loop: step response scope plot

AKM 42 with ServoStar 300

–

no load

Figure 3: Velocity loop: closed loop bode plot

AKM 42 with ServoStar 300

–

no load

Figure 4: Velocity loop: open loop bode plot

AKM 42 with ServoStar 300 – no load

(controller integral part is switched off)

PCIM EUROPE 2004 • PROCEEDINGS • 383

1. Set the filter parameters by experience of a servo

expert

Figure 5: Bode plot of motor / load plant with

coupled load [6]

black: motor only

red: motor with ideal coupled load

blue: motor with compliantly coupled

load (low damped)

2. Set the filter parameters by more or less trial and

error

3. Use the measured or calculated inverse transfer

function to set the filter

Most applications are not just compliant two mass

systems. We find often compliant systems with

distributed mass or multiple mass configurations.

Here is the calculation or the estimation of the inverse

transfer function with an acceptable order very

difficult or not possible. For these applications setting

the filters is the real task. To get an understanding of

the behavior of these systems we can look at an ideal

two mass system. It is always assumed that the

feedback system is mounted on the motor side as

usual in the industry.

Detailed analyses of a two mass system in the

frequency domain

In the following 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 stiff

as possible.

Figure 5 shows a Bode plot of a low damped non

inertia-matched system with J

L

~ 9J

M

. Three different

sections characterize the plot:

• In the frequency range well below the anti-

resonant frequency the plant acts like a scaled

(low frequency) inertia with

J

. The

frequency of the anti-resonance (zero) is defined

by:

ML

JJ +=

Hz

2

1

L

AR

J

c

f

π

=

(1)

• In the frequency range well above the low

damped resonant frequency the plat acts like a

scaled inertia with only the value of the motor

(high frequency) inertia . The frequency

of the resonance (pole) is defined by:

M

JJ =

ML

ML

JJ

JJR

c

f

+

=

π2

1

(2)

• The inertia ratio, the spring constant and the

system damping are defining the plot in the

transition area of the resonant frequencies.

The transfer function can be formed in a two-part

notation [6]:

( )

( )

++

++

+

=

+

cdss

cdssJ

JJ

J

sJST

sV

ML

ML

JJ

JJ

L

ML

M

M

2

2

1

(3)

The term on the left is just the motor behavior and the

term on the right is the effect of the compliant

coupled load. An ideal filter would be similar or equal

to the inverse function of the right part.

The high frequency gain (

∞=

s

) of the right part is

one and the low frequency gain (s = 0) is

( )

ML

M

JJ

J

sG

+

== 0

(4)

Figure 6 shows the low frequency gain drop as a

function of the load to inertia ratio. This gain drop

results also in a bandwidth drop. Figure 7 shows the

resonance frequencies as a function of the load to

motor inertia ratio. The anti-resonance is strictly

moving with the square root of the inertia ratio. The

resonance frequency is only slightly moving.

M

ML

AR

R

J

JJ

f

f +

=

(5)

In a servo system with matching inertia ( ) the

resonance frequency is

LM

JJ =

2

higher than the anti-

resonance frequency. If the load inertia is significant

higher than the motor inertia, the anti resonance

frequency will drop down. To achieve an appropriate

bandwidth a stiffer coupling c is required (for

example a direct drive solution).

By comparing the anti-resonance (nominator from the

second part of equation 3) with the standard

oscillation equation:

384 • PCIM EUROPE 2004 • PROCEEDINGS

1 10 100 1 k Hz 10 kf

0

-90

-270°

ϕ

( )

ωF

0

+10

-20

-30 dB

-40

-180

- 17 dB

- 34 dB

38 Hz

265 Hz

1 10 100 1 k Hz 10 kf

0

-90

-270°

ϕ

( )

ωF

0

+10

-20

-30 dB

-40

-180

- 17 dB

- 34 dB

38 Hz

265 Hz

( )

0

ωω<<∆F

0

+10

-20

-30 dB

-40

0.01 0.1 1 10 10

0

J

L

/J

M

( )

0

ωω<<∆F

0

+10

-20

-30 dB

-40

0.01 0.1 1 10 10

0

J

L

/J

M

Figure 6: Low frequency “gain drop” as function of

the load to inertia ratio

Figure 9: Bode plot of a first order and second order

lag filter f

z

= 38 Hz, f

p

= 265 Hz

and D

AR

=D

R

=0.5

02

2

0

2

0

=++ ssD

ZZZ

ωω

(6)

we get for the anti-resonance damping:

(

)

cJdD

LZ

2=

(7)

and the resonance damping:

+

= c

JJ

JJ

dD

ML

ML

P

2

(8)

Interesting is the relation between these equations:

M

ML

Z

P

J

JJ

D

D +

=

(9)

The equation shows that in an ideal two mass system

the resonance frequency (pole) is always better

damped than the anti-resonance (zero).

Figure 8 shows several idealized Bode plots

(damping not shown) with different load inertia

configurations. Parameter for the plots is the inertia

ratio.

Wrapped up the key behaviors of the compliantly

coupled two mass system are:

100

1

0.1

10

0.1 0.3 1 3 10f / f

0

J

L

/J

M

• Low gain at “low” frequencies < f

AR

• High gain at “high” frequencies >f

R

• Phase lead between f

AR

and f

R

(figure 5)

• Significantly influenced by the inertia ratio

The desired filter to compensate a two inertia

compliantly coupled system should own high gain at

low frequencies and low gain at high frequencies.

The phase lag between the corner frequencies should

be less than the phase lead of the system in this area.

If the gain drop at low frequencies is low (< 15 dB)

and the system is well damped a two parameter

(

PZ

ω

ω

,

) first order lag filter is a good approach:

( )

f

s

s

sF

Z

P

P

Z

πω

ω

ω

ω

ω

2;

1

=

+

+

=

(10)

In case of higher inertia mismatches – which results in

a higher gain drop – or a low damped system a four

parameter (

PZPZ

DD,,,

ω

ω

) second order lag filter

(bi quad) could be better fitting:

( )

2

2

22

22

2

2

2

Z

p

PPP

ZZZ

sDs

sDs

sF

ω

ω

ωω

ωω

++

++

=

(11)

Figure 9 shows two Bode plots of these filters.

Setting these four 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 auto-

tuning

The task is to determine the two

parameters of the velocity loop PI-

controller and - if necessary - to select

and parameterize the first or second

order lag filter to compensate the

resonance

resonance

anti-resonance

100

1

0.1

10

0.1 0.3 1 3 10f / f

0

J

L

/J

M

anti-resonance

Figure 7: resonance / anti-resonance frequency as

function of the load to motor inertia ratio

( )

ωF

+10

+20

-20 dB

-30

0

-40

0.1 0.3 1 3 10 30

0.1

0.3

1

3

10

J

L

= 100 J

M

( )

ωF

+10

+20

-20 dB

-30

0

-40

0.1 0.3 1 3 10 30

0.1

0.3

1

3

10

J

L

= 100 J

M

Figure 8: several idealized bode plots with different load inertia

configurations – damping is not shown – see figure 5

PCIM EUROPE 2004 • PROCEEDINGS • 385

described compliant coupled load behavior. This is

executed in three major steps:

1. Measure system transfer function

2. Select and parameterize anti resonance lag filter

3. Set PI-gains according to Nyquist stability

criteria

Measure system transfer function

In the shown plots we can see that the system has

mainly the behavior of a PI-controlled single

integrator with delay. Due to the integral part of the

velocity loop controller the phase shift in the low

frequency area is close to 180°. The behavior in the

high frequency area is significantly influenced by the

current loop performance and the velocity estimation

method. By using an observer to eliminate the digital

differentiation delay this observer has to be adapted to

the system. The digital drive ServoStar 300 can run a

simple – first order - inertia estimation at a defined

frequency (usually 300-500Hz). This frequency

should be set above of the system resonance

frequency to adapt the observer to the high frequency

(motor only) inertia.

According to this constrains the system transfer

function can be best measured when:

• all internal filters are switched off,

• the current loop is well tuned [5],

• the velocity PI-integral part is off or very low and

• the velocity observer is well adapted [4].

Figure 10: Belt driven linear motion system produced

by Montech –

www.montech.ch

Figure 11 shows such a measured open loop Bode

plot (green) of an industrial linear motion system,

figure 10. The red marked plot shows the gain

increasing

effect of the compliant coupled load.

Select and parameterize anti resonance lag filter

Figure 12 shows more detailed the gain increase

effect of the compliant coupled load (red). The blue

plot shows the gain approximation of a 2

nd

order lead

filter. The parameters ( and

dc-gain = -1.5dB) are fitted by using a simple least

square adoption to the measured plot. The fitting

works considerably good as shown in the Bode plot.

The highlight is that here a compliant coupled three-

inertia system (belt gear box + linear motion belt) is

compensated quite well.

HzfHzf

PZ

146,74

==

The requested current set-point filter is the inverse

transfer function of the approximated 2

nd

order filter

(

HzfHzf

ZP

146,74

=

=

). The dc-gain can be ignored

and the first approximation of the damping is set to:

2

1

==

PZ

DD

The frequency with the highest phase lag is here:

Hzfff

ZPlag

105==

This is fitting perfectly with the phase lead (at 105

Hz) of the measured open loop bode plot, figure 13.

According to the Nichols diagram here is a minimum

gain requested. Due to digital delays and system time

constants the phase shift crosses 180° in an area of

several 100 Hz. Here is a maximum gain requested.

The frequency dependent minimum or maximum

value is defined by the corresponding open loop phase

shift.

Nyquist stability criteria and the Bode plot

The easiest interpretation of the Nyquist stability

criteria is to check of the gain margin (at 180° phase)

and the phase margin (at 0dB gain) in the open loop

Bode plot, Figure 13.

But this is a verification at only two points. What is

beside these two points? We can see the detailed

behavior much better in the Nichols diagram, figure

14. Here we see the phase-gain relation for each

frequency.

Figure 11: Open loop Bode plot (green) and

the gain increase effect of the

compliant coupled load (red)

Montech linear motion system

-4

-2

0

2

4

6

8

10

12

14

20 - 600 Hz

dB

Figure 12: Measured gain increase effect of the

compliant coupled load (red)

2

nd

order filter approximation (blue)

Montech linear motion system

386 • PCIM EUROPE 2004 • PROCEEDINGS

( )

ωF

+10

+20

-20 dB

-30

0

-40

270 180 90 0360°

+30

+40

ϕ

0.5 dB

1 dB3 dB

6 dB

12 dB

f

I-Part

velocity observer

& digital delay

( )

ωF

+10

+20

-20 dB

-30

0

-40

270 180 90 0360°

+30

+40

ϕ

0.5 dB

1 dB3 dB

6 dB

12 dB

( )

ωF

+10

+20

-20 dB

-30

0

-40

270 180 90 0360°

+30

+40

ϕ

0.5 dB

1 dB3 dB

6 dB

12 dB

f

I-Part

velocity observer

& digital delay

Figure 14: Nichols plot - always open loop –

here with 0.5

dB peaking

doted line: without integral part

10 30 100 300 Hz 1 kf

0

-90

-270

-360°

ϕ

( )

ωF

+10

+20

-10

-20 dB

-30

-180

0

0 dB Gain

50° phase margin

10 30 100 300 Hz 1 kf

0

-90

-270

-360°

ϕ

( )

ωF

+10

+20

-10

-20 dB

-30

-180

0

0 dB Gain

50° phase margin

Figure 13: Open loop Bode plot with 50° phase

margin and 10 dB gain margin

Accepting 1dB peaking (red oval) we can see that the

gain is limited if the phase shift is between 115° to

245°. For low frequencies – due to the phase shift - a

minimum gain is requested, for high frequencies the

maximum is shown. The gain margin would be 5 dB;

phase margin would be 65°.

Set PI-gains according to Nyquist stability

criteria

By switching the integral part off we get the doted

line in figure 14. Now the gain can be increased until

the plot touches the lower side of 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 of the oval (the plot is moving left).

Figure 15 shows the Bode plot of the tuned velocity

loop.

Summary

Instead of tuning a servo drive system by highly

educated servo experts the Bode plot based auto-tune

can provide reliable results in a short time frame. The

result is documented with an open and closed loop

Bode plot that shows key performance indices like

bandwidth and stability margin.

The key advantages of using a Bode plot based auto-

tuning are:

• Ease of use

• Very good results in a short time frame

• Less set up cost

This results in advantages for the designed product:

• Higher reliability and lower fault risk

• Higher productivity

• Fast development time

References

[1] V. Wesselak, M. Köhler and G. Schäfer, „Robust

Speed Control Based on the Identification of

Mechanical Parameters“, PCIM, Germany 2003.

[2] J. O. Krah and K. Neumayer, “Motorsteuerung –

kompakt und flexibel”, Elektronik Heft 3/2004,

Weka Verlag.

[3] J. O. Krah, S. Geiger and G. Jaskowski, “Free

Programmable Signal Processing inside a High

Performance Servo Amplifier”, PCIM, Germany,

1998.

[4] J. O. Krah, “Software Resolver-to-Digital

Converter for High Performance Servo Drives”,

PCIM, Germany 1999.

[5] J. O. Krah, J. Holtz, “High Performance Current

Regulation for Low Inductance Servo Motors”, IEEE

Industry Appl. Soc. Annual Meeting, St Louis, Oct.

1998.

[6] G. Ellis, Cures for Mechanical Resonance in

Industrial Servo Systems, PCIM Germany 2001.

10 dB gain margin

180° phase

10 dB gain margin

180° phase

Figure 15: Open loop (green) and

closed loop (red) bode plot of the

Montech linear motion system with 2

nd

order lag filter and switched on integral

PCIM EUROPE 2004 • PROCEEDINGS • 387

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