High Order Control Design

8/2/2019 High Order Control Design

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Elmo Position Control

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High Order Control Design AdvantageOver PI

and PID Controllers referenceYaniv O.1, Theodor Y. and Safonov S.

Abstract

A robotic application is used to show that advanced controllers are much capable than PI

controllers. They can achieve higher bandwidth, lower settling time and better disturbancerejection. The increased performance costs little in sensor noise amplification. We showusing true-life design examples that advanced control algorithms improve equally wellboth speed and position controllers.

Introduction

Consider an electrical motor with shaft angle )(t , driven by the current )(ti . We want

the shaft speed, )(t , to follow a given trajectory, )(tT . For this purpose, we embed the

motor in a feedback structure as described schematically in Figure 1. The controller in

Figure 1generates a correcting current command, )(ti , so as to keep the speed error,

)(te , minimal.

Figure 1: Speed control feedback structure around a motor

The controller is required to minimizing the speed error and in the same time thesynthesized current command must remain smooth enough so that (i) no excessivestresses will shorten the system life, and (ii) the current amplifier will be able to

effectively follow the current command.The controller design must consider both small and large signals behavior. The smallsignal design cares for the behavior when the tracking error is small, and thus the

required correction current (torque) is within the amplifier limits. Large signal (nonlinear)design must maintain good stability and performance while the current (torque)requirement goes beyond the amplifier limits. Out of range current (torque) requirementsmay develop due to extreme reference signal changes or due to extreme disturbances.This article focuses on the small signal (linear) design.

1 Address for correspondence: O. Yaniv, Elmo Position Control, Shidlovskey 1, Yavne, 81101, POB 13081, Israel

and Faculty of Eng. Tel-Aviv University, Tel-Aviv 69978, Israel. [email protected]

-Motor

+Load

Sensor

ControllerT

Sensornoise

ie Amplifier


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The controller must have a parameterization, so that users will be able to tune it to theirspecific applications. The most common controller parameterizations are P, PIand PID. A

P(proportional)controller keeps the current )(ti proportional to the speed error,

)()( ttT . A PIcontroller generates )(ti as the sum of two terms. The Pterm which is

proportionalto the speed error, and the Iterm which is proportional to the integralof thespeed. A PID controller is a PIcontroller plus the D term, that is, a term proportional to

the speed error derivative. The worst drawback ofPIand PID controllers is their poor highfrequency attenuation. Some commercial motion controllers add low-pass filters to their PIcontrollers, to improve the high frequency attenuation.The traditional P, PIor PID controller have one big advantage they are very simple, anda technician can tune them effectively using simple "cut and try" methods. These simplecontrollers suffice for simple applications moderate or low performance requirements,and good enough mechanics and sensors. A very simple control problem is, however, a

symptom of too generous mechanics and sensors design. More complicated controllers can

push the tracking and disturbance attenuation performance to the physical limits of thesystem. An advanced controller can get the desired performance out of a lighter structure,or within degraded, cheaper sensors. For the same mechanics-sensors set, an advanced

controller can increase the speed range in which accurate enough motions are possible.We use the term advanced-controllers for controllers of almost free structure and order.Advanced controllers do not preserve the PIsimplicity. They have many parameters, and

require an automated design suite for effective tuning. The decision to use advancedcontrollers is psychologically not easy. You have to trust the tuning suite of the Ph.D. guybetter than you trust your senses. Moreover, you have to believe that the tuning suitedoes take appropriate design margins, so that you wont have vibrations when the load

changes a bit.In this paper, we compare the performance of advanced and traditional controllers,

controlling a robotic arm. The advanced controllers are shown to do much better than theP PI or PID controllers. Section 2 compares PIcontrollers and advanced controllers, by alaboratory test. Section 3 extends the comparison of Section 2 to frequency domain.Section 4 shows similar comparison results for cascaded position control. Embedding thespeed controller ofFigure 1in an outer position feedback loop makes a cascaded position

controller see Figure 2. The position controller is required to follow a trajectory )(tPT .

Figure 2: Cascaded position control feedback loop

Position

Controller-

Speed

Controller dtPlant

dt

d

PT P

-


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Speed Control Comparison by Laboratory Tests

Our design example deals with a two-join robot, see Figure 3. The robot is lightweight,and pays for the lightweight with high link compliances. An electrical motor drives each

joint. For each motor, a tachometer measures the motor shaft speed and an encodermeasures the motor shaft angle. The upper motor (motor 1) drives the internal link, andthe lower motor (motor 2) drives the external link.For this robot, a PIspeed controller proved useless, since the robot became unstable forvery low gains. We helped the PIwith an additional high frequency low-pass pole. The PI

plus low-pass performance shown in the next figures is probably better than what anexperienced technician could achieve. This is since this robot exhibits high couplingbetween its articulated axes. If one use traditional PItuning methods to optimize eachaxis when the other axis is inactive, the integrated system may become unstable or mightloose some of its gain and phase margins due to the two axes interaction. If on the otherhand, one use traditional PItuning methods to optimize each axis when the other axis isactive, stability of the integrated system is guaranteed but again the closed loop might

loose some of its gain and phase margins.

Motor 1

Motor 2

Link 1

Link 2

Figure 3: Robot for laboratory testsa two joint robot with two motors, two tachometers and two encoders.


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The robot was tested for several speed reference commands. Figure 4and Figure 5show the step response ofPIplus low-pass controller and of an advanced controlleragainst the reference step, for motors 1 and 2, respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-200

0

200

400

600

800

1000

Cnt/sec.

reference

PI+low pass

Advanced

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-10

-6

-2

2

6

10

Ampere

Sec.

PI+low pass

Advanced

Figure 4: Comparison between PI plus low-pass controller and advanced

controller.

The step command to motor 1 is 600[cnt/sec], and to motor 2 zero.

Clearly for motor 1 (Figure 4), the tracking error, rise time and settling time of the

advanced controller are much lower than the corresponding results of the PIplus low-pass. The rise time of the advanced controller is 0.017seconds, about 43% of the 0.04seconds rise time of the PIplus low-pass. The same relation holds for the settling time.The prices for the higher performance of the advanced controller are twice the current

peak and larger high frequency noise.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-200

600

1400

2200

3000

Cnt/sec

reference

PI+low passAdvanced

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-10

-6

-2

2

6

10

Ampere

Sec.

PI+low pass

Advanced

Figure 6: Comparison between PIplus low-pass controller and advanced

controller.

Trajectory command to motor 1 is 2000[cnt/sec], acceleration limitation20000[cnt/sec^2]. Motor 2 is commanded to stop.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-200

600

1400

2200

3000

Cnt/sec

reference

PI+low pass

Advanced

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-2

-1

0

1

2

3

Ampere

Sec.

PI+low pass

Advanced

Figure 7: Comparison between PIplus low-pass controller and advancedcontroller.

Trajectory command on motor 2 is 2000[cnt/sec], acceleration limitation

20000[cnt/sec^2]. Motor 2 is commanded to stop.


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Speed Control Comparison by Frequency Domain Analysis

In section 2 the robot's tracking performance for different controllers has been studied.We subjected the controller to abrupt reference waveforms, which expose the transientbehavior of the closed loop. The time domain tests of Section 2 show the final result, butthey offer no explanation to the difference in the results achieved. The frequency domain

analysis of this section grants insight to questions such as the feasibility of better designs,and the design margins taken. The frequency plots provide an estimate for settling timeand overshoot. This estimate confirms the result of Section 2.

Open Loop

The robot has two motors and four sensors, two tachometers and two encoders,generating eight transfer functions from the current commands introduced into each of the

motors to each of the sensors. Let ijp denote the transfer function from current introduced

into motorjto the integral of the angle (integral of speed) measured by the tachometer

on the shaft of motor i; and ijr denote the transfer function from the current command

introduced into motorjto the encoder coupled to the shaft of motor i. Figure 8andFigure 9depict these eight discrete Bode plots.

-120

-100

-80-60

-40

-20

p11

dB

-120

-100

-80-60

-40

-20

p12

101

102

103

-120

-100-80

-60

-40

-20

p21

[rad/sec]

101

102

103

-120

-100-80

-60

-40

-20

p22

[rad/sec]

Figure 8: Bode plot of ijp

, from input j to integral of the tachometer on link i


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-8-

-120

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-80

-60-40

-20

r11

dB

-120

-100

-80

-60-40

-20

r12

101

102

103

-120

-100

-80

-60

-40

-20

r21

[rad/sec]

101

102

103

-120

-100

-80

-60

-40

-20

r22

[rad/sec]

Figure 9: Bode plot of ijr , from input j to the encoder located on link i

Figure 8shows that link 1 has four dominant resonances; with frequencies ranging from

200 to 3000 rad/sec. Link 2 has a dominant resonance at about 100 rad/sec. Theseresonance frequencies would limit the performance of any controller, but their effect on PI

controllers is most marked. Advanced controllers can attenuate the resonant frequenciesusing notch or low-pass filters; or they can actively damp some resonant modes. Thetransfer functions from current commands to encoders, Figure 9, differ from the transferfunction ofFigure 8since the encoders are mounted on flexible couplings, whereas thetachometers are mounted rigidly on the motor shaft. In some frequencies the couplingbetween the axes is so large that the reaction to current injected to motor 1 on its shaft,is much lower than the reaction of the shaft of motor 2. The ratio between these reactions

is depicted in Figure 10. For currents injected to motor 1, whose spectral densities aremainly around 200Hz and/or 300Hz, the tachometer located on motor 2 reads a signal upto 5 times larger than the tachometer located on motor 1. A similar phenomenon, but

much lower in size, happens when motor 2 is driven.


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101

102

103

-45

-30

-15

015

30

p21

/p11

dB

101

102

103

-45

-30

-15

0

15

30

p12

/p22

dB

[rad/sec]

Figure 10: Bode plot relative cross talk.

The upper plot is the ratio of tachometer 2 to tachometer 1 due to current injected to

motor 1. Lower plot is the ratio of tachometer 1 to tachometer 2 due to current injected tomotor 2.

Controller Design

Advanced control design techniques are based on the controlled plant transfer function;the process of achieving this transfer function is called identification. Following the plant

identification process, the control engineer designs a controller using his own experience,knowledge and skills. The major properties of a closed loop feedback system can beconcluded from the open loop transfer function, for example, rise time, settling time,

robustness to plant changes, amplification of sensor noise, and if it is possible to improvethe closed loop performance. Bode plots of the open loop for motor 1 and motor 2 for the

PIplus low-pass controllers are shown in Figure 11and Figure 12, respectively. Bodeplots of the open loop for motor 1 and motor 2 for the advanced controllers are shown in

Figure 13and Figure 14, respectively.


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101

102

103

-50

-30

-10

10

30

dB

101

102

103

-360

-270

-180

-90

0

90

[rad/sec]

Phase[deg]

Figure 11: Open loop Bode plot of motor 1 (PIand low-pass)

101

102

103

-50

-30

-1010

30

dB

101

102

103

-360

-270

-180

-90

0

90

[rad/sec]

Phase[deg]

Figure 12: Open loop Bode plot of motor 2 (PIand low-pass)


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101

102

103

-45

-30

-15

015

30

dB

101

102

103

-360-300-240-180-120

-600

60

[rad/sec]

Phase[deg]

Figure 13: Open loop Bode plot of motor 1 (advanced controller)

101

102

103

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-30

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0

15

30

dB

101

102

103

-360-300-240-180

-120-60

060

[rad/sec]

Phase[deg]


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101

102

103

-45

-30

-15

015

30

dB

101

102

103

-360-300-240-180-120

-600

60

[rad/sec]

Phase[deg]

Figure 14: Open loop Bode plot of motor 2 (advanced controller)

Comparing Figure 11,Figure 12,Figure 13and Figure 14, we have the followingconclusion:

1. The bandwidth of the advanced controller for motor 1 is about 15Hz, almost 2.5larger than the 6.2Hz of the PIcontroller.

2. The bandwidth of the advanced controller for motor 2 is about 13Hz, almost 2.5larger than the 5.7Hz of the PIcontroller.

3. The low frequency disturbance attenuation of the advanced controller for motor 1 is

5 times better than that of the PIcontroller.

4. The low frequency disturbance attenuation of the advanced controller for motor 2 is10 times better than that of the PIcontroller.

We present a Nichols chart in order to convince the reader that a fair comparison was

made, in the sense that similar gain and phase margins were taken for the PIplus low-pass and the advanced controllers. Figure 16compares the open loop on motor 2 foradvanced controller (left) and PIplus low-pass (right). Clearly both have the same phase

and gain margins, about 8dB and 35deg. Figure 15is the same comparison for motor 1.The margins of the advanced controller are similar. It is impossible to increase the gain ofthe PI(right) and maintain the same margins since: (i) the phase margin will be less thanthe required 35deg and (ii) the resonance whose gain is about 9dB is highly phase

uncertain.


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-360 -270 -180 -90 0-30

-20

-10

0

10

20

30

Phase[Deg]

dB

co=14

-360 -270 -180 -90 0-30

-20

-10

0

10

20

30

Phase[Deg]

dB

co=6.2

Figure 15: Comparison by open loop Nichols plot

of speed controller of motor 1, left advanced, right PI plus low-pass. Crossover requenciesare 14[Hz] and 6.2[Hz], respectively.

-360 -270 -180 -90 0-30

-20

-10

0

10

20

30

Phase[Deg]

dB

co=16

-360 -270 -180 -90 0-30

-20

-10

0

10

20

30

Phase[Deg]

dB

co=5.2

Figure 16: Comparison by open loop Nichols plot

of speed controller of motor 2, left advanced, right PIplus low-pass. Gain and phasemargins are about the same, crossover frequencies are 16[Hz] and 5.2[Hz], respectively.


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Cascaded Position Control Comparison Laboratory Tests

measures the motor shaft speed and an encoder measuring the motor shaft angle. Acascaded position controller has been designed where the speed loop is the PIplus low-pass or the advanced controllers of sections 2 and 3 and the position controller is a simplegain. Figure 17and Figure 18show test results for that cascaded position controller.

The comparison shows that the advanced controller tracks the reference command much

better than the PIplus low-pass. The current consumed by both controllers is about thesame with about the same peak value.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

150

300

450

600

Cnt

reference

PI+low p ass

Advanced

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1

-0.5

00.5

1

1.5

2

Amp

ere

Sec.

PI+low p ass

Advanced

Figure 17: Comparison between PIplus low-pass controller and advanced controller.

The trajectory command for motor 1 is 500[cnt] with speed and acceleration limitation of2000[cnt/sec] and 20000[cnt/sec^2], respectively. Motor 2 commanded to stop.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

150

300

450

600

Cnt

reference

PI+low pass

Advanced

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1

-0.5

0

0.5

1

1.5

2

Ampere

Sec.

PI+low pass

Advanced

Figure 18: Comparison between PIplus low-pass controller and advanced controller.

The trajectory command on motor 2 is 500[cnt] with speed and acceleration limitation of

2000[cnt/sec] and 20000[cnt/sec^2], respectively. Motor 1 commanded to stop.

Conclusions

We used a robotic application to compare the performance of a traditional PI-PIDcontroller versus more advanced controllers. For this compliant mechanic system, the PIcontrollers were left behind the more advanced controllers, in the criteria of bandwidth,settling time and low frequency disturbance rejection. This is just another case, where to

get the most out a mechanical system, PIcontrollers are not enough. Complex controlproblems deserve an advanced controller. To effectively design an advanced controller, weneed a frequency domain system model, and an automated controller design system. Wedeveloped an identification & design environment that can identify the dynamics of

complex mechanical systems, including inter-axis coupling. The identification results aredirectly fed to an automatic controller design environment, and the results of thecontroller design are directly fed to program the motion controller. The same environment

also designs automatically the large signal control policy. Large signal advancedcontrollers are out of the scope of this paper they deserve their own paper.

Copyright 2001 Elmo Position Control. All rights reserved.

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High Order Control Design