Practical PID Controller Tuning for Motion Control

Ozhan Ozen Emre Sariyildiz, Haoyong Yu Kenji Ogawa, Kouhei Ohnishi Asif Sabanovic
Faculty of Engineering and Department of Biomedical Department of System Design Faculty of Engineering and
Natural Sciences Engineering Engineering Natural Sciences
Sabanci University National University of Singapore Keio University Sabanci University
Istanbul, Turkey Singapore Yokohama, Japan Istanbul, Turkey
ozhanozen@sabanciuniv.edu {bieemre,bieyhy}@nus.edu.sg {kenji,ohnishi}@sum.sd.keio.ac.jp asif@sabanciuniv.edu

Abstract— The most popular control method in the industry Apart from these most common methods, there are also
is PID control due to its simple structure and effective more advanced PID tuning methods such as Rivera’s method
performance. Although huge numbers of PID controller tuning which uses the advantages of 2 degree of freedom (2-DOF)
methods have been proposed so far, existing ones still do not have control approach [10] and many intelligent tuning algorithms
the desired performances and the simplicity. Complex system such as Genetic Algorithms (GA) and Fruit Fly Optimization
dynamics make it challenging for engineers and students to apply (FOA) [4], [6]. Although these methods enhance the
these methods on their applications especially in the motion capabilities of the conventional algorithms, they are too
control and robotics areas. Such systems generally include complex to be used by engineers, students and even most of
nonlinearity, friction, varying inertia and unknown disturbances
researches. Therefore, these methods are not used in the
which make the conventional tuning methods ineffective and too
complex to be used. There is need for simple and effective PID
industry and even not common in the academia.
tuning methods in these areas. In order to solve this problem, this Complex system dynamics are the main problem of the
paper proposes two novel practical PID tuning methods for existing conventional tuning methods. Especially in motion
motion control systems. These methods bring the superiority of control and robotics, where the main usage of PID controllers
the 2 degree of freedom control approach to simple PID is to control servo position and force, this problem is faced
controller structures analytically. They are very effective in more frequently. Such systems generally have nonlinearities,
motion control and robust both to parameter uncertainty and
uncertainties, friction, varying system parameters and
unknown disturbances, yet very simple. They can be easily used
unknown disturbances. These properties complicate the use of
by the engineers in the industry and the students with very basic
control knowledge, so little effort and time. The tuning methods the existing PID tuning methods [11]. The PID controllers
of robust PID and PI controllers with velocity feedbacks are tuned according to these methods, may behave very differently
proposed, for position and force control problems of servo when the inertia changes or an unknown load disturbance is
systems, respectively. The validities of the proposals are verified applied. This makes the motion control an even harder area for
by the experimental results. proper PID controller tuning since one must consider these
complex system dynamics and uncertainties when tuning.
Keywords—PID controller, tuning, robustness, motion control Therefore there is need for effective but very simple PID
robotics, position control, force control tuning methods in motion control area.
I. INTRODUCTION In order to address this problem, this paper proposes two
novel practical PID tuning methods for motion control; tuning
Proportional-Integral Derivative (PID) control is the most of a PID controller with velocity feedback for servo position
widely used control in the industry [1]–[2]. The reasons for its control and tuning of a PI controller with velocity feedback for
popularity are its simple usage and effectiveness [3]. So far, servo force control. Although these methods are applied on
several PID tuning methods have been proposed [4]. However, basic PID controller structures, they use the advantages of 2-
these existing methods do not have the desired performances DOF control approach. They provide high robustness to
and simplicity for many applications i.e., an important part of varying system parameters, unmodelled dynamics and
the PID controllers in the industry have bad performances due unknown disturbances. These methods are very effective for
to their poor tunings [1]. motion control and very simple to use. They can be easily
The most common tuning method was proposed by Ziegler used by engineers and students with so little effort and time.
and Nichols [5]. This method needs many iterations for tuning, Their validity are verified by the experimental results.
gives big overshoot and the robustness to varying system The rest of the paper is organized as follows. In Section II,
parameters is low [6]. There is also the popular Cohen-Coon the direct tuning steps are given with explanations. In Section
formula which is derived to give good robustness against load III, the analytical derivations of the methods are presented. In
disturbances, but this method is not satisfactory because of its Section IV, the experimental results are given. The Section V is
overshoot and oscillatory response [7]–[9]. the conclusion part.

l-))) 
 II. CONTROLLER TUNING
In this section, tuning algorithms are described for both
position and force control problems. A PID controller with
velocity feedback and a PI controller with velocity feedback
are tuned for position and force control, respectively. The block
diagrams of such servo control systems are shown in Fig. 1 and
Fig. 2.
In these figures and the following tuning steps, the
following definitions apply;

‫ݔ‬ Position; Fig. 1. Block diagram of the PID control system with velocity feedback for
position control.
‫ݔ‬ሶ Velocity;
߬௘௫௧ External force;
‫ ݔ‬௥௘௙ Position reference;
௥௘௙ External force reference;
߬௘௫௧
݁ Error;
‫ܬ‬ Inertia;
ܾ Viscous friction coefficient;
௣ ௙ Proportional gain;
‫ܭ‬௣ ǡ ‫ܭ‬௣
௣
‫ܭ‬ௗ Derivator gain;
௣ ௙ Integrator gain; Fig. 2. Block diagram of the PI control system with velocity feedback for
‫ܭ‬௜ ǡ ‫ܭ‬௜ force control.
௣ ௙ Velocity feedback gain;
‫ܭ‬௩ ǡ ‫ܭ‬௩
߬ Control signal;
ௗ௜௦ ௗ௜௦ ௗ௜௦ Interactive, external and friction ௣ ‫݌‬ ‫݌‬
߬௜௡௧ ǡ ߬௘௫௧ ǡ ߬௙௥௖ x ‫ܭ‬௣ ൌ ‫ܭ‬௣ௗ௘௦ ൅ ‫ܭ‬ௗௗ௘௦ ܴǢ
disturbances; ௣ ‫݌‬
௣ ௙ General disturbances; x ‫ܭ‬௜ ൌ ‫ܭ‬ௗௗ௘௦ ܴǢ
߬ௗ ǡ ߬ௗ ௣ ‫݌‬
௣ ௙ Desired proportional gain; x ‫ܭ‬ௗ ൌ ‫ܭ‬ௗௗ௘௦ Ǣ
‫ܭ‬௣ௗ௘௦ ǡ ‫ܭ‬௣ௗ௘௦ ௣
௣ x ‫ܭ‬௩ ൌ ‫ܬ‬௡ ܴǤ
‫ܭ‬ௗௗ௘௦ Desired derivator gain;
‫ܭ‬௘௡௩ Environmental stiffness coefficient; Increase R while updating the controller gains according to
‫ܦ‬௘௡௩ Environmental damping coefficient; it, until the system starts to be influenced negatively by
‫ܬ‬௡ Nominal inertia; practical constraints such as noise.
߱௡ Natural frequency; B. Force Control
Ƀ Damping coefficient;
Similarly, consider an ideal force control system with P
ܴ Robustness variable. controller, assuming that there is no external or interactive
disturbance, the system inertia is nominal and close to the
A. Position Control upper limit of the exact varying inertia. Select the desired
proportional gain accordingly;
Consider an ideal position control system which is linear
and having no friction or any disturbance, having nominal ௙ ܾʹ
inertia and PD controller. Select the nominal inertia as close as x ‫ݏ݁݀ܭ‬
‫݌‬ ൌ Ǥ
Ͷ‫ ݒ݊݁ܭ ݊ܬ‬Ƀʹ
possible to the upper limit of the exact varying inertia, for the
reason that selecting the nominal inertia higher makes the
‫ܭ‬௘௡௩ and ‫ܦ‬௘௡௩ are the environment parameters that are
system more stable [12], [13]. Select the desired proportional
defining the material properties of the contact point in force
and derivator gains according to the desired natural frequency
and damping coefficient; control. The force controlled, ߬௘௫௧ , is a function of these
parameters and position. If these parameters are unknown,
௣ they can be estimated by an adaptive algorithm such as in
x ‫ݏ݁݀ܭ‬
‫݌‬ ൌ ‫ ݊ʹ߱ ݊ܬ‬Ǣ [14]. In practice, it is harder to tune the controller gain in force
௣ control due to unknown environment. Therefore, selecting
x ‫ݏ݁݀ܭ‬
݀ ൌ ‫ʹ ݊ܬ‬Ƀ߱݊ Ǥ ௙
‫ܭ‬௣ௗ௘௦ lower at first and increasing it slowly, until the system
Select ܴ value considering that the higher this value, the performance deteriorates, is a safer method in terms of
more robust the system will be to the unmodeled dynamics, stability. Increase R until the system is affected by the noise
friction and external disturbances. Set the real controller gains and set the real controller gains according to the following
according to the following relations; relations;


 ߬ ൌ ݁൫‫ܭ‬௣ ൅ ‫ܭ‬ௗ ݃൯ ൅ න ݁ ൫‫ܭ‬௣ ݃൯  ൅
݁ሶ ‫ܭ‬ௗ  െ  ‫ܬ‬௡ ݃‫ݔ‬ሶ ǤሺͶሻ

If the controller gains of the DOB based control system

௣ ௣
and ݃ are selected as‫ݏ݁݀ܭ‬
‫݌‬ ǡ ‫ݏ݁݀ܭ‬
݀ andܴ, respectively, (4) is
equal to the control signal of the PID controller for position
control which is tuned according to the proposal. This control
signal can be expressed as;

Fig. 3. Block diagram of a DOB control system. ߬ ൌ ݁‫ܭ‬௣ ൅ න ݁ ‫ܭ‬௜ ൅ ݁ሶ ‫ܭ‬ௗ  െ  ‫ܭ‬௩ ‫ݔ‬ሶ Ǥሺͷሻ

݂ ௙
x
௙
‫ܭ‬௣ ൌ ‫ܭ‬௣ௗ௘௦ Ǣ Similarly, if proportional gain and ݃ are selected as ‫ݏ݁݀ܭ‬
‫݌‬
௙ ݂ andܴ, respectively, the control signal of a DOB based force
x ‫ܭ‬௜ ൌ ‫ܭ‬௣ௗ௘௦ ܴǢ control system can be written as;
௙
x ‫ܭ‬௩ ൌ ‫ܬ‬௡ ܴǤ

߬ ൌ ݁൫‫ܭ‬௣ௗ௘௦ ൯ ൅ න ݁ ൫‫ܭ‬௣ௗ௘௦ ܴ൯  െ  ‫ܬ‬௡ ܴ‫ݔ‬ሶ Ǥሺ͸ሻ

III. ANALYTICAL DERIVATION
Disturbance observer (DOB), which was proposed by K.
Ohnishi, is a robust control tool that is widely used in motion This is the same as the control signal of the PI controller for
control systems due to its simplicity and efficiency [15]. In a force control which is tuned according to proposal.
DOB based control system, robustness and performance goals
IV. EXPERIMENTS
are achieved in the inner and outer loops, respectively. As the
bandwidth of the DOB is increased, not only external Position and force control experiments were conducted
disturbances are suppressed but also stability and performance using two linear motors which are shown in Fig. 4. These
are improved [12], [14]. The block diagram of a DOB based motors have constant inertia which are 0.4 kg each. Motor 1
position control system is shown in Fig. 3. was used for control experiments, and Motor 2 was used to
apply external sinusoidal disturbance on Motor 1. A KYOWA
These additional definitions apply in the figure and LUR-A-50NSA1 force sensor was used to measure the external
derivation; force applied on Motor 1, in the force control experiment. The
݃ Bandwidth of the DOB; specifications for the experimental setup are given in Table I.
߬ ௗ௜௦ ൌ ߬ ௗ ൅ ο‫ݔܬ‬ሷ ൅ ܾ‫ݔ‬ሶ Total disturbance; A. Position Control Experiment
߬Ƹ ௗ௜௦ Estimated disturbance; The first part of the position control experiment was
߬ ௗ௘௦ Desired torque. performed without the external sinusoidal disturbance. A ramp
reference input, which increases to 0.03 meters from 0 meters
According to the DOB system, the equation of motion is; in 0.2 seconds, is applied at 1 second. The position control
responses with the PD controller, which have
௣ ௣
‫ܭ‬௣ௗ௘௦ and ‫ܭ‬ௗௗ௘௦ as the controller gains and the PID controller,
‫ܬ‬௡ ‫ݔ‬ሷ ൌ ߬ െ ߬ ௗ௜௦ Ǥሺͳሻ which was tuned according to the proposal with the same
gains and 600 asܴ, are shown in Fig. 5. As it is seen from the
Where control signal is; figure, there is some steady state error due to the disturbance
of the system such as friction. The PD controller alone cannot
eliminate this steady state error due to the lack of an
߬ ൌ ߬ ௗ௘௦ ൅ ߬Ƹ ௗ௜௦ Ǥሺʹሻ integrator. On the contrary, the PID controller eliminates this
steady state error while having the same dynamic response.
The estimated disturbance can be expressed as;

݃ ௗ௘௦
߬Ƹ ௗ௜௦ ൌ ሺ߬ െ ‫ܬ‬௡ ‫ݔ‬ሷ ሻǤሺ͵ሻ
‫ݏ‬

By considering the figure, (1), (2) and (3), the control

signal can be rewritten as;

Fig. 4. Experimental setup.


 TABLE I
EXPERIMENTAL SETUP SPECIFICATIONS
Parameters Descriptions Values
݀ܶ Sampling period ͳͲିସ ‫ݏ‬

‫ܬ‬௡ Nominal motor inertia ͲǤͶ ݇݃

௣
‫ܭ‬௣ௗ௘௦ Desired proportional gain for position control ͺͲͲ
experiment
௣
‫ܭ‬ௗௗ௘௦ Desired derivator gain for position control ͸Ͳ
experiment
௙ Desired proportional gain for force control ͲǤʹ
‫ܭ‬௣ௗ௘௦
experiment
Fig. 6. PID position control responses for ramp input with different values for
݃௩௘௟ Cut-off frequency of the velocity measurement ‫݀ܽݎ‬ R.
filter ͸ͲͲ
‫ݏ‬
݃௦௘௡ Cut-off frequency of the force measurement ‫݀ܽݎ‬
filter ͸ͲͲ
‫ݏ‬

The second and third parts of the position control

experiment were conducted with the external sinusoidal
disturbance applied to Motor 1. This disturbance was started
to be applied at 2 seconds in 2nd part, and at 4 seconds in 3rd
part. The frequency of the disturbance is increased every two
seconds to the following values respectively; 0.1 Hz, 0.5 Hz, 2
Hz, 4 Hz. In the second part, the same reference input as in the
first part was applied. The position control responses of the
PID controller, which was tuned with different ܴ values, are
shown in Fig. 6. As it is seen, increasing the ܴ value improves Fig. 7. PID position control responses for sinusoidal input with different
the system robustness. While for low values of ܴ only the low values for R.
frequency disturbance is suppressed, for high values of ܴ also
the high frequency part of the disturbance is suppressed,
making the system position response closer to the disturbance- B. Force Control Experiment
free position response. The same force control experiments were conducted for
environments with high and low stiffness. The first part of
In the third part of the position control experiment, instead
these experiments was performed without the external
of the a ramp reference input, a sinusoidal reference input,
sinusoidal disturbance and on an environment with high
with 0.03 m as the offset, 1 Hz as the frequency and 0.01 m as
stiffness. A step force reference, which has a magnitude of 2N,
the amplitude, was applied to the Motor 1. The same external
was applied at 0 second. A distance between the motor and
disturbance is also applied. The position control responses
environment contact points was kept in order to observe the
with different values for ܴ are shown is Fig. 7. As in the ramp
impact. Force control responses with the P controller, which
input case, increasing ܴ improves system robustness, making
has ‫ܭ‬௣ௗ௘௦ ௙ as the controller gain, and the force response with the
the position control response converge to the disturbance-free
PI controller, which is tuned according to the proposal, having
control response.
the same controller gain value and 100 asܴ, are shown in Fig.
8. As it is seen, the P controller is influenced by the
disturbances too much and it cannot eliminate the huge steady
state error due to the lack of the integrator. On the other hand,
there is no steady state error in the PI controller case.
Moreover, response is faster. The huge overshoot magnitude is
due to the distance between motor and environment contact
points and can be reduced by decreasing this distance.
In the second and third parts of the force control
experiment, which were also conducted on environment with
high stiffness, an external sinusoidal disturbance was applied.
Again, this disturbance was started to be applied at 2 seconds
in 2nd part, and at 4 seconds in 3rd part. The frequency of the
disturbance is increased every two seconds to the following
values respectively; 0.1 Hz, 0.5 Hz, 2 Hz, 4 Hz. In the second
part, the same step reference input as in the first part was
Fig. 5. PD and PID position control responses without external disturbance.
applied. The force responses of the PI controller, which was


tuned with different ܴ values, are shown in Fig. 9. As it is
seen, the higher theܴ, the more robust the system becomes,
and the response becomes closer to the disturbance-free case.
In the third part, a sinusoidal force reference input was
applied instead of a step input, with an offset of 3N, amplitude
of 1.5N, and frequency of 1Hz. The external disturbance
properties were kept the same. Like in the second part, high ܴ
values give more robust control performances, as expected.
The fourth, fifth and sixth parts of the force control
experiment are almost same as with the first, second and third
parts, respectively. However, the only difference is that
instead of an environment with high stiffness, these
experiments were conducted on an environment with low
Fig. 10. PI force control responses for sinusoidal input with different values
stiffness. Force control responses with the P controller and the
for R on an environment with high stiffness.
PI controller, are shown in Fig. 11. As in the environment with
high stiffness, the steady state error is eliminated also in this
case due to the robustness of the controller tuning method.
Force control responses of the PI controller, in the presence
of the external sinusoidal disturbance, for varying ܴ values,
with step and sinusoidal reference inputs are shown in Fig. 12
and Fig. 13, respectively. The results are similar to the high
stiffness cases. As ܴ is increased, the system becomes more
robust, eliminating the external disturbances with higher
success. However, the response becomes more oscillatory for
high values ofܴ. This is due to the fact that low environmental
stiffness value changes the system dynamics. If this situation is
creating a problem for applications, ‫ܭ‬௣ௗ௘௦ ௙ or ܴ can be reduced.

Fig. 11. P and PI force control responses without external disturbance on an

environment with low stiffness.

Fig. 8. P and PI force control responses without external disturbance on an

environment with high stiffness.

Fig. 12. PI force control responses for step input with different values for R
on an environment with low stiffness.

Fig. 9. PI force control responses for step input with different values for R on
an environment with high stiffness.


 ACKNOWLEDGEMENT
This research was supported in part by the Ministry of
Education, Culture, Sports, Science and Technology of Japan
under Grant-in-Aid for Scientific Research (S), 25220903,
2013.

REFERENCES
[1] C. C. Yu, “Introduction,” in Autotuning of PID Controllers, 2nd ed.
German: Springer, 2006, ch. 1, sec. 2, pp. 3-4.
[2] S. Skogestad, “Probably the best simple PID tuning rules in the world,”
J Process Contr, vol 13, pp. 291-309, Sep. 12, 2001.
[3] K. H. Ang, G. Chong, “PID Control System Analysis, Design, and
Technology,” IEEE Trans.Control Systems Technology, vol. 13, no. 4,
Fig. 13. PI force control responses for sinusoidal input with different values pp. 559-576, July, 2005.
for R on an environment with low stiffness. [4] P. Cominos, N. Munro, “PID controllers: recent tuning methods and
design to specification,” IEE Proc.-Control Theory Appl., vol. 149, no.
1, pp. 46-53, Jan., 2002.
V. CONCLUSION [5] G. Ziegler and N. B. Nichols, "Optimum setting for automatic
controllers," Trans. ASME, vol. 64, pp. 759-768, 1942.
In this paper, two novel practical PID tuning algorithms [6] J. Han, P. Wang, X. Yang, “Tuning of PID controller based on fruit fly
for motion control systems are proposed. These tuning optimization algorithm,” in Internaional Conference on Mechatronics
methods are for robust PID and PI controllers with velocity and Automation (ICMA), 2012, pp. 409-413.
[7] W. K. Ho, 0. P. Gan, E. B. Tay, E. L. Ang, “Performance and Gain and
feedbacks, for position and force control problems of servo Phase Margins of Well-Known PID Tuning Formulas,” IEEE
systems, respectively. The analytical derivations of these Transactions on Control Systems Technology, vol. 4, no. 4, pp. 473-477,
methods are based on the 2-DOF control approach which July, 1996.
considers the performance and robustness separately. This [8] A. Abbas, “A new set of controller tuning relations,” ISA Transactions,
superior property is achieved on simple PID structures with vol. 36, no. 3, pp. 183-187, 1997.
[9] R. Gorez, “New design relations for 2-DOF PID-like control systems,”
these algorithms in a way that the tuning takes so less effort
Automatica, vol. 39, pp. 901-908, Jan, 2003.
and time. Therefore, there are very effective for motion [10] D. E. Rivera, M. Morari, S. Skogestad, “Internal Model Control. 4. PID
control and robust both to parameter uncertainty and unknown Controller Design,” Chemical Engineering, vol. 25, pp. 253-26, 1986.
disturbances, yet very simple. They can be used by the [11] G. P. Liu, S. Daley, “Optimal-tuning PID control for industrial systems,”
engineers and the students with very basic control knowledge Control Engineering Practice, vol. 9, pp. 1185-1194, 2001.
just by following the direct tuning steps described in Section [12] E. Sariyildiz, K. Ohnishi, “Stability and Robustness of Disturbance
Observer Based Motion Control Systems,” IEEE Trans. Industrial
II. Moreover, this approach can be applied into different Electronics, vol. 62, no. 1, pp. 414-422, Jan., 2015.
problems. Hence, it has a high impact on the industry and the [13] E. Sariyildiz, K. Ohnishi, “On the Explicit Robust Force Control via
academia. The validity of the proposal is verified with the Disturbance Observer,” IEEE Trans. Industrial Electronics, to be
experiments. published.
[14] E. Sariyildiz, K. Ohnishi, “An Adaptive Reaction Force Observer
Design,” IEEE/ASME Trans.Mechatronics, vol. 20, no. 2, pp. 750-760,
Apr., 2015.
[15] K. Ohnishi, M. Shibata, and T. Murakami, “Motion control for advanced
mechatronics,” IEEE/ASME Trans.Mechatronics, vol. 1, no. 1, pp. 56–
67, Mar. 30, 1996.