Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 118-121___________________________________________________________________________________________________________Tracking/Robust Trade-off Design of a Sampled-data PID Controller for Second-order Plus
Dead-time Systems
Ryo Kurokawa and Takao Sato

Department of Mechanical Engineering, University of Hyogo
2167 Shosha, Himeji, Hyogo 671-2280 Japan
E-mail: tsato@eng.u-hyogo.ac.jp
Ramon Vilanova

Department of Telecommunications and Systems Engineering, Universitat Aut𝑜̀ nomade Barcelona
Edifici Q-Campus de la UAB, 08193 Bellaterra, Barcelona, Spain
E-mail: ramon.vilanova@uab.cat
Yasuo Konishi

Department of Mechanical Engineering, University of Hyogo
2167 Shosha, Himeji, Hyogo 671-2280 Japan
E-mail: konishi@eng.u-hyogo.ac.jp

Abstract
In this paper, we propose a new design method of a second-order plus dead-time (SOPDT) sampled-data ProportionalIntegral-Derivative (PID) control system, where the continuous-time plant is controlled using the discrete-timecontroller. The proposed control system is designed so that the tracking performance is optimized subject to thestability margin constraint. In the present study, the servo and regulation optimal controllers are designed. Finally,
the effectiveness of the proposed method is demonstrated through numerical examples.
Keywords: PID control, Sampled-data system, SOPDT system, Sensitivity function, Robust

to a prescribed robust stability. In this method, theoptimal PID parameters are decided based on a first-orderplus dead-time (FOPDT) continuous-time system. Inorder to design a discrete-time control system, Tajika etal.5 proposed a design method for controlling a discretetime FOPDT system. The present study discusses adesign method of the PID controller for controlling asecond-order plus dead-time (SOPDT) system, in which
the continuous-time plant is controlled using the discretetime controller. In the proposed method, both servo andregulation optimized control methods are designed.

1. Introduction
Proportional-Integral-Derivative (PID)1,2 control has
been widely used in industry. Since the performance of
PID control depends on the tuning parameters, additionaltuning methods have been studied recently. Although thestability of a control system is critical, its tracking
performance is also important. However, because of thetrade-off relationship between stability and tracking
performance, they cannot be optimized simultaneously.
Arrieta and Vilanova3,4 proposed a simple PID tuning
method that optimizes the tracking performance subject

Copyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 118-121___________________________________________________________________________________________________________

where 𝑀 is the maximum value of the sensitivity
function, and 𝑀 is the desired value selected by thedesigner. The recommended range of 𝑀 is from 1.4 to2.01. The smaller the value of 𝑀 , the larger the stability
margin. On the other hand, the larger the value of 𝑀 ,the better the tracking performance, although the stability
margin becomes small.

Finally, the effectiveness of the proposed method isdemonstrated through numerical examples.
2. Description of the Control System
Consider the continuous-time controlled plant given as
follows:
𝑃 𝑠𝐾𝜔
2𝜁𝜔 𝑠𝑠𝜔𝑒(1)3.2. Evaluation function
In the present study, the evaluation function 𝐽 is definedas the integral absolute error:

where 𝐾 is the plant gain, 𝜔 is the natural angularfrequency, 𝜁 is the damping coefficient, and 𝐿 is thedead-time. In the present study, we discuss the design
method of the sampled-data control system using thefollowing discrete-time PID control law:
𝑢 𝑘

𝐶 𝑧

𝑒 𝑘

𝐶 𝑧

𝑦 𝑘

𝐶 𝑧

𝐶 𝑧

𝐶 𝑧

𝐶 𝑧

𝐾 1

𝐶 𝑧

𝐾𝐽𝑇 1

The PID parameters are optimized for a normalized
system, and hence, dimensionless parameters are definedas 𝜏 𝐿𝜔 , ℎ 𝑇 𝜔 , 𝜅𝐾 𝐾, 𝜏𝑇 𝜔 , and𝜏𝑇 𝜔 . The range of these parameters are set as0.1 𝜏 1.0, 0.01 ℎ 0.10, and 0.3 𝜁 1.2.
In the proposed method, the constrained optimal problem
is preliminarily solved for a designated finite plant,
which is defined by discrete 𝜏, ℎ, and 𝜁, and the data set
in which the optimal normalized PID parameters fordiscrete 𝜏, ℎ, and 𝜁, is obtained. In Fig. 1, the obtained
normalized PID parameters are plotted by ∘, where𝑀
1.4 and 𝑇
0.01.

3. Definition of the Optimization Problem
As the constraint condition, the stability margin isdefined using the sensitivity function, and the evaluation
function for the tracking performance is also defined.
3.1. Constraint condition
is defined as follows:
11𝑃 𝑧

𝐶 𝑧

(3)

is the discrete-time controlled plant.
where 𝑃 𝑧
Using the sensitivity function, the constraint condition isdefined as follows:
|𝑀𝑀

𝑀 |

0(5)

4. Controller Design𝑧

where 𝑢 𝑘 is the control input, 𝑦 𝑘 is the plant
output, 𝑒 𝑘𝑟 𝑘𝑦 𝑘 is the control error, and𝑟 𝑘 is the reference. Moreover, 𝑇 , 𝐾 , 𝑇 , and 𝑇
are the sampling time, the proportional gain, the integraltime, and the differential time, respectively.

𝑆 𝑧

𝑦 𝑘 |𝑧𝑇The sensitivity function 𝑆 𝑧|𝑟 𝑘

A trade-off relationship exists between the servo
performance and the regulation performance. In thepresent study, the PID parameters are optimized for theservo and regulation control, respectively.

(2)

𝑇𝑇 1

|𝑒 𝑘 |(4)max 𝑆 𝑒119

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 118-121___________________________________________________________________________________________________________The desired normalized PID parameters for an arbitrary
plant are decided by the linear interpolation from the dataset. Practically speaking, the interpolated parameters arecalculated using the nearest four points, as shown in Fig.
2. From this figure, the vector equation is obtained asfollows:OP⃗𝛼OA⃗𝛽OB⃗

0𝛼

10𝛽

10𝛾1𝛾OC⃗(6)

where point O [𝜏 , 𝜁 , ℎ , 𝜅 , 𝜏 , 𝜏 ], A [𝜏 , 𝜁 ,ℎ , 𝜅 , 𝜏 , 𝜏 ], B [𝜏 , 𝜁 , ℎ , 𝜅 , 𝜏 , 𝜏 ], and C[𝜏 , 𝜁 , ℎ , 𝜅 , 𝜏 , 𝜏 ] are the nearest points of thedesired [𝜏 , 𝜁 , ℎ ]. Then, Eq. (6) is rearranged asfollows:𝜅𝜏𝜏𝜅𝜏𝜏𝛼 𝜅𝛼 𝜏𝛼 𝜏𝜅𝜏𝜏𝛽 𝜅𝛽 𝜏𝛽 𝜏𝜅𝜏𝜏𝛾 𝜅𝛾 𝜏𝛾 𝜏𝜅𝜏𝜏Solving these equations, the desired 𝜅 , 𝜏 , and 𝜏 for[𝜏 , 𝜁 , ℎ ] are obtained, where α, β, and γ are decided
based on the following equations:𝜏𝜁ℎFig. 1. Relationships among 𝜏, 𝜁, and 𝜅 , 𝜏 and 𝜏
1.4 and 𝑇0.01)
design, 𝑀𝜏𝜁ℎ𝛼 𝜏𝛼 𝜁𝛼 ℎ𝜏
𝜁ℎ𝛽 𝜏𝛽 𝜁𝛽 ℎ𝜏
𝜁ℎ𝛾 𝜏𝛾 𝜁𝛾 ℎ𝜏𝜁ℎIn Fig. 1, the interpolated parameters are plotted over thediscrete calculated optimal parameters. Furthermore, 𝑀
is calculated for both the preliminarily solved andinterpolated systems using the approximation method,
and the obtained 𝑀 values are shown in Table 1. This
result reveals that the proposed decision method issufficiently effective.

(servoTable 1. Obtained𝑀

Servo design
𝑀
1.4
1.6
1.8
2.0

Fig. 2. Image of the linear interpolation

120

Min
1.398
1.599
1.790
1.996

Mean
1.403
1.605
1.807
2.010

Regulation design
Max
1.440
1.668
1.909
2.156

Min
1.399
1.597
1.798
1.997

Mean
1.403
1.605
1.807
2.009

Max
1.453
1.663
1.897
2.137

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 118-121___________________________________________________________________________________________________________1.3, 𝜁 0.43, and 𝐿 0.43 as the model variation.
Furthermore, 𝑀 is varied as 1.4, 1.6, 1.8, and 2.0,
respectively, and the control results are compared. The
obtained results are shown in Fig. 4. The reference valueis 1.0, and the unit step disturbance signal is added after
20 s. The model variation is caused at 40 s. Figure 3
shows that the smaller the value of 𝑀 , the larger thestability margin, and vice versa. On the other hand, thelarger the value of 𝑀 , the better the tracking
performance, and vice versa.

5. Numerical Simulation
In this section, the effectiveness of the proposed method
is confirmed.
5.1. Control performance for various values of 𝜻First, the control performance is confirmed for 𝜁. The
controlled plant is defined as 𝐾 4.2, 𝜔1.13, and𝐿 0.44 in Eq. (1), and 𝑇
0.018. Here, we consider
four pattern damping coefficients: 𝜁0.451 , 𝜁0.69, 𝜁1.0, and 𝜁1.199. The control results areshown in Fig. 3. The reference value is set to 1.0, and theunit step disturbance signal is added after 20 s. Figure 3
shows that the proposed method is effective for underand over-damping systems.

Conclusion
In the present study, we have proposed a new design
method for controlling an SOPDT sampled-data system,
where the continuous-time plant is controlled by thediscrete-time PID control law. In the proposed method,
the PID parameters are designed for the normalized
system, and the tracking performance is optimized
subject to the assigned𝑀 . Finally, the effectiveness ofthe proposed method is demonstrated through numericalexamples.

5.2. Verification of stability margin
Next, the stability margin is confirmed. Here, thecontrolled plant is defined as 𝐾 2.02 , 𝜔0.91 ,𝜁 0.33 , and 𝐿 0.98 in Eq. (1), and 𝑇
0.05 .
After 40 s, the dynamics is changed to 𝐾 2.6, 𝜔Acknowledgements
The present study was supported by JSPS KAKENHI
Grant Number JP16K06425. The authors would like toexpress their sincere thanks for the support from theJapan Society for the Promotion of Science.
References
1. K. J. Åström and T. Hägglund, Advanced PID Control,
ISA-Instrumentation, Systems, and Automation Society,2006.
2. R. Vilanova and A. Visioli, PID Control in the Third
Millennium (Springer 2012, UK).
3. O. Arrieta and R. Vilanova, Simple PID Tuning Rules
with Guaranteed 𝑀 Robustness Achievement, IFAC
Proceedings 44(1), 2011, pp. 12042-1204.
4. O. Arrieta and R. Vilanova, Simple Servo/Regulation
Proportional-Integral-Derivative (PID) Tuning Rules forArbitrary 𝑀 -Based Robustness Achievement, Industrial& Engineering Chemistry Research 51(6), 2012, pp.
2666-2674.
5. H. Tajika, T. Sato, R. Vilanova and Y. Konishi, Optimal
PID Control in Discrete Time Using a Sensitivity Function,
23th Mediterranean Conference on Control andAutomation (MED), 2015, pp. 249-254.

Fig. 3. Output responses for each damping coefficient 𝜁1.4)(servo design and 𝑀

Fig. 4. Output responses for each 𝑀

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