Tracking/Robust Trade-off Design of a Sampled-data PID Controller for Second-order Plus Dead-time Systems

Ryo Kurokawa; Takao Sato; Ramon Vilanova; Yasuo Konishi

doi:10.2991/jrnal.2018.5.2.10

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Volume 5, Issue 2, September 2018, Pages 118 - 121

Tracking/Robust Trade-off Design of a Sampled-data PID Controller for Second-order Plus Dead-time Systems

Authors

Ryo Kurokawa, Takao Satotsato@eng.u-hyogo.ac.jp

Department of Mechanical Engineering, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280 Japan

Ramon Vilanovaramon.vilanova@uab.cat

Department of Telecommunications and Systems Engineering, Universitat Autònomade Barcelona, Edifici Q-Campus de la UAB, 08193 Bellaterra, Barcelona, Spain

Yasuo Konishikonishi@eng.u-hyogo.ac.jp

Department of Mechanical Engineering, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280 Japan

Available Online 30 September 2018.

DOI: 10.2991/jrnal.2018.5.2.10 How to use a DOI?
Keywords: PID control; Sampled-data system; SOPDT system; Sensitivity function; Robust
Abstract: In this paper, we propose a new design method of a second-order plus dead-time (SOPDT) sampled-data Proportional-Integral-Derivative (PID) control system, where the continuous-time plant is controlled using the discrete-time controller. The proposed control system is designed so that the tracking performance is optimized subject to the stability 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.
Copyright: Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

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, additional tuning methods have been studied recently. Although the stability of a control system is critical, its tracking performance is also important. However, because of the trade-off relationship between stability and tracking performance, they cannot be optimized simultaneously. Arrieta and Vilanova^3,4 proposed a simple PID tuning method that optimizes the tracking performance subject to a prescribed robust stability. In this method, the optimal PID parameters are decided based on a first-order plus dead-time (FOPDT) continuous-time system. In order to design a discrete-time control system, Tajika et al.⁵ proposed a design method for controlling a discrete-time FOPDT system. The present study discusses a design method of the PID controller for controlling a second-order plus dead-time (SOPDT) system, in which the continuous-time plant is controlled using the discrete-time controller. In the proposed method, both servo and regulation optimized control methods are designed. Finally, the effectiveness of the proposed method is demonstrated through numerical examples.

2. Description of the Control System

Consider the continuous-time controlled plant given as follows:

(1)P(s)=Kωn2s2+2ζωns+ωn2e−Ls

where K is the plant gain, ω_n is the natural angular frequency, ζ is the damping coefficient, and L is the dead-time. In the present study, we discuss the design method of the sampled-data control system using the following discrete-time PID control law:

(2)u(k)=Ce(z−1)e(k)+Cy(z−1)y(k)Cd(z−1)=Ce(z−1)+Cy(z−1)Ce(z−1)=Kp{1+TsTi(1−z−1)}Cy(z−1)=Kp{Td(1−z−1)Ts}

where u(k) is the control input, y(k) is the plant output, e(k)(=r(k) − y(k)) is the control error and r(k) is the reference. Moreover, T_s, K_p, T_i and T_d are the sampling time, the proportional gain, the integral time, and the differential time, respectively.

3. Definition of the Optimization Problem

As the constraint condition, the stability margin is defined using the sensitivity function, and the evaluation function for the tracking performance is also defined.

3.1. Constraint condition

The sensitivity function S_f(z⁻¹) is defined as follows:

(3)Sf(z−1)=11+Cd(z−1)Pd(z−1)

where P_d(z⁻¹) is the discrete-time controlled plant. Using the sensitivity function, the constraint condition is defined as follows:

(4)|Ms−Msd|=0Ms=maxω|Sf(e−jω)|

where M_s is the maximum value of the sensitivity function, and Msd is the desired value selected by the designer. The recommended range of Msd is from 1.4 to 2.0¹. The smaller the value of M_s, the larger the stability margin. On the other hand, the larger the value of M_s, the better the tracking performance, although the stability margin becomes small.

3.2. Evaluation function

In the present study, the evaluation function J is defined as the integral absolute error:

(5)J=∑k=0∞|e(k)|=∑k=0∞|r(k)−y(k)|

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

4. Controller Design

The PID parameters are optimized for a normalized system, and hence, dimensionless parameters are defined as τ = Lω_n, h = T_sω_n, κ_p = K_pK, τ_i = T_iω_n, and τ_d = T_dω_n. The range of these parameters are set as 0.1 ≤ 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 for discrete τ, h, and ζ, is obtained. In Fig. 1, the obtained normalized PID parameters are plotted by ∘, where Msd=1.4 and T_s = 0.01.

The desired normalized PID parameters for an arbitrary plant are decided by the linear interpolation from the data set. Practically speaking, the interpolated parameters are calculated using the nearest four points, as shown in Fig. 2. From this figure, the vector equation is obtained as follows:

(6)OP→=αOA→+βOB→+γOC→0≤α≤10≤β≤10≤γ≤1

where point O [τ^O, ζ^O, h^O, κpO , τiO , τdO ], A [τ^A, ζ^A, h^A, κpA , τiA , τdA ], B [τ^B, ζ^B, h^B, κpB , τiB , τdB ], and C [τ^C, ζ^C, h^C, κpC , τiC , τdC ] are the nearest points of the desired [τ^P, ζ^P, h^P]. Then, Eq. (6) is rearranged as follows:

κpP=κpO+α(κpA−κpO)+β(κpB−κpO)+γ(κpC−κpO)τiP=τiO+α(τiA−τiO)+β(τiB−τiO)+γ(τiC−τiO)τdP=τdO+α(τdA−τdO)+β(τdB−τdO)+γ(τdC−τdO)

Solving these equations, the desired κpP , τiP , and τdP for [τ^P, ζ^P, h^P] are obtained, where α, β, and γ are decided based on the following equations:

τP−τO=α(τA−τO)+β(τB−τO)+γ(τC−τO)ζP−ζO=α(ζA−ζO)+β(ζB−ζO)+γ(ζC−ζO)hP−hO=α(hA−hO)+β(hB−hO)+γ(hC−hO)

In Fig. 1, the interpolated parameters are plotted over the discrete calculated optimal parameters. Furthermore, M_s is calculated for both the preliminarily solved and interpolated systems using the approximation method, and the obtained M_s values are shown in Table 1. This result reveals that the proposed decision method is sufficiently effective.

Msd	Servo design			Regulation design

	Min	Mean	Max	Min	Mean	Max
1.4	1.398	1.403	1.440	1.399	1.403	1.453
1.6	1.599	1.605	1.668	1.597	1.605	1.663
1.8	1.790	1.807	1.909	1.798	1.807	1.897
2.0	1.996	2.010	2.156	1.997	2.009	2.137

Table 1.

Obtained M_s

5. Numerical Simulation

In this section, the effectiveness of the proposed method is confirmed.

5.1. Control performance for various values of ζ

First, the control performance is confirmed for ζ. The controlled plant is defined as K = 4.2, ω_n = 1.13, and L = 0.44 in Eq. (1), and T_s = 0.018. Here, we consider four pattern damping coefficients: ζ¹ = 0.451, ζ² = 0.69, ζ³ = 1.0, and ζ⁴ = 1.199. The control results are shown in Fig. 3. The reference value is set to 1.0, and the unit step disturbance signal is added after 20 s. Figure 3 shows that the proposed method is effective for under-and over-damping systems.

5.2. Verification of stability margin

Next, the stability margin is confirmed. Here, the controlled plant is defined as K = 2.02, ω_n = 0.91, ζ = 0.33, and L = 0.98 in Eq. (1), and T_s = 0.05. After 40 s, the dynamics is changed to K = 2.6, ω_n = 1.3, ζ = 0.43 and L = 0.43 as the model variation. Furthermore, Msd 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 value is 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 Msd , the larger the stability margin, and vice versa. On the other hand, the larger the value of Msd , the better the tracking performance, and vice versa.

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 the discrete-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 assigned Msd . Finally, the effectiveness of the proposed method is demonstrated through numerical examples.

Acknowledgements

The present study was supported by JSPS KAKENHI Grant Number JP16K06425. The authors would like to express their sincere thanks for the support from the Japan Society for the Promotion of Science.

References

1.KJ 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, UK, 2012.

3.O Arrieta and R Vilanova, Simple PID Tuning Rules with Guaranteed Ms Robustness Achievement, IFAC Proceedings, Vol. 44, No. 1, 2011, pp. 12042-1204.

4.O Arrieta and R Vilanova, Simple Servo/Regulation Proportional-Integral-Derivative (PID) Tuning Rules for Arbitrary Ms-Based Robustness Achievement, Industrial & Engineering Chemistry Research, Vol. 51, No. 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, in 23th Mediterranean Conference on Control and Automation (MED) (2015), pp. 249-254.

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Journal: Journal of Robotics, Networking and Artificial Life
Volume-Issue: 5 - 2
Pages: 118 - 121
Publication Date: 2018/09/30
ISSN (Online): 2352-6386
ISSN (Print): 2405-9021
DOI: 10.2991/jrnal.2018.5.2.10 How to use a DOI?
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Ryo Kurokawa
AU  - Takao Sato
AU  - Ramon Vilanova
AU  - Yasuo Konishi
PY  - 2018
DA  - 2018/09/30
TI  - Tracking/Robust Trade-off Design of a Sampled-data PID Controller for Second-order Plus Dead-time Systems
JO  - Journal of Robotics, Networking and Artificial Life
SP  - 118
EP  - 121
VL  - 5
IS  - 2
SN  - 2352-6386
UR  - https://doi.org/10.2991/jrnal.2018.5.2.10
DO  - 10.2991/jrnal.2018.5.2.10
ID  - Kurokawa2018
ER  -

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