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6th International Conference on Control Engineering and Information Technologies, 25-27 October 2018, Istanbul, Turkey

TECHNICAL PROGRAM COMMITTEE Abdellah Kouzou, DZ

Besbes Moez, TN

Abdullah Ersan Oğuz, TR

Bilal Erol, TR

Abdulrahman Bajodah, SA

Bülent Özkan, TR

Abdurrahman Karamancıoğlu, TR

Cemil Öz, TR

Adam Fifth, US

Cengiz Hacızade, TR

Agustin A. Sanchez de la Nieta Lopez, AN

Cenk Ulu, TR Ceren Gülra Melek, TR

Ahmed Rhif, TN

Cihan Tayşi, TR

Ahmet Denker, TR

Cihan Tunç, US

Ahmet Serbes, TR Claudia Fernanda Berrio, TR Ahmet Yiğit Arabul, TR

Cristiane Lionço, BR

Akin Tascikaraoglu, TR

Cüneyt Yılmaz, TR

Ali Fuat Ergenc, TR

Çağrı Güngör, TR

Ali İhsan Çanakoğlu, TR Daniel Feliu Talegon, ES Ali Kara, TR David Pozo Camara, RU Ali Rıfat Boynueğri, TR Defoort Michael, FR Alpaslan Parlakci, TR Dimah Dera, US Altuğ İftar, TR

Doğan Onur Arısoy, TR

Amaç Güvensan, TR Duygun Erol Barkana, TR Aytaç Altan, TR

Ecir Uğur Küçüksille, TR

Azeem Khan, ZA

Elena Battini Sönmez, TR

Bahar İlgen, TR

Elif Çiçek, TR

Banu Ataşlar Ayyıldız, TR Elizabet Gonzales, ES Banu Diri, TR

Emir Öztürk, TR

Bedri Kekezoğlu, TR

Engin Özdemir, TR

Berk Altıner, TR Enver Tatlicioglu, TR

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FRACTIONAL ORDER SIGNALS, SYSTEMS AND CONTROL

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Root-Locus Analysis of Fractional Order Transfer Functions Using LabVIEW: An Interactive Application

245

Tuning of Fractional Order PID Controller using CS Algorithm for Trajectory Tracking Control

251

Online fractional-order PIα controller tuning scheme for a class of First Order Plus Time Delay (FOPTD) plants

257

Designing an Undamped Oscillator Using Fractional Order Delayed Systems

263

Online Tuning of Derivative Order Term in Fractional Controllers

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273

PID CONTROL Intelligent mixed H2/H∞ FOPID controller optimized for radar guided missile

274

PID Control of DC Servo Motor using a Single Memory Neuron

279

Stochastic Optimization of PID Parameters for Twin Rotor System with Multiple Nonlinear Regression

284

High Performance PID Control Design for Second Order Systems Based on Time-Scale Separation - a Case Study

289

Hyperbolic Tangent Adaptive LQR+PID Control of a Quadrotor

295

301

CONTROL & INSTRUMENTATION Chaotic Lorenz Synchronization Circuit Design for Secure Communication

302

Detection of Leak Site in Sloped Transmission Lines

308

Fuzzy rules adaptive digital controller for nonlinear-switching LED current drive circuit

314

Enhancement of Full Coverage Markov Model for Diverse Systems with Common Cause Failures

320

A Comparative Study of the Friction Models with Adaptive Coefficients for a Rotary Triple Inverted Pendulum

326

Online Fractional order PID Controller tuning for a Class of Fractional Order Plants based on Bode’s Ideal Transfer Function, FRIT and RLS

332

Contribution to PI Control Speed of a Doubly Fed Induction Motor

340

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Stochastic Optimization of PID Parameters for Twin Rotor System with Multiple Nonlinear Regression Barkın Büyükçakır Faculty of Engineering and Architecture Izmir Katip Celebi University Izmir, Turkey [email protected]

Furkan Elmaz Faculty of Engineering and Architecture Izmir Katip Celebi University Izmir, Turkey [email protected]

Savaş Şahin Faculty of Engineering and Architecture Izmir Katip Celebi University Izmir, Turkey [email protected]

Levent Aydın Faculty of Engineering and Architecture Izmir Katip Celebi University Izmir, Turkey [email protected]

Twin Rotor Multi Input Multi Output System (TRMS) [4] is a dynamic system that can perform pitch and yaw motions (Fig. 1). Because of its nonlinear inherent behavior and high cross coupling effect between its two rotors, the controller design in order to control TRMS is accepted as a challenging work. For PID controller design and optimization for TRMS, Particle Swarm [5], and Genetic Algorithm [6] methods are appropriate for the optimization of PID parameters. It has been observed that these optimization processes are effective in bringing system performance to the desired level.

Abstract—This paper demonstrates finding the PID controller parameters which are used to acquire the desired settling time of the horizontal and vertical position variables of the Twin Rotor experiment setup. Modeling and optimization processes have been achieved based on multiple nonlinear regression analysis and Differential Evolution algorithm, respectively. In the modeling part a hybrid approach combining the ideas of Artificial Neural Network and regression have been introduced. For this process, training and test phases are executed on Mathematica environment with constraints on settling time, rising time and maximum overshoot. In order to provide comparison of the output parameters, the PID parameters based on the Ziegler-Nichols tuning method is used. The experimental performance of the system is tested, thus an appropriate set of methods for tuning PID parameters is introduced.

In this paper, adjustment of PID coefficients has been performed in the Mathematica environment with nonlinear regression (NLR) modeling [7] and Differential Evolution [8] algorithm for minimizing settling time criteria of TRMS. The study was applied experimentally on TRMS. The proposed Nonlinear Multiple Regression method and the Differential Evolution algorithm have been tested for correct identification of the system's responses and determination of the appropriate PID coefficients for TRMS.

Keywords — Twin Rotor system; PID; optimization; nonlinear regression; Differential Evolution.

I.

INTRODUCTION

Proportional-Integral-Derivative (PID) controller is a conventional control method widely used in industry and education [1]. In PID controller design, parameters called Kp (Proportional coefficient), Ki (Integral coefficient) and Kd (Derivative coefficient) are the main factors affecting controller performance. Methods such as Ziegler-Nichols [2] and Good Gain [3] are used to determine and tune these coefficients experimentally. Although these methods give satisfactory results in simple systems, they can’t provide robustness at the desired level in complex and highly nonlinear systems.

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In the present paper, the main steps can be summarized as:

284

i.

Data set acquisition on TRMS for implementation of the optimization process.

ii.

Application of the Ziegler-Nichols tuning method on TRMS and recording of the performance of the obtained PID parameters.

iii.

Explanation of the optimization process and application of the process on the obtained dataset.


III. ZIEGLER-NICHOLS TUNING In order to compare the results obtained at the end of the proposed optimization process, the PID parameters of the TRMS system were tuned by the Ziegler-Nichols method. According to the Ziegler-Nichols method, the values of Kp, Ki and Kd are equalized to 0 and Kp values increased from 0 until the pitch and yaw movements are separately oscillated. These critical Kp values (Kc) are recorded and oscillation periods (Tc) are measured for two motions and the Ziegler-Nichols lookup table (Table I) is used to determine PID parameters. The obtained PID parameters are applied to the system. The response graph (Figures 3,4), applied PID parameters and outputs (ts, tr, µ0) of the system were recorded (Table II). The errors have been calculated by Normalized Mean Square Error (NMSE) [10], Integral Square Error (ISE) [11] and Integral Time Weighted Absolute Error (ITAE) [12]. These results are also given in Table III for future comparison.

Fig. 1. TRMS experiment setup [9]

iv.

v.

vi.

Adjustment of PID coefficients with multiple nonlinear regression (NLR) modeling and Differential Evolution algorithm by minimizing settling time criteria of TRMS

TABLE I.

Testing of the results based on the proposed nonlinear multiple regression model and Differential Evolution algorithm for correct identification responses of the system. Therefore, determination of the appropriate PID coefficients for TRMS.

ZIEGLER-NICHOLS LOOKUP TABLE

Controller Type P

Kp 0.5*Kc

PI PID

Ki -

0.45*Kc

1.2*Kp/Tc

-

0.6*Kc

2*Kp/Tc

Kp*Tc/8

Performance comparison between Ziegler-Nichols method and optimization process and the discussion of the results II. DATASET ACQUISITION

In order to apply the proposed method of optimization, a data set was acquired experimentally from TRMS. PID parameters Kp, Ki and Kd which are regarded as the dominant parameters on the system response, were gradually incremented and system outputs rising time (tr), settling time (ts) and maximum overshoot (µ0) are recorded within each iteration. All PID parameters and outputs are recorded for pitch and yaw motions independently. The pseudocode for data set generation is given in Fig. 2.

Fig. 3. Ziegler-Nichols tuning performance on pitch motion

Fig. 4. Ziegler-Nichols tuning performance on yaw motion Fig. 2. Pseudocode of dataset generation algorithm

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Kd

-


TABLE II.

ZIEGLER-NICHOLS TUNED PID PARAMETERS AND OUTPUTS

SSE = ∑ (Y – Ŷ)2

(2)

SST = ∑ (Y – Ȳ)2

(3)

R2 = 1- SSE/SST

(4)

Value Parameter

Pitch

Kp

8.4739

Yaw

8.7404

Ki

7.3848

23.2512

Kd

0.20723

1.0575

tr

19.98 s

8.4 s

ts

29.37 s

8.82 s

µ0

3%

1.73%

TABLE III.

The formula given in (2) is the sum of squared errors. where Y is the set of real output values, and Ŷ is the set of outputs of the model. Eq. (3) is the sum of all squares. In this equation, Ȳ is the mean of the set of real outputs. The surface plots of trained pitch and yaw models for Kd = 7 are shown in Figs. 5 and 6, respectively. showing the suitability of the model for ts, μ0 and tr outputs.

ERROR CRITERIA EVALUATION OF ZIEGLER-NICHOLS TUNING Value Error Criterion

Pitch

The models based on training sets were applied to the test sets of the outputs ts, tr and µo. The performances of the models on test sets are given in Table IV, and the surface plots of pitch and yaw models for Kd = 7 are also given in Figures 7 and 8, respectively. The fitness values obtained are considered to be satisfactory in this study.

Yaw

NMSE

0.18

ISE

17.89

0.24 24.74

ITAE

77.18

111.095

Although there is a dominant linear relationship between the inputs and outputs of the system, as seen in Fig. 5, Fig. 6, Fig. 7 and Fig. 8, there are also higher-order, regions with moderately low-dominance. The selected fourth order nonlinear model showed higher conformity compared to the linear model in these regions.

IV. MODELING AND OPTIMIZATION A. Modeling In this section, the mathematical modeling of the ts, tr and μ0 responses of the pitch and yaw motions to the PID coefficients is discussed. ts, tr, μ0 outputs are functionalized according to the input parameters, assuming that the parameters affecting the system are Kp, Ki, Kd. known, the PID controller is linear, but the system can display nonlinear responses due to disturbances. In order to predict these responses by an appropriate model, it is possible to introduce more efficient models by combining nonlinear multiple regression analysis and artificial neural network modeling procedure at this stage. The data set acquired in Section II is used for the calculation of 53 regression constants of the multiple non-linear regression model (see (1)).

R2 VALUES OF OUTPUTS

TABLE IV.

Pitch Outputs

(1)

Training R2 Values

Yaw Testing R2 Values

Training R2 Values

Testing R2 Values

ts

0.99

0.98

0.99

0.98

tr

0.98

0.97

0.99

0.98

µ0

0.91

0.89

0.88

0.87

ck = {0,5,13,25}, dk = {2,3,4,5} 80% of the data sets of the pitch and yaw movements were selected randomly as training sets. The remaining data are assigned as test sets. Thus, the model will be tested on this data sets that are not considered in the traning step. The R2 values of the regression models obtained for the outputs ts, tr and μ0 are calculated according to the formulations given in (2-4) and the results are given in Table IV. These fitness values indicate that the model is fitted to the training set adequately.

Fig. 5. Surface plots for ts, µ0 and tr outputs of the pitch motion for training phase

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TABLE VI. Motion

CALCULATED OPTIMIZATION RESULTS Kp

Ki

Kd

µ0

tr

ts

Pitch

4.24

69.97

9.57

%3

5s

8s

Yaw

1

96.62

6.4

%1

4s

4.6 s

Fig. 6. Surface plots for ts, µ0 and tr outputs of the yaw motion for training phase

V. PERFORMANCE EVALUATION The PID parameters obtained at the end of the optimization stage were applied on the real system and comparisons with the performance of the PID parameters acquired in Section III were made as shown in Fig. 6 for pitch and in Fig. 7 for yaw motions. The performance chart and evolution of error criterion of the optimized PID parameters are given in Table VII and Table VIII respectively.

Fig. 7. Surface plots for ts, µ0 and tr outputs of the pitch motion for testing phase

When the optimized PID parameters are applied on the real system, it is observed that for both pitch and yaw motions, the rising time, settling times and maximum overshoot values of the system are lower than those of the Zeigler-Nichols coefficients. It is also seen that the cross coupling effect was less dominant in the optimized PID parameters because the maximum exceeding percentages are lower and the oscillation is smaller in magnitude at the rising and settling stages. Moreover, it is determined that the system behaved more ideally with optimized parameters compared to those of the Ziegler-Nichols method.

Fig. 8. Surface plots for ts, µ0 and tr outputs of the yaw motion for testing phase

B. Optimization Process The optimization process is carried out based on Differential Evolution method in Mathematica environment. In this process all output models receive PID coefficients as inputs. Minimization of ts was the aim of the optimization process. In order for the optimization process to conform to the behavior of the real system, upper and lower constraints have been introduced for tr and μ0. This prevents the optimization process from giving unrealistic results. Constraints on PID parameters are determined for this specific system by expert opinion. Optimization inputs, outputs and constraints are given in Table V. The PID parameters and outputs given by the optimization process are shown in Table VI. TABLE V.

CONSTRAINTS OF INPUTS AND OUTPUTS Constraints Pitch

Inputs

Outputs

Fig. 9. Performance comparison on pitch motion

Yaw

1≤ Kp ≤ 10

1≤ Kp ≤ 10

10≤ Ki ≤ 100

10≤ Ki ≤ 100

1≤ Kd ≤ 10

1≤ Kd ≤ 10

5≤ tr ≤ (Ts-3)

1≤ tr ≤ (ts-0.3)

%0 ≤ µ0 ≤ %5

%0 ≤ µ0 ≤ %2

6≤ ts ≤ 17

3≤ ts ≤ 5

Fig. 10. Performance comparison on yaw motion

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TABLE VII.

EXPERIMENTAL OPTIMIZATION RESULTS

[9]

Feedback Instruments Ltd., "Twin Rotor MIMO Datasheet" Mathworks,2013.[Online].Address: http://www.feedbackinstruments.com/pdf/brochures/33-007-PCI.pdf [Available 2018]. [10] C. J. Willmott, “Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance ” Clim. Res., vol. 30, no. 1, pp. 79–82, 2005.” [11] H. Górecki and L. Popek, “Integral square error (ISE) criterion for systems with time-delay,” Arch. Autom. i Robot., vol. 36, 1991. [12] F. G. Martins, “Tuning PID Controllers using the ITAE Criterion*,” Int. J. Eng. Educ., vol. 21, no. 5, 2005.

Value Parameter

Pitch

Yaw

ts

16.66 s

7.28 s

tr

8.901 s

7.015 s

µ0

%2.4

%1.13

TABLE VIII.

ERROR CRITERIA EVALUATION OF OPTIMIZATION RESULTS



Value Error Criterion

Pitch

Yaw

NMSE

0.13

0.15

ISE

14.78

18.15

ITAE

72.19

94.25

VI. CONCLUSION In this study, the optimization of the PID parameters for TRMS were performed with nonlinear regression method and differential evolution algorithm. Results of the optimization process are experimentally applied to the system. It has been observed that the parameters obtained from the result of optimization process enabled the system to behave closer to the desired performance compared to Ziegler-Nichols parameters according to performance indexes calculated in Chapter III and V. The nonlinear mathematical model used in the optimization process also includes unpredictable behavior resulting from high coupling effects between rotors and noise, so it can also be used in the optimization of maximum overshoot and rising time regarding the high R2 obtained in the presence of these behaviors. The presented series of methods can be used to optimize a specific characteristic of PID controlled systems where a data set can be obtained. REFERENCES [1]

[2] [3] [4] [5]

[6]

[7] [8]

H. Wu, W. Su and Z. Liu, "PID controllers: Design and tuning methods," 2014 9th IEEE Conference on Industrial Electronics and Applications, Hangzhou, 2014, pp. 808-813. Feedback Instruments, "Twin Rotor MIMO," LD Didactic, 2013. Haugen, Finn Aakre & , Techteach. (2010). The Good Gain method for PI(D) controller tuning. Feedback Instruments, "Twin Rotor MIMO," LD Didactic, 2013. R. Maiti, A. Kolay, K. Das Sharma and G. Sarkar, "PSO based PID controller design for twin rotor MIMO system," Proceedings of The 2014 International Conference on Control, Instrumentation, Energy and Communication (CIEC), Calcutta, 2014, pp. 56-60. J. G. Juang, M. T. Huang and W. K. Liu, "PID Control Using Presearched Genetic Algorithms for a MIMO System," in IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), vol. 38, no. 5, pp. 716-727, 2008. Shouhong Wang,Nonlinear regression: a hybrid model,Computers & Operations Research,Volume 26, Issue 8,1999,Pages 799-817 R. Storn and K. Price, "Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces," Journal of Global Optimization 11(4), pp. 341-359, 1997.

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