experimental implementation of cdm based two mode

International Journal of Pure and Applied Mathematics Volume 118 No. 18 2018, 2241-2251 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue

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EXPERIMENTAL IMPLEMENTATION OF CDM BASED TWO MODE CONTROLLER FOR AN INTERACTING 2*2 DISTILLATION PROCESS JANANI.R1, *I. THIRUNAVUKKARASU2, VINAYAMBIKA. S. BHAT3 1

Department of Electronics and Instrumentation Engineering,Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya, Kanchipuram, Tamil Nadu, India. 2,3

Department of Instrumentation and Control Engineering,Manipal Institute Of Technology, Manipal University, Manipal, Karnataka, India. 1

ABSTRACT: The article illustrates the implementation of CDM based two mode controller for an interacting 2*2 distillation process. The decouplers are designed for the system to minimize the interaction between the loops, and first order plus dead time model is achieved for each decoupled subsystem. By using the graphical relationship between the controller and performance criteria the tuning parameters of two mode controller is obtained.This control strategy is demonstrated both in simulation and real-time environment using MATLAB/Simulink software. Also the closed loop performance indices are tabulated.

KEYWORDS: MIMO Process, Gain Margin, Phase Margin, Coefficient Diagram Method, Decouplers 1.

INTRODUCTION All chemical processes in process industries usually have two or more controlled outputs requiring two or more

manipulated variables such process generally called as MIMO (Multi Input Multi Output) process. Despite the considerable work that has been done on advanced multivariable controller for MIMO systems, multiloop PI controllersare favored in most commercial process control applications.The controller is designed and implemented on each loop by considering the loop interactions and internal coupling with time delay. Multiloop controllers have been widely used because of their better performance and robustness. In the current research, a simple decoupler plus decentralized PI controller is proposed for an interacting 2*2 distillation process based on the desired gain and phase margin.The article is organized as follows: Section 2 presents Decoupler design; Section 3 gives a brief summary on Coefficient Diagram Method (CDM), Section 4 gives the expressions for PI Controller design. Section 5 describes the experimental setup. The simulation and implementation results of CDM based two mode controller is shown in Section 6, which is then followed by conclusions.

2.

DECOUPLER DESIGN Decoupler design is one of the widely accepted techniques to diminish the interactions between the control loops. The

essence of decoupler is to compensate the effect of interaction between the loops. The design of decoupling controller matrix is very important. The decoupling controller can be obtained by the method proposed by Wang et al [6]. In order to design a simple decoupling controller [7]the loop transfer function is a first order plus dead time model then the diagonal elements are

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unity and in some cases the principal diagonal may degenerate into time delay element. The structureof unity feedback closedloop control system with decoupler is shown in Fig 1.

Figure.1. Structure of MIMO system with Decoupler

The generalized TITO process model with dead time is given by 𝐺𝑃 𝑠 =

𝑔11 (𝑠)𝑒 −𝜏 11 𝑠 𝑔21 (𝑠)𝑒 −𝜏 21 𝑠

𝑔12 (𝑠)𝑒 −𝜏 12 𝑠 𝑔22 (𝑠)𝑒 −𝜏 22 𝑠

(1)

The decoupling controller matrix can be given as 𝐷 𝑠 =

𝑑11 𝑑21

𝑑12 𝑑22

(2)

The elements of the decoupling controller matrix can be designed as follows 𝑑11 = 𝑑22 = 1 𝑔12

𝑑12 = − 𝑔 and𝑑21 = 11

3.

(3) −𝑔21

(4)

𝑔22

CO-EFFICIENT DIAGRAM METHOD Coefficient Diagram Method (CDM) is an algebraic approach over polynomial ring in the parameter space, where a

special diagram called “Coefficient diagram” is used to carry the necessary information and as the criteria of good design. In CDM the characteristic polynomial and the controller are designed simultaneously with the help of the coefficient diagram [9]. The characteristic polynomial specifies stability and response.

In this approach, the coefficients of the CDM

controllerpolynomials can be determined more easily.There are explicit relations between the performance parameters specified before the design because of simultaneous design structure, the designer is able to keep a good balance between the

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rigor and requirement and the complexity controller. For higher model system CDM requires higher order controller. The basic block diagram of a CDM control system is shown in Fig, 2. N(s) is the numerator and D(s) is the denominator polynomial of the plant transfer function G(s). A(s) is the forward denominator polynomial of the controller transfer function while F(s) and B(s) are the feedback numerator polynomials of the controller transfer function respectively. The characteristic polynomial of the closed loop system is given in the following form. 𝑛 𝑖 𝑖=0 𝑎𝑖 𝑠

𝑃 𝑠 = 𝑎𝑛 𝑠 𝑛 + ⋯ + 𝑎1 𝑠 + 𝑎0 =

(5)

The CDM design parameters namely, the stability index𝛾𝑖 , the equivalent time constant τ, are defined as follows. 𝛾1 = 𝑎𝑖2 𝑎𝑖+1 . 𝑎𝑖−1 , 𝜏 = 𝑎𝑖 𝑎0

i=1 ~ (n-1)

(6) (7)

Then characteristic polynomial will be expressed by 𝑎0 , 𝜏, 𝛾𝑖 as follows 𝑃 𝑠 = 𝑎0

𝑛 𝑖=2

𝑖−1 1 𝑗 =1

𝑗 𝛾𝑖−𝑗

𝜏𝑠

𝑖

+ 𝜏𝑠 + 1

(8)

Figure.2. Block diagram of CDM control design

The stability of the system is dependent on stability indices and the stability limit indices.The robustness of the plant is specified by variations of the stability indices.From the Frequency Time Domain Performance map in the (Kp, Ki) plane, the tuning parameters of the controller can be selected such that the desired specifications get satisfied.

4.

CONTROLLER DESIGN For design of suitable PI controller, the diagonal elements of 𝐻 𝑠 i.e., ℎ11 𝑠 and ℎ22 𝑠 are approximated into a first

order plus dead time (FOPDT) model as

ℎ𝑖𝑖 𝑠 =

𝐾𝑒 −𝜃𝑠

(9)

𝑇𝑝 𝑠+1

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Where 𝐾, 𝑇𝑝 and θ are process gain, time constant and delay time respectively [5]. The exponential term can be approximated by Taylor series such that the denominator term of the model is not affected. The above FOPDT model represented in Eq. (9) becomes 𝑁(𝑠)

ℎ𝑖𝑖 𝑠 = 𝐷(𝑠) =

−𝑘𝜃𝑠 +𝑘

(10)

𝜏 𝑝 𝑠+1

The CDM based two mode controller is

𝐺𝑐 𝑠 =

𝐾𝑝 𝑠+𝐾𝑖

(11)

𝑠

The characteristic polynomial of the model transfer function represented by Eq. (12) is P s = τp − kθK p s2 + 1 − θkK i + kK p s + kK i

(12)

τ = 1 + kK p − kθK i

(13)

kK i

From Eq. (13), K i can be calculated as K i = 1 + kK p

kτ + kθ

(14)

By substituting s = jω in the loop transfer function, the numerator and the denominator of the polynomial can be decomposed into their even and odd parts as given in Eq. (15) hii jω =

N e −ω 2 +jωN 0 (−ω 2 )

(15)

D e −ω 2 +jωD 0 (−ω 2 )

The characteristic polynomial of the closed loop system can be written as ∆ s =

K i Ne − K p ω2 No cos ωθ + ω K p Ne + K i No sin ωθ − ω2 Do + j ω K p Ne + K i No cos ωθ − K i Ne −

Kpω2No sinωθ+ωDe=0

(16)

Equating the real and imaginary parts of the characteristic equation to zero K p −ω2 No cos ωθ + ωNe sin⁡ωθ K p ωNe cos ωθ + ω2 No sin⁡ωθ Kp =

+ K i Ne cos ωθ + ωNo sin⁡ωθ + K i ωNo cos ωθ − Ne sin⁡ωθ

= ω 2 D0 = − ωDe

ω 2 N o D o +N e D e cos ωθ +φ +ω N 0 D e −N e D o sin ωθ +φ

(17) (18)

−A∗ N 2e +ω 2 N 2o

The values of Kp can be obtained from the above Eq. (18) and can be substituted in Eq. (14). The point of interaction in the FTDPM gives the value of Kp and Ki for the desired gain and phase margin.

5.

EXPERIMENTAL SETUP

Distillation is a process for the separation of two different mixtures based on differences in the boiling points of its constituents. In the current research, a mixture of Isopropyl alcohol and water in the ratio of 30% and 70% are considered for

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the distillation process here we consider the transfer function model for simulation. The reflux flow rate (L) is measured as LPH and reboiler power rate (Q) measured in KW are identified as the manipulated variables and temperature of tray 5 (T5) and tray 1 (T1) in Deg. Cel are the controlled variable (PV). Also the research presents the simulation of control algorithm using MATLAB/Simulink software with and without load disturbance. All the four performance indices such as Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Absolute Error (ITAE) and Integral Square Time Error (ISTE) are analyzed.

Figure.3. Distillation Column Setup

6.

SIMULATION & RESULTS

The mathematical model identified by Vinaya and Arasu for interacting distillation column shown in Fig.3 is considered for the simulation studies and for real-time implementation [2, 3]. The process transfer function model is

𝐺 𝑠 =

−0.13𝑒 −0.03𝑠

0.18𝑒 −0.03𝑠

1.14𝑠+1 −0.34𝑒 −1.22𝑠

0.64𝑠+1 0.18𝑒 −0.03𝑠

1.23𝑠+1

0.32𝑠+1

(19)

Note that the time constants and dead times in the pilot plant distillation column model are measured in terms of hours and the process gain unit is given by ˚C/%. Since for the sampling time of 0.01 sec is used in the VDPID-03 DAQ card, the open loop data has huge data collection. Hence, the data were converted into hours to get the plant model. The decoupler is

𝐷 𝑠 =

1 (0.604 𝑠+1.89)𝑒 −1.19𝑠 1.23𝑠+1

1.5789 𝑠+1.385 0.64𝑠+1

(20)

1

The diagonal elements of open loop transfer function of the process is obtained as

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𝑞11 =

0.21𝑒 −2𝑠 1.35𝑠+1

and𝑞22 =

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−0.29𝑒 −1.545𝑠

(21)

0.735 𝑠+1

The above loop transfer function can be decomposed into numerator and denominator terms.On equating the real and imaginary terms of the characteristic polynomial of the system to zero, we can able to obtain the frequency dependent stability boundary locus in plane (Kp, Ki) by assuming gain margin = 3db and phase margin =60 deg as shown in Fig.4(a) and Fig.4(b)

0

2 1.8

-0.5

1.6 1.4

Ki

Ki

X: -0.7446 Y: -1.253

-1

X: 1.205 Y: 1.148

1.2 1

-1.5

0.8 -2

0.6 0.4

-2.5

0.2 0 -4

-2

0

2

4

6

8

-3 -6

10

-5

-4

Kp

-3

-2 Kp

-1

0

1

2

(b)

(a) Figure 4(a) and (b) FTDPM for Loop 1 and Loop 2 For Loop 1, For gain margin =2db 𝐾𝑝11 = −2.38 cos 2𝜔 + 3.214𝜔 sin 2𝜔

𝐾𝑖11 = 3.214𝜔 2 cos 2𝜔 + 2.38𝜔 sin 2𝜔

(22a)

For phase margin = 60 deg 𝐾𝑝11 = −4.762 cos 2𝜔 + 1.047 + 6.43𝜔 sin(2𝜔 + 1.047)

𝐾𝑖11 = 6.428𝜔 2 cos⁡ ( 2𝜔 + 1.047) + 4.762𝜔 sin( 2𝜔 + 1.047)

For 𝜏 = 1.8 𝐾𝑝11 = 6.428𝜔 sin 2𝜔 − 4.762 cos 2𝜔

𝐾𝑖11 =

1+𝑘𝑘 𝑝 𝑘𝜏 +𝑘𝜃

(22b)

(22c)

For Loop 2, For gain margin =2db 𝐾𝑝22 = 1.724 cos 1.545𝜔 − 1.267𝜔 sin 1.545𝜔

𝐾𝑖22 = −1.267𝜔 2 cos 1.545𝜔 − 1.724𝜔 sin 1.545𝜔

(23a)

For phase margin = 60 deg 𝐾𝑝22 = 3.45 cos 1.545𝜔 + 1.047 − 2.535𝜔 sin(1.545𝜔 + 1.047)𝐾𝑖22 = −2.545𝜔 2 cos⁡ ( 1.545𝜔 + 1.047) − 3.45𝜔 sin( 1.545𝜔 + 1.047)

For 𝜏 = 1.8 𝐾𝑝22 = 3.45 cos 1.545𝜔

− 2.535𝜔 sin( 1.545𝜔 )

𝐾𝑖22 =

1+𝑘𝑘 𝑝 𝑘𝜏 +𝑘𝜃

(23b)

(23c)

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The resulting PI Controller is 1.204 0 1.148 0 𝐾 = (24) 0 −0.7446 𝑖 0 −1.253 The frequency time domain performance is obtained by plotting frequency domain stability region in (Kp, Ki) plane. From 𝐾𝑃 =

which Kp& Ki values can be chosen which satisfy the desired properties.

Figure. 5. Servo Response of Y1 and Y2 when r1=1 and r2=0

Figure 6. Servo Response of Y1 and Y2 when r1=0 and r2=1

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Figure.7. Servo Response of Y1 and Y2 when r1=67deg and r2=55deg

Figure 8.Implementation of Controller Designed Based Coefficient Diagram Method for SP-T1=68 Deg. Cel and SP-T5=75 Deg. Cel on a Pilot Plant Binary Distillation Column

7.

CONCLUSION Fig.5 to Fig. 7 shows the simulation studies for the obtained controller with good tracking of the servo response.

The simulation result shows that the closed loop performance is achieved without any oscillation. The response shows that the controller provides proper tracking with the givensetpoint. Table 1 shows the performance index for the servo response by CDM Method. The performance indices such as IAE, ISE, ITAE, and ISTE for the nominal plant is determined and compared with plant with uncertainty of +30% in process gain(K), process gain(K) and time constant (τ) and the third process gain (K), time constant (τ) and delay time (θ). .

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Level of Uncertainty Nominal Plant 30 % Uncertainty in Kp 30 % Uncertainty in Kp and τ 30 % Uncertainty in all the parameters

8.

Y1 5.299 5.933

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Table 1. Performance Index for the Servo Response by CDM Method IAE ISE ITAE Y2 ∑(Y1+Y2) Y1 Y2 ∑(Y1+Y2) Y1 Y2 ∑(Y1+Y2) 5.53 10.837 4.405 4.542 8.947 17.41 21.32 38.73 8 10.5 16.503 4.612 7.758 12.37 26.46 89.95 116.41 7

Y1 8.788

ISTE Y2 ∑(Y1+Y2) 8.798 17.586

10.27

32.39

42.66

7.496

15.3 6

22.856

5.338

10.76

16.098

46.29

189.4

235.69

15.35

73.34

88.69

12.6

90.7

103.3

6.608

226.2

232.808

225.9

2733

2958.9

58.59

7850

7908.59

ACKNOWLEDGEMENT

The authors would like to thank,Dept. of Electronics and Instrumentation, SCSVMV University, Kanchipuram for providing the online library facility and workspace for doing the simulation studies and Dept. of Instrumentation and Control Engineering, MIT, Manipal University for providing the real-time experimental facility for carrying out the experimental work 9.

FUTURE WORK

The above presented algorithm can be extended to the non-square systems. Presented we are in the process of getting the pressure models of Tray 1 and Tray 5. The pressure transmitters were installed in the pilot plant distillation column. The pneumatic control valve fixing on the distillate tank is also in progress.

REFERENCES I. II.

III. IV.

V.

VI.

M. Senthilkumar, S. Abraham Lincoln. (2015), “Multiloop PI controller for achieving simultaneous time and frequency domain specifications”, Journal of Engineering Science and Technology, Vol.10., Issue.8, pp-1103-1115. Vinayambika. S. Bhat, I. Thirunavukkarasu, S. Shanmuga Priya. (2016), “An Experimental Study on Implementation of Centralized PI Control Techniques on Pilot Plant Binary Distillation Column”, International Journal of Chem. Tech Research, Vol.9., no. 11, pp. 244-251. Vinayambika. S. Bhat, S. Shanmuga Priya, I. Thirunavukkarasu (2016), “Design and Implementation of Decentralized PI Controller for Pilot Plant Binary Distillation Column”, International Journal of ChemTech Research, vol.9, no.12, August. Janani. R, Vinayambika.S. Bhat, I. Thirunavukkarasu, Shreesha. C, “Multi-variable PI Controller based on Gain and Phase Margins for a Pilot Plant Binary Distillation Column”, Submitted in the conference CHEMCON 2017, organized by Dept. of Chemical Engineering, Haldia Institute of Technology, Haldia, West Bengal, India to be held on 27-30th Dec 2017 Janani. R, Vinayambika.S. Bhat, I. Thirunavukkarasu, “ Admissible set of PI Controllers based on Gain and Phase Margin for a Pilot Plant Binary Distillation Column”, Submitted in 30th Symposium IChemE 2017, organized by Monash University, Malaysia, to be held on 6-7th Dec. 2017 Wang. Q. G., Huang. B., Guo. X.(2000), “Auto-tuning of TITO decoupling controllers from step tests”. ISA Transactions, 39(4), pp. 407-418

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VII. T.K.Sethuramalingam, B.Nagaraj, “A Comparative Approach On PID Controller Tuning Using Soft Computing Techniques”, International Journal of Innovations in Scientific and Engineering Research (IJISER), Vol.1, No.12, pp.460465, 2014. VIII. W.K. Ho, C.C. Hang and J.H. Zhou (1995), “Performance and Gain and Phase Margins of well-known PI tuning Formulas”, IEEE Transactions on Control Systems Technology, Vol.3, No.2, June IX. S. Manabe. (1998), “Coefficient Diagram Method”, IFAC Automatic Control in Aerospace. X. Tan,N.,I.Kaya and D.P.Atherton (2003) ,”Computation of stabilizing PI and PID controllers,”Proc. Of the IEEE Intern. Conf. on the Control Applications (CCA2003), Istanbul, Turkey.

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