Optimal tuning of decentralized fractional order PID controllers for TITO process using equivalent transfer function

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ScienceDirect Cognitive Systems Research 58 (2019) 292–303 www.elsevier.com/locate/cogsys

Optimal tuning of decentralized fractional order PID controllers for TITO process using equivalent transfer function S.K. Lakshmanaprabu a,⇑, Mohamed Elhoseny b, K. Shankar c a

Department of Electronics and Instrumentation Engineering, B.S. Abdur Rahman Crescent Institute of Science and Technology, Chennai, India b Faculty of Computers and Information, Mansoura University, Egypt c Department of Computer Applications, Alagappa University, Karaikudi, India Received 25 September 2018; received in revised form 9 April 2019; accepted 21 July 2019 Available online 24 July 2019

Abstract This paper presents a method of designing independent fractional order PI/PID controller for two interacting conical tank level (TICTL) process based on the Equivalent Transfer Function (ETF) model and simplified decoupler. The TICTL process is decomposed into independent single input single output (SISO) model using ETF. A bat optimization algorithm is utilized to independently design a diagonal fractional order PI/PID controller based on ETF model. The effectiveness of the proposed method is illustrated with simulation examples and also the experimental TICTL process utilized to validate the proposed method. Ó 2019 Published by Elsevier B.V.

Keywords: Fractional order PI/PID control; TITO process; Optimal control; Decoupler; Equivalent transfer function; Bat algorithm

1. Introduction Controlling the multi input multi output interconnected process is the most challenging problems for control system engineer. Several authors have been proposed the control designing procedures for multivariable systems, but still, the researchers working on this problem to enhance the control performance for Multi Input Multi Output (MIMO) process. There are three major control schemes such as decentralized (multiloop), decoupled control, centralized control have been presented for MIMO process. In general, decentralized (multiloop) control scheme has been widely used in the industrial process because of its advantages in failure tolerance and easy implementation. However, Model predictive control (MPC) is capable of ⇑ Corresponding author.

E-mail addresses: [email protected] (S.K. Lakshmanaprabu), [email protected] (M. Elhoseny). https://doi.org/10.1016/j.cogsys.2019.07.005 1389-0417/Ó 2019 Published by Elsevier B.V.

handling MIIMO control problem effectively; But MPC is used mostly used on higher level to provide setpoints to the lower level PID controllers (Xiong, Cai, & He, 2007). The MPC is employed in supervisory level, where the sampling time of MPC is higher than the lower level PID control loops. Managing the coupling problem using MPC can be troublesome due to the limited bandwidth of MPC (Garrido, Va´zquez, & Morilla, 2012). Hence, the PI/PID based controllers are used to control the process at the lower level. The improvement in the lower level control system increases the overall performance of multivariable control. The multiloop PI/PID controllers generally utilized technique in the multivariable process. The purpose of utilizing the PI/PID in multi loop because of its simple structure, failure tolerance and capacity of meeting user specification. But the tuning of PI/PID is difficult due to the interaction effect between input and output, because the design of one loop depends on another loop. There has been many designing procedure proposed in the literature

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such as detuning (Besta & Chidambaram, 2016), sequential loop closing, independent loop (Vu & Lee, 2010b) and relay auto tuning method. In the detuning method, the controller parameters are found for most important loop transfer function without considering interaction effect and then the controller gains are detuned by considering interaction effect to meet some user control specification. But the performance and stability of closed loop system has not been discussed properly in the detuning procedures. In the sequential tuning methods, controllers are tuned while closing the loop one after other, but the final controller design completely depends on the order of other controller. The multiloop PI/PID Controller provides better control performance for the process with modest interaction. It fails to provide reasonable control performance when the interaction effects between loops are significant. In such a case, the decoupler based control scheme is preferred for MIMO process. Three types of decouplers are available in the literature such as ideal, inverted, and simplified decoupler (Cai, Ni, He, & Ni, 2008). The ideal decoupler formed using the inversion of process transfer function which may result in complex dynamics. The simplified decoupler is widely used to develop an ETF model and decentralized controller is designed for corresponding ETF model. Vu and Lee (2010a) have designed independent IMC based PI/PID controller using Effective Open Loop Transfer Function (EOTF) model, where the higher order EOTF model is approximated into reduced order model using maclaurin series. The EOTF resulted in a higher order model, it requires model reduction techniques to form reduced order model which make the controller design easier. Hajare and Patre (2015) have approximated the higher order EOTF model to reduced order model using frequency response fitting. The formulation of EOTF for higher order process is complex and decoupler design also makes controller complicated control structure. Xiong and Cai (2006) has demonstrated a method of designing decentralized PI/PID controller using effective transfer function. The ETF model is developed using the information of effective relative gain array, relative gain array (RGA), relative frequency of open loop transfer function model. Vijaykumar, Rao, and Chidambaram (2012) has designed ETF model using normalized relative gain array (RNGA) and relative average time array (RARTA) to form ETF model and then controller is tuned using maclaurin series for corresponding EFT model. Wang, Huang, and Guo (2000) has developed systematic design for full dimensional PID control for higher order process using approximated decoupler but the robustness of closed loop control system cannot be guaranteed. Shen, Sun, and Xu (2014) expanded the normalized decoupling method for higher order process. Rajapandiyan and Chidambaram (2012) have proposed new method of decoupled process approximation for higher order process, where the simplified decoupler with ETF approximation method is combined to form new approxi-

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mation procedure for decoupled process, and then the diagonal controller is designed using simplified internal model control (SIMC) method for approximated decoupled process. The integer order PI/PID controller still dominated in the industries. Recently, the fractional order controller has received extensive consideration in industries and academia. The three hundred years old fractional calculus has some interesting history, however last few decades the fractional calculus has been gain popularity in control system engineering and other engineering application. The researcher have demonstrated that the FOPID controller outer performs than the integer order PID controller for many application. However there is lack of tuning methods available compared to PI/PID controller. The analytical and numerical based tuning method for FOPI/FOPID is reported in the literature. The complete review of FOPID tuning methods are discussed in Shah and Agashe (2016). Generally, the FOPID controller tuned to meet user defined specification (time, frequency domain) by analytical method and optimization method. Mostly, the optimization based tuning methods are used to obtain the FOPI/FOPID controller parameters. In Agababa (2015), the particle swarm optimization, differential evolutionary optimization, bat optimization used to tune the fractional order PI/PID controller for SISO process. The design of multiloop FOPID controller is difficult for MIMO process because of its coupling effect between input and output. The main objective of this paper is to make the student to understand about the real time industrial control problem. Also, this papers helps control system student community to understand about the fractional order control system in simple manner. The design of coupled system is difficult due to its coupling effect; hence, the coupled interconnected system is separated into equivalent single input single output system to make design of control system easier. The design of one loop controller depends on the other loop, so it is always complicated to design a FOPID controller for multivariable system. By using independent loop method, the MIMO process is decomposed into independent single input single output (SISO) for tuning of controller easily. In this paper, the FOPID controller is used as a diagonal controller with simplified decoupler. The diagonal controller parameters are obtained using bat optimization algorithm for minimum values of time weighted integral absolute error. 2. Literature survey Chen, Tang, Li, and Lu (2018) proposed a synthesis tuning method for PIk to satisfying the frequency domain specification such as gain crossover frequency and phase margin. In general, the synthesis methods for tuning controllers are proposed for reduced order model. The optimal PIk controllers are tuned for higher order process according to the specified phase margin and crossover frequency (Chen et al., 2018).

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Bingul and Karahan (2018) compared the PID and FOPID controller performance where the controller parameters are tuned using artificial bee colony and particle swarm optimization. The controller tuned with three different cost function such as settling time, rise time, overshoot, and steady-state error (Bingul & Karahan, 2018). Shah and Agashe (2017) demonstrated the effect of order of differentiation (m) and integration (k) on time domain specification. The PID and FOPID controller implemented for the quadruple tank level system (Shah & Agashe, 2017). Haji and Monje (2018) presented multiloop FOPID controller for distillation column process using improved bat algorithm. The multivariable FOPID controllers are designed with and without decoupler, the unknown controller parameters are obtained using different optimization method with different time domain specifications as a objective function. The distillation column process parameters such methanol in the distillate and bottom compositions is controlled using two feedback loops, where the two loops contains ten FOPID controller parameters. The unknown ten parameters are optimally tuned for minimizing the integral error criteria (Haji & Monje, 2018). Zhang, Yang, Zhou, and Gui (2018) presented FOPID tuning method using state transition algorithm. The effect of sample time on the objective function of optimization for FOPID is discussed (Zhang et al., 2018). Idamakanti, Nasir, and Singh (2018) developed a IMC based integer and non integer controller for automatic generation control of two area power system using the method of simplified decoupling techniques (Idamakanti et al., 2018). Jin, Wang, and Liu (2016) developed a IMC based controller for Two Input Two Output (TITO) system, the complex model is decomposed into independent SISO model and then the higher order SISO model is reduced using Maclaurin series expansion. Then, the robust IMC based controller is tuned for reduced order model (Jin et al., 2016). Lakshmanaprabu, Banu, and Hemavathy (2017) developed IMC based Fractional Order PID controller for TITO process where the complex model is decomposed into SISO model and Controller parameters tuned independently using novel bat algorithm (Lakshmanaprabu et al., 2017). San-Millan, Feliu-Talego´n, Feliu-Batlle, and RivasPerez (2017) demonstrated an effectiveness of PI and PID controller for the real time system, and the TITO real model is identified as the first order model by closing the other loops. The effective open loop transfer function is developed from the real time laboratory prototype of a hydraulic canal. The reduced effective model was utilized for design of PI and PID controller using frequency domain specifications (San-Millan et al., 2017). Moradi (2014) proposed centralized FOPID controller for TITO process and the controller parameters are tuned using the genetic algorithm. The feedback control for

TITO system is designed with centralized fractional order PID controllers which contains 20 tuning parameters. The tuning of controller with multi objective function is requires lot of computation power for finding the optimal centralized FOPID controller parameters (Moradi, 2014). Chaib, Choucha, and Arif (2017) demonstrated Bat Algorithm (BA) based FOPID controller for power system stabilizer. The BA is applied to find the optimal controller values for the four different fitness function such as IAE, ISE, Integral Absolute Time Error (ITAE) and Integral Square Time Error (ITSE). The comparison of different fitness function for FOPID tuning are recorded and concluded that the overshoot with Integral Absolute Error (IAE), Integral Time weighted Absolute Error (ITAE) is bigger than Integral Squared Error (ISE), Integral Time weighted Squared Error (ITSE). However, IAE and ITAE have better response and settling time than ISE, ITSE (Chaib et al., 2017). Nasirpour and Balochian (2017) developed a FOPID controller tuning method for TITO air-conditioning system. The air-conditioning system is decomposed into SISO system using decoupling techniques and the FOPID controller parameters are optimally tuned using PSO algorithm through minimization of multi objective function which includes settling time, rise time, overshoot and integral time squared error (Nasirpour & Balochian, 2017). Katal and Narayan (2017) proposed a optimal fractional order PID controller for two tank liquid level system. The controller is tuned using multi-objective variant of particle swarm optimization. The multi objective minimization problem is solved using Pareto optimal set which is a simplified illustration of data. It is very difficult to use Pareto optimal set for multi dimensional optimization problems. The level diagrams are utilized for helping the selection of better solution from Pareto optimal set (Katal & Narayan, 2017). 3. Basic fractional calculus Fractional calculus is an emerging technique in engineering and sciences, it has some unique features and it represents system completely. Non integer order differentiator and Integrator are represented by differ-integral operators a Dqt . The fractional derivative and fractional integral combined and expressed in generalized form, 8 dq q>0 > < dtq q 1 q¼0 ð1Þ a Dt ¼ > :Rt q ðdsÞ q<0 a where q is a fractional order and ‘a’ is an initial conditions. Many definitions are proposed for fractional differ- integral. The Riemann and Liouville (R-L) definition for fractional derivative is the most popular, this is defined as follows,

S.K. Lakshmanaprabu et al. / Cognitive Systems Research 58 (2019) 292–303 q a D t f ðt Þ ¼

d q f ðtÞ 1 dn q ¼ d ð t  aÞ Cðn  qÞ dtn

Z

t

ðt  sÞnq1 f ðsÞds 0

ð2Þ where n is the integer which is not less than q i.e. n1  q < n and C is the Gamma function. Z inf tz1 et dt ð3Þ CðzÞ ¼ 0

The R-L definition for fractional integral is given by Z t 1 dn q D f ð t Þ ¼ ðt  sÞq1 f ðsÞds ð4Þ a t CðqÞ dtn 0 The fractional order transfer function can be obtained easily for the system with non integer order differential equation. q a Dt y ðt Þ

¼

þ    þ an1 Dct n1 y ðtÞ þ an Dct n y ðtÞ

b1 Dgt 1

uð t Þ þ    þ

bm Dgt m

uð t Þ

ð5Þ

The fractional order transfer function of a single variable system with zero initial condition can be defined by Lf 0 Dat f ðtÞg ¼ sa F ðsÞ

ð6Þ

Therefore, GðsÞ ¼

b1 sg1 þ b2 sg2 þ . . . þ bm sgm a 1 s c1 þ a 2 s c2 þ . . . þ a m s cn

ð7Þ

gi ; ci are order of numerator and ai, bi are real numbers. In this paper, the outloup approximation for the fractional order is used for simulation. The fractional derivative or integral sa is approximated into integer order transfer function using a recursive distribution of poles and zeros. sa  k

N Y 1 þ ðs=xz;n Þ   1 þ s=xp;n n¼1

ð8Þ

The approximated transfer function assigned in the pre specified frequency range. K is a regulated gain, it is adjusted in both sides of equation for a unit gain at 1 rad/s. The frequencies of poles and zeros are given by, pffiffiffi xz;1 ¼ xl g; xp;n ¼ xz;n e ; n ¼ 1 . . . N; xz;nþ1 pffiffiffi ¼ xp;n g ; n ¼ 1 . . . N  1 ð9Þ e ¼ ðxh =xl Þ

m=N

; g ¼ ðxh =xl Þ

ð1 mÞ=N

4. Equivalent transfer function The TITO model of this process is defined by the G(s),   g11 ðsÞ g12 ðsÞ G ðsÞ ¼ ð10Þ g21 ðsÞ g22 ðsÞ The multiloop controller transfer function matrix is,   gc1 ðsÞ 0 G c ðsÞ ¼ ð11Þ 0 gc2 ðsÞ

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The effective open loop transfer function is developed by incorporating decouplers in the TITO process. The open loop transfer function between input and output is developed while other loop is closed. It is not easy to develop an EOTF model by closing other loops, because closing other loops requires controller, so the EOTF model is completely depends on the other loop controller. The EOTF model between u1 and y1 is developed while loop 2 is closed with the feedback controller gc2. It is assumed that the gc2 gives perfect control, where the output of y2 attains setpoint with no transient. geotf 11 ðsÞ ¼ g 11 ðsÞ 

g12 ðsÞg21 ðsÞ g22 ðsÞ

ð12Þ

geotf 22 ðsÞ ¼ g 22 ðsÞ 

g21 ðsÞg12 ðsÞ g11 ðsÞ

ð13Þ

The EOTF model resulted in higher order dimension, which is difficult to use directly in the controller design, hence the model has to be reduced using any approximation method. To avoid this computational complexity, the equivalent transfer function (ETF) model obtained using relative gain array (RGA), relative normalized gain array (RNGA), relative average resitance time array (RARTA) methods. The steady state information and transient response information of TITO process are incorporated to form ETF model, which is used to develop a controller separately. The normalized gain (KN,ij), RGA(K), RNGA(/), Relative average resistance time(c) and RARTA (C) are found to develop a ETF model. Consider the First Order Plus Dead Time (FOPDT) transfer function matrix (G(s)) for TITO process is given below, " k11 h s # k 12 e 11 eh12 s s11 þ1 s12 þ1 GðsÞ ¼ ð14Þ k 21 eh21 s sk2222þ1 eh22 s s21 þ1 where Kij is the process gain between ith output and jth input; sij is the time constant of FOPDT model between ith output and jth input; hij is the dead time of FOPDT model between ith output and jth input (i, j = 1,2). The process gain at steady state,   k11 k12 K ¼ Gðs ¼ 0Þ ¼ ð15Þ k21 k22 Relative gain array RGA,   k11 k22 k12 k21 1 K¼ jK j k12 k21 k22

ð16Þ

The normalized gain array for TITO process G(s) is,   KN;11 KN;12 KN ¼ ð17Þ KN;21 KN;22 where KN;ij ¼

kij rij

¼

kij , sij þ hij

the rij is the average residence

time of ith output to jth input.

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RNGA,



/ ¼ K N  K T N ; where / ¼

/11

/12

/21

/22



RARTA,   c11 c12 C¼ c21 c22 r ^

ð18Þ

ð19Þ /

where cij ¼ rijij ¼ Kijij is a relative average residence time, C ¼ /  K, where /is RNGA and K is the RGA. The ETF model is developed using RGA, RNGA and RARTA as, b g ij ðsÞ ¼

kij 1 ecj hj s Kij cij sij s þ 1

The ETF model for TITO process is,   g^11 ðsÞ g^12 ðsÞ b G ðsÞ ¼ g^21 ðsÞ g^22 ðsÞ

ð20Þ

ð21Þ

In the presence of decoupler, the TITO system behaves like two independent loops. There are many decoupling methods are available such as ideal decoupler, inverted decoupler, normalized decoupler. Wang et al. (2000) decoupler is a recommended realizable decoupler where the extra time delay is incorporated. 2 3 evðh22 h21 Þs  gg1211 ððssÞÞ evðh12 h11 Þs 5 DðsÞ ¼ 4 g ðsÞ ð22Þ vðh21 h22 Þs vðh11 h12 Þs  g21 e e 22 ðsÞ where,  1 if h P 0 v ð hÞ ¼ 0 if h < 0 5. Fractional order PI/PID control The FOPID is a special case of PID controller where it is less sensitive to parameter variation of controller The fractional order PID controller provides more flexibility in tuning of controller with additional two tuning parameters. The transfer function of FOPID controller Gfopid(s) is given below,  1 Gfopid ðsÞ ¼ Kp 1 þ k þ td sl ð23Þ ti s where Kp is the proportional gain, ti is the integral time constant, k is the order of integrator, td is the derivative time and m order of derivative. The performance FOPID has been demonstrated by many researchers that the FOPID controller outer performs than conventional PID controller due to the additional tuning parameter such as integrator order and differentiator order. The addition two parameters provide better flexibility in adjustment of gain and phase characteristics in closed loop control system.

The FOPID controller tuned to meet the time domain performance criteria such as integral square error (ISE), Integral absolute error (IAE), integral time absolute error (ITAE). The ISE based tuning provides the lower overshoot in the closed control response than IAE based controller setting. But the ISE based controller may tends to settle in large settling time. The ITAE based controller provides less overshoot response with faster settling time. ITSE is less sensitive and it is not comfortable computationally. In this paper, fractional order PI/PID controller tuned using bat optimization algorithm. The controllers are independently designed using each ETF model. 6. Bat optimization algorithm In evolutionary computation techniques, Bat optimization has gained popularity in all engineering nonlinear multimodal problems (Yang & Gandomi, 2012). The natural forging strategies of bat is inspired and mimicked into bat optimization algorithm. The bat emits sounds pulses and receive back to identify the prey. The pulse emission rate varies depending on the location of species. Initially bats flies randomly with velocity Vi at position xi with fixed frequency fmin to search prey. It updates its frequency and pulse emission rate depending on the position of prey and then the bat position is updated to move near to prey. The loudness varies from maximum to minimum when the bat moves towards the prey. The bats starts with position xti and velocity vti at frequency fi to find the prey. The bat founds new solution (prey) by adjusting frequencies and loudness, then the quality of the solution is ranked by loudness and pulse emission rate which is developed based on the global solution (prey). The bat optimization algorithm (BOA) flow chart is shown in the Fig. 1. The position xti and velocity vti at frequency fi are given by, The frequency of ith bat, f i ¼ f min þ ðf max  f min Þ g

ð24Þ

Here g is a random vector drawn from uniform distribution. The velocity of ith bat, t

vti ¼ vt1 þ ðxi  xtbest Þ f i i

ð25Þ

where xtbest is the global best location. The bat new position, þ vti xti ¼ xt1 i

ð26Þ

The best position is found by random walk, xnew ¼ xold þ wAt

ð27Þ

where ‘w’ is a random number from 1 to 1. A is an average loudness of all bats. The loudness variation directly proportional to the closeness of prey position, when the bat moves towards to the prey then it reduces its loudness and increases the pulse emission rate. That is given as, T

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297

Start

Initialize the bat population, pulse frequency rates (r i) and loudness A0 Calculate fitness value for each bat location (J RMSE)

Generate new locations by adjusting f i and updating vi Consider First bat

rand> ri

Generate local solution around the best location Replace new temporary location with the solution of local search

Evaluate fitness of the new temporary location of this bat

Next iteration

Consider next bat

If the new location

A0> rand

Accept temporary location as new Location increase ri and decrease A0

Keep old location as new location

All the bats are considered?

Rank the location, select best location and save the r i and A0

Is termination criteria satisfied?

End

Fig. 1. Flow chart of bat optimization algorithm.

Ati þ 1 ¼ la Ati ;

rti þ 1 ¼ r0i ½1  exp ð  ctÞ

ð28Þ

where la is a constant in the range of [0, 1] and c is a positive constant. When time reaches infinite the rti equal to roi . For any case, 0 < la < 1 and c > 0 Ati ! 0;

rti ! r0i ; as

t!1

ð29Þ

For the easy implementation of BOA, the standard recommended la, c values are chosen as 0.9. 6.1. Tuning of -FOPID controller using BOA The minimization of integrated error performance indices are the general specific control requirement. The

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Integral Time Absolute Error (ITAE) which yields faster settling time with less overshoots. The optimization for controller design problem is formulated and the constraints bounds are given below, Z 1



y ðtÞ  y ðtÞ dt J¼ ð30Þ sp 1

" DðsÞ ¼

7. Simulation study 7.1. Case 1: VL column system The bench mark example of VL column given by luyben (1986) is a TITO process with higher interaction between input and output. The Transfer function model of VL column is given below, " # s 0:3s Gp ðsÞ ¼

2:2e 7sþ1

2:8 e1:8s 9:5 sþ1

1:3e 7sþ1

4:3 e0:35s 9:2sþ1

ð31Þ

The normalized gain matrix, RGA, RNGA, RARTA are found using Eqs. (16)–(18) and then the ETF model is obtained using Eq. (18). " # 1:353 0:959s 2:078 0:2656s e e 6:91sþ1 6:91sþ1 b GðsÞ ¼ 4:4769 1:593s 2:6455 0:334s ð32Þ e e 8:41sþ1 8:794sþ1 The EOTF/ETF models are developed by assuming the perfect control assumption, but this assumption can be validated only by incorporating decoupler in the open loop model. The simplified decoupler D(s) is developed using equation (22).

0:5909

ð5:9907sþ0:6512Þ 1:45s e 9:5sþ1

e0:7s

# ð33Þ

Additional time delay included in the decoupler D(s), 

GðsÞ:DðsÞ ¼

0

where ysp is the set points of controller and y1 are the output of ETF model, t is the simulation time (in s). The ‘t’ is the time weighting factor which give less weight to initial error and large weightage to steady state error. The minimum value of ‘J’ produce controller response with less overshoot and faster settling time. The tuning of FOPID controller parameters using optimization techniques are depends on the selection of objective function for controller tuning. The different kind of objective function gives different values of FOPID controller parameters. The ITAE based indices provide better controller with faster servo tracking and disturbance rejection without offset. The main reason for selection of ITAE for the FOPID controller design is that the order of integrator produce offset for a long time. Hence, the ITAE is utilized for providing more penalties to the offset error. The ITAE is a better performance indicator of the closed loop control scheme where overshoot, setting time, rise time, offset are considered. Therefore, the ITAE is used as the objective function of multiloop FOPID controller design. The controller with small values of ITAE considered as best controller for system and also it indicates that the controller response is fast.

1

g11 ðsÞ g12 ðsÞ g21 ðsÞ g22 ðsÞ



   g11 ðsÞ 0 1 d 12 ðsÞe0:7s ¼ 0 g 22 ðsÞ d 21 ðsÞ e0:7s ð34Þ

The decoupled processes are, 1:3535 0:9559s e 6:691s þ 1 2:6455 1:034s e g 22 ðsÞ ¼ 8:794s þ 1

g 11 ðsÞ ¼

ð35Þ ð36Þ

Rajapandiyan and Chidambaram (2012) obtained the multiloop PI controller parameter using simplified internal model control (SIMC) with EOTF model. The SIMC based PI controller gc(s) for FOPDT process gp(s) is given by,  k 1 hs e ; gc ¼ k c 1 þ g p ðsÞ ¼ ð37Þ ss þ 1 ti s þ 1  s ð38Þ kc ¼ ; ti ¼ minðs; 8hÞ 2Kh The reported (Rajapandiyan & Chidambaram, 2012) PI controller setting is, " #   1 0 2:5858 1 þ 6:691s   Gc ðsÞ ¼ ð39Þ 1 0 1:6066 1 þ 8:276s The optimal FOPI/FOPID controller settings are obtained via minimization of ITAE for ETF model (g11*(s), g22*(s)) using BOA. The optimal controller values are tabulated in Table 1. The closed loop response for V-L column is shown in Fig. 2 and Fig. 3. The unit step changes are applied at both references. At t = 50, the setpoint of loop2 is changed from 0.25 to 1. The proposed controller is compared with FOPI controller and SIMC based controller proposed by Rajapandiyan and Chidambaram. From the Fig. 2, Fig. 3, it is observed that the proposed FOPID controller produce better performance than the FOPI, PI controller. The interaction effect reduced effectively with faster regulatory response by the proposed FOPID controller.

Table 1 The optimal FOPID values of ETF model based on IAE.

PI FOPI FOPID

Loop1 Loop2 Loop1 Loop2 Loop1 Loop2

Kc

ti

k

td

m

ITAE

2.588 1.6066 2.772 1.955 3.91 2.7

6.691 8.276 6.7 8.96 7.1689 9.20

1 1 1 1.002 1.005 0.98

– – – – 0.32 0.59

– – – – 0.96 0.979

2.661 3.473 2.618 3.326 1.521 3.041

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299

1.4 setpoint(yr1)

1.2

PI(SIMC) FOPI FOPID

1

y1

0.8

0.6

0.4

0.2

0

0

10

20

30

40

50

60

70

80

90

100

time(sec)

Fig. 2. Servo and regulatory response for the VL column (loop1). 1.4 setpoint(y ) r2

PI(SIMC) FOPI FOPID

1.2

1

y2

0.8

0.6

0.4

0.2

0

0

10

20

30

40

50

60

70

80

90

100

time(sec) Fig. 3. Servo and regulatory response for the VL column (loop 2).

7.2. Case 2: The TICTL process experimental setup To demonstrate the practical applicability, the two interacting conical tank level process is considered and model parameters are identified from the experimental data. This obtained model is further used for simulation study and resulted controller is implemented on the laboratory two interacting conical tank level process (TICTLP) setup. The TICTLP is shown in Fig. 4 which consist of two tanks in series with variable speed pump. The laboratory setup of TICTLP is interfaced to personnel computer using ADuC841 micro controller based data acquisition (DAQ) card. The DAQ has 8 channel analog inputs and 2 channel analog outputs with 12 bit resolution. The input and output voltage range of DAQ is 0–5 V. The water levels in the tanks 1, 2 are measured using ABB 2600 series differential pressure transmitters. The output of transmitter

is 4–20 mA current signal, which is converted into 0–5 V range for interfacing with ADuC841 DAQ card. The variable speed pumps P1, P2 with thyristor driver circuit are fitted to provide liquid inflow to tanks 1 and 2 respectively for the input ranges from 0 to 5 V. The input voltage is given to tyrister driver circuit (TE10 A) to produce varying voltage (0–230 V) for running a pump at different speed. PWM signal is generated and given to tyristor driver circuits (TE10 A) to produce the varying voltage (0–230 V) for running a pump at different speed. The change in applied voltages is directly proportional to the speed and inflow rate of liquid. The technical specification of experimental setup is given in Table 2. 7.2.1. Black box modeling Obtaining the transfer function model of the plant is not a simple task for TITO process. The accurate modeling can

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Table 2 Technical Specification. Part name

Description

DPT Pump Rotameter TE10A

ABB 2600T SERIES pressure transmitter Maximum height of Tank 1 and 2 Input Voltage to pump 1 and 2 Thyristor power control

The RGA, RNGA, average residence time (Tar) and RARTA are found to develop ETF model. The gain array (K), relative gain array (K) of TICTLP is,     1:8361 0:7234 1:182 0:182 K ¼ ;K¼ 0:74 1:89 0:182 1:182 The normalized gain matrix found using Eq. (17),

improve the controller performance and it can help the controller to reduce the effect of uncertainties. The aim is to make the water level in conical tank remained stable, and then first order model is to be captured by applying step changes in each input. The system is allowed to take new steady state at fixed input flow rate. The TICTLP is highly nonlinear and coupling process. The nonlinearity is due to the variation in the area of conical tank, hence, the linearized transfer function matrix developed around single operating points. The operating points are selected based on the domain knowledge and input output characteristics. In this study, the nominal operating points are fixed as [V1 = 50% (2.5 V), V2 = 50% (2.5 V), h1 = 49% (24.5 cm), h2 = 51.2% (25 cm)]. The nominal input voltages v1 = 50%, v2 = 50% are applied to pumps and then tank level allowed to take steady state output. The both tanks took 1000 s to reach steady state, and then the step change is introduced at input voltage v1 from 50% (2.5 V) to 60% (3 V) while the input 2 (V2) voltage remained at constant 50% (2.5 V). The output level h1, h2 are recorded with the sampling time of 1 s, from this recorded data the first order plus dead time (FOPDT) transfer function (g11(s)) between input V1 and h1 and transfer function (g22(s) between input v1 and h2 is identified using process reaction curve method. Similarly, the same steps are followed to obtain the transfer function g21 (s) between V2 and h1 and the g22 (s) between V2 and h2. The identified transfer function model of TICTLP at operating condition [V1 = 50% (2.5 V), V2 = 50% (2.5 V), h1 = 49% (24.5 cm), h2 = 51.2% (25 cm)] is, " GP ðsÞ ¼

1:8361 e11:5s 340:7 sþ1 0:74 e19:1s 407:3 sþ1

0:723 e19:2s 415:4 sþ1 1:89 e12:4s 365:6sþ1

#

 KN ¼

0:0052 0:0017 0:0017 0:0050



The RNGA, RARTA found using Eqs. (18) and (19),  /¼

1:1246

- 0:1246

- 0:1246

1:1246



 ;C¼

0:9511

0:6833

0:6833

0:9511



The ETF model transfer function parameter is found using Eq. (20) " # 1:553 3:966 e10:9s 283:9 e13:1s 324:1 sþ1 sþ1 b G P ðsÞ ¼ 4:057 1:598 e12:6s 347:7sþ1 e11:8s 278:3 sþ1 The Wang et al. (2000) decoupler is used in the system to incorporate the extra time delay in the system. " # 134:167 s0:3938 7:7s 1 e 415:4 sþ1 DðsÞ ¼ 143:145 s0:3915 6:7s e 1 407:3 sþ1 The independent decomposed process is considered for controller design. The controller is designed independently using, g 11 ðsÞ ¼

1:553 1:598 e10:9s ; g 22 ðsÞ ¼ e11:8s 324:1 s þ 1 347:7 s þ 1

The controller setting obtained using the BOA as discussed in the Section 4 and 6.1. The controller settings are tabulated in Table 3. The BOA optimization is adopted with population size 20, iteration size 100. The initial guess of lower and upper bounds of loop1 controller Kp1 is fixed between the range of [0–20], Ki values between the range of [0–10], Kd values between the range of [0–10]. Similarly, loop2 controller parameters kp2, ki2 and kd2 are fixed.

Fig. 4. Real time experimental setup of TICTP.

S.K. Lakshmanaprabu et al. / Cognitive Systems Research 58 (2019) 292–303

301

Table 3 The optimal FOPID values of ETF model based on ITAE.

PI

Loop1 Loop2 Loop1 Loop2 Loop1 Loop2

FOPI FOPID

Kc

ti

k

Kd

m

IAE

ITAE

ISE

0.623 0.605 6.82 7.05 10.1 9.74

87.2 94.4 116.7 121.5 105.3 112.8

– – 0.991 0.995 0.998 1.00

– – – – 10.32 9.81

– – – – 0.752 0.673

11.41 12.53 10.92 12.04 10.14 11.89

846.6 941.4 821.1 902.82 798.3 891.05

6.18 6.69 5.92 6.20 5.47 5.84

The BOA is implemented in MATLAB 2014a, the simulation were carried out using the personnel computer with 3.2 GHz with Intel V process with 4 GB of random access memory. The optimal values of FOPID controller parameters obtained by BOA with IAE, ISE and ITAE are tabulated in Table 3. The proposed controller is implemented for TICTLP and experimental results of closed loop response are shown in Figs. 5–8. The step variation is introduced in closed loop system, where the setpoint of liquid level in tank 1 (h1) is applied at 500 sec and liquid level in tank 2 (h2) is applied

at 1000 sec. At t = 1000 sec, the setpoint of h1 is changed from 20 cm to 30 cm while the liquid level in tank 2 is unchanged. The setpoint variation in tank 1 level will affects the liquid level in tank 2 due to the interaction flow. But, the proposed controller compensates the interaction effect faster than the FOPI, PI controller. The servo tracking of proposed controller is faster than the FOPI, PI Controller with minimum settling time and overshoot. It can be clearly observed that the proposed controller track and regulate the desired liquid level faster than the PI, FOPI controller.

50 Setpoint (h )

45

1

PI FOPI FOPID

1

% Level in Tank 1 (h )

40 35 30 25 20 15 10 5 0

0

500

1000

1500

2000

2500

3000

3500

4000

time(sec)

Fig 5. Output response h1 for TICTLP.

100

% Controller output (V1)

90 80 PI FOPI FOPID

70 60 50 40 30 20 10 0 0

500

1000

1500

2000

2500

time(sec)

Fig. 6. Controller response for TICTLP.

3000

3500

4000

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S.K. Lakshmanaprabu et al. / Cognitive Systems Research 58 (2019) 292–303 60

2

% Level in Tank 2(h )

50 Setpoint (h 2 )

40

PI FOPI FOPID

30

20

10

0 0

500

1000

1500

2000

2500

3000

3500

4000

time(sec)

Fig. 7. Output response for TICTLP.

100 90

PI FOPI FOPID

2

% Controller output (V )

80 70 60 50 40 30 20 10 0

0

500

1000

1500

2000

2500

3000

3500

4000

time(sec)

Fig. 8. controller response for TICTLP.

8. Conclusion

Appendix A. Supplementary material

Generally, it is quite difficult to design a fractional order PI/PID controller for multivariable process. In this paper, the independent design of FOPI/FOPID controller is designed with decoupler for TITO process. The ETF model is developed for TITO system using the information of RGA, RNGA and then the FOPID controller parameters are independently designed for ETF model. The FOPID controller parameters are tuned using bat optimization algorithm to achieve minimum value of ITAE. The proposed controller validated for simulation example. In addition, the controller is implemented for the real time laboratory TIFCTL process. It is concluded that proposed FOPID controller has enhance performance than the centralized PI and centralized FOPI controller. The FOPID controller provides optimal performance when the order of integrator is close to 1. This paper provides insightful understanding of multivariable controller design for complex industrial control problem. For a dynamical and nonlinear system, a new dynamical decoupling method can be designed with improved robustness and adaptability.

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cogsys.2019.07.005.

Declaration of Competing Interest The authors declared that there is no conflict of interest.

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