Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment

Engineering Science and Technology, an International Journal xxx (2018) xxx–xxx

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Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment Mohit Jain a,⇑, Asha Rani a, Nikhil Pachauri a, Vijander Singh a, Alok Prakash Mittal b a b

Division of Instrumentation and Control Engineering, Netaji Subhas Institute of Technology, University of Delhi, Delhi 110078, India All India Council for Technical Education (AICTE), New Delhi, India

a r t i c l e

i n f o

Article history: Received 23 January 2018 Revised 17 June 2018 Accepted 5 July 2018 Available online xxxx Keyword: 2-DOF FOPI Controller Heat flow experiment Water cycle algorithm Optimization

a b s t r a c t This paper presents a two degree of freedom fractional order PI controller for temperature control of a real-time Heat Flow Experiment (HFE). The introduction of two degrees of freedom and fractional calculus to PI controller enhances its flexibility at the cost of large number of tuning parameters and controller design becomes a combinatorial problem. Controller parameters are therefore optimized by a metaheuristic algorithm called as water cycle algorithm (WCA) which leads to WCA tuned two degree of freedom fractional order PI (W2FPI) controller. The present work also explores the potential of WCA as an effective controller tuning technique. The convergence analysis of WCA justifies its effectiveness with respect to state-of-the-art optimizers i.e. genetic algorithm, simulated annealing, dragonfly algorithm, flower pollination algorithm, cuckoo search algorithm, particle swarm optimization, differential evolution and artificial bee colony algorithm. Investigation reveals the superiority of W2FPI controller in comparison to its WCA tuned integer order variant W2PI and conventional PI controller. Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Temperature is a dominant variable and its control is always a challenge in various classes of industries to regulate several manufacturing processes and their operations. It directly affects the product quality [1] as well as safety of the plant. As an illustration chemical reactors suffer from serious problem of ‘‘thermal runaway” during exothermic reaction [2] which may lead the reactor temperature towards unsafe region of operation. This problem can be avoided with the help of precise temperature control. Further heat treatment of materials also requires precise temperature profile tracking and set point regulation. The issues of precise and efficient temperature control may be handled by intelligent control schemes. A conventional PI/PID controller is generally employed in various applications [3–5] due to its simple structure, robustness and ease of implementation, however it lacks flexibility in tuning parameters. A revolutionary change is observed in the field of control engineering with the advent of fractional calculus leading to fractional order controller. In this control strategy the orders of regular integrator and differentiator terms are replaced by

⇑ Corresponding author.

non-integer values [6–8]. The FOPI/PID controllers combine the benefits of traditional PI/PID and fractional mathematics and hence provide better performance [9,10]. As an additional benefit, the fractional order controllers can handle delicate as well as complicated processes at the cost of enhanced computation [11]. Ahn et al. [12] proposed a unique scheme for tuning of a fractional order integral derivative (Ia Db ) controller for HFE on the basis of gain and phase margins and a satisfactory temperature profile tracking is claimed. Malek et al. [13] presented two variations of FOPI controller and verified them experimentally on HFE. The fractional order controllers are also tested in robotics [10,14–16] and power system engineering [17,18]. The literature claims the effectiveness of FOPI/PID controllers over traditional PI/PID controllers. The conventional PI/PID and FOPI/PID controllers are one degree of freedom (1-DOF) type i.e. only one closed loop is available and due to this reason set point tracking and disturbance rejection cannot be handled simultaneously. This issue of multiple objectives motivated the researchers to develop two degree of freedom (2-DOF) control algorithm [19,20]. Several authors explored this control structure in various fields of engineering [21–23]. Ghosh et al. [24] proposed a 2-DOF PID controller for a highly nonlinear and unstable magnetic levitation system. It is concluded that proposed controller performs better in the presence of transients in comparison to 1-DOF PID controller due to an extra gain

E-mail address: [email protected] (M. Jain). https://doi.org/10.1016/j.jestch.2018.07.002 2215-0986/Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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parameter. Sahu et al. [25] investigated a parallel 2-DOF PID control scheme for load frequency control and differential evolution (DE) is employed for optimal tuning of controller parameters. Recently, Dash et al. [26] claimed that 2-DOF PID controller is very effective for automatic generation control of an interconnected thermal systems and firefly algorithm (FA) is applied to get optimal gain settings of the controller. Flexibility of integer order 2-DOF PI/PID controller may be enhanced by incorporating fractional order mathematics, but the controller tuning becomes complicated. The problem may be efficiently handled by nature inspired meta-heuristics such as genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO), artificial bee colony (ABC) and DE [27–32]. In the past few years, several nature inspired optimization algorithms are proposed such as squirrel search algorithm (SSA) [33], owl search algorithm (OSA) [34], cuckoo search algorithm (CSA) [35], flower pollination algorithm (FPA) [36], dragonfly algorithm (DA) [37], grasshopper optimization algorithm (GOA) [38], whale optimization algorithm (WOA) [39], salp swarm algorithm (SSWA) [40], crow search algorithm (CS) [41] and water cycle algorithm (WCA) [42] etc. Recently, CSA is employed for effective tuning of 2-DOF FOPID controller designed for robotic manipulator [9]. It is claimed that the proposed controller is superior to its integer order variant 2-DOF PID and PID controller. Debbarma et al. [43] also designed a firefly algorithm optimized 2-DOF FOPID controller for automatic generation control of power systems. Feng and Xiao-ping [44] proposed an effective pitch control of unmanned air vehicle by PSO optimized 2-DOF FOPID controller. Pachauri et al. suggested NSGA-II for better tuning of 2-DOF FOPID control scheme implemented on bio-reactor [21]. It is revealed from the literature that improved performance of 2-DOF FOPI/PID controller is achieved at the cost of large number of tuning parameters and it becomes important to optimize the controller parameters. It is also reported that WCA is a highly efficient meta-heuristic as compared to other optimization methods. In this paper WCA is employed to handle the crucial problem of tuning 2-DOF FOPI controller for precise temperature control of

real-time HFE. The present work contributes an application and validation of WCA based optimization for tuning of 2-DOF FOPI controller. Moreover various meta-heuristic algorithms i.e. GA, SA, DA, FPA, CSA, PSO, DE and ABC are also attempted for optimal tuning of controller. The rest of the paper is organized as follows. Section 2 describes the process, laboratory setup and system identification. Section 3 discusses the design considerations and optimization of 2-DOF FOPI controller. In Section 4 experimental results including performance evaluation of the proposed controller are presented. Section 5 discusses the outcomes of present work and finally work is concluded in Section 6. 2. Description of system The Heat Flow Experiment (HFE) under consideration consists of a fiberglass chamber fitted with blower, coil based heater at one end and three equally spaced temperature sensors at the other end (Fig. 1). Fan speed is measured by a tachometer placed on the blower. Temperature of the chamber is sampled at three distinct locations by platinum temperature transducers having fast settling time. Basic input output model of HFE is shown in Fig. 2. Two analog voltage signals V h heater voltage and V b blower voltage are generated and applied to HFE through data acquisition (DAQ) device to control the heater temperature and blower speed respectively. The temperature inside the chamber varies with the change in two input voltage signals. The sensor output signal is acquired from three analog input channels of DAQ device. The value of these voltage signals is constrained in the range 0–5 V to ensure safe operation and life span of heating element is increased. The temth

perature of the chamber at i lowing equation:

sensor can be described by the fol-

d T i ðtÞ ¼ f ðV h ; V b ; T a ; xi Þ i ¼ 1; 2; 3 dt

ð1Þ th

where T a is the ambient temperature and xi is the distance of i sensor from heater. The heater voltage is manipulated and blower voltage is maintained at 5 V for complete experimentation so as to control temperature profile of the duct. 2.1. Laboratory setup of the plant

Fig. 1. Apparatus of Heat Flow Experiment.

Fig. 3 shows the snapshot of experimental setup of HFE used in this work. The HFE apparatus is interfaced to personal computer with WinCon 5.2 software which operates the plant in real-time through MATLAB Simulink environment. MATLAB is configured with real-time workshop (RTW) and Microsoft Visual C++ is used as compiler.

Fig. 2. Basic input output model of the HFE.

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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70

Temperature (°C)

60 50 40 Response of Sensor 1 Response of Sensor 2 Response of Sensor 3 Random input voltage for Heater

30 20 10

Fig. 3. Laboratory set up of HFE.

0 2.2. WinCon interface for real-time control of HFE WinCon is a rapid prototyping software that forms a convenient interface between Simulink and hardware. The WinCon suite comprises WinCon server and WinCon client. The server generates realtime C code using MATLAB RTW on the basis of Simulink model created by the user. Further C code is compiled to WinCon controller library file (.wcl) using Visual C++ and assigned to the client for execution. WinCon client is designed to run Simulink based code on a real-time system at a predefined sampling rate. TCP/IP protocol is used as the communication protocol between client and server. The WinCon client and HFE are linked using data acquisition and control board (DACB) (Fig. 4).

0

10

20

30 Time (Sec)

40

50

60

Fig. 5. Open loop response of the plant.

10 Simulated model output Experimentally measured output

5

0

2.3. System identification The complete thermodynamic model of HFE is not available and difficult to derive, therefore system identification is carried out. The mathematical model of HFE is approximated by applying a random voltage sequence to the heater and open loop response of three sensors is recorded for 60 s (Fig. 5). It is observed that the sensor close to heater, experiences high temperature and less delay in comparison to the sensors placed farther. The plant is estimated through rigorous experimentations as a first order time delayed (FOTD) model defined as follows:

Ks Pest ðsÞ ¼ eðT d sÞ T ps þ 1

ð2Þ

where K s ¼ 8:3867, T p ¼ 5:4666 and T d ¼ 0:5973 are evaluated using gradient search technique. It is evident from the measured and simulated response (Fig. 6) that FOTD model is a satisfactory approximation of HFE. The identified model is further employed for optimal tuning of designed controller as discussed in the subsequent sections.

Fig. 4. WinCon interface for real-time control of HFE.

−5

−10

0

10

20

30 Time (Sec)

40

50

60

Fig. 6. Measured and simulated response of plant.

3. 2-DOF FOPI control scheme The primary aim of suggested controller is to offer a simple and robust control algorithm to provide a commercially feasible and efficient solution to process industries. The structure of 2-DOF FOPI controller is shown in Fig. 7 and its output is defined as follows:


 1 uc ðsÞ ¼ K p brðsÞ  yðsÞ þ a frðsÞ  yðsÞg T is Ki ¼ K p fbrðsÞ  yðsÞg þ a frðsÞ  yðsÞg s

ð3Þ

Fig. 7. Basic structure of 2-DOF FOPI controller.

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where K p is proportional gain, b is set-point proportional weight, T i is the integral time constant, K i ¼ K p =T i is integral gain and a is the fractional order parameter of integrator with a > 0. Appropriate values of these four parameters result in optimized and stable performance of the plant. In the present work a simple approximation in frequency domain of fractional order derivative is considered [45]. a

CðsÞ ¼ ks ; a 2 R

ð4Þ

Further polynomial approximation is made using a recursive distribution of N poles and zeros: 0

CðsÞ ¼ k

N Y 1 þ s=xzn n¼1

ð5Þ

1 þ s=xpn

0

0

where k is the gain and jCðsÞj ¼ 0dB in 1 rad/s for k ¼ 1. The poles and zeros are found in the range ½xl ; xh  and defined for positive value of a: a

1a

v ¼ ðxh =xl ÞN ; w ¼ ðxh =xl Þ N ; pffiffiffiffi

ð6Þ

xz1 ¼ x w; xzn ¼ xp;n1 w; n ¼ 2; . . . ; N;

ð7Þ

xpn ¼ xz;n1 v; n ¼ 1; . . . ; N:

ð8Þ

The fractional order integration operator is designed using negative value of a which exchanges the roles of zeros and poles. The block diagram of 2-DOF FOPI based closed loop control of HFE is shown in Fig. 8. The control signal of designed controller is passed through a limiter, which acts as voltage signal to heater for temperature control of HFE. The effect of incorporating fractional mathematics to 2-DOF structure is presented in Fig. 9. Conventional PI controllers have 1-DOF control structure which can realize only the hatched area and cannot optimize both set point tracking as well as disturbance rejection simultaneously. However in case of 2-DOF

PI controller there are two closed loops which can realize Pareto optimal solution at point c. Further introduction of fractional mathematics shifts this Pareto optimal solution to c0 , thus more optimized and precise control solution is achieved. 3.1. Water cycle algorithm As discussed previously, an effective optimization technique is needed to tune 2-DOF FOPI controller for efficient performance of the system. Therefore a recently proposed nature inspired water cycle algorithm is employed for the purpose. WCA utilizes the concept of earth’s water cycle which consists of evaporation, condensation, precipitation and collection phases. It starts with an initial population of raindrops N pop called as solution of optimization problem. Every raindrop in the population is associated with an objective function value (Obj) which is calculated as follows:

Objk ¼ f ðxk1 ; xk2 ; xk3 ; . . . :xkn Þ k ¼ 1; 2; 3; . . . :Npop

ð9Þ

where n is the number of parameters to be optimized. The following steps are involved in WCA for a minimization problem: 1. Define N SR i.e. number of rivers + sea, from the best individuals i.e. raindrops with minimum Obj values. 2. Raindrop with least Obj value is called as sea. Only one sea is assumed in the work. 3. Remaining raindrops are declared as streams i.e. N Streams ¼ N pop  N SR . 4. Streams are assigned to specific rivers or sea as per Eq. (10):

 ( )   Obj   m  NStreams ; NSm ¼ round PN  SR  Objq 

m ¼ 1; 2; . . . ; NSR

q¼1

ð10Þ

Fig. 8. Block diagram of 2-DOF FOPI based temperature control of HFE.

Fig. 9. Conceptual visualization of combining fractional mathematics with 2-DOF structure.

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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where NSm is total number of streams which discharge into a particular river or sea. 5. New probable position of streams and rivers is achieved by the following equations: k k k X kþ1 Stream ¼ X Stream þ R  D  ðX Riv er  X Stream Þ

ð11Þ

k k k X kþ1 Riv er ¼ X Riv er þ R  D  ðX Sea  X Riv er Þ

ð12Þ

where R is a uniformly distributed random number lying between 0 and 1 whereas D is a constant in the range [1,2]. 6. Exchange the position of river with a stream if it provides improved solution than river. 7. Similarly position of a river is exchanged with sea if river finds better solution than sea. 8. Check the following evaporation condition:

jX kSea  X kRiv er j < lmax

k ¼ 1; 2; 3; . . . ; NSR  1

where lmax is defined as a small value near to zero. 9. If the evaporation condition is satisfied then precipitation process is started and given by Eq. (13):

X new Stream ¼ X lb þ R  ðX ub  X lb Þ

ð13Þ

where X lb and X ub are lower and upper bounds of parameter under optimization, respectively. 10. lmax is decreased adaptively to promote the exploitation by following equation: k

kþ1

k

lmax ¼ lmax 

lmax max iteration

ð14Þ

11. If stopping criterion is satisfied the algorithm is terminated, otherwise go to step 5. 3.2. Optimal design of 2-DOF FOPI controller The hardware validation is depicted via block diagram shown in Fig. 10. The system is switched to position-1 for offline tuning and real-time control is achieved at position-2. WCA is used for offline tuning of controller parameters with identified plant model in the closed loop. It generates suitable controller parameters so that

5

error between reference input and output of the identified plant is minimized. The switch is then put to position-2 and optimized controller parameters obtained offline are used in 2-DOF FOPI controller for temperature control of HFE in real-time. Almost every optimization technique demands a suitable objective function which is designed as per specifications and requirements of control system. The most frequently used objective functions for single objective optimization problem are given as:

J 1 ¼ IAE

ð15Þ

J 2 ¼ ISE

ð16Þ

J 3 ¼ ITAE

ð17Þ

where IAE is integral absolute error, ISE is integral square error and ITAE is integral time absolute error. However precise temperature control of HFE is characterized on the basis of time response specifications i.e. the controlled output of HFE must possess least rise time and settling time with minimal deviation from set point. Hence, HFE presents a multi-objective control system design problem i.e. several specifications must be satisfied simultaneously like rise time (t r ), settling time (t s ) and previously mentioned single objectives. A weighted sum objective function may be designed for the purpose. In the present work following objective function is used:

Obj ¼ w1 IAE þ w2 t r þ w3 t s

ð18Þ

where w1 ; w2 and w3 are weights to be allocated optimally as per the desired system response. Any combination of these weighing factors can be considered while satisfying the following constraint [46]: k X wi ¼ 1

ð19Þ

i¼1

It observed experimentally that large value of w2 causes fast response with higher overshoot, larger w3 leads to sluggish response and w1 is obtained using Eq. (19). After rigorous experimentation for various combinations of the three weights, it is found that w1 ¼ 0:30; w2 ¼ 0:25 and w3 ¼ 0:45 are the best suitable

Fig. 10. WCA based 2-DOF FOPI controller for temperature control of HFE.

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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weights for the desired temperature control of HFE. Further, the controller parameters K p ; K i ; a and b (Eq. (3)) are selected optimally for efficient performance of the system. The advantages of including fractional mathematics are also investigated by linearly varying a between 0 and 1 while keeping other controller parameters constant and its effect on objective function is observed (Fig. 11). It is evident that fusion of fractional mathematics significantly affects the objective function value. Thus a provides a means for fine tuning of the controller and motivates to design optimized 2-DOF FOPI controller for HFE. The design steps of WCA optimized 2-DOF FOPI control scheme for temperature control of HFE are given as follows: Step 1 Encode the raindrop as a row vector consisting of different controller gain parameters and randomly initialize such N pop raindrops using uniform distribution. The encoded raindrops are stored in controller gain parameter (CP) matrix:

2

K 1p

6 6 K2 6 p CP ¼ 6 6 . 6 .. 4 K Np pop

3

a1 7 a2 7 7

K 1i

b1

K 2i

b2

.. .

.. .

.. .

N

bNpop

aNpop

K i pop

7 7 7 5

The aim is to find an optimal solution set of controller gain n o parameters CP  ¼ K p ; K i ; b ; a for which the defined objective function (Eq. (18)) attains minimum value. Step 2 Evaluate each raindrop by simulating the identified plant model (Eq. (2)) in closed loop with 2-DOF FOPI controller (Eq. (3)) for 60 s. Compute integral absolute error (IAE) and time domain specifications tr ; t s from the simulated closed loop response. Finally the objective function (Eq. (18)) value is calculated corresponding to each raindrop and stored in ascending order as follows:

2

1

Obj

3

7 6 6 Obj2 7 7 6 Obj ¼ 6 . 7 6 . 7 4 . 5 N Obj pop Step 3 Categorize the raindrops as sea, river and stream on the basis of their objective function values.

Objective function value

70 60

Kp=0.7258 Ki=0.0546

50

β=0.9993

40

Step 4 Generate new raindrops i.e. controller gain parameter vectors using Eq. (11) and Eq. (12). Step 5 If evaporation condition found true, start precipitation using Eq. (13). Step 6 If the iteration count exceeds the predefined maximum limit, terminate the algorithm, else repeat the process from step 2. Fig. 12 presents a detailed flowchart of the proposed tuning method for 2-DOF FOPI controller. One major drawback of metaheuristics is large number of governing parameters and their selection significantly affects the performance of algorithm. Apart from this there is no thumb rule for optimal selection of parameters except rigorous experimental analysis. WCA may be the best choice among other meta-heuristics as it offers less number of parametric settings. Rigorous sensitivity analysis is carried out for selection of N pop ; lmax and N SR parameters. Some significant results for variations in N pop ; lmax and N SR are presented in Tables 1–3 respectively. It is also revealed that WCA succeeds in finding the optimal region in most of the cases which indicates low parametric sensitivity of the algorithm. Lower population size (N pop ) leads to unstable performance as standard deviation (SD) is much high (Table 1). In contrary higher population size makes it more accurate and stable but computationally expensive. Thus large value of population size makes WCA unsuitable for online tuning. In the present work a moderate value of N pop =24 (Table 2) is considered. lmax helps in achieving necessary balance between exploration and exploitation phases of WCA. Higher values of lmax promote exploration and lower values enhance exploitation about optimal solution [42]. Further extremely low values of lmax may provide more accurate solution, however it may perturb the necessary balance of exploration and exploitation. In this work lmax =1E-5 provides sufficient level of accuracy for tuning 2-DOF FOPI. N SR in WCA is a user defined term which divides the search space into small regions. Higher value of N SR slows down the search process. Experimental analysis (Table 3) reveals that N SR ¼ 4 provides satisfactory performance for the current optimization problem with less computational effort. The governing parameters of WCA, considered in the present work are given in Table 4. The lower and upper bounds of tuning parameters like K p ; K i ; b and a are found after rigorous experimentation. Apart from WCA, other state-ofthe-art optimization techniques are also used to minimize the above defined objective function (Eq. (18)). The convergence rate of WCA and other optimization algorithms are recorded as shown in Fig. 13. It is observed that the specified objective function is best minimized by WCA as compared to other algorithms. WCA achieves the optimum value of objective function earlier in comparison to other optimization techniques. It converges to the optimal point in 30 iterations and may prove to be a better tuning technique for 2-DOF FOPI controller. Further, fast convergence rate of WCA makes it suitable for online controller tuning and is thus an obvious choice for tuning 2-DOF FOPI controller. Table 5 presents the optimized values of objective function and the corresponding controller parameters obtained after 30 independent runs from different optimization techniques.

30 4. Experimental results

20

After obtaining the optimum controller parameters from the mentioned optimization algorithms, the switch is connected to

10 0

0

0.2

0.4

α

0.6

0.8

Fig. 11. Effect of variation in a on objective function.

1

2nd position (Fig. 10) and the parameters derived from GA, SA, DA, FPA, CSA, PSO, DE, ABC and WCA are used in 2-DOF FOPI controller which leads to G2FPI, S2FPI, D2FPI, F2FPI, C2FPI, P2FPI, DE2FPI, A2FPI and W2FPI controllers respectively. Step response analysis is carried out to examine the effectiveness of designed

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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Fig. 12. Flowchart for WCA based tuning of 2-DOF FOPI controller.

Table 1 The effect of variation of N pop on the performance of WCA with lmax ¼ 1E  5 and N SR ¼ 4. Function

Parameter

N pop ¼ 10

N pop ¼ 15

N pop ¼ 20

N pop ¼ 24

N pop ¼ 30

Obj

Best Mean SD

3.6136E+01 3.9114E+01 3.3625E+00

3.5921E+01 3.6236E+01 3.9454E01

3.5922E+01 3.7551E+01 3.1251E+00

3.5923E+01 3.6002E+01 1.1701E01

3.5910E+01 3.5942E+01 2.5291E02

controllers. The set point temperature at first monitoring location inside the duct is increased to 45 °C and the temperature sensor readings of HFE are recorded for 60 s. The recorded temperature variations, corresponding error and IAE by various controllers are shown in Fig. 14, Fig. 15 and Fig. 16 respectively. It is observed from the results that W2FPI controller exhibits superior perfor-

mance in comparison to other controllers as WCA achieves optimal solution in the defined stopping criterion. Therefore for further experimental studies, only W2FPI controller is considered. Conventional Tyreus-Luyben tuned PI (TLPI) and WCA tuned integer order two degree of freedom PI (W2PI) controllers are also designed for performance comparison of proposed control scheme.

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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Table 2 The effect of variation of lmax on the performance of WCA with N pop ¼ 24 and N SR ¼ 4. Function

Parameter

lmax ¼ 1E  1

lmax ¼ 1E  2

lmax ¼ 1E  3

lmax ¼ 1E  4

lmax ¼ 1E  5

Obj

Best Mean SD

3.5910E+01 3.6006E+01 1.0660E01

3.5974E+01 3.6133E+01 1.1889E01

3.6005E+01 3.6052E+01 5.5409E02

3.5922E+01 3.5954E+01 4.4044E02

3.5923E+01 3.6002E+01 1.1701E01

Table 3 The effect of variation of NSR on the performance of WCA with N pop ¼ 24 and lmax ¼ 1E  5. Function

Parameter

N SR ¼ 2

N SR ¼ 3

N SR ¼ 4

N SR ¼ 5

N SR ¼ 6

Obj

Best Mean SD

3.5935E+01 3.5966E+01 3.8643E02

3.5928E+01 3.6085E+01 1.3140E01

3.5923E+01 3.6002E+01 1.1701E01

3.5926E+01 3.6022E+01 1.0309E01

3.5926E+01 3.5973E+01 5.7368E02

the results that proposed controller provides superior performance for set point tracking. The results of W2PI and W2FPI are almost coinciding but quantitative comparison (Fig. 19) shows that W2FPI significantly reduces the IAE. It is also observed that the proposed controller makes precise changes in the control effort (Fig. 20). This is due to the reason that 2-DOF control scheme copes up with set point tracking and disturbance rejection simultaneously whereas fractional calculus improves the performance of conventional PI due to additional design parameters. The control effort of W2FPI controller is very close to its integer order variant, which shows that fractional order parameter provides fine tuning.

Table 4 Governing parameters of WCA. WCA parameters

Value

Size of population (N pop ) Number of rivers and sea (N SR ) Evaporation condition controlling parameter (lmax ) Convergence criteria (Maximum iterations) Number of variables Parameter bounds

24 4 1e-5

100 4 K p 2 ½0; 5; K i 2 ½0; 2; b 2 ½0:2; 1:5; a 2 ½0; 1

4.2. Disturbance rejection 4.1. Set point tracking The effectiveness of designed W2FPI controller is tested for servo problem commonly known as set point tracking. In this experimental study, initially the target temperature is raised to 45 °C and after 30 s an increment of approximately 10% of the operating point i.e. 5 °C is introduced for next 30 s. Temperature variations of HFE and corresponding error for various controllers are shown in Fig. 17 and Fig. 18 respectively. It is observed from

Practically, external random disturbances deviate the plant output from desired operating point, known as regulatory problem. Therefore designed controllers are tested for regulatory problem. This analysis is performed on an initial set point of 45 °C and after steady state is reached, simulated dynamic disturbances are introduced in the system for small duration. Two types of disturbances are considered and defined mathematically as follows:

d1 ðtÞ ¼ m1 ð1  eðtt0 Þ Þuðt  t 0 Þ

GA SA DA FPA CSA PSO DE ABC WCA

Objective function value

42 40

36.6

2

10

38

36.4

X: 30 Y: 35.92

36.2

36 29

30

31

32

35.8 96

0

10

20

30

X: 100 Y: 35.92

36 98

ð20Þ

100

40 50 60 Number of iterations

70

80

90

100

Fig. 13. Convergence rate comparison of various optimization techniques.

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M. Jain et al. / Engineering Science and Technology, an International Journal xxx (2018) xxx–xxx Table 5 Optimized tuning parameters of 2-DOF FOPI controller using various algorithms. Tuning technique

Kp

Ki

b

a

Obj

GA SA DA FPA CSA PSO DE ABC WCA

0.9593 3.3027 1.5497 1.3591 1.0121 0.1342 0.5305 0.5110 1.2211

0.5259 0.0176 0.01 0.2559 0.2263 1.6110 0.0101 0.8222 0.01

0.9103 0.9452 1.0225 0.7530 0.7885 0.9022 1.0746 1.0785 1.0324

0.6444 0.3970 0.6938 0.9999 0.9998 0.7179 0.1646 0.0000 0.0022

40.1628 44.5616 36.4556 36.3788 36.0279 42.7140 35.9404 36.2215 35.9227

Temperature (°C)

45

40 Set point G2FPI S2FPI D2FPI F2FPI C2FPI P2FPI DE2FPI A2FPI W2FPI

35

30

25

0

10

20

30 Time (Sec)

40

50

Fig. 16. IAE for 2-DOF FOPI controller tuned by various algorithms.

60

55 50 Temperature (°C)

Fig. 14. Step response of 2-DOF FOPI controller tuned by various optimization techniques.

18 G2FPI S2FPI D2FPI F2FPI C2FPI P2FPI DE2FPI A2FPI W2FPI

16

Temperature (°C)

14 12 10 8

45 40

48

35 46 44

25 20

6 4

Set Point TLPI W2PI W2FPI

30

15 20 25 0

10

20

30 Time (Sec)

40

50

60

Fig. 17. Temperature variations of HFE by designed controllers for set point tracking.

2 0 −2

0

10

20

30 Time (Sec)

40

50

60

Fig. 15. Error in temperature for step input.

d2 ðtÞ ¼ m2 ½uðt  t 1 Þ  uðt  t2 Þ t 2 > t 1

ð21Þ

where d1 ðtÞ is an exponentially decaying spike of m1 strength at time instant t 0 and d2 ðtÞ is a short duration pulse with magnitude m2 at instant t 1 for a time interval t 2  t 1 . These simulated disturbances model the sensor noise (in volts) and is added to the output of first temperature sensor. The response, error and control effort of

TLPI, W2PI and W2FPI for disturbance d1 ðtÞ introduced at 50 s with m1 ¼ 0:15 at an operating point of 45 °C are depicted in Fig. 21–23 respectively. It is observed that W2FPI controller reduces error significantly and temperature settles at the desired operating point earlier in comparison to other designed controllers. IAE is calculated for a duration of 50 s i.e. from t = 50 s to 100 s for all the three controllers (Fig. 27) and is best minimized by W2FPI controller. The performance of controllers is also tested for another disturbance d2 ðtÞ with m2 ¼ 0:20 and system response is shown in Fig. 24. It is inferred from the results that performance of W2FPI controller is significantly better than the other designed controllers. It is also observed that fast and precise changes are made by W2FPI controller in the manipulated variable (Fig. 26) which leads to minimum error in the operating temperature (Fig. 25 and 27).

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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M. Jain et al. / Engineering Science and Technology, an International Journal xxx (2018) xxx–xxx

25

50 TLPI W2PI W2FPI

Temperature (°C)

20

Error (°C)

15 10 5

46

40

45

Set Point TLPI W2PI W2FPI

35

0 −5

45

44 50 0

10

20

30 Time (Sec)

40

50

30 40

60

55

50

60 60

70 Time (Sec)

80

90

100

Fig. 18. Error in response of HFE by designed controllers for set point tracking. Fig. 21. Temperature variations of HFE controlled by various controllers in presence of d1 ðtÞ disturbance.

1 0.5 0 Error (°C)

−0.5 −1 −1.5 −2 −2.5

TLPI W2PI W2FPI

Fig. 19. IAE comparison of designed controllers for set point tracking.

−3 −3.5 40

5.5 TLPI W2PI W2FPI

Voltage (Volts)

5

50

60

70 Time (Sec)

80

90

100

Fig. 22. Error in actual and set point temperature for the designed controllers in presence of d1 ðtÞ disturbance.

4.5 4

3.5

3.5

3

2.5

0

10

20

30 Time (sec)

40

50

60

Fig. 20. Control effort applied by designed controllers for set point tracking.

Voltage (Volts)

3

2.5 2 1.5 1 TLPI W2PI W2FPI

4.3. Robustness testing

0.5 Conventional controllers are still found in process industries due to their robustness towards parametric and operating point variation. Therefore, the robustness testing of the designed controllers is performed for two types of temperature profiles at operating point different from the one used for optimization. In the first case, 48 C is the set point around which a sinusoidal temperature

0 40

50

60

70 Time (Sec)

80

90

100

Fig. 23. Control effort applied by designed controllers in presence of d1 ðtÞ disturbance.

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

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M. Jain et al. / Engineering Science and Technology, an International Journal xxx (2018) xxx–xxx

50 Set Point TLPI W2PI W2FPI

Temperature (°C)

49 48 47 46 45 44 43 50

Fig. 27. IAE comparison in the presence of d1 ðtÞ and d2 ðtÞ disturbances.

60

70 80 Time (Sec)

90

100

Fig. 24. Temperature variations of HFE controlled by various controllers in presence of d2 ðtÞ disturbance.

Temperature (°C)

2 1 0 Error (°C)

50

49

40

50

48 47

48

46

30

80 85 90 95

35

Set Point TLPI W2PI W2FPI

35

−1 −2

30

−3 TLPI W2PI W2FPI

−4 −5 50

60

70 80 Time (Sec)

90

100

40 60 Time (Sec)

80

100

Type of set point tracking

TLPI

W2PI

W2FPI

Sinusoidal Random

118.5435 111.7152

102.4333 86.3803

87.9524 84.9691

20

TLPI W2PI W2FPI

3.5

20

Table 6 IAE comparison for sinusoidal and random set point tracking.

4.5 4

0

Fig. 28. Temperature control of HFE under sinusoidal set point tracking.

Fig. 25. Comparison of error produced by designed controllers for d2 ðtÞ disturbance.

TLPI W2PI W2FPI

15

3 2.5

Error (°C)

Voltage (Volts)

45

2 1.5

10

5

1 0

0.5 0 50

60

70 80 Time (Sec)

90

100

Fig. 26. Comparison of control effort applied by designed controllers for d2 ðtÞ disturbance.

−5

0

20

40 60 Time (Sec)

80

100

Fig. 29. Error in temperature control of HFE under sinusoidal set point tracking.

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M. Jain et al. / Engineering Science and Technology, an International Journal xxx (2018) xxx–xxx

6

20 TLPI W2PI W2FPI

15

4

Error (°C)

Voltage (Volts)

5

TLPI W2PI W2FPI

3

10

5

2

0 1

−5 0

0

20

40 60 Time (Sec)

80

0

20

100

40 60 Time (Sec)

80

100

Fig. 32. Error in temperature control of HFE under random set point tracking.

Fig. 30. Control effort used in temperature control of HFE under sinusoidal set point tracking.

5 4.5

50

45 52

49

51

40

30

35

35

40

4 3.5 3 2.5

50

30

Voltage (Volts)

Temperature (°C)

50

TLPI W2PI W2FPI

49 10

20

30

Set Point TLPI W2PI W2FPI

2 1.5 0

0

20

40 60 Time (Sec)

80

100

Fig. 31. Temperature control of HFE under random set point tracking.

variation is examined. Fig. 28 shows the recorded response of the three controllers under sinusoidal profile tracking. It is observed that the performance of W2FPI controller is superior to its integer order variant as well as conventional PI controller. Table 6 also depicts that IAE is minimum in case of W2FPI controller. Fig. 29 and Fig. 30 show the corresponding error generated and control effort applied by the designed controllers under sinusoidal temperature profile tracking. In the second case a random temperature profile tracking is considered about 50 °C. Fig. 31, Fig. 32 and Fig. 33 show the comparative analysis of recorded responses, error caused and control effort applied by designed controllers respectively. It is observed that performance of W2FPI controller is better and IAE is significantly reduced (Table 6) in comparison to its integer order variant and conventional PI controller. 5. Discussion Precise temperature control may be achieved with the help of a simple, efficient and robust control scheme which is the primary aim of this work. Therefore a 2-DOF PI control scheme is implemented to handle servo and regulatory problem simultaneously. The performance of 2-DOF PI controller is further enhanced by

20

40 60 Time (Sec)

80

100

Fig. 33. Control effort used in temperature control of HFE under random set point tracking.

introducing fractional calculus at the cost of slightly increased computational effort. The number of parameters in the designed controller is increased which demands an efficient optimization algorithm for optimal performance of the system. In this paper WCA is used for tuning 2-DOF FOPI controller and the designed scheme is tested for temperature control of HFE. The system under consideration possesses significant nonlinearity in the form of actuator saturation. This is due to the 0–5 V limits imposed on input voltage signals for safe plant operation and increased life span of heating element. Further direct interaction of plant with the surroundings also increases the difficulty in precise and accurate control. The effect of environmental disturbances is observed in the initial temperature for experimentations of temperature control and set point tracking made on different days. Thus effect of ambient conditions on the HFE offers an opportunity to test the robustness of proposed controller. According to No Free Lunch (NFL) [47] theorem a single optimization algorithm cannot solve all optimization problems. Therefore, a rigorous comparative analysis of various potentially acclaimed meta-heuristics such as GA, SA, DA, FPA, CSA, PSO, DE, ABC and WCA is carried out to select the most suitable optimizer for current controller tuning problem. Apparently, WCA is quite

Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002

M. Jain et al. / Engineering Science and Technology, an International Journal xxx (2018) xxx–xxx

close to PSO from the perspective of their governing equations. However, literature claims [42] that there exist several technical differences among the two algorithms which makes WCA superior in comparison to PSO and other competitive optimizers like DE and ABC. WCA uses the mechanism of multiple guidance points (rivers) which guides other candidate solutions for seeking better solutions in search space. However the search process of PSO relies on the best and personal experiences of particles. WCA employs evaporation condition and raining process to prevent the algorithm from premature convergence [42]. Such conditions are not present in PSO. The commonly used parameters like population, lower bound, upper bound of the tuning parameters and stopping criteria are considered same for all algorithms for fair comparison. As per the industrial requirements of online tuning, a metaheuristic must possess fast convergence rate. Therefore convergence rates of the considered optimization techniques are analyzed (Fig. 13). Results reveal that WCA achieves minimum value of objective function very fast. It may prove to be a better alternative for online tuning of controllers. It is also observed that GA got stuck in local optima and shows premature conversion. SA could not reach the optimal region in the defined stopping criterion and demands more iterations. This is due to the requirement of a large number of parameters need to be initialized in both algorithms and there are no guidelines in the literature regarding their proper selection. It is also revealed that DA, FPA and CSA are very close to optimal region but could not achieve the global optimum and hence these algorithms are not suitable for this particular application. Improper selection of parameters of optimization techniques may lead to premature or sluggish convergence. As an illustration, there are several types of selection, crossover and mutation strategies available in GA and their choice affects its performance. The most crucial task in SA is to set initial temperature and annealing schedule. On contrary, meta-heuristics like DA, FPA, CSA and WCA do not have these limitations, which motivate to employ these techniques for parameter tuning in the present work. As there is no thumb rule for optimal selection of WCA parameters a rigorous sensitivity analysis is carried out. The number of parameters to be selected is less and therefore various possible combinations of parametric settings are tried. It is revealed from the analysis that WCA successfully finds the optimal region in almost all the cases and is thus less sensitive to parameter variations. This indicates that WCA inherently possesses a good balance between exploration and exploitation phases which is a mandatory requirement for an efficient meta-heuristic. WCA is thus found more suitable and effective for optimal tuning of designed control scheme. The optimization of controller parameters and testing are performed at the same operating point and sufficient time delay is provided in order to attain normal conditions before the next experiment. It is observed from the results that WCA effectively tunes the 2-DOF FOPI as it not only achieves the desired operating point but also provides very tight temperature control. It is observed in the step response analysis that WCA effectively tunes the 2-DOF FOPI with minimum IAE = 57.4246. The results of W2PI and W2FPI are almost coinciding in set point tracking, however quantitatively W2FPI provides least IAE. This is due to the reason that 2-DOF control scheme copes up with set point tracking and disturbance rejection simultaneously whereas fractional calculus improves the performance of conventional PI due to additional design parameters. The control effort of W2FPI controller is very close to its integer order variant, which shows that fractional order parameter provides fine tuning. Set point tracking, disturbance rejection and robustness analysis is also carried out to evaluate the consistency, precision and robustness of designed controllers. In each experimental study, initially the system is operated at a pre-decided operating point and on reaching the steady state, set point is changed or disturbance is added. The simulated distur-

13

bances are considered to ensure repeatability of input for all the controllers, as efficient reproduction of hardware disturbances is difficult. It is observed that fast and precise changes are made by W2FPI controller in the manipulated variable for all cases which leads to minimum error in the operating temperature. Thus WCA offers better global search ability as it not only offers fast convergence rate but also provides highly accurate and optimal solution in comparison to PSO, ABC and DE for controller tuning problem.

6. Conclusion The present work focuses on the efficient and precise temperature control of HFE with the help of WCA tuned 2-DOF FOPI controller. The designed controller is tested for the first time on a laboratory setup of real-time heat flow experiment. It is observed from the results that WCA provides better optimization of controller parameters in comparison to GA, SA, DA, FPA, CSA, PSO, DE and ABC. WCA tuned integer order PI (W2PI) and conventional PI controllers are also designed for comparative analysis. Results reveal that W2FPI controller significantly reduces the IAE as compared to the other designed controllers. Set point tracking and disturbance rejection analysis also confirm the superiority of W2FPI controller. Hence, it is concluded from the results that introduction of 2-DOF and fractional order to PI controller leads to a quite effective, robust and precise temperature control of HFE. In future a more effective control scheme may be proposed with dynamic controller gains i.e. adaptive 2-DOF FOPI control scheme. It might be useful for plants like chemical reactors having parametric variations. Fuzzy and neural network may be used for implementation of adaptive 2-DOF FOPI control scheme.

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Please cite this article in press as: M. Jain et al., Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment, Eng. Sci. Tech., Int. J. (2018), https://doi.org/10.1016/j.jestch.2018.07.002