PID Controller Based Nelder Mead Algorithm For Electric .

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PID Controller Based Nelder Mead Algorithm for Electric Furnace System with Disturbance71PID Controller Based Nelder Mead Algorithmfor Electric Furnace System with DisturbanceVunlop Sinlapakun1 and Wudhichai Assawinchaichote2 , Non-membersABSTRACTThis paper presents a design of PID controller forfurnace temperature control system with disturbance.Currently, PID controller has been used to operate inelectric furnace temperature control system becauseits structure is simpler compared to others. However,the issue of tuning and designing PID controller adaptively and efficiently is still open. This paper presentsan improved PID controller efficiency from tuning byNelder Mead method. The parameters of PID controller shall be obtained from the Nelder Mead optimization procedure. Errors between desired magnitude response and actual magnitude response are calculated by using the Integral of Absolute Error (IAE).The proposed Nelder Mead based PID design methodis simpler, more efficient and effective than the existing traditional methods included Ziegler Nichols,Cohen-Coon and Direct Synthesis. Simulation resultshows that the performance of PID controller usingthis proposed method is better than traditional methods and resistant to disturbance.Keywords: Electric Furnace, Disturbance, PIDController, PID Parameters, Nelder Mead Optimizationrise time, overshoot and settling time still occur andmay be not suitable for electric furnace temperaturesystems.This paper proposed Nelder Mead-based PID controller for solving these problems. It is used to determine the optimal parameters of PID controller usingthe calculation of Integral of Absolute Error (IAE),which is traditional method for finding the best valuein form of nonlinear. After applying Nelder MeadAlgorithm, then the parameters kp , ki and kd areobtained. These results will be compared with traditional methods included Ziegler-Nichols [10], CohenCoon [11] and Direct Synthesis [12] and different disturbances.2. PID CONTROLLERPID controller consists of Proportional, Integraland Derivative control. Proportional control is responsible for faster enter steady state, Integral controlis responsible for reducing overshoot in steady stateand Derivative control is responsible for making thesystem more stable.This paper introduces a single-input single-output(SISO) PID controller, which consists of PID controller D(s) and controlled plant G(s) are shown inFig. 1 which is simple and effective.1. INTRODUCTIONElectric furnace is one of many furnaces availabletoday. It uses electricity as its main power sourceto generate heat which widely uses in various industrial production processes. However, the current controller design that is popular for use with electric furnace, such as PID control [1,2], neural network [3]and adaptive fuzzy control [4-8]. The PID controldesign is popular and easiest way for electric furnace,but it is also a problem for the design is nonlinear system [9], time delay and disturbance. Nowadays, thereare many methods for tuning PID, such as ZieglerNichols [10], Cohen-Coon [11], Direct Synthesis [12],Genetic algorithm (GA) [13], particle swarm optimization (PSO) [14], differential evolution (DE) [15],and multi-objective optimization algorithms [16]. Allof these methods do not deliver good tuning sinceManuscript received on July 14, 2015.Final manuscript received on January 11, 2016.1,2 The authors are with King Mongkut’s University of Technology Thonburi, 126 Pracha-utid Road Bangmod, Toongkru,Bangkok, Thailand, E-mail: vunlops45@gmail.com and wudhichai.asa@kmutt.ac.thFig.1: A control system with PID controller.where D(s) is transfer function of PID Controller,G(s) is transfer function of controlled plant, r(t) is input signal to controlled plant, e(t) is the system error,u(t) is controlled input and y(t) is output signal.From Fig.1, the equation of standard PID Controller is de(t)(1)u(t) kp e(t) ki e(t)dt kadt, and can be written in the form of transfer function isD(s) kiU (s) kp kd sE(s)s(2)

72ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.10, NO.1 May 2016where U (s) is transfer function of controlled input, E(s) is transfer function of the system error e(t),kp ,ki and kd are proportional gain, integral gain andderivative gain, respectively.From (2), PID controller can be written asApproximation of (4) isH(s) 1 0.75s1 0.75s(6)Then, from (5) and (6) will beG(s) 0.75s3 0.1125s 0.15 1.825s2 1.25s 0.2(7)Hence, (7) is transfer function of electric furnace,which is used for experiment in this paper.4. NELDER MEAD OPTIMIZATION FORPID CONTROLLERFig.2: Block diagram of PID controller.3. ELECTRIC FURNACE TEMPERATURECONTROL SYSTEMThe compositions of electric furnace temperaturecontrol system [17] are electrical furnace, controllerand thermocouple which controller is used to controlthe temperature in electrical furnace is shown as Fig.3.In this paper, Nelder Mead optimization is usedfor searching the best parameters of PID controllerfor use with the furnace temperature control system. This method had been introduced by Nelder andMead in 1965. It is a basic principle for determiningminimum of nonlinear multiple variable equations.Structure of control system by using Nelder MeadOptimization for PID controller is shown in Fig. 4.Fig.4: Structure of Nelder Mead with a control system and PID Controller.Fig.3: Block diagram of electric furnace control.where r is input voltage, U is output voltage fromcontroller, y is output voltage from thermocouple andR is armature resistance.In this paper, transfer function of electric furnaceis chosen as [17]In this paper, the result of the optimization isbased on the error from the calculation of IAE. Result is shown in (8) and (9), which is based on the desired magnitude response and the actual magnituderesponse.Error(K) f (K) n e(t) , t(8)t 0G(s) 0.15s2 1.1s 0.2(3)e(t) 1 y(t),, transfer function of a 1.5 time delay isH(s) e 1.5s(4)Then, transfer function of electric furnace with a1.5 time delay isG(s) 0.15e 1.5ss2 1.1s 0.2(5) 0, ts , 2ts , . . . , nt 0, ts , 2ts , . . . , n(9)where ts is sampling time, n is maximum time foroptimization, Error(K) or f (K) is IAE, K is parameters of PID controller, e(t) is system error, y(t)is control output or actual magnitude response and 1is desired magnitude response.Nelder Mead Optimization consists of B (Bestpoint), G (Good point), W (Worse point), M (Mid

PID Controller Based Nelder Mead Algorithm for Electric Furnace System with Disturbance73point), E (Expansion Point), R (Reflect point), C(Construction point) and S (Shrink point).4. 1 Initial Triangle BGWLet f (K) be the function that used for minimizingwhich Nelder Mead method will find the three pointsof a triangle asB f (K1 ), G f (K2 ), and W f (K3 )(10)That B is the best point (value less than G andW), G is good point (next to best), and W is theworst point.Fig.5: All points that used for Nelder Mead Method.(2) Calculate f (K1 ), f (K2 ), f (K3 ) for finding B,G, W, where B G W.(3) Compute M, E and f (E).(4) Compare f (E) and f (G), if f (E) f (G) replace W with E, go to step 8; else Compute Rand f (R) go to step 5.(5) Compare f (R) and f (W ), if f (R) f (W )replace W with R go to step 6.(6) Compare f (R) and f (G), if f (R) f (G)Compute C and f (C) go to step 7; else go tostep 8.(7) Compare f (C) and f (W ), if f (C) f (W )replace W with C go to step 8; else computeS, replace G with M and replace W with S goto step 8.(8) (8) Rearrange the B, G, W, where B G W and repeat step (3) until some predefinedstopping criteria.4. 2 Mid pointThe building process uses the Mid point of the linefrom B and G asB G2M (11)4. 3 Expansion pointThe Expansion point is calculated from Mid pointand Worst point asE 3M 2W(12)4. 4 Reflection pointThe Reflection point is calculated from Mid pointand Expansion point asR M E2(13)The Pseudo code of Nelder Mead is shown in Fig.6.4. 5 Contraction pointThe Contraction points that used on this paperhave 2 points. The first point is calculated fromWorst point and Mid point and the second point iscalculated from Reflection point and Mid point asC1 W MR Mor C2 22(14)4. 6 Shrink PointThe Shrink point is constructed from Best pointand Worst point asB W(15)2All points that used for Nelder Mead method areshown as Fig.5According to the calculation, the algorithm stepsare shown as below:(1) Generate an initial configuration K randomly,where K1 [kp1 ki1 kd1 ], K2 [kp2 ki2 kd2 ],and K3 [kp3 ki3 kd3 ].S Fig.6: Pseudo code of Nelder Mead method for optimization the PID controller.

74ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.10, NO.1 May 2016where i is the iteration for optimization which setsthe maximum number of iterations imax 100.5. DESIGN EXAMPLE AND SIMULATIONRESULTThe input signal r(t) that used on this section isunit step function.{0, t 0r(t) (16)1, t 0Setting the ranges of kp , ki and kd are between0 to 30, maximum time for optimization n 25s,sampling time ts 0.05s and maximum number ofiterations imax 100.The step response for linear system under differentmethods based PID controller is compared in Fig.9., from (16) is shown as Fig.7.Fig.9: The step response for linear system underdifferent methods based PID controller.The error of step response for linear system underdifferent methods based PID controller is comparedin Fig.10.Fig.7: The input signal r(t).The disturbance n(t) that used on this section issquare wave signals from -0.1 to 0.1, -0.2 to 0.2, -0.3to 0.3, -0.4 to 0.4, -0.5 to 0.5 and -0.6 to 0.6 that isshown as Fig.8.Fig.10: The error of step response for linear systemunder different methods based PID controller.Fig.8: The disturbance n(t).5. 1 Optimized PID controller design for linearsystem with Nelder Mead AlgorithmThe transfer function of linear system isG(s) 1s2(17)The performances of these methods are evaluatedby these indices including rise time, %overshoot, settling time and Error (IAE) that are shown as Table1.From Table 1, rise time and Error(K) of NelderMead is smaller than Ziegler Nichols; settling timeand %overshoot of Nelder Mead is close to ZieglerNichols.Then, the results show that the transient response and steady-state performances obtained by

PID Controller Based Nelder Mead Algorithm for Electric Furnace System with Disturbance75Nelder Mead for linear system are better than ZieglerNichols [10].Table 1: Comparative performance of step responsefor linear system under different methods. Method Ziegler Nichols Nelder MeadPerformances kpkikdRise time%overshootSettling timeError(K),n 25 s,ts 0.05 0.497728.32160.06993.05870.68361.91515. 2 Optimized PID controller design for nonlinear system with Nelder Mead AlgorithmThe transfer function of nonlinear system isG(s) 1s2 1(18)Setting the ranges of kp , ki and kd are between0 to 30, maximum time for optimization n 25 s,sampling time ts 0.05 s and maximum number ofiterations imax 100.The step response for nonlinear system under different methods based PID controller is compared inFig.11.Fig.12: The error of step response for nonlinearsystem under different methods based PID controller.From Table 2, the results show that the transientresponse and steady-state performances obtained byNelder Mead for nonlinear system are better thanZiegler-Nichols [10].Table 2: Comparative performance of step responsefor nonlinear system under different methods. Method Ziegler Nichols Nelder MeadPerformances kpkikdRise time%overshootSettling timeError(K),n 25 s,ts 0.05 7414.757514.99130.146500.26052.23045. 3 Optimized PID controller design for electric furnace temperature system withNelder Mead AlgorithmFig.11: The step response for nonlinear system under different methods based PID controller.The error of step response for nonlinear systemunder different methods based PID controller is compared in Fig.12.The performances of these methods are evaluatedby these indices including rise time, %overshoot, settling time and Error (IAE) that are shown as Table2.In this experiment, the transfer function of electric furnace from (7) will be chosen for simulatingthe design of PID controller which uses Nelder Meadoptimization to determine the best parameters of PIDcontroller by setting the ranges of kp , ki and kd arebetween 0 to 30, maximum time for optimizationn 25 s, sampling time ts 0.05 s and maximumnumber of iterations imax 100.The step response for electric furnace under different methods based PID controller is compared inFig.13.The error from step responses of electric furnaceunder different methods based PID controller is compared in Fig.14.The performances of these methods are evaluatedby these indices including rise time, %overshoot, set-

76ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.10, NO.1 May 2016Table 3: Comparative performance of Nelder Meadwith traditional methods. Nelder Method Ziegler Cohen- DirectPerformances Nichols Coon Synthesis MeadkpkikdRise time%overshootSettling timeError(K),n 25 s,ts 0.05 sFig.13: The comparison of step response of closedloop system under PID .63245.59411.31157.00077.651846.8696ment presents about optimized PID controller designfor electric furnace temperature systems with disturbance n(t) that it is shown in Fig. 15.Fig.15: A control system and PID Controller withdisturbance.From Fig.15, the control output y(t) is calculatedfrom y1 (t) and y2 (t), theny(t) y1 (t) y2 (t)Fig.14: The comparison of error of closed loop system under PID controller.tling time and Error (IAE) that are shown as Table3.From Table 3, rise time of Nelder Mead is closeto Ziegler Nichols but smaller than Cohen-Coon andDirect Synthesis; settling time and Error(K) of NelderMead are smaller than Ziegler-Nichols, Cohen-Coonand Direct Synthesis; %overshoot of Nelder Mead isbigger than Direct Synthesis but smaller than ZieglerNichols and Cohen-Coon.Then, the results show that the transient responseand steady-state performances obtained by NelderMead for electric furnace are better than ZieglerNichols [10], Cohen-Coon [11] and Direct Synthesis[12].5. 4 Optimized PID controller design for electric furnace temperature system with disturbanceThe experimental results from 5.3 showed PIDcontroller based on Nelder Mead for Electric furnaceare better than traditional methods, then this experi-(19), and can be written in the s-domain isY (s) Y1 (s) Y2 (s)()G(s)D(s)Y1 (s) R(s)1 Gs D(s)()G(s)Y2 (s) N (s)1 G(s)D(s)(20)(21)(22)where y(t) is the control output, y1 (t) is controloutput from input signal, y2 (t) is control output fromdisturbance, D(s) is transfer function of PID Controller, G(s) is transfer function of controlled plant,r(t) is input signal to controlled plant, e(t) is the system error and u(t) is controlled input.In this experiment, the transfer function of electric furnace from (7) will be chosen for simulatingthe design of PID controller which uses Nelder Meadoptimization to determine the best parameters of PIDcontroller by setting the ranges of kp , ki and kd arebetween 0 to 30, maximum time for optimizationn 25 s, sampling time ts 0.05 s and maximumnumber of iterations imax 100.The step response of control output from inputsignal y1 (t) of electric furnace under different distur-

PID Controller Based Nelder Mead Algorithm for Electric Furnace System with Disturbance77bances based PID controller is compared in Fig.16.Fig.16: The step response of control output frominput signal y1 (t) of closed loop system under differentdisturbances.Fig.18: The error of step response of control outputfrom input signal y1 (t) of closed loop system underdifferent disturbances.The step response of control output from disturbance y2 (t) of electric furnace under different disturbances based PID controller is compared in Fig.17.Fig.19: The step response of control output y(t) ofclosed loop system under different disturbances.Fig.17: The step response of control output fromdisturbance y2 (t) of closed loop system under differentdisturbances.Fig.17 shows that the disturbance n(t) is effectiveonly in the initial state. After the initial state, thedisturbance will not affect to the control output y(t).The error of step response of control output frominput signal y1 (t) of electric furnace under different disturbances based PID controller is comparedin Fig.18.The step response of control output y(t) of electric furnace under different disturbances based PIDcontroller is compared in Fig.19.The performances of step response from Fig.16 areevaluated by these indices including rise time, %overshoot, settling time and Error (IAE) that are shownin Table 4 and Table 5.From Table 4 and Table 5, they are comparativeperformance of Nelder Mead with low disturbanceand high disturbance. The low disturbance consistsof square wave signals from -0.1 to 0.1, -0.2 to 0.2 and-0.3 to 0.3. The high disturbance consists of squarewave signals from -0.4 to 0.4, -0.5 to 0.5 and -0.6 to0.6 that the performances from rise time, %overshootand settling time is not much different but Error(K)varied according the increased of disturbance.From Table 4 and Table 5, the comparative performances of Nelder Mead with disturbances, the proposed controller can well operate although the electricfurnace system exists the disturbances in the systemprocess.5. 5 Comparison PID controller design forelectric furnace temperature system withvery high disturbanceThe experimental results from 5.4 showed PIDcontroller based on Nelder Mead for Electric furnace

78ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.10, NO.1 May 2016Table 4: Comparative performance of Nelder Meadwith traditional methods.hhhhn(t)hh DisturbancesPerformances hhhhh -0.1 to 0.1 -0.2 to 0.2 -0.3 to 0.3kpkikdRise time%overshootSettling timeError(K),n 25 s,ts 0.05 66.35681.111611.17178.985748.4078The PID parameters of the traditional methodsfrom Table 3 will be chosen for comparison the performance of PID controller design for electric furnacetemperature system with very high disturbance whichshows a comparison as Fig.21.Table 5: Comparative performance of Nelder Meadwith high disturbance.hhhn(t)hhh DisturbancesPerformances hhhhh -0.4 to 0.4 -0.5 to 0.5 -0.5 to 0.5kpkikdRise time%overshootSettling timeError(K),n 25 s,ts 0.05 36.86800.990216.477412.877250.2561with disturbance, it has shown that the proposed controller can well operate although the electric furnacesystem exists the disturbances in the system process.In this experiment, the very high disturbance issquare wave signal from -1 to 1 that it is shown asFig.20.Fig.21: The comparison the performance of PIDcontroller design for electric furnace temperature system with very high disturbance.From Fig.21, the comparative performances ofNelder Mead with very high disturbance, the transient and steady-state performances are more robust to disturbance and better than the traditionalmethods with very high disturbance included ZieglerNichols [10], Cohen-Coon [11] and Direct Synthesis[12].6. CONCLUSIONSIn this paper, Nelder Mead based PID controllerdesign method for Electric furnace temperature control system was simulated in MATLAB. The key operations of this method include maximum time foroptimization, sampling time and maximum numberof iterations that the performance of this method isdepended on disturbance. The obvious advantages ofthe proposed approach are that 1) the transient andsteady-state performances are better than the traditional methods included Ziegler-Nichol

Currently, PID controller has been used to operate in electric furnace temperature control system because its structure is simpler compared to others. However, the issue of tuning and designing PID controller adap-tively and efficiently is still open. This paper presents an improved PID controller effic

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