Turbojet Engine Industrial Min–Max Controller Performance Improvement Using Fuzzy Norms

Home , Aircraft engine controls, Engine pressure ratio, Fuel control unit, Turbojet, Turbomachinery

electronics

Article Turbojet Engine Industrial Min–Max Controller Performance Improvement Using Fuzzy Norms

Soheil Jafari * and Theoklis Nikolaidis

Centre for Propulsion Engineering, School of Aerospace Transport and Manufacturing (SATM), Cranfield University, Cranfield MK43 0AL, UK; t.nikolaidis@cranfield.ac.uk * Correspondence: s.jafari@cranfield.ac.uk

Received: 29 October 2018; Accepted: 9 November 2018; Published: 11 November 2018

Abstract: The Min–Max control strategy is the most widely used control algorithm for gas turbine engines. This strategy uses minimum and maximum mathematical functions to select the winner of different transient engine control loops at any instantaneous time. This paper examines the potential of using fuzzy T and S norms in Min–Max selection strategy to improve the performance of the controller and the gas turbine engine dynamic behavior. For this purpose, different union and intersection fuzzy norms are used in control strategy instead of using minimum and maximum functions to investigate the impact of this idea in gas turbine engines controller design and optimization. A turbojet engine with an industrial Min–Max control strategy including steady-state and transient control loops is selected as the case study. Different T and S norms including standard, bounded, Einstein, algebraic, and Hamacher norms are considered to be used in control strategy to select the best transient control loop for the engine. Performance indices are deﬁned as pilot command tracking as well as the engine response time. The simulation results conﬁrm that using Einstein and Hamacher norms in the Min–Max selection strategy could enhance the tracking capability and the response time to the pilot command respectively. The limitations of the proposed method are also discussed and potential solutions for dealing with these challenges are proposed. The methodological approach presented in this research could be considered for enhancement of control systems in different types of gas turbine engines from practical point of view.

Keywords: Min–Max control strategy; gas turbine engine; fuzzy norms; control engineering; T-norm; S-norm

1. Introduction The control system in Gas Turbine Engines (GTEs) is to provide the stable and safe operation of the engine components at the operability and performance level for which it is designed. The control system should be able to satisfy all engine control modes simultaneously. These control modes are steady state control mode, transient control mode, and physical limitation control modes [1]. The steady state control mode is to track the pilot lever angle (PLA) command precisely and with the minimum error. The transient control mode is to satisfy the thrust requirement with the minimum possible response time. The physical limitation control mode is to protect the engine from malfunctions or exceeding limits like over-speed, over-temperature, surge and stall. These control modes should be satisﬁed simultaneously to guarantee the safe and precise operation for the engine [2]. There are several control strategies proposed to deal with the above-mentioned requirements dating back to 1952. Each of these algorithms has its advantages and disadvantages [3,4]. Some of them are not capable to satisfy all engine control modes simultaneously and some of them are weak in some modes and strong in some other modes. A comprehensive review and analysis on the history of GTEs control strategies could be found in [5,6].

Electronics 2018, 7, 314; doi:10.3390/electronics7110314 www.mdpi.com/journal/electronics Electronics 2018, 7, x FOR PEER REVIEW 2 of 14

There are several control strategies proposed to deal with the above-mentioned requirements dating back to 1952. Each of these algorithms has its advantages and disadvantages [3,4]. Some of Electronicsthem are2018 not, 7capable, 314 to satisfy all engine control modes simultaneously and some of them are weak2 of 14 in some modes and strong in some other modes. A comprehensive review and analysis on the history of GTEs control strategies could be found in [5,6]. The Min–Max control strategy is known as a practical algorithm to satisfy all engine control The Min–Max control strategy is known as a practical algorithm to satisfy all engine control modes simultaneously without any error and malfunction [7]. Results of design and implementation modes simultaneously without any error and malfunction [7]. Results of design and implementation of Min–Max controller were presented within the Basic Research in Industrial Technologies for of Min–Max controller were presented within the Basic Research in Industrial Technologies for Europe-Europe/America (BRITE-EURAM) project OBIDICOTE (On Board Identification, Diagnosis Europe-Europe/America (BRITE-EURAM) project OBIDICOTE (On Board Identification, Diagnosis and Control of gas Turbine Engines) and confirmed by all OBIDICOTE partners (SNECMA, Rolls-Royce and Control of gas Turbine Engines) and confirmed by all OBIDICOTE partners (SNECMA, Rolls- plc, MTU Aero Engines, Volvo Aero Corporation, Fiat Avio, Techspace Aero S.A., Lufthansa Technik Royce plc, MTU Aero Engines, Volvo Aero Corporation, Fiat Avio, Techspace Aero S.A., Lufthansa AG, Aerospatiale, Chalmers University of Technology AB, National Technical University of Athens, Technik AG, Aerospatiale, Chalmers University of Technology AB, National Technical University of Technische Universität München, Universität Stuttgart, Université Catholique de Louvain). Thus, this Athens, Technische Universität München, Universität Stuttgart, Université Catholique de Louvain). strategy is the most practical control method for gas turbine engines [8–10]. Thus, this strategy is the most practical control method for gas turbine engines [8–10]. A Min–Max controller contains different control loops: A Min–Max controller contains different control loops: •  Steady-stateSteady-state control control loop loop to tocalculate calculate the steady the steady state fuel state flow fuel according flow according to the engine to the operating engine operating condition, condition, • TransientTransient control control loop loop to to control control the the engine engine acceleration acceleration and decelerationand deceleration in response in response to the power to the leverpower angle lever (PLA), angle (PLA), • PhysicalPhysical limitation control loops to satisfy the engine constraints including the the engine engine stall, stall, flameout,flameout, over speed,speed, and over temperature limitations.limitations. The main idea of a Min–MaxMin–Max control structure for a single spool turbojet engine is shown in Figure 11.. As As shown shown in in this this figure, figure, the the fuel fuel flow flow to the to engine the engine could could be divided be divided into two into parts: two s parts:teady- steady-statestate fuel flow fuel calculated flow calculated by the steady by the state steady control state controlloop, and loop, transient and transient fuel flow fuel calculated flow calculated by one byof the one transient of the transient (PLA loop) (PLA or loop) physical or physical limitation limitation control controlloops. Th loops.e transient The transient fuel flow fuel will flow then will be thenadded be to added the steady to the state steady fuel state flow fuel to calculate flow to calculate the total thefuel totalflow fuelfor the flow engine for the combustion engine combustion chamber chamberat any instantaneous at any instantaneous time. time.

Figure 1. MMin–maxin–max control structure for jet engines.engines.

In order to select the appropriate appropriate control loop for transient transient fuel flowflow calculation, calculation, a selection strategy should be defined defined in the controller structure. The Min Min–Max–Max control strategy is proposed and confirmedconfirmed by well-knownwell-known industry and engine manufacturersmanufacturers during last decades [[7]7].. The min min–max–max control loop selection strategy uses the following logicallogical algorithm:algorithm: 푊푓,푡푟푎푛푠푖푒푛푡 = min (min (max (푊푓,푑푒푐, 푊푓,푃퐿퐴), 푊푓,푎푐푐), 푊푓,푁푚푎푥) (1) Wf ,transient = min min max Wf ,dec , Wf ,PLA), Wf ,acc) , Wf ,Nmax (1) where 푊푓,푃퐿퐴 , 푊푓,푑푒푐 , 푊푓,푎푐푐 and 푊푓,푁푚푎푥 are the transient fuel flow rates calculated by the PLA, wheremaximumWf , PLAdeceleration,, Wf ,dec, W maximumf ,acc and W accelerationf ,Nmax are theand transient maximum fuel speed flow loops, rates respectively. calculated by All the details PLA, maximum deceleration, maximum acceleration and maximum speed loops, respectively. All details about design, development, and implementation of Min–Max control strategies could be found in [2,11,12]. The schematic of Min–Max selection strategy in the controller structure is shown in Figure2. Electronics 2018, 7, x FOR PEER REVIEW 3 of 14 aboutElectronics design,2018, 7 ,development, 314 and implementation of Min–Max control strategies could be found3 of in 14 [2,11,12]. The schematic of Min–Max selection strategy in the controller structure is shown in Figure 2.

FigureFigure 2. 2. SchematicSchematic of of engine engine fuel fuel controller controller with with min min–max–max selection selection strategy strategy..

AfterAfter formation formation of of the the Min Min–Max–Max control control algorithm, algorithm, many many st studiesudies have have done done for for performance performance improvementimprovement of of this this strategy. strategy. Control Control loops loops g gainain tuning tuning in in single single objective objective [1,13 [1,13]] and and multi multi-objective-objective featurefeature [14,15 [14,15]],, additional additional control control modes modes definition definition for for engine engine constraints constraints [2,16 [2,16],], emission emission level level redureductionction [17,18 [17,18],], adding adding start start-up-up phase phase modelling modelling and and optimization optimization [19,20 [19,20]],, multi multi criteria criteria decision decision makingmaking strategy strategy [21 [21],], integrated integrated flight/propulsion flight/propulsion control control strategy strategy [22,23 [22],, 23real],- real-timetime simulation simulation test [24,25test [24], ,and25], andperformance performance optimization optimization of engine of engine and and cont controllerroller from from different different points points of of view view [15,17 [15,17]] areare some of of these these milestones milestones in in Min–Max Min–Max control control strategy strategy improvements. improvements. However, However, in all these in all studies these studiesthe final the selection final selection strategy strategy between between the transient the transient control control loops are loops kept are fixed. kept fixed. OnOn the the other other hand, hand, advanced advanced control control algorithms algorithms like like fuzzy fuzzy logic logic [26 [26––2828],], model model predictive predictive controlcontrol [29,30 [29,30],], and and sliding sliding mode mode control control [31,32 [31,32],], are are investigated investigated theoretically theoretically in in recent recent years. years. AlthoughAlthough these these studies studies present present successful successful approach approacheses in in designing designing c controllerontroller f foror GTEs, GTEs, they they are are usuallyusually focused focused on on a a specific specific j jetet e enginengine and and do do not not use use a a methodological approach approach like like Min Min–Max–Max controlcontrol system system as as well well.. Moreover, Moreover, these these approach approacheses are are relying relying more more on on computer computer simulations simulations rather rather thanthan real real-world-world applications applications resp respectect to to stability stability and/or reliability reliability issues. issues. TheThe main main contribution contribution of of th thisis paper paper is is to to use use advanced mathematical mathematical concepts concepts in in the the practical practical MinMin–Max–Max control control algorithm algorithm to to start start filling filling the the existing existing gap gap between between the the two two above above-mentioned-mentioned approachesapproaches.. For For this purpose, sectionsection twotwo willwill focusfocus on on problem problem formulation formulation with with a turbojeta turbojet engine engine as asa casea case study study where where the the main main idea idea is to is change to change the Min–Maxthe Min–Max selection selection strategy strategy (Equation (Equation (1)) by (1 using)) by usingdifferent different fuzzy fuzzy T-norms T-norms and S-norms and S-norms to optimize to optimize a pre-defined a pre-defined performance performance index for index the for engine. the engineSection. Section three presents three presents the simulation the simulation results by results applying by applying different different norms in norms the selection in the selection strategy. strategy.Section four Section is to four discuss is to and discuss analyze and the analyze simulation the simulation results, to raise results the, limitationsto raise the and limitations challenges and of challengesthe proposed of approach, the proposed and appto proposeroach, and potential to propose solutions potential for dealing solutions with future for dealing challenges. with Finally,future challenges.conclusion Finally, remarks conclusion are presented remarks in section are presented five. in section five.

2.2. Problem Problem Formulation Formulation TheThe main main idea of thethe paperpaper isis to to use use fuzzy fuzzy norms norms in in the the Min–Max Min–Max selection selection strategy strategy of Equationof Equation (1). (From1). From mathematical mathematical point point of view, of view, for unionfor union and and intersection intersection operators operators it could it could be written be written that: that:

µ휇A퐴∪B퐵(푥x) = max⁡[[휇µ퐴A((푥x)), ,휇µ퐵B(푥(x)])] (2(2))

휇퐴∩퐵(푥) = min⁡[휇퐴(푥), 휇퐵(푥)] (3) µA∩B(x) = min[µA(x), µB(x)] (3) It means that the set defined by Equation (2) is the smallest set containing both A and B and the It means that the set deﬁned by Equation (2) is the smallest set containing both A and B and the set defined by Equation (3) is the largest set contained by both A and B [32]. In the literature, only the set deﬁned by Equation (3) is the largest set contained by both A and B [32]. In the literature, only the

Electronics 2018, 7, x FOR PEER REVIEW 4 of 14 Electronics 2018, 7, 314 4 of 14 above type of operation is used in the GTEs control strategy. However, other possibilities exist in fuzzyabove sets type definition. of operation In other is used words, in the we GTEs could control define strategy. 퐴 ∪ 퐵 as However, any fuzzy other set possibilitiescontaining both exist A in and fuzzy B andsets definition.not necessarily In other the smallest words, we set. could The aim define of thisA ∪ paperB as any is to fuzzy study set other containing types of both operA atorsand B forand unionnot necessarily and intersection the smallest of fuzzy set. The sets aim in ofthe this Min paper–Max is controlto study strategy other types and of to operators investigate for union the performanceand intersection of this of contribution. fuzzy sets in The the Min–Maxmain reason control is that strategy the operators and to investigatein (2) and (3) the may performance not be satisfactoryof this contribution. in some situations. The main For reason instance, is that an the operator operators tha int (2) calculates and (3) maylarger not fuzzy be satisfactory sets for intersectionin some situations. may have For a positive instance, impact an operator on the that final calculates performance larger of fuzzy the controller sets for intersection and the GTE may simulationhave a positive results. impactMoreover, on theit is finalinteresting performance from theoretical of the controller point of andview the to explore GTE simulation the possibility results. of Moreover,using fuzzy it issets interesting in industrial from control theoretical strategies point as of it view may to result explore in higher the possibility and/or more of using flexible fuzzy performancesets in industrial for the control controller strategies and the as it engine. may result For non in higher-fuzzy and/or sets only more one flexible type of performance operation is for possiblethe controller for union and and the intersection. engine. For However, non-fuzzy fuzzy sets onlysets give one us type the of flexibilit operationy of is defining possible different for union operatorand intersection. in the context However, of industrial fuzzy Min sets–Max give control us the structure. flexibility Thus of defining, the main different advantage operator is to keep in the thecontext ability ofof industrialsatisfying Min–Maxall control controlloops simultaneously structure. Thus, (having the main the advantageMinMax structure) is to keep while the ability trying of to satisfyingimprove the all performance control loops w simultaneouslyithout using huge (having computational the Min–Max efforts structure) of meta- whileheuristic trying optimization to improve algorithms.the performance The T and without S norms using are huge functions computational mapping efforts T/S: [0, of 1 meta-heuristic] × [0, 1] → [0, optimization 1] which satisfies algorithms. the mathematicalThe T and S commutativity, norms are functions monotonicity, mapping T/S:and [0,associativity 1] × [0, 1] conditions.→ [0, 1] which The satisfies fuzzy S the-Norms mathematical and T- Normscommutativity, that could monotonicity,be used for union and and associativity intersection conditions. values calculation The fuzzy are S-Norms listed in andTable T-Norms 1 [33,34]. that could be used for union and intersection values calculation are listed in Table1[33,34]. Table 1. Different fuzzy sets for T and S norms. Table 1. Different fuzzy sets for T and S norms. Norms Title Formulation Normst1 Bounded Titledifference t1(A,B) = max Formulation (0, A + B − 1) t1s1 BoundedBounded difference sum s1(A,B)t1(A,B) = min = max (1, (0,A + A B) + B − 1) s1t2 EinsteinBounded product sum t2(A,B) = (AB/(2s1(A,B) − = [ minA + (1,B − A AB]) + B) t2s2 EinsteinEinstein productsum s2(A,B)t2(A,B) = =(A (AB/(2 + B)/(1− +[ AAB) + B − AB]) s2t3 AlgebraicEinstein product sum s2(A,B)t3(A,B) = = (A AB + B)/(1 + AB) t3 Algebraic product t3(A,B) = AB s3s3 ProbabilisticProbabilistic sum sum s3(A,B)s3(A,B) = A + = B A − + AB B − AB t4t4 HamacherHamacher product product t4(A,B)t4(A,B) = (AB)/(A = (AB)/(A + B − + AB) B − AB) s4s4 HamacherHamacher sum sum s4(A,B)s4(A,B) = (A =+ (AB − + 2AB)/(1 B − 2AB)/(1 − AB)− AB) t5t5 MinimumMinimum t5(A,B)t5(A,B) = min = (A,B) min (A,B) s5 Maximum s5(A,B) = max (A,B) s5 Maximum s5(A,B) = max (A,B)

ThusThus,, the the Equation Equation (1) (1) could could be be re re-written-written in inthe the following following form: form:

푊푓,푡푟푎푛푠푖푒푛푡 = 푡푖(푡푖 (푠푖(푊푓,푑푒푐, 푊푓,푃퐿퐴), 푊푓,푎푐푐), 푊푓,푁푚푎푥) (4) Wf ,transient = ti(ti (si Wf ,dec , Wf ,PLA), Wf ,acc) , Wf ,Nmax (4) where i could be selected from Table 1 (1 to 5). It means that any T-norm and S-norm from Table 1 couldwhere be iusedcould to beselect selected the winner from Table between1 (1 todifferent 5). It means transient that control any T-norm loops andin the S-norm Min–Max from control Table1 systemcould structure. be used to In select other the words, winner the between transient different control transientloops structure control for loops the inGTE the could Min–Max be plotted control assystem Figure structure.3. In other words, the transient control loops structure for the GTE could be plotted as Figure3.

Figure 3. Transient control loops for a turbojet engine with fuzzy sets. Figure 3. Transient control loops for a turbojet engine with fuzzy sets.

Electronics 2018, 7, 314 5 of 14

Since there are 5 different possibilities deﬁned for each norm in this paper, all different possibilities for Min–Max selection strategy in the Equation (4) would be: 5 × 5 × 5 = 125. The idea is to simulate these 125 scenarios to investigate the effectiveness of using fuzzy norms in gas turbine engines Min–Max controllers.

3. Simulation Results In order to investigate the effects of fuzzy sets for union and intersection in gas turbine engine performance, a Matlab/Simulink program is generated including an engine and an industrial Min–Max controller. A single spool turbojet engine is selected as a case study. The engine characteristics are listed in Table2. A reduced order Wiener model based on experimental results is generated to predict the dynamic behavior of the engine [35]. The engine is modelled and analyzed with details by the authors in [12], and [22]. The results of the modelling are also validated against experimental data. The Min–Max control strategy is also designed for the engine in [1,2] and the gains of the controller are optimized using genetic algorithm in [1], using particle swarm optimization in [12], and using invasive weed optimization in [15,21]. The schematic of the engine and Min–Max control strategy is shown in Figure1. The developed Matlab/Simulink program block diagram is shown in Figure4.

Table 2. Turbojet engine characteristics [36,37].

Characteristics Value Type Single Spool Turbojet Length 851 mm (33.5 in) Diameter 348 mm (13.7 in) Dry weight 61.2 kg (135 lb) Compressor 4 stage axial Combustors Annular Turbine Single Stage Maximum thrust 5.33 kN (1200 lbf) Overall Pressure Ratio 6.3:1 Air mass ﬂow 8.14 kg/s (17.94 lb/s) Speciﬁc fuel consumption: 1.1 kg/(daN h) (1.03 lb/(lbf h)) Maximum rotational Speed 29,700 rpm Thrust-to-weight ratio 8.9:1

A brief description about the operating procedure of the model and controller is as follow:

• Firstly, a predefined PLA profile is defined for the model. This command is a percentage of the available thrust that the pilot requires at any instantaneous time. Since the thrust is not measurable directly, other parameters (engine rotational speed or engine pressure ratio [38–40]) are used to translate the required thrust to a measurable parameter in Min–Max algorithm. In this study, the engine pressure ratio is used for this purpose. • The Min–Max controller gets the PLA command as well as engine situation (acceleration, deceleration, rotational speed) and using the strategy described earlier calculates the appropriate fuel flow for the engine. The control law in each loop could be Proportional (P), Proportional-Integral (PI), or Proportional-Integral-Derivative (PID) algorithm [2]. • The calculated fuel flow will satisfy all engine control modes simultaneously and protect the engine from malfunctions and physical limitations. More details and simulations of the engine and Min–Max control system could be found in [1,2].

Since a reduced order Wiener model is used for the engine, it is fast and enough to use the direct search approach for engine and controller simulation to get the results. Therefore, the model is run 125 times with different T and S norms (from Table1) to generate the results. Electronics 2018, 7, 314 6 of 14 Electronics 2018, 7, x FOR PEER REVIEW 6 of 14

PLA Loop

Limitation Loops

Figure 4. Engine and controller program block diagram developed in Matlab/Simulink. Figure 4. Engine and controller program block diagram developed in Matlab/Simulink. The pre-defined PLA profile that should be tracked by the engine is shown in Figure5. As can be seenSince from a reduced this figure, order for Wienert ≤ 15 model s, the PLA is used value for isthe constant engine, at it 0.6.is fast It meansand enough that the to use pilot the requires direct searchthe 60 percentapproach of for the engine available and thrust controller from simulat the engine;ion to it get then the increases results. Therefore, to the value the of model 1 at t =is 15run s, 125and times remains with at different this level T for and a furtherS norms 15 (from s; it is Table associated 1) to generate with a sudden the results. change in the pilot command to testThe the pre ability-defined of transient PLA profile and that physical should limitation be tracked control by the modes engine simultaneously; is shown in Figure and 5 then. As willcan bereturn seen to from the valuethis fig ofure, 0.7 atfort =t ≤ 30 15 s ands, the remains PLA value constant is constant until the at end 0.6. ofIt themeans simulation. that the Itpilot will requires also test the 60 deceleration percent of controlthe ava loop,ilable transient thrust from behavior the engine; of the it engine then increases and controller to the in value deceleration of 1 at t = procedure 15 s, and remainsand the capabilityat this level of for the a controller further 15 in s; trackingit is associated the pilot with command. a sudden The change predefined in the pilot PLA command profile is the to testworst the case ability scenario of transient containing and physical the sudden limitation change control (acceleration modes simultaneously; and deceleration) and in then pilot will command return towith the a value very largeof 0.7 amplitudeat t = 30 s and (from remains idle to constant maximum until power). the end Therefore, of the simulation. the different It will control also test modes the decelerationsatisfaction capability control loop, of the transient controller behavior will be of tested the engine using thisand scenario.controller in deceleration procedure and theMoreover, capability to investigateof the controller the effects in tracking of different the pilot norms command. quantitatively, The predefined a performance PLA indexprofile should is the worstbe defined. case scenario The indices containing defined the insudden this paperchange are (acceleration error of PLA and tracking deceleration) and thein pilot response command time. withThe formera very large satisfies amplitude the steady (from state idle control to maximum mode and power). the latter Therefor satisfiese, the the different transient control control modes mode. satisfactionThese indices capability are formulated of the controller as follow: will be tested using this scenario. Moreover, to investigate the effects of different norms quantitatively, a performance index should be defined. The indices definedZ insim this_time paper are errorEPR of PLA tracking and the response time. J1 = PLA − dt (5) The former satisfies the steady state control0 mode and theEPR lattermax satisfies the transient control mode. These indices are formulated as follow: tacc + tdec 푠푖푚_푡푖푚푒 J2 = 퐸푃푅 (5) (6) 퐽1 = ∫ |푃퐿퐴 − sim⁄_time | 푑푡 0 퐸푃푅푚푎푥 1 J3 = 푡 +푡 {β1 J1 + β2 J2} (6) (7) 퐽 =β{ 푎푐푐+ β 푑푒푐} 2 1푠푖푚_푡푖푚푒2 where the EPR is the instantaneous engine pressure ratio, EPRmax is the maximum engine pressure 1 (7) ratio during the simulation, sim_time퐽3 = is the{훽 simulation1퐽1 + 훽2퐽2} time, and t is the time index. The tacc, t 훽1+훽2 dec are the acceleration and deceleration times which the engine requires to follow the PLA command where the EPR is the instantaneous engine pressure ratio, 퐸푃푅푚푎푥 is the maximum engine pressure (settling times with ±2% error). The performance criteria are normalized. The index J1 is focused on ratio during the simulation, sim_time is the simulation time, and t is the time index. The ttacc, dec are the acceleration and deceleration times which the engine requires to follow the PLA command

Electronics 2018, 7, x FOR PEER REVIEW 7 of 14

Electronics 2018, 7, 314 7 of 14 (settling times with ±2% error). The performance criteria are normalized. The index 퐽1 is focused on the tracking ability of the controller to follow the pilot requirements. 퐽2 will consider the controller response time and 퐽3 is a weighted combination of 퐽1 and 퐽2 in which criteria are weighted the tracking ability of the controller to follow the pilot requirements. J2 will consider the controller according to their importance by the coefficients of i . In this paper, the weight factors 1  0.5 , response time and J3 is a weighted combination of J1 and J2 in which criteria are weighted according to   0.5 퐽 their2 importance are selected by the coefﬁcients for 3. It means of βi. In that this the paper, importance the weight of factors the twoβ1 objectives= 0.5, β2 = is 0.5 equal are selected in the foroptimizationJ3. It means process. that the importance of the two objectives is equal in the optimization process.

Figure 5. Variation of PLA value for the simulation. Figure 5. Variation of PLA value for the simulation. The maximum steady state error of 10 percent and the maximum acceleration/deceleration time The maximum steady state error of 10 percent and the maximum acceleration/deceleration time of 60 s are set to deﬁne the boundary of solution space for acceptable combinations. The values of of 60 s are set to define the boundary of solution space for acceptable combinations. The values of J1, J2 and J3 are normalized against these predeﬁned values to make the comparison fair. 퐽1, 퐽2 and⁡퐽3 are normalized against these predefined values to make the comparison fair. Figure6a–c show the normalized value of the performance indices ( J1, J2 and J3) for all 125 Figures 6a–c show the normalized value of the performance indices (퐽 , 퐽 and⁡퐽 ) for all 125 possibilities. It can be seen that there are three cases with high potential for1 2 using3 in Min–Max possibilities. It can be seen that there are three cases with high potential for using in Min–Max algorithm: number 24 and 76 and 125. These cases are discussed here. algorithm: number 24 and 76 and 125. These cases are discussed here. Case I: This case has the minimum value of the tracking error in all combinations (number 24 Case I: This case has the minimum value of the tracking error in all combinations (number 24 in in Figure6a). In other words, the best PLA tracking ability is achieved with this Min–Max selection Figure 6a). In other words, the best PLA tracking ability is achieved with this Min–Max selection strategy: strategy: W = t (t (s W , W ), W ) , W (8) f ,transient 5 2 5 f ,dec f ,PLA f ,acc f ,Nmax (8) 푊푓,푡푟푎푛푠푖푒푛푡 = 푡5(푡2 (푠5(푊푓,푑푒푐, 푊푓,푃퐿퐴), 푊푓,푎푐푐), 푊푓,푁푚푎푥) Thus, by using Einstein T-norm (t2 in Table1) in the second step of the Min–Max control strategy,Thus the, by tracking using Einstein ability T of-norm the controller (푡2 in Table will 1) be in enhancedthe second in step comparison of the Min with–Max the control conventional strategy, Min–Maxthe tracking controller. ability of the controller will be enhanced in comparison with the conventional Min–Max controller.CaseII: This case has the minimum response time in all combinations (number 76 in Figure6b). In otherCase words, II: This the case best has maneuverability the minimum response will be achieved time in all with combinations this Min–Max (number selection 76 strategy:in Figure 6b). In other words, the best maneuverability will be achieved with this Min–Max selection strategy: = ) ) (9) 푊푓,푡푟푎푛푠푖푒푛푡Wf ,transient = 푡5(푡4 t(5푠(5t(4푊(푓s,5푑푒푐W, 푊f ,dec푓,푃퐿퐴, W), f푊,PLA푓,푎푐푐,),W 푊f푓,acc,푁푚푎푥, W) f ,Nmax (9)

Equations (9) states that using Hamacher T-norm ( 푡4 in Table 1) will enhance the Equations (9) states that using Hamacher T-norm (t4 in Table1) will enhance the maneuverability ofmaneuverability the Min–Max controlof the Min strategy.–Max control strategy. Case III: III: ThisThis case case has has the the minimum minimum weighted weighted indices indices in all in combinations all combinations (number (number 125 in Figure 125 in Figure6c). In6 c). other In other word words,s, both both track track-ability-ability and and maneuverability maneuverability will will be be considered considered with with the same same importance with this Min–MaxMin–Max strategy: (10) 푊푓,푡푟푎푛푠푖푒푛푡 = 푡5(푡5 (푠5(푊푓,푑푒푐, 푊푓,푃퐿퐴), 푊푓,푎푐푐), 푊푓,푁푚푎푥) Wf ,transient = t5(t5 (s5 Wf ,dec , Wf ,PLA), Wf ,acc) , Wf ,Nmax (10)

Electronics 2018, 7, 314 8 of 14 Electronics 2018, 7, x FOR PEER REVIEW 8 of 14

TheThe above above combination combination is the is thestandard standard Min Min–Max–Max control control algorithm. algorithm. It means It that means if we that want if weto considerwant to considera weighted a weighted combination combination of tracking of tracking and maneuverability and maneuverability into account into account (with equal (with importance),equal importance), the conventional the conventional Min– Min–MaxMax algorithm algorithm with with standard standard minimum minimum and and maximum maximum mathematicalmathematical functions functions will will perform perform w well.ell.

(a)

(b)

Figure 6. Cont.

Electronics 2018, 7, 314 9 of 14 Electronics 2018, 7, x FOR PEER REVIEW 9 of 14

(c)

FigureFigure 6.6. PerformancePerformance indicesindices valuesvalues forfor engineengine andand controllercontroller simulationssimulations withwith differentdifferent TT andand NN

norms.norms. NormalizedNormalized valuesvalues forfor J1퐽1( a(a),), J 2퐽(2b ),(b), and andJ3⁡퐽(3c ).(c)

ToTo conﬁrmconfirm thethe abovementionedabovementioned conclusions,conclusions, thethe resultsresults ofof engineengine andand controllercontroller simulationssimulations withwith thethe aboveabove achievedachieved threethreecases casesare areshown shownin inFigures Figures7 –79–.9 Figure. Figure7 shows7 shows the the PLA PLA tracking tracking with with thethe threethree differentdifferent optimal cases. As As it it can can be be seen seen in in this this figure, ﬁgure, case case I (using I (using Einstein Einstein T-norm T-norm in the in theselection selection strategy) strategy) has has the the best best tracking tracking ability ability (minimum (minimum steady steady state state error). error). Thus Thus,, it could could be be concludedconcluded that that if if the the winner winner between between the the maximum maximum acceleration acceleration rate rate control control loop loop and and the the maximum maximum of PLAof PLA control control loop loop and and maximum maximum deceleration deceleration rate controlrate control loop loop will bewill selected be selected using u Einsteinsing Einstein t-norm t- formulation,norm formulation, the controller the controller would performwould perform better than better the than standard the standard Min–Max Min controller–Max controller from tracking from PLAtracking command PLA command point of view. point Moreover, of view. Moreover, it could also it could be seen also that be caseseen IIthat has case the minimumII has the minimum response timeresponse between time the between three simulated the three cases. simulated It means cases that. using It means Hamacher that using t-norm Hamacher in the selection t-norm strategy in the Electronics 2018, 7, x FOR PEER REVIEW 10 of 14 willselection enhance strategy the maneuverability will enhance the of maneuverability the engine. of the engine. Finally, case III results fall between cases I and II results. It means both error and response time will be considered simultaneously when the standard Min–Max selection strategy will be used. This strategy is not the best in command tracking nor in maneuverability. However, if the performance index is defined as weighted combination of tracking and maneuverability, it would perform better than the other cases. Figure 7 confirms the satisfaction of steady state and transient control modes by the optimal controllers. In other words, the pilot requirements are tracked with minimum error (steady state control mode) and in a short response time (transient control mode). In order to confirm the satisfaction of the physical limitation control mode, the variation of engine rotational speed for three cases are shown in Figure 8. This figure shows the smooth and safe variation of the engine rotational speed in all three cases without any over-speed for the engine. Therefore, the physical limitation control mode is also satisfied by the enhanced Min–Max control structures. Finally, to confirm the feasibility of the obtained results, the total fuel flow to the engine (actuator output) is shown in Figure 9 for the three cases. This figure confirms that the outputs of modified

controllers are smooth, feasible, and implementable without any fluctuation and disturbance. Thus, this figure confirms theFigureFigure practicalit 7 7.. TrackingTrackingy of the PLAproposed command approach by different in design optimal and cases.cases optimization. of industrial Min–Max control strategies for gas turbine engines.

Figure 8. Variation of engine rotational speed in different optimal cases.

Figure 9. Total fuel flow to the engine in different optimal cases.

Electronics 2018, 7, x FOR PEER REVIEW 10 of 14

Figure 7. Tracking the PLA command by different optimal cases. Electronics 2018, 7, 314 10 of 14 Figure 7. Tracking the PLA command by different optimal cases.

Figure 8. Variation of engine rotational speed in different optimal cases. FigureFigure 8 8.. VariationVariation of engine rotational speed in different optimal cases.cases.

Figure 9. Figure 9. TotalTotal fuel fuel flow flow to to the the engine engine in in different different optimal cases cases.. Finally, case IIIFigure results 9. fall Total between fuel flow cases to the I engine and II in results. different It optimal means cases both. error and response time will be considered simultaneously when the standard Min–Max selection strategy will be used. This strategy is not the best in command tracking nor in maneuverability. However, if the performance index is defined as weighted combination of tracking and maneuverability, it would perform better than the other cases. Figure7 confirms the satisfaction of steady state and transient control modes by the optimal controllers. In other words, the pilot requirements are tracked with minimum error (steady state control mode) and in a short response time (transient control mode). In order to confirm the satisfaction of the physical limitation control mode, the variation of engine rotational speed for three cases are shown in Figure8. This figure shows the smooth and safe variation of the engine rotational speed in all three cases without any over-speed for the engine. Therefore, the physical limitation control mode is also satisfied by the enhanced Min–Max control structures. Finally, to confirm the feasibility of the obtained results, the total fuel flow to the engine (actuator output) is shown in Figure9 for the three cases. This figure confirms that the outputs of modified controllers are smooth, feasible, and implementable without any fluctuation and disturbance. Thus, Electronics 2018, 7, 314 11 of 14 this figure confirms the practicality of the proposed approach in design and optimization of industrial Min–Max control strategies for gas turbine engines.

4. Discussion The presented results show that using fuzzy norms in Min–Max control algorithm for gas turbine engines can enhance the performance of the controller and the engine. However, there are some challenges and issues in this regard that should be addressed before this approach could be considered as a methodological approach for all types of gas turbine engines. The main challenges can be summarized as follow:

• The number of variables: the control modes for GTE can be summarized as a steady-state control mode (to satisfy the pilot demand), transient control mode (to satisfy an acceptable response time) and physical limitation control mode (to protect the engine against malfunctions). In an industrial control strategy for a turbojet engine, four different transient control loops are designed. One of them is in charge of the pilot demand (pilot lever angle (PLA) control loop) and three of them give guarantee about safe operation of the engine to protect the GTE against physical damages. However, the number of transient control loops would be increased in turbofan engines (two and three spool engines) and industrial gas turbine engines respect to many more constraints and parameters that should be considered for these types of GTEs. Therefore, the number of required runs would be increased noticeably, and it may not be affordable to solve the problem using direct search method. • Control loop gains: another issue is that when the Min–Max selection strategy is changed, it could not be claimed that the tuned gains are optimal. In other words, after changing the selection strategy, control loop gains should be tuned again to get the maximum potential of the control structure. In this approach, we could get a Pareto Front from a set of possible cases rather than having just two or three cases. However, adding these parameters to the problem will increase the complexity of the problem noticeably and may not be affordable easily. • Objective functions: the other issue is the performance indices defined for the problem. This paper considered the steady state error and the response time to the pilot command. However, there are many more objectives that could be considered for the controller enhancement like total fuel consumption in a mission, emission level, robustness and responses to the uncertainties etc. Addressing these objectives also needs a methodological approach and a problem formulation. • Effects of uncertainty: the main structure of the proposed controllers are the same with the industrial Min–Max controllers which is robust and reliable. However, after replacing the simple mathematical functions with fuzzy norms, the controller should be tested again under uncertain conditions (e.g., different weather and flight conditions) to confirm the ability of the proposed approach in dealing with uncertainties.

A potential solution to deal with the above-mentioned issues is to formulate a methodological approach for a framework development for this problem. This approach will be able to formulate the problem as a comprehensive engineering optimization problem with all parameters included in objective function formulation for any type of GTEs. Then, using a meta-heuristic optimization algorithm, both controller loops gains and Min–Max strategy norms could be optimized simultaneously. Moreover, different indices could be defined for the objective function in both weighted single objective and multi-objective features. However, the size of the problem and the mixed variables types (discrete for norms and continuous for gains) would be main issues for development of this framework. The main achievement of this study is to confirm that considering fuzzy norms for the Min–Max selection strategy is a high potential field for the performance optimization of GTE control systems. It could be the first step to connect the fuzzy sets with the industrial Min–Max control approach. Electronics 2018, 7, 314 12 of 14

5. Conclusions This paper focuses on using union and intersection fuzzy sets in the Min–Max selection strategy of gas turbine engine control systems to enhance the controller and engine performance. The main idea is to use S-norms for union and T-norms for intersection sets to calculate the appropriate fuel flow for the GTE to satisfy all engine control modes simultaneously. A single spool turbojet engine is selected as a case study with a Min–Max control system including four transient control loops. The Min–Max selection strategy has two unions and one intersection to investigate. Thus, 125 different cases simulated and tested on track-ability and maneuverability indices. The results of the simulation show that use of Einstein and Hamacher norms will enhance the PLA tracking and response time of the engine respectively. The limitations of the used method are the lower number of control loops in turbojet engines in comparison with other types of GTEs, fixed control loop gains, and limited objective indices tested in simulations. The potential solution suggested for these issues is to define a framework for defining the idea of using fuzzy sets in GTEs industrial controller as an engineering optimization problem. In this framework the fuzzy sets as well as control loops gains could be defined as optimization variables and more indices like emission level and total fuel consumption could be added to the objective function. The main achievement of the paper is to confirm the high potential of fuzzy norms to be used in industrial GTEs Min–Max control algorithms.

Author Contributions: Conceptualization, Methodology, Investigation, S.J. and T.N.; Writing-Original Draft Preparation, S.J.; Writing-Review & Editing, T.N. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Jafari, S.; Montazeri-Gh, M. Evolutionary Optimization for Gain Tuning of Jet Engine Min-Max Fuel Controller. J. Propul. Power 2011, 27, 1015–1023. [CrossRef] 2. Richter, H. Advanced Control of Turbofan Engines; Springer Science+Business Media: New York, NY, USA, 2012. [CrossRef] 3. Spang III, H.A.; Brown, H. Control of jet engines. Control Eng. Pract. 1999, 7, 1043–1059. [CrossRef] 4. Chirgwin, R. Airbus Conﬁrms Software Brought down A400M Transport Plane. The Register, 31 May 2015; Sponsored: A Layman’s Guide on How to Operate Your SIEM Under the GDPR. 5. Jaw, L.C.; Mattingly, J.D. Aircraft Engine Controls; American Institute of Aeronautics and Astronautics: New York, NY, USA, 2009; ISBN 978-1-60086-705-7. 6. Moir, I.; Seabridge, A. Engine Control Systems; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2008. [CrossRef] 7. Kreiner, A.; Lietzau, K. The Use of Onboard Real-Time Models for Jet Engine Control; MTU Aero Engines: Munich, Germany, 2002. 8. Dambrosio, L.; Camporeale, S.M.; Fortunato, B. Performance of Gas Turbine Power Plants Controlled by One Step Ahead Adaptive Technique; ASME: München, Germay, 2000. 9. Van Essen, H.; Lange, H. Nonlinear Model Predictive Control Experiments on a Laboratory Gas Turbine Installation; ASME: München, Germany, 2000. 10. Orme, J.; Schkolnik, G. Flight Assessment of the Onboard Propulsion System Model for the Performance Seeking Control Algorithm on an F-15 Aircraft. In Proceedings of the 31st Joint Propulsion Conference and Exhibit, San Diego, CA, USA, 10 July–12 July 1995. 11. Mohammadi, E.; Montazeri-Gh, M. Active Fault Tolerant Control with self-enrichment capability for gas turbine engines. Aerosp. Sci. Technol. 2016, 56, 70–89. [CrossRef] 12. Montazeri-Gh, M.; Jafari, S.; Ilkhani, M.R. Application of particle swarm optimization in gas turbine engine fuel controller gain tuning. Eng. Optim. 2012, 4, 225–240. [CrossRef] 13. Zeng, D.; Zhou, D.; Tan, C.; Jiang, B. Research on Model-Based Fault Diagnosi for a Gas Turbine Based on Transient Performance. Appl. Sci. 2018, 8, 148. [CrossRef] 14. Chipperﬁeld, A.; Fleming, P. Gas turbine engine controller design using multi-objective genetic algorithms. IEEE Trans. Ind. Electron. 1995, 43, 214–219. [CrossRef] Electronics 2018, 7, 314 13 of 14

15. Jafari, S.; Montazeri-Gh, M. Invasive Weed Optimization for Turbojet Engine Fuel Controller Gain Tuning. Int. J. Aerosp. Sci. 2013, 2, 138–147. [CrossRef] 16. Salehi, A.; Montazeri-Gh, M. Hardware-in-the-loop simulation of fuel control actuator of a turboshaft gas turbine engine. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ. 2018.[CrossRef] 17. Aghasharifian Esfahani, H.; Montazeri-Gh, M. Designing and Optimizing Multi-Objective Turbofan Engine Control Algorithm Using Min-Max Method. Int. J. Comput. Sci. Inf. Secur. 2016, 14, 1040–1050. 18. Salehi, A.; Montazeri-Gh, M. Black box modelling of a turboshaft gas turbine engine fuel control unit based on neural NARX. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ. 2018.[CrossRef] 19. Montazeri-Gh, M.; Fashandi, S.A.M.; Jafari, S. Theoretical and experimental study of a micro jet engine start-up behavior. Tehnicki Vjesnik 2018, 25, 839–845. [CrossRef] 20. Rossi, L.; Sorce, A.; Traverso, A. Gas turbine combined cycle start-up and stress evaluation: A simplified dynamic approach. Appl. Energy 2017, 190, 880–890. [CrossRef] 21. Jafari, S.; Khalaf, P.; Montazeri-Gh, M. Multi-Objective Meta Heuristic Optimization Algorithm with Multi Criteria Decision Making Strategy for Aero-Engine Controller Design. Int. J. Aerosp. Sci. 2014, 3, 6–17. [CrossRef] 22. Montazeri-Gh, M.; Jafari, S.; Nasiri, M. Application of Particle Swarm Optimization in Gain Tuning of Integrated Flight and Propulsion Control. Int. J. Aerosp. Sci. 2013, 2, 55–70. [CrossRef] 23. Zhang, J.; Kang, W.; Li, A.; Yang, L. Integrated flight/propulsion optimal control for DPC aircraft based on the GA-RPS algorithm. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 2015, 230, 157–171. [CrossRef] 24. Montazeri-Gh, M.; Nasiri, M.; Jafari, S. Real-time multi-rate HIL simulation platform for evaluation of a jet engine fuel controller. Simul. Modell. Pract. Theory 2011, 19, 996–1006. [CrossRef] 25. Martucci, A.; Volponi, A.J. Fuzzy fuel flow selection logic for a real time embedded full authority digital engine control. J. Eng. Gas Turbines Power 2003, 125, 909–916. [CrossRef] 26. Montazeri-Gh, M.; Yousefpour, H.; Jafari, S. Fuzzy logic computing for design of gas turbine engine fuel control system. In Proceedings of the 2nd IEEE International Conference on Computer and Automation Engineering (ICCAE), Singapore, 26–28 February 2010. 27. Mohammadi, E.; Montazeri-Gh, M. A fuzzy-based gas turbine fault detection and identification system for full and part-load performance deterioration. Aerosp. Sci. Technol. 2015, 46, 82–93. [CrossRef] 28. Bazazzadeh, M.; Badihi, H.; Shahriari, A. Gas Turbine Engine Control Design Using Fuzzy Logic and Neural Networks. Int. J. Aerosp. Eng. 2011, 2011, 1–12. [CrossRef] 29. Mu, J.; Rees, D. Approximate Model Predictive Control for Gas Turbine Engines. In Proceedings of the 2004 American Control Conference, Boston, MA, USA, 30 June–2 July 2004. 30. Pandey, A.; Oliveira, M. Model Predictive Control for Gas Turbine Engines. In Proceedings of the ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition, Oslo, Norway, 11–15 June 2018. 31. Bonfiglio, A.; Invernizzi, A.; Lanzarotto, D.; Palmieri, A.; Procopio, R. Definition of a sliding mode controller accounting for a reduced order model of gas turbine set. In Proceedings of the 52nd International Universities Power Engineering Conference (UPEC), Heraklion, Greece, 28–31 August 2017. 32. Yang, S.; Wang, X.; Yang, B. Adaptive sliding mode control for limit protection of aircraft engines. Chin. J. Aeronaut. 2018, 31, 1480–1488. [CrossRef] 33. Alkhazaleh, S.; Salleh, A.R. Fuzzy Soft Multiset Theory. Abstr. Appl. Anal. 2012, 2012, 350603. [CrossRef] 34. Salleh, A.R.; Alkhazaleh, S. An application of soft multiset theory in decision making. In Proceedings of the 5th Saudi Science Conference, Makkah, Saudi Arabia, 16–18 April 2012; pp. 16–18. 35. Lu, F.; Ye, Y.; Huang, J. Gas Turbine Engine Identification Based on a Bank of Self-Tuning Wiener Models Using Fast Kernel Extreme Learning Machine. Energies 2017, 10, 1363. [CrossRef] 36. Rideau, J.F.; Guyader, G.; Cloarec, A. MICROTURBO Families of Turbojet Engine for Missiles and UAV’s From the TR60 to the new bypass turbojet engine generation. In Proceedings of the 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Hartford, CT, USA, 21–23 July 2008. 37. Microturbo TRI 40/TRI 60 (2008). The Market for Missile/Drone/UAV Gas Turbine Engines; Forecast International: Newtown, CT, USA, 2009. 38. Adibhatla, S. Propulsion control law design for the NASA STOVL controls technology program. In Proceedings of the AIAA International Powered Lift Conference, Santa Clara, CA, USA, 13–15 November 1993. Electronics 2018, 7, 314 14 of 14

39. Adibhatla, S.; Cooper, P.; Pajakiwski, A.; Romine, B.; Virnig, J.; Bodden, D. STOVL Controls Technology, Vol. I Integrated Flight/Propulsion Control Design; NASA Contractor Report 195361; NASA: New York, NY, USA, 1994. 40. Dadd, G.J.; Sutton, A.E.; Greig, A.W.M. Multivariable control of military engines. In Proceedings of the Advanced Aero-Engine Concepts and Controls, Defense Research Agency, Farnborough, England, AGARD Presentation, Seattle, WA, USA, 1995.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).