Soft Computing Applications on SR-30 Turbojet Engine

AIAA 2004-6444II. SR-30 TURBOJET ENGINE AND FACILITYTurbine Technologies’ SR-30 Turbojet EngineThe SR-30 engine is a turbojet engine with asingle-stage radial-flow compressor with a maximumpressure ratio of, PR=3.4, a single-stage axial-flowturbine, and a reverse-flow annular combustionchamber and it operates obeying the Braytonthermodynamic cycle in the same fashion as largeturbojet engines (Figure 1). The engine, as produced byTurbine Technologies, includes five pressuretransducers, five thermocouples, a load-cell for thrustmeasurements, a custom motor winding for reading theengine RPM, and a fuel-flow-rate measurement systemto monitor/measure the operating parameters of theengine. The engine generates . 20 lbs of thrust at 90,000RPM while ingesting m = 1.1 lb/sec of air. The enginehas a length of 10.75 in., and the exit exhaust diameterof D exit =2.25 inches. Further details can be found in thepaper by Watanabe et. al [1].the A/D board using cables. While one of these units(NI-SC-2345) houses the thermocouple input modules(NI-SCC-TC02), the millivolt range input modules (NI-SCC-AI06), a connector block for digital outputsignals, and the analog output modules, the other unit(NI-CA-1000) houses a connector block (NI-CB-68LPR) to connect other analog inputs. Each inputmodule includes self-contained signal-conditioningunits such as low pass filters and instrumentationamplifiers.While the thermocouple input modules are used toread the temperatures, the millivolt input range modulesare used to read the load-cell output voltage and one ofthe pressure signals. Digital I/O lines are used togenerate TTL signals to turn on and off the relays.Relays (BASCO Company, ELK 924) are used toreplace manual switches, which are used to start and tostop the ignition, the fuel flow and to turn on and offthe valve for high-pressure air.Fuel Flow Valve ControllerThe control output signal for the Single-inputsingle-output(SISO) system under study is the signal tocontrol the fuel-flow rate. The fuel-flow rate in theengine is adjusted by controlling valve opening toobtain a desired thrust value. The operation of thecontrol-valve is automated with the use of aCompumotor-ES22BS stepper motor. The ES22BSstepper motor is mounted on a simple stand to hold themotor steadily. The stepper motor has a maximumtorque of 9.4 lb-in, a maximum resolution of 128,000Step/rev with the use of the GT-L5 drive, and the shaftinertia is 1.1 oz-in 2 .Figure 1 SR-30 Turbojet EngineThe engine system is equipped with a dataacquisition board, addition of connection panels for thesensors, a stepper motor as fuel-flow rate valvepositioner, and several relays to control severalswitches for engine operation.Data Acquisition SystemThe current system at the University of Alabamaincludes a National Instruments NI PCI-6031E A/Dboard, which has a 16-bit resolution, 100 kSamples/secsampling rate, 64 analog input, 2 analog output, and 8digital I/O ports. Signal connections to the A/D boardare made using two enclosures, which are attached to2American Institute of Aeronautics and AstronauticsIII. PID CONTROLLER DESIGN USING THEFREQUENCY-RESPONSE METHODA PID type classical controller was designed andimplemented using the frequency-response method toidentify and learn the engine characteristics duringclosed-loop operations. This section outlines the stepby-stepdesign of the PID controller and the results ofits implementation.A series of data sets for the frequency-responsedesign was obtained using a LabVEW program. Thechosen input and output for the plant, the SR-30turbojet engine, are the fuel flow valve position indegrees and the thrust force in pounds, respectively. Togive the sinusoidal input excitation, the valve positionwas excited in a sine-wave manner with 5 degrees of amean position, corresponding to a steady-state thrustvalue of about 5.3 pounds, and 4 degrees of peak-to-

AIAA 2004-6444peak amplitude. Different frequencies of the excitationwere attempted to cover a frequency range of interest.In order to construct a pair of Bode Plots for thefrequency-response design method, gains and phasesneed to be calculated in the region of frequencies.However, raw measurements of signals fromtransducers usually contain noise. The load-cell systemof the SR-30 engine is not an exception and the thrustoutput signal contains such noise. Therefore, the thrustoutput signal was conditioned using the “filtfilt”command of MATLAB is used to condition the thrustoutput signals. After the signal conditioning, the peaksof the thrust output were found by seeking sign changesof signal derivatives. A Gain at each frequency wascalculated finding an average of the peak-to-peakvalues. An example of the raw measurement and thefiltered signals of the thrust output with the input signalfrequency of 6.0 rad/sec are shown in Figure 2.Thrust Output [lbs]0.30.20.10.00 1 2 3 4 5-0.1-0.2-0.3Time [sec]Figure 2 Raw and Filtered ThrustIII-A. PLANT IDENTIFICATIONfilteredThe frequency region of our interest ranges from0.1 rad/sec to 11.0 rad/sec. The data obtained from thesine-wave data collection is plotted on the pair of BodeDashed/SL App.Solid/ModelStar/F-R DataFigures 3a & 3b Open-Loop Bode PlotsrawDashed/Model without DelaySolid/Model with DelayStar/F-R Data-180plots in Figures 3a and 3b. The plant is modeled withMATLAB. Assuming a second order system, cornerfrequencies of 0.5 and 9.0 rad/sec were identified. Theassumed open-loop transfer function of the plant is:4.5P =(1)() s( s + 0.5)( s + 9.0)Although, the first-estimated model, the red solidline, matches in a satisfactory manner in the magnitudeBode plot, there is an obvious discrepancy between thefirst-estimated model (black-dashed) and the frequencyresponsedata in the phase Bode plot. Therefore, thepresence of a time-delay in the model was suspected.After several attempts of time-delay modeling, adding0.2 seconds of time-delay model with the MATLAB“pade” command to the plant (red-solid) showed aconsistent matching with the frequency-response data.The estimated model with the time delay is plotted inthe Bode plots with dashed-blue lines. The assumedopen-loop transfer function of the plant with the timedelayis:P() s4.5−0.2s= e (2)( s + 0.5)( s + 9.0)The assumed open-loop transfer function of theplant with the 4 th order “pade” approximation timedelay is:( )( )( )( )( )( )224.5 s −57.9s+ 914 s − 42.1s+ 1149P()s =22s + 0.5 s + 9.0 s + 57.9s+ 914 s + 42.1s+ 1149….(3)III-B. PID CONTROLLER DESIGN AND TESTProportional-Integral-Derivative (PID) controlapproach was chosen for the engine control since it isone of the popular closed-loop control approaches thatcan be applied to a wide range of engineering problems.One customary representation of such controller can beexpressed as:⎛ 1()⎟ ⎞G⎜cs = Kp1 + + Tds (4)⎝ Tis ⎠After attempting several combinations of numbersfor the PID gains using MATLAB SISO tool program,a combination of K p = 3.68, T i = 2.16 sec, and T d =0.061 sec was found to be a reasonable controller forthe application.3American Institute of Aeronautics and Astronautics

AIAA 2004-6444Figure 4, shows that the proposed controller gives apositive gain margin of 12.6 db and a positive phasemargin of 65.8 deg, which make the system stable.Since the system has a positive gain margin, there is atolerance of raising the proportional gain of the PIDcontroller.Cutoff frequency is the maximum frequency at whichthe output of a system will track an input sinusoid in asatisfactory manner.6.05.55.0Thrust [lbs]4.54.03.53.02.52.00 1 2 3 4 5 6Time [sec]Figure 7 Step Responses from SR-30 EngineFigure 4 Open-Loop Bode PlotsFigure 5 is the closed-loop Bode plots for thesystem. From the gain plot, the proposed closed-loopmodel has a cutoff frequency of about 3.01 rad/sec.Step input tests were conducted to test the designedPID controller. Figure 6 is the result from estimatedmodel using MATLAB. It shows a rise time of 0.65 secand a settling time of 1.15 sec. Figure 7 shows theexperimental results of step input tests using the SR-30engine. In the figure, three cases were plotted. For eachcase, a step command was issued at t = 1.0 sec. Thethrust values for the three cases were 3.5, 4.0, and 5.0lbs. For all cases the results show about 2 seconds ofsettling time with near zero steady-state errors.The large initial jump observed in the 3 to 5 lbincrease thrust run is undesirable and it is due tounforeseen nonlinear effects. This jump does not appearwhen using the fuzzy controller.IV. FUZZY LOGIC CONTROLLER DESIGN ANDAPPLICATION [2]Figures 5 Closed-Loop Bode PlotsThe employment of Bayesian and fuzzy techniquesderive from the “soft computing” family of algorithms.Here “soft computing” refers to computationalmechanisms that can determine suitable relationships(in a system data set) to assess and determine aquantitative opinion(s) based on future conditions.Since the scope of this paper is not on the depth ofunderstanding Bayesian and fuzzy techniques,application of them is presented. In short, withinMSFC, such computational mechanisms are viewed asa collection of algorithms that can achieve optimal ornear-optimal results in the presence of imprecise data,uncertainty [3], unknown physics, and probabilisticoutcomes. The central goal in soft computing is toattain more robust response.Figure 6 Simulated Unit Step Response4American Institute of Aeronautics and Astronautics

AIAA 2004-6444Similar to the PID design, for main-stage control, afuzzy logic controller was designed and implemented tocontrol the SR-30 engine and the response of the engineto a step input using the PID and Fuzzy controllers wascompared. The use of fuzzy logic was seen suitablesince it accommodates the uncertainties associated withcontroller architecture. Fuzzy logic is a branch ofmathematics that deals with approximate reasoning.Fuzzy logic, developed by Lotfi A. Zadeh of theUniversity of California at Berkely, combines the topicsof multi-valued logic, probability theory, and artificialintelligence for simulation of human thoughts by usingcomputer software as a medium. The technology offuzzy logic enables a computer to make decisions basedon vagueness or imprecision intrinsic in most physicalsystems. Fuzzy logic also provides a convenient way tointroduce useful nonlinearities into the control law toachieve specific effects, such as reducing largeovershoots. In depth details of fuzzy logic are outside toscope of this paper. A more detailed description onfuzzy logic can be found in [5].In the rest of this section the fuzzy logic controllerdesign and related programming process are discussed.A brief description of the design of the fuzzy logiccontrol algorithm is introduced. A simulation programwritten to test the fuzzy logic controller and the resultsof the simulation are discussed. Implementation of thefuzzy logic control algorithm on the SR-30 engineusing LabVIEW is detailed and the hardware-in-thelooptest results are discussed.IV-A.FUZZY LOGIC ALGORITHMDESCRIPTIONFuzzy Logic algorithm differs from the PID controlalgorithm since it can be used for non-linear-timevariantsystems. The algorithm intends to mimic humanthoughts in the sense that the relation between an inputand its output is assessed by a user, depending on theuser’s experience. In the fuzzy logic terminology,vague terms used such as hot, and cold are calledlinguistic terms and range of real numbers for the inputand the output is referred to as universe of discourse.Range of the input values divided into sections by itslinguistic terms is called antecedence and range ofoutput values divided into sections by its linguisticterms is called consequence.In a fuzzy logic control system, the fuzzificationprocess translates real input values to linguistic inputvalues, next the fuzzy inference system drives a5American Institute of Aeronautics and Astronauticsconclusion in a form of linguistic output valuesaccording to its rule base, which resembles IF-THENstructure used in many text-based programminglanguages. The final process called the defuzzificationprocess translates back the linguistic output values tocorresponding real output values for subsequent controlactions.IV-B. FUZZY LOGIC CONTROLLERARCHITECTUREThe NI LabVIEW Fuzzy Logic Controller Toolkitwas used to develop fuzzy logic controllers for the SR-30 engine control application. The toolkit allows theuser to interface with LabVIEW programs. The FuzzySet Editor specifies the details of the fuzzification step,and the details of defuzzification step. The Rule BaseEditor of the Fuzzy Logic Controller Toolkit enablesthe user to specify the details of the fuzzy inferencesystem. Before testing a fuzzy logic algorithm with theSR-30 engine, a LabVIEW-fuzzy simulation programwas developed using the plant model identified in thedesign process of the PID controller. The fuzzysimulation program had similar inputs to the ones of theLabVIEW-PID controller and the controller outputagain was the fuel-flow valve position. The fuzzysimulation program calculates appropriate valveposition in every loop execution for adjusting the fuelflowvalve opening to obtain desired thrust. A picture ofthe Labview fuzzy simulation program developed forthis purpose is shown in Figure 10. The input variablesused in the design of the PD part of the controller werethrust error calculated at node (1) and thrust error ratecalculated at node (2). As for the integral part of theprogram, the thrust error was the only input variable.The PD part and the integral part evaluate their outputsto the stepper motor separately from their inputs. Thesum of the angle outputs calculated at node (3) fromthe two parts is the total angle output to the steppermotor to control the fuel-flow valve position.The Figure 11-A shows the architectures of thefuzzy sets associated with the thrust error and the thrusterror rate as the controller inputs and the Figure 11-Bshows the architectures of the fuzzy sets associated withthe valve position and the valve position error (SeeTable 1 for the corresponding linguistic values for eachoutput). The use of the thrust error and thrust error rateeach with five linguistic terms results in 5 2 = 25 rulesfor the PD part. The use of the thrust error with fivelinguistic variables results in five more rules for theintegral part bringing the number of the total rules tothirty. The choice of using thirty rules gives areasonable number of rules and allows the user to

AIAA 2004-6444Integral Part1PD Part32Engine ModelFigure 10: Fuzzy Logic Simulation VI (wiring diagram)1.0N MN Z MP P1.0N MN Z MP P0.50.50.00.0-11.0 -5.5 0.0 5.5 11.0a b c d e f gA) INPUTSB) OUTPUTSFigure 11: Fuzzy Sets Associated with Thrust Error and Thrust Error Rate Inputs and FuzzySets associated with Valve Position and Valve Position Error Outputsbenefit from the capabilities of the fuzzy logiccontroller. Since the aim of the present study was todevelop a test-bed and demonstrate the hardware-inthe-loopanalysis, a relatively simple fuzzy logic designwas used. Table 2 explains the meanings of the symbolsto express each linguistic term. Table 3 shows a pair ofthe rule bases used in the design process.Among the defuzzification methods such as theCenter-of-Maximum, the Center-of-Gravity, and theMean-of-Maximum that are available in the FuzzyLogic Toolkit, the Center-of-Maximum method wasused in the design. The Center-of-Maximum methodresults in a lesser computation time and it also results ina linear variation of the output variable once fullyoverlapping linguistic terms were used.IV-C. FUZZY LOGIC CONTROLLER TESTUSING SIMULATED ENGINEThe LabVIEW-fuzzy logic controller design wasfirst tested in a LabVIEW simulation model using theplant transfer function identified in the PID controllerdesign process. It was assumed that the fuzzy logiccontrol algorithm would be applicable to this enginecontrol application because the architecture of thealgorithm is capable of handling non-linearity. In thedesign of the fuzzy logic algorithm, several differentuniverse of discourse ranges were tested for the6American Institute of Aeronautics and Astronautics

AIAA 2004-6444Thrust [lbs]andValve Position [deg]antecedence and the consequence ranges. The bestranges were identified which gave the least rise timewithout an overshoot.Step Input Test: Figure 12 shows a unit step responsefrom the simulation model, which includes the plantmodel and the fuzzy logic controller. The valve positionvery quickly rises to 3.6 degrees within 0.33 seconds.The simulated time delay is visible with about 0.2seconds. The thrust output from the plant modelincreases without an overshoot to the desired thrustvalue within 3.0 seconds. The number below thehorizontal axis in Figure 12 represents the number ofloops executed and the rate of the loop execution was25 time/sec.ThrustThrust DesiredValve PositionFigure 12: Step Response from Simulation Model.Twenty-five steps correspond to 1 second.stepsSystem Response with Limits on the Valve position:The fuzzy controller design was also tested to observethe effects on the system responses if valve positionlimits were set. The valve position limits were requiredbecause the controller tries to move the linkage out ofthe possible range of the motion. The reason of thenecessity for the limits is shown in Figure 13. Once thedesired thrust is obtained and subsequently decreasedback to a lower value, the valve position needs to bereduced below zero degrees, which is not achievabledue to the physical constraints of the fuel-flow valvesystem. Additionally, changing the valve position tobelow a certain angle would result in the engine to shutdown. Although the effect of the minimum angle limitset results in a much slower response of the engine, it isrequired to avoid any damage to the engine. In order toeliminate such control scenarios occurring, limits on theintegral controller was placed. The integrated controlangle and the control angle sent to the engine were resetto zero once the controller required an angle belowzero. The number below the horizontal axis in Figure13 also represents the number of loops executed and therate of the loop execution was 25 /sec.IV-D. FUZZY LOGIC CONTROLLER TESTWITH SR-30 TURBOJET ENGINEThe Fuzzy Logic Controller LabVIEW-Virtual-Instrument was next tested to control the SR-30 engine.In order to compare the response of the simulated andthe actual SR-30 engine responses, the same type of astep input test was attempted. The results are shown inFigure 14 (Case 1), Figure 15 (Case 2). Different rangesof the linguistic variable for the valve position outputwere used for each case while the same ranges of thelinguistic variables for the inputs were used with theones in the simulation. Refer to Figure 11 and Table 1for the numbers used for each case.A step command of 4.5 pounds was issued fromthe idling operation status for each case. Figure 14shows the result from the first attempt and the samefuzzy inference system used in the test simulation wasused for this case. There was an oscillation in the valveposition trying to achieve the desired thrust numbersince the thrust was not getting a steady state. In FigureThrust [lbs] and Valve Position [deg]5.0desired thrust4.54.03.5thrust3.02.5valve position2.01.51.0145 147 149 151 153 155 157 159Time [sec]Figure 14: Step Response from SR-30 Engine Case 17American Institute of Aeronautics and Astronautics

AIAA 2004-644415, the result from the second attempt is shown. Asteady state was achieved in about 5.0 seconds with anear no-steady error. Although the settling time waslonger than the one observed in the simulation using thePID controller, the performance of the fuzzy controllerfor this case showed an improvement over the first case.Figure 15 shows a good agreement with the simulationresult shown in Figure 12.to Pearl’s treatment of the subject [4]. The specificBBN approach to be employed in this effort is shown inFigure 16 and its basic procedure is beyond the scope ofthis paper. The basic steps are listed below.λ X (u)Uπ X (u)Thrust [lbs] and Vavle Position [deg]5.04.54.03.53.02.52.01.51.0desired thrustthrustvalve position161 163 165 167 169 171 173 175Time [sec]Figure 15: Step Response from SR-30 Engine Case 2V. BAYESIAN BELIEF NETWORKSCONTROLLER DESIGN [2]For the engine start phase, the primary softcomputing technology to be utilized is Bayesian beliefnetworks (BBN). The sole intent of the BBN is toqualify each of the states during engine start-up prior toreaching main-stage. This will further assure certaintyin the health of the engine and proceeding into mainstagein addition to providing added assurance intopreventing any premature engine shutdowns.BBN’s have been proven to be good predictive anddiagnostic mechanisms for reasoning about the state ofevents in environments where uncertainty is universal.Suppressing the details, the genealogy of this SCT isstrongly rooted in classic statistical Bayesian inferencetheory where a subjectivist viewpoint is taken. In shortBayesian inference uses a different interpretation ofprobability where one’s degree of belief in some eventis part of the reasoning. BBN’s are computationalarchitectures that permit declarative (prior conditionalprobabilistic values) and subjective opinions (posteriorprobabilistic values) about world (factual) knowledge tobe part of the reasoning and assessment through avisual network representation and a unique syntacticmessage-passing feature. Furthermore, the visualnetwork makes it easier to view the top-down cause andeffect (or condition to consequence) relationships. For amore detailed account of this SCT, the reader is referred8American Institute of Aeronautics and AstronauticsV X WYλ Y (x)π Y (x)π Z (x)λ Z (x)Figure 16 Bayesian Belief Network1. Belief Updating – When node X is activated toupdate its parameters for belief updating, it firstinspects all messages transmitted to it by its parent (π)and its children nodes (λ). Then using all input, itupdates its belief.2. Bottom-up Propagation – Using messagestransmitted by Y&Z, compute message to transmit toparent node U.3. Top-down Propagation – Node X then computesnew messages to be sent to its children nodes Y&Z.A first design of the Bayesian Belief Networks forthe SR-30 Engine is presented in the Figure 17a and17b. Table 4 gives the initial design for the BBNalgorithm.LegendVirtual NodeUnidirectional BeliefPropagationBidirectional BeliefPropagationContinuationBoundary BetweenSequence &Belief NetworkAirSwitchFuelSwitchEngineVibrationOil/FuelLeakageSwitchStatusIgnitionSwitch3 Sec.TimerTempLimitsMasterKeySwitchEngineStartInitiatedTerminateEngineStart Seq.Figure 17a Bayesian Belief Networks forSR-30 Turbo Jet Engine Start-UpThe design required analysis of the engine-start dataand careful choice of the parameters for the design. ForZOilPressureFuelSwitchElectricMasterSwitchEngineStartInitiated

AIAA 2004-6444this purpose six sets of engine-start data were acquiredand analyzed. This initial design will allow applicationof the BBN on the existing test bed.Virtual Nodes: SensorsP1 Compressor InletP2 Compressor ExitP3 Combustion ChamberP4 Turbine Stage ExitP5 Thrust Nozzle ExitRPM Engine Shaft SpeedFFS Fuel Flow SensorTf Thrust ForceT1 Compressor ExitT2 Compressor ExitT3 Turbine Stage InletT4 Turbine Stage ExitT5 Thrust Nozzle ExitEngineStartInitiatedP1…………………EngineStartTerminateEngineStart Seq.T05Select VirtualNodesAnomalyDetectionFigure 17b Bayesian Belief Networks forSR-30 Turbo Jet Engine Start-UpVI. CONCLUSIONSDevelopment of a test-bed for soft-computingstudies has been presented. Modifications on anexisting SR-30 Turbine Technologies small-scaleturbojet engine included the integration of a new dataacquisitionboard, replacement of the manual switchesand manual lever with computer controlled switchesand a stepper motor. The operation and control of theengine has been automated using a PC using NationalInstruments LabVIEW program.Two different control algorithms have beendesigned, tested in a simulated environment andintegrated into the test bed. Both the PID and the fuzzylogicalgorithms have been shown to successfullycontrol the operation of the engine.Additionally a design of the BBN algorithm ispresented. The algorithm will be integrated into thesystem in a future study.REFERENCES123456Watanabe, A., Davis, R. N., Ölçmen, S. M., Polites, M. E., andTrevino, L. C., “A Test bed for Evaluating <strong>Soft</strong> <strong>Computing</strong>Technologies for Rocket Engine Control.” 42nd AIAAAerospace Sciences Meeting and Exhibit, Reno, NV, January05-08, 2004.Trevino, L. C., Ölçmen, S. M., and Polites, M. E., “Use of <strong>Soft</strong><strong>Computing</strong> Technologies for Rocket Engine Control.” 22ndDigital Avionics Systems Conference, Indianapolis, IN, October12-16, 2003.Steincamp, J., First Annual Report, Marshall Space Flight Center,Huntsville, AL, September 2001.Pearl, J., “Probabilistic Reasoning in Intelligent Systems:Networks of Plausible Inference,” Morgan KaufmannPublishers, Inc., San Mateo, CA, 1988.Zadeh, L. A., “Fuzzy Logic,” Internal Report, University ofCalifornia, Berkeley, CA, 1988.Watanabe, A. “Development of a Test-Bed for Rocket EngineControl Algorithm Improvement.” Masters Thesis, TheUniversity of Alabama, Tuscaloosa, AL, 2004.9American Institute of Aeronautics and Astronautics

AIAA 2004-6444Table 1: Linguistic Values in Figure 11Case Output A B C D E F GSimulation andCase 1 with SR-30Valve position -20.0 -12.5 -10.0 0.0 10.0 12.5 20.0Case 2 with SR-30 Valve position -30.0 -18.8 -15.0 0.0 15.0 18.8 30.0Case 3 with SR-30 Valve position -35.0 -21.9 -17.5 0.0 17.5 21.9 35AllValve positionerror-1.6 -0.8 -0.2 0.0 0.2 0.8 1.6Table 2: Linguistic TermsN1I thrust error is negative N1O valve position change will be negativeMN1I thrust error is medium negative MN1O valve position change will be medium negativeZ1I thrust error is near zero Z1O valve position change will be near zeroMP1I thrust error is medium positive MP1O valve position change will be medium positiveSP1I thrust error is positive P1O valve position change will be positiveN2I thrust error rate is negative N2O adjustment for valve position error will be negativeMN2I thrust error rate is medium negative MN2O adjustment for valve position error will be medium negativeZ2I thrust error rate is near zero Z2O adjustment for valve position error will be near zeroMP2I thrust error rate is medium positive MP2O adjustment for valve position error will be medium positiveP2I thrust error rate is positive P2O adjustment for valve position error will be positiveTable 3: Rule Bases for Valve Position and Valve Position Error OutputsN1I MN1I Z1I MP1I P1IN2I N1O MN1O MN1O MN1O Z1O N1I MN2OMN2I N1O MN1O Z1O Z1O Z1O MN1I MN2OZ2I MN1O MN1O Z1O MP1O MP1O Z1I Z2MP2I Z1O Z1O Z1O MP1O P1O MP1I MP2OP2I Z1O MP1O MP1O MP1O P1O P1I MP2OValve Position Output Valve Position Error Output10American Institute of Aeronautics and Astronautics

AIAA 2004-6444Table 4: SR-30 Turbo Jet Engine Bayesian Design ParametersLink Matrices (M): All link matrices (conditional probabilities) are unity for convenienceMaster Key Switch Prior: π MKS = [0.95 0.05 ] [Probability True Probability False]Electric Master Switch Prior: π EMS = [0.99 0.01]Engine Start Initiated Start Node Prior: π ESI = [0.99 0.01]Engine Start Node Prior: π ES = [0.99 0.01]Temperature (T) LikelihoodProbabilitiesT1 & T2 DisabledT3 Compressor Exit(Normalization Factor 1200)T4 Turbine ExitT5 Nozzle Exitλ T3 [-∞ → e/s] = [ 0 1 ][e/s → 5 ] = [ 0.15 0.85 ]λ T4 [-∞ → e/s] = [0 1][e/s → 5 ] = [ 0.15 0.85 ]λ T5 [-∞ → e/s] = [0 1][e/s → 5 ] = [ 0.10 0.9 ]Probability Probability[ Too High Too Low ]e/s = engine startPressure LikelihoodProbabilitiesP1 Compressor InletP2 Compressor ExitP3 Combustion ChamberP4 Turbine Stage ExitP5 Thrust Nozzle Exitλ P2 [-∞ → e/s] = [ 0 1 ][e/s → 9 ] = [ 0.05 0.95 ]λ P3 [-∞ → e/s] = [0 1][e/s → 9 ] = [ 0.1 0.9 ]λ P4 [-∞ → e/s] = [0 1][e/s → 9 ] = [ 0.05 0.95 ]λ P5 [-∞ → e/s] = [0 1][e/s → 9 ] = [ 0.1 0.9 ]λ RPM [-∞ → e/s] = [0 1][e/s → 9 ] = [ 0.1 0.9 ]λ Thrust [-∞ → e/s] = [0 1][e/s → 9 ] = [ 0.10 0.9 ]λ Flow [-∞ → e/s] = [0 1][e/s → 9 ] = [ 0.05 0.95 ]Probability Probability[ Too High Too Low ](Note: > 0.65)(Note: > 0.625)(Note: > 1)(Note: > 4800)(Note: > 3.5)(Note: > 1.7)11