1、NASA CONTRACTORREPORTNASA CR-220000!ZMAXIMUM LIKELIHOOD IDENTIFICATIONAND OPTIMAL INPUT DESIGN FORIDENTIFYING AIRCRAFT STABILITYAND CONTROL DERIVATIVESby David E. Stepner and Raman K. MehraPrepared bySYSTEMS CONTROL, INCPalo Alto, Calif. 94306or Langley Research CenterNATIONAL AERONAUTICS AND SPACE
2、ADMINISTRATION WASHINGTON, D. C. MARCH 1973Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. Report No, 2. Government Accession No. 3. Recipients Catal
3、og No.NASA CR-22004. Title and Subtitle 5. Report DateMarch 1973MAXIMUM LIKELIHOOD IDENTIFICATION AND OPTIMAL INPUT DESIGNFOR IDENTIFYING AIRCRAFT STABILITY AND CONTROL DERIVATIVES7. Author(s)David E. Stepner and Raman K. Mehra9. Performing Organi_tionNemeendAddres=Systems Control, Inc.260 Sheridan
4、AvenuePslo Alto, California 9430612. SpomoringAgencyNameandAddressNational Aeronautics and Space AdministrationWashington, D.C. 2054615. Su_e._,iary Notes6. Performing Organization Code8. Performing Organization Report No.10. Work Unit No.1. Contract or Grant No.NAS i- 1070013. Type of Report and Pe
5、riod CoveredContractor Report14, Sponsoring Agency Code16. Absl_actA new method of extracting aircraft stability and control derivatives from flighttest data is developed based on the maximum likelihood criterion. It Is shown that thisnew method is capable of processing data from both linear and non
6、linear models, bothwith and without process noise and includes output error and equation error methods asspecial cases. The first application of this method to flight test data is reportedfor lateral maneuvers of the HL-10 and M2/F3 lifting bodies, including the extractionof stablllty and control de
7、rivatives in the presence of wind gusts. All the problemsencountered in this identification study are discussed. Several different methods(includlng a priori weighting, parameter fixing and constrained parameter values) fordealing with Identlflabillty and uniqueness problems are introduced and the r
8、esultsgiven.The method for the design of optimal inputs for Identifying the parameters oflinear dynamic systems is also given. The criterion used for the optimization is thesensitivity of the system output to the unknown parameters. Several simple examplesare first given and then the results of an e
9、xtensive stability and control derivativeidentification simulation for a C-8 aircraft are detailed.17. Key Wor_ (suggested byAuthor($)Parameter identificationMaximum likelihoodAerodynamic derivativesAircraft modelingControl design for parameter extraction18. Distribution StatementUnclassified - Unli
10、mited19. Security CMmif. (of this report) 20. Security Clamif. (of this Page) 21. No. of PagesUnclassified Unclas sl fled 205*For Rleby the Natiorml Technical Information Service, Springfield, Virginia 2215122. Price*3 .ooProvided by IHSNot for ResaleNo reproduction or networking permitted without l
11、icense from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-III.IV.TABLE OF CONTENTSINTRODUCTIONOBJECTIVES AND SUMMARY OF RESULTS2.1 Study Objectives2.2 Maximum Likelihood Identification Technique2.2.1 X-22 VTOL Simulated Data2.2.2 HL-10 Flig
12、ht Data2.2.3 M2/F3 Plight Data2.3 Optimal Input Design2.3.1 Optimal Input for C-8 Aircraft Identification2.3.2 Monte Carlo SimulationBACKGROUND FOR AIRCRAFT PARAMETER IDENTIFICATION3.1 Previous Identification Methods3.1.1 Time Vector Method3.1.2 Analog-Matching Methods3.1.3 Equation Error Methods3.1
13、.4 Output Error Methods3.1.5 Advanced MethodsMAXIMUM LIKELIHOOD (ML) IDENTIFICATION4.1 Linear Systems4.2 Nonlinear Systems4.3 Numerical Optimization4.4 Relationship to Output Error and Equation ErrorMethods4.5 Identifiability and Uniqueness Problems in Extractionof Stability and Control Derivatives4
14、.5.1 Symptoms and Causes of Identifiability Problems4.5.2 Approaches to Identifiabillty Problems4.6 Maximum Likelihood Identification ProgramPagei667891011121314151516171818202227303334343638iiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE OF
15、 CONTENTSV. RESULTS OF IDENTIFYING AIRCRAFT STABILITY ANDCONTROL DERIVATIVES5.1 X-22 Simulated Data5.1.1 Generation of X-22 Simulated Data5.1.2 Program Descriptlon5.1.3 Limitations of Previous Results5.1.4 Comparison with Single Step and Multi-StepInput Sequences5.1.5 Comparison of Forward and Backw
16、ard Correlatlon5.1.6 Addltlonal Performance Index5.1.7 Accounting for Correlation Between Processand Measurement Noise5.1.8 Incluslon of Additional Partial Derivatives5.1.9 Aerodynamic DerivativeEstlmates5.2 HL-IO Flight Test Data5.2.1 Dynamical Equations of Motion and ObservatoryEquations5.2.2 Char
17、acteristics of HL-10, Fllght 19-25.2.3 Results of Fllght 19-25.2.4 Output-Error with Y and Y Identifiedp r5.2.5 Output Error with Constrained Parameter Values5.2.6 Output Error with Different Initial Conditions5.2.7 Output Error with A Priori Weighting5.2.8 Parameter Estimates Used for Prediction5.3
18、 M2/F3 Fllght Test Data5.3.1 Output Error - No Wind Gusts Included5.3.2 Perfect Measurement of SidesllpAngle5.3.3 Wind Gusts Included: Direct Identificationof Process Noise Covarlance and Time Constantof Correlated GustsPage464747495O525759606264686871748080818186869194102ivProvided by IHSNot for Re
19、saleNo reproduction or networking permitted without license from IHS-,-,-TABLE OF CONTENTS5.3.4 Three State Model with A Priori Weighting5.3.5 Three State Model with Fixed Parameters5.3.6 Three State Model with Rank Deficient SolutionVl. BACKGROUND FOR LINEAR SYSTEM INPUT DESIGN6.1 Related Work on I
20、nput Design in System IdentificationVII. THEORY OF INPUT DESIGN FOR LINEAR SYSTEM IDENTIFICATION7.1 Problem Formulation7.2 Optimal Energy - Constrained Input Using MaximumPrinciple7.2.1 Transition Matrix Method7.2.2 Riccati Equation Method7.3 Application of Functional Analysis7.4 Examples7.4.1 First
21、 Order System with Unknown Gain7.4.2 Levadis Example7.4.3 Second Order Example7.5 Extension to Systems with Process Noise7.5.1 Example7.6 State-Variable Constralnts7.7 Steps in Optimal Input Program7.8 Specialized AlgorithmsVIII. APPLICATIONS OF OPTIMAL INPUT DESIGN TO C-8 AIRCRAFT8.1 ShOrt Period (
22、Two-State) Longitudinal Dynamicsof C-8 Aircraft8.1.1 Optimal Input for Two State Model8.1.2 Fourier Transform of the Optimal Input8.1.3 Comparison with a Doublet Input of EqualDuration and Energy8.1.4 Effect of Small Parameter Value Changes onOptimal Input10610711111611811912212512612712913113113313
23、6138140141141145147147147152152154vProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE OF CONTENTS8.1.5 Weighted Trace Criterion8.2 C-8 Monte Carlo Simulation8.2.1 Optimal and Suboptimal Inputs for Monte CarloSimulation8.2.2 Generation of Simulated
24、 Flight Data8.2.3 Description of Monte Carlo IdentificationSimulation8.2.4 Results of Monte Carlo Simulation8.3 Optimal Input Through First-Order FilterCONCLUSIONSAREAS FOR FURTHER INVESTIGATIONAPPENDIX A - EQUATIONS OF MOTION FOR X-22 VTOLAPPENDIX B - GRADIENT AND INFORMATION MATRIX CALC_TION WITHA
25、DDED PARTIAL DERIVATIVE TERMS FOR X-22REFERENCES156161162163165167178182184185188192viProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FIGURESi.i4.14.25.15.25.35.45.55.65.75.85.9The Integrated Aircraft Identification ProcessImplementation of Maximum L
26、ikelihood EstimatorMaximum Likelihood Program Flow ChartInput Sequence Used in Generating Cornell DataMultistep Inputv and _ Originals Used in Calculating _, nx, nyv and _ Used in Calculating q, nx, n After ChangeYX-22 Estimated and Actual (Simulated) Stability and ControlDerivative Time HistoriesHL
27、-IO Observed Data and Control Sequence Time HistoriesHL-10 Observed Data and Estimates: Output ErrorHL-10 Fit Errors in p and r Measurements - Output ErrorHL-10 Observed Data and Estimates: Output Error withA Priori Weighting5.10 HL-IO Output Error with A Priori Weighting and Biases5.11 HL-IO Predic
28、tion of Final 2 Seconds of Data5.12 M2/F3: Observed Data and Control Sequence Time Histories5.13 M2/F3: Observations and Estimates - Output Error5.14 M2/F3: Observations and Estimates - Kalman Filter withz8 = 8 + 8N5.15 Time History of B +n nB5.16 M2/F3:, Direct Identification of Wind Gust Model5.17
29、 M2/F3 Time Histories with A Priori Weighting5.18 Performance Criterion as a Function of the Numbers ofModel Parameters321395255575865727778838487899299i00104108ii0viiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FIGURES8.15 Parameter Estimate Hist
30、ograms for5.19 M2/F3 Time Historieswith Dependent Parameter at FixedValues 1125.20 M2/F3 Time Histories with Rank Defieient Solution 1157.1 Flow Chart of Optimal Input Computer Program 143-i8.1 _max Vs. T Curve for a 2-State/5 Parameter Model 1498.2 Optimal Input for Short Period Longitudinal Dynami
31、cs 1508.3 Pitch Rate and Angle-of-Attack Time Histories withOptimal Input 1518.4 Fourier Transform of Optimal Input 1538.5 Suboptimal Doublet Input 1528.6 Optimal Input for System with i0% Parameter Variation 1558.7 Optimal Elevator Deflection with Unity Weights 1578.8 Optimal State Time Histories f
32、or Unity Weights 1588.9 Optimal Input and State Time Histories - with Weighted Trace 1608.10 Optimal and Suboptimal Input for Monte Carlo Simulation 1648.11 Block Diagram of Monte Carlo Simulation 1661718.12 Parameter Estimate Histograms for Mq8.13 Parameter Estimate Histograms for M 1728.14 Paramet
33、er Estimate Histograms for Z 173174M6eZ_e8.16 Parameter Estimate Histograms for 175viiJProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8.178.188.198;20FIGURESHistogram of Estimation Errors for n8Histogram of Estimation Errors for nqTwo-slded Optimal
34、Input from Output of First Order ServoTwo-sided Sub-Optimal Input from Output of First OrderServo176177179180ixProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLES5.1 X-22 Identification Results5.2 Standard Deviation of Process and Measurement Nois
35、e5.3 HL-IO Parameter Estimates and Standard Deviations5.4 M2/F3 Parameter Estimates and Standard Deviations8.1 Monte Carlo Results Based on Identification for 50 Setsof Simulated Data53557595168XProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MAXIMUM
36、 LIKELIHOOD IDENTIFICATION AND OPTIMAL INPUT DESIGNFOR IDENTIFYING AIRCRAFT STABILITY AND CONTROL DERIVATIVESBy David E. Stepner and Raman K. MehraSystems Control, Inc.INTRODUCTIONAircraft parameter identification is the process of extractingnumerical values for the aerodynamic stability and control
37、 derivatives,and other subsidiary parameters (wind gusts, sensors errors, etc.),from a set of flight test data (a time history of the flight controlinputs and the resulting aircraft response variables). The field ofidentification is one that has been pursued by diverse interests formany years. The p
38、ractical application of this work to aircraftflight testing has existed for over 25 years. In spite of the wealth ofexperience which has been accumulated in this span of time, importantrequirements still exist for improving the techniques for extractingstability and control derivatives.First, there
39、exists today a greater need for stability and controlderivatives. There are currently two principal requirements for themathematical models that these coefficients provide. These are (i) toprovide inputs to simulators*, and (2) to provide a basis for thedesign of flight control systems. A third pote
40、ntial may also exist.Because the stability and control derivatives define a given aircraftmore uniquely than the response mode criteria such as those in the*This may include digital computer simulations, fixed and moving base groundsimulators, and in-flight simulators such as variable stability airc
41、raft.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Flying Qualities Military Specification MIL-F-8785 there is reason tobelieve that these parameters will ultimately play more of a major rolein the design, testing, and certification of aircraft.Sec
42、ond, with the continuing advances in aircraft design and perform-ance capabilities, the ability to extrapolate wind tunnel test resultsis diminishing and the importance of flight testing is growing. This isaided by the new Department of Defense pollcy of building prototype air-craft and thoroughly f
43、light testing them before a production conm_tmentis made.The principal elements of the aircraft identification process(see Fig. i.I) are: (i) the identification algorithm, (2) the flightcontrol input and (3) the instrumentation. The ultimate success ofthe identification process is totally dependent
44、on all three of theseelements, not Just one of them alone. This study was concerned withthe first two points, namely, the development of a general advanceddigital identification technique based on the maximum likelihood criterionand the design of control inputs which will enhance the ability toident
45、ify specific aircraft stability and control derivatives. Digitalparameter identification techniques have already reached a stage wherethey are being used increasingly over analog matchlng techniques forextracting stability and control derivatives from flight test data.Systems Control, Inc., (SCI) un
46、der this present contract developedthe maximum likelihood identification technique, which was usedsuccessfully to reduce data from flight tests where gusts were present.In such cases both the measurement noise and process noise statisticswere identified.The importance of the control input in the ide
47、ntlfiabillty ofstability and control derivatives from response data has been apparentfor a long time. Under this contract, SCI has developed and appliedan efficient computational technique to design the optimal inputs foridentifying specific stability and control derivatives.Provided by IHSNot for R
48、esaleNo reproduction or networking permitted without license from IHS-,-,-Flight ControlillInputsExternal Disturbances _ Aircraft Model .-.J Flight Control _=_,_str_,_ent S_ecificationInput Design I_eent_catio_,_lgorithmActual ResponseVariablesJ /Flight Control InputsMeasurement ,_ System I_,_.,Identificatin Algorithm.81Stability andControl Derivatives.Sensor Errors, ModelingErrors, Identification /ErrorsFlight RecordData Processing(IdentificationAlgorithm)IIIAircraft ModelsM
