Formal specification of requirements for analytical redundancy-ba

Graduate Theses, Dissertations, and Problem Reports 2000 Formal specification of requirements for analytical redundancybased fault -tolerant flight control systems Diego Del Gobbo West Virginia University Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Del Gobbo, Diego, "Formal specification of requirements for analytical redundancy-based fault -tolerant flight control systems" (2000) Graduate Theses, Dissertations, and Problem Reports 2378 https://researchrepository.wvu.edu/etd/2378 This Dissertation is protected by copyright and/or related rights It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s) You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU For more information, please contact researchrepository@mail.wvu.edu Formal Specification of Requirements for Analytical Redundancy based Fault Tolerant Flight Control Systems Diego Del Gobbo Dissertation submitted to the College of Engineering and Mineral Resources at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Marcello Napolitano, Ph.D., Chair Larry Banta, Ph.D Robert Bond, Ph.D Ali Mili, Ph.D Gary Morris, Ph.D Department of Mechanical and Aerospace Engineering Morgantown, West Virginia 2000 Keywords: Fault Tolerance, Flight Control System, Analytic Redundancy, System Requirements Specification, Relational Algebra Copyright 2000 Diego Del Gobbo Abstract Formal Specification of Requirements for Analytical Redundancy based Fault Tolerant Flight Control Systems By Diego Del Gobbo Flight control systems are undergoing a rapid process of automation The use of Fly-By-Wire digital flight control systems in commercial aviation (Airbus 320 and Boeing FBW-B777) is a clear sign of this trend The increased automation goes in parallel with an increased complexity of flight control systems with obvious consequences on reliability and safety Flight control systems must meet strict fault-tolerance requirements The standard solution to achieving fault tolerance capability relies on multi-string architectures On the other hand, multi-string architectures further increase the complexity of the system inducing a reduction of overall reliability In the past two decades a variety of techniques based on analytical redundancy have been suggested for fault diagnosis purposes While research on analytical redundancy has obtained desirable results, a design methodology involving requirements specification and feasibility analysis of analytical redundancy based fault tolerant flight control systems is missing The main objective of this research work is to describe within a formal framework the implications of adopting analytical redundancy as a basis to achieve fault tolerance The research activity involves analysis of the analytical redundancy approach, analysis of flight control system informal requirements, and re-engineering (modeling and specification) of the fault tolerance requirements The USAF military specification MIL-F-9490D and supporting documents are adopted as source for the flight control informal requirements The De Havilland DHC-2 general aviation aircraft equipped with standard autopilot control functions is adopted as pilot application Relational algebra is adopted as formal framework for the specification of the requirements The detailed analysis and formalization of the requirements resulted in a better definition of the fault tolerance problem in the framework of analytical redundancy Fault tolerance requirements and related certification procedures turned out to be considerably more demanding than those typically adopted in the literature Furthermore, the research work brought up to light important issues in all fields involved in the specification process, namely flight control system requirements, analytical redundancy, and requirements engineering Acknowledgments I would like to thank Dr Marcello Napolitano, my advisor, for his support during this research I am thankful to Dr Ali Mili for having brought some light in the chaos that characterized the early phases of this work His guidance was crucial to the successful completion of this project I am also thankful to Dr Francesco Nasuti for his friendship and for the numerous helpful discussions on the many faces of analytical redundancy I wish to thank all of the Drs., researchers, and students who played a role in this research work Among them I would like to cite Dr Wu Wen, Dr Jack Callahan, Dr Steve Easterbrook, Dr Bojan Cukic, Dr Mark Shereshevsky, Dr Harjinder Sandhu, and Dr Vittorio Cortellessa In the years of meetings and discussions since the start of the project, they helped me understand the hidden truth behind a multidisciplinary research work I have no words to thank my wife Teresa, without her love and support I would not be here now Of course, I am grateful to my parents for their unbounded love, and to my grandmother for ”being the origin of the family”, as she says I am also grateful to my brothers for not hanging me upside down this time, and to my sister for her unforgettable shout of joy Finally, I wish to thank all of those who kept asking: ”So have you done?” I have! iii Contents Introduction Background information 2.1 Issues on the analytical redundancy approach in fault tolerant flight control systems 2.1.1 Analytical redundancy 2.1.2 Analytical redundancy in flight control systems 2.2 Formal specification of system requirements 2.2.1 Requirements engineering 2.2.2 Advantages of adopting a formal specification language 6 12 12 16 Research framework 3.1 FTC: the system to be specified 3.1.1 FTC environment 3.1.2 Main functions of the FTC system 3.1.3 FTC interface with its environment 3.2 DHC-2 aircraft 3.3 Military specification for AFCS 18 18 19 21 26 29 31 35 36 45 51 54 56 60 62 62 62 66 67 71 71 74 Formal specification of the FTC environment 4.1 Relational specification of elementary requirements 4.2 Composition of elementary requirements 4.3 Formal specification of the FTC environment 4.3.1 Performance requirement composition 4.3.2 DHC-2 detail-specification 4.3.3 Correctness of AFCS design Formal requirements specification of the FTC 5.1 FTC requirements 5.1.1 FTC functional requirements 5.1.2 FTC non-functional requirements 5.1.3 FTC-AR requirements 5.2 Formal specification of FTC-AR 5.2.1 Components partitioning 5.2.2 Formal specification of fault hypotheses iv 5.2.3 Relational specification of the FTC-AR requirements 5.3 Feasibility analysis 5.3.1 Traditional interpretation of detectability and identifiability 5.3.2 Formal definition of detectability and identifiability 75 77 78 80 Conclusions 82 A Predicate Logic and Relational Algebra A.1 Logic A.1.1 Propositional logic A.1.2 Predicate logic A.2 Relational algebra and requirements specification A.2.1 Basics of relational algebra A.2.2 Relational specifications 92 92 92 93 95 95 97 B Elementary specifications of the AR-FTC environment B.1 Elementary requirements of AFCS performance specification B.2 Elementary requirements of DHC-2 detail-specification B.2.1 DHC-2 airplane dynamics B.2.2 DHC-2 Flight Control System Hardware B.2.3 DHC-2 Flight Control System Software B.3 Fault modes B.3.1 Control-surface fault modes B.3.2 Engine fault modes B.3.3 Actuator fault modes B.3.4 Rate gyro fault modes B.3.5 Accelerometer fault modes B.3.6 Air data sensor fault modes B.3.7 Angle of attack sensor fault modes B.3.8 Attitude and heading sensor fault modes B.4 DHC-2 requirement space restriction sets B.5 Elementary requirements of interface blocks to AR-FTC system C Support tables of the specification v 102 103 113 113 119 126 130 130 131 131 132 133 133 134 134 136 138 141 List of Tables 3.1 DHC-2 autopilot functions and related controls 4.1 4.2 30 Constants used within the specification of the HH function Domain and image variables used within the specification of the HH function 4.3 Quantified variables used within the specification of the HH function 4.4 Predicates and functions used within the specification of the HH function 41 A.1 Syntax of propositional logic A.2 Semantics of propositional logic A.3 Syntax of predicate logic 93 94 94 41 41 41 B.1 Minimum acceptable control accuracy for ALH function 111 C.1 Elementary requirements C.2 Composed requirements C.3 Fault modes C.4 Restriction sets C.5 Spaces used within the requirements specification C.6 Domain and image variables C.7 Constants C.8 Quantified variables C.9 Auxiliary terms C.10 Data-types vi 143 149 150 151 152 154 161 172 176 182 List of Figures 3.1 3.2 3.3 Environment of the FTC system FTC within its environment Block diagram of DHC-2 aircraft and its FCS 19 27 32 4.1 4.2 4.3 Sample requirements specification structure Structure of the AFCS performance requirements specification Structure of the DHC-2 detail-specification 46 55 58 5.1 DHC-2 and FTFCS requirements specification structure 68 vii List of Symbols and Abbreviations ACT ADC AFCS ALH AR AR-FTFCS CP Cin Cout DAC DHC-2 DP FBW FCC FCL FCS FCSw FDC FTC FTC-ADC FTC-AR FTC-CP FTC-DAC FTC-DP FTC-IN FTC-OUT FTC-SW FTFCS HH HS MFCS PAH RAH UAV Actuator Analog to Digital Converter Automatic Flight Control System Altitude Hold Analytical Redundancy Analytical Redundancy based Fault Tolerant Flight Control System Control Panel Computer input Computer output Digital to Analog Converter De Havilland DHC-2 aircraft Display Panel Fly-By-Wire Flight Control Computer Flight Control Law Flight Control System Flight Control Software Flight Dynamics and Control Fault Tolerance Capability Fault Tolerance Capability – Analog to Digital Converter Fault Tolerance Capability – Analytical Redundancy module Fault Tolerance Capability – Control Panel module Fault Tolerance Capability – Digital to Analog Converter Fault Tolerance Capability – Display Panel module Fault Tolerance Capability – software Input interface Fault Tolerance Capability – software Output interface Fault Tolerance Capability – Safety switch Fault Tolerant Flight Control System Heading Hold Heading Select Manual Flight Control System Pitch Attitude Hold Roll Attitude Hold Unmanned Aerial Vehicle viii ,
Infallible / fail-operational software component Infallible / fail-operational hardware component Subsystem made of more than one component Fallible hardware component Directional data stream Infallible / fail-operational software component of FTC system Infallible / fail-operational hardware component of FTC system Physics law Note: Symbols used in the document are collected in the tables of Appendix C ix Table C.7: Constants (continued) Symbol and Value − ls1 = −10 · deg2SI + ls1 = 10 · deg2SI − ls2 = −20 · deg2SI + ls2 = 20 · deg2SI Type angle-T angle-T angle-T angle-T psi2SI = 6894.757 f oot2SI = 0.3048 deg2SI = π/180 g2SI = 9.80665 RP M 2SI = 0.1047198 inHg2SI = 3386.389 hp2SI = 745.6999 float float float float float float float Description inferior limit of symmetric autopilot integrator superior limit of symmetric autopilot integrator inferior limit of ALH autopilot superior limit of ALH autopilot Conversion factors conversion factor from psi to P a conversion factor from foot to m conversion factor from degree to rad conversion factor from g to m · s−2 conversion factor from RPM to rad · s−1 conversion factor from inHg to P a conversion factor from hp to J · s−1 171 Table C.8: Quantified variables Symbol CX0 () CXα () CXq () CXαδf () CXα2 () CXα3 () CXδf () CXδr () CY0 () CYβ () CYp () CYr () CYαδr () CYδa () CYδr () CYβ˙ () CZ0 () CZα () CZq () CZαδf () CZα3 () CZβ2 δe () CZδe () CZδf () Cl0 () Clβ () Clp () Clr () Type time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T time-T → → → → → → → → → → → → → → → → → → → → → → → → → → → → Description actual value of stability derivative of force along XB -axis actual value of stability derivative of force along XB -axis actual value of stability derivative of force along XB -axis actual value of control derivative of force along XB -axis actual value of stability derivative of force along XB -axis actual value of stability derivative of force along XB -axis actual value of control derivative of force along XB -axis actual value of control derivative of force along XB -axis actual value of stability derivative of force along YB -axis actual value of stability derivative of force along YB -axis actual value of stability derivative of force along YB -axis actual value of stability derivative of force along YB -axis actual value of control derivative of force along YB -axis actual value of control derivative of force along YB -axis actual value of control derivative of force along YB -axis actual value of stability derivative of force along YB -axis actual value of stability derivative of force along ZB -axis actual value of stability derivative of force along ZB -axis actual value of stability derivative of force along ZB -axis actual value of control derivative of force along ZB -axis actual value of stability derivative of force along ZB -axis actual value of control derivative of force along ZB -axis actual value of control derivative of force along ZB -axis actual value of control derivative of force along ZB -axis actual value of stability derivative of moment about XB -axis actual value of stability derivative of moment about XB -axis actual value of stability derivative of moment about XB -axis actual value of stability derivative of moment about XB -axis stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T controlDerivative-T stabilityDerivative-T stabilityDerivative-T controlDerivative-T controlDerivative-T stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T controlDerivative-T controlDerivative-T controlDerivative-T stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T controlDerivative-T stabilityDerivative-T controlDerivative-T controlDerivative-T controlDerivative-T stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T stabilityDerivative-T 172 Table C.8: Quantified variables (continued) Symbol Clαδa () Clδa () Clδr () Cm0 () Cmα () Cmq () Cmr () Cmα2 () Cmβ2 () Cmδe () Cmδf () Cn0 () Cnβ () Cnp () Cnq () Cnr () Cnβ3 () Cnδa () Cnδr () Ix () Iy () Iz () Jxy () Jxz () Jyz () P () SAx SAy Type time-T → controlDerivative-T time-T → controlDerivative-T time-T → controlDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → controlDerivative-T time-T → controlDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → stabilityDerivative-T time-T → controlDerivative-T time-T → controlDerivative-T time-T → momentOfInertia-T time-T → momentOfInertia-T time-T → momentOfInertia-T time-T → productOfInertia-T time-T → productOfInertia-T time-T → productOfInertia-T time-T → power-T accelerometerGain-T accelerometerGain-T Description actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of actual value of engine power actual value of actual value of 173 control derivative of moment about XB -axis control derivative of moment about XB -axis control derivative of moment about XB -axis stability derivative of moment about YB -axis stability derivative of moment about YB -axis stability derivative of moment about YB -axis stability derivative of moment about YB -axis stability derivative of moment about YB -axis stability derivative of moment about YB -axis control derivative of moment about YB -axis control derivative of moment about YB -axis stability derivative of moment about ZB -axis stability derivative of moment about ZB -axis stability derivative of moment about ZB -axis stability derivative of moment about ZB -axis stability derivative of moment about ZB -axis stability derivative of moment about ZB -axis control derivative of moment about ZB -axis control derivative of moment about ZB -axis moment of inertia along XB -axis moment of inertia along YB -axis moment of inertia along ZB -axis product of inertia in XB YB -plane product of inertia in XB ZB -plane product of inertia in YB ZB -plane Ax accelerometer gain Ay accelerometer gain Table C.8: Quantified variables (continued) Symbol SAz ST Sα Sφ Sψ Sθ Sps Sp Sqdyn Sq Sr xa () xe () xr () νAx () νAy () νAz () νT () να () νφ () νψ () νθ () νps () νp () νqdyn () νq () νr () νuwt () Type accelerometerGain-T TSensorGain-T AOTSensorGain-T attitudeSensorGain-T attitudeSensorGain-T attitudeSensorGain-T pressureSensorGain-T gyroGain-T pressureSensorGain-T gyroGain-T gyroGain-T array of float array of float array of float time-T → accelerometerOutput-T time-T → accelerometerOutput-T time-T → accelerometerOutput-T time-T → TSensorOutput-T time-T → AOTSensorOutput-T time-T → attitudeSensorOutput-T time-T → headingSensorOutput-T time-T → attitudeSensorOutput-T time-T → pressureSensorOutput-T time-T → gyroOutput-T time-T → pressureSensorOutput-T time-T → gyroOutput-T time-T → gyroOutput-T time-T → float 174 Description actual value of Az accelerometer gain actual value of temperature sensor gain actual value of angle of attack sensor gain actual value of roll attitude sensor gain actual value of heading sensor gain actual value of pitch attitude sensor gain actual value of static pressure sensor gain actual value of roll rate gyro gain actual value of dynamic pressure sensor gain actual value of pitch rate gyro gain actual value of yaw rate gyro gain state vector of small perturbation aileron model state vector of small perturbation elevator model state vector of small perturbation rudder model instance of Ax accelerometer output noise instance of Ay accelerometer output noise instance of Az accelerometer output noise instance of temperature sensor output noise instance of angle of attack sensor output noise instance of roll attitude sensor output noise instance of heading sensor output noise instance of pitch attitude sensor output noise instance of static pressure sensor output noise instance of roll rate gyro output noise instance of dynamic pressure sensor output noise instance of pitch rate gyro output noise instance of yaw rate gyro output noise instance of noise driving Dryden turbulence model Table C.8: Quantified variables (continued) Symbol νvwt () νwwt () a, b biasAx biasAy biasAz biasT biasα biasφ biasψ biasθ biasps biasp biasqdyn biasq biasr dpt() k m() t, t1 , t2 , t3 , t4 Type time-T → float time-T → float float accelerometerBias-T accelerometerBias-T accelerometerBias-T TSensorBias-T AOTSensorBias-T attitudeSensorBias-T headingSensorBias-T attitudeSensorBias-T psSensorBias-T gyroBias-T qdynSensorBias-T gyroBias-T gyroBias-T time-T → float integer time-T → force-T time-T Description instance of noise driving Dryden turbulence model instance of noise driving Dryden turbulence model coefficients used within the trim-condition requirement actual value of Ax accelerometer bias actual value of Ay accelerometer bias actual value of Az accelerometer bias actual value of temperature sensor bias actual value of angle of attack sensor bias actual value of roll attitude sensor bias actual value of heading sensor bias actual value of pitch attitude sensor bias actual value of static pressure sensor bias actual value of roll rate gyro bias actual value of dynamic pressure sensor bias actual value of pitch rate gyro bias actual value of yaw rate gyro bias non-dimensional pressure increase in propeller slipstream counter actual airplane weight continuous time instant 175 Table C.9: Auxiliary terms Symbol and expression in terms of base quantities Fx () = Xasd () + Xacd () + Xp () + Xgr () + Xw () Fy () = Yasd () + Yacd () + Yp () + Ygr () + Yw () Fz () = Zasd () + Zacd () + Zp () + Zgr () + Zw () () I1 () = Iy ()Iz () − Jyz I2 () = Jxy ()Iz () + Jyz ()Jxz () I3 () = Jxy ()Jyz () + Iy ()Jxz () () I4 () = Ix ()Iz () − Jxz I5 () = Ix ()Jyz () + Jxy ()Jxz () () I6 () = Ix ()Iy () − Jxy |I()| = Ix ()Iy ()Iz () − 2Jxy ()Jxz ()Jyz () − () − I ()J () − I ()J () Ix ()Jyz y z xz xy ˙ Jtc (( V˙ a (0), α(0), ˙ β(0), p(0), ˙ q(0), ˙ r(0) ˙ )= CVa V˙ a2 (0) + Cαβ ( α˙ (0) + β˙ (0) ) +Cpqr ( p˙2 (0) + q˙2 (0) + r˙ (0) ) KH (Vˆa ) = (−1 · 10−4 Va2 + 0.015Vˆa − 0.5975)π/180 Kφ (Vˆa ) = 9.75 · 10−4 Vˆa2 − 0.108Vˆa + 2.335625 Kψ (Vˆa ) = 0.05Vˆa − 1.1 Kθ (Vˆa ) = 1.375 · 10−3 Vˆa2 + 0.1575Vˆa − 4.8031 ¯ θ (Vˆa ) = 1.375 · 10−3 Vˆa2 + 0.1575Vˆa − 4.8031 K Kd (Vˆa ) = −2.5 · 10−3 Vˆa + 0.2875 Type Description Auxiliary terms time-T → force-T time-T → force-T time-T → force-T time-T → inertiaParam-T time-T → inertiaParam-T time-T → inertiaParam-T time-T → inertiaParam-T time-T → inertiaParam-T time-T → inertiaParam-T time-T → inertiaParam-T total force along XB -axis total force along YB -axis total force along ZB -axis inertia parameter for moment inertia parameter for moment inertia parameter for moment inertia parameter for moment inertia parameter for moment inertia parameter for moment inertia parameter for moment (acceleration-T, angularVelocity-T, angularVelocity-T, angularAcceleration-T, angularAcceleration-T, angularAcceleration-T) float airspeed-T → float airspeed-T airspeed-T airspeed-T airspeed-T airspeed-T → → → → → 176 float float float float float equations equations equations equations equations equations equations cost function for trim-condition → proportional gain of ALH autopilot proportional proportional proportional proportional proportional gain gain gain gain gain of of of of of asymmetric autopilot HH/HS autopilot PAH autopilot ALH and ALS autopilot ALH and ALS autopilot Table C.9: Auxiliary terms (continued) Symbol and expression in terms of base quantities Kq (Vˆa ) = −4.75 · 10−4 Vˆa2 + 0.0540Vˆa − 1.593 KH˙ (Vˆa ) = (−3.875 · 10−4 Vˆa2 + 0.04025Vˆa − 1.1041)π/180 Ktc (Vˆa ) = · 10−4 Vˆa2 − 0.03Vˆa + 0.9375 ¯ tc (Vˆa ) = 0.03Vˆa + 0.25 K L() = Lasd () + Lacd () + Lp () M () = Masd () + Macd () + Mp () N () = Nasd () + Nacd () + Np () OF FADC = 2ADCnb −1 OF FDAC = 2DACnb −1 Pl () = I1 ()/|I()| Pm () = I2 ()/|I()| Pn () = I3 ()/|I()| Ppp () = −(Jxz ()I2 () − Jxy ()I3 ())/|I()| Ppq () = (Jxz ()I1 () − Jyz ()I2 () − (Iy () − Ix ())I3 ())/|I()| Ppr () = −(Jxy ()I1 () + (Ix () − Iz ())I2 () − Jyz ()I3 ())/|I()| Pqq () = (Jyz ()I1 () − Jxy ()I3 ())/|I()| Pqr () = −((Iz () − Iy ())I1 () − Jxy ()I2 () + Jxz ()I3 ())/|I()| Prr () = −(Jyz ()I1 () − Jxz ()I2 ())/|I()| Ql () = I2 ()/|I()| Qm () = I4 ()/|I()| Qn () = I5 ()/|I()| Qpp () = −(Jxz ()I4 () − Jxy ()I5 ())/|I()| Type Description airspeed-T → float airspeed-T → float proportional gain of symmetric autopilot proportional gain of ALS autopilot airspeed-T → float airspeed-T → float time-T → moment-T time-T → moment-T time-T → moment-T integer integer proportional gain of PAH autopilot proportional gain of ALH and ALS autopilot total rolling moment total pitching moment total yawing moment ADC internal representation of ADC zero input DAC internal representation of DAC zero output inertia parameter for moment equations inertia parameter for moment equations inertia parameter for moment equations inertia parameter for moment equations inertia parameter for moment equations time-T time-T time-T time-T time-T → → → → → inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T time-T → inertiaParam-T inertia parameter for moment equations time-T → inertiaParam-T time-T → inertiaParam-T inertia parameter for moment equations inertia parameter for moment equations time-T time-T time-T time-T time-T → → → → → inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T 177 inertia inertia inertia inertia inertia parameter parameter parameter parameter parameter for for for for for moment moment moment moment moment equations equations equations equations equations Table C.9: Auxiliary terms (continued) Symbol and expression in terms of base quantities Qpq () = (Jxz ()I2 () − Jyz I4 () − (Iy () − Ix ())I5 ())/|I()| Qpr () = −(Jxy ()I2 () + (Ix () − Iz ())I4 () − Jyz ()I5 ())/|I()| Qqq () = (Jyz ()I2 () − Jxy ()I5 ())/|I()| Qqr () = −((Iz () − Iy ())I2 () − Jxy ()I4 () + Jxz ()I5 ())/|I()| Qrr () = −(Jyz ()I2 () − Jxz ()I4 ())/|I()| Rl () = I3 ()/|I()| Rm () = I5 ()/|I()| Rn () = I6 ()/|I()| Rpp () = −(Jxz ()I5 () − Jxy ()I6 ())/|I()| Rpq () = (Jxz ()I3 () − Jyz ()I5 () − (Iy () − Ix ())I6 ())/|I()| Rpr () = −(Jxy ()I3 () + (Ix () − Iz ())I5 () − Jyz ()I6 ())/|I()| Rqq () = (Jyz ()I3 () − Jxy ()I6 ())/|I()| Rqr () = −((Iz () − Iy ())I3 () − Jxy ()I5 () + Jxz ()I6 ())/|I()| Rrr () = −(Jyz ()I3 () − Jxz ()I5 ())/|I()| ADCnb −1 SADC = DAC2 + −DAC − Type Description time-T → inertiaParam-T inertia parameter for moment equations time-T → inertiaParam-T inertia parameter for moment equations time-T → inertiaParam-T time-T → inertiaParam-T inertia parameter for moment equations inertia parameter for moment equations SDAC = V time-T time-T time-T time-T time-T time-T → → → → → → inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T inertiaParam-T inertia inertia inertia inertia inertia inertia parameter parameter parameter parameter parameter parameter for for for for for for moment moment moment moment moment moment equations equations equations equations equations equations time-T → inertiaParam-T inertia parameter for moment equations time-T → inertiaParam-T time-T → inertiaParam-T inertia parameter for moment equations inertia parameter for moment equations time-T → inertiaParam-T float inertia parameter for moment equations ADC resolution float DAC resolution time-T → turbulence-T filter implementing Dryden turbulence model time-T → turbulence-T filter implementing Dryden turbulence model V 2DACnb −1 DACV + −DACV − huwt () = F −1 σu 2Lu Va (t) hvwt () = F −1 σv Lv Va (t) 1+ VLu jω a (t) √ L v 1+ V (t) jω a v jω 1+ VL(t) a 178 Table C.9: Auxiliary terms (continued) Symbol and expression in terms of base quantities hwwt () = F −1 σw √ L w Lw 1+ Va (t) jω Va (t) 1+ Lw jω V (t) b p() T 2Va () T u˜e () = δ˜e () Vc¯ aq() () b r() T u˜r () = δ˜r () 2V a () T b p() uˆa () = δˆa () 2Va () T uˆe () = δˆe () Vc¯ aq() () b r() T uˆr () = δˆr () 2V a () drr(Vˆa ) = −7.5 · 10−5 Vˆa2 Type Description time-T → turbulence-T filter implementing Dryden turbulence model array of float input vector to aileron model array of float input vector to elevator model array of float input vector to rudder model array of float input vector to aileron model array of float input vector to elevator model array of float input vector to rudder model airspeed-T → float turn-coordination loop gain of asymmetric autopilot airspeed component along XB -axis component along XB -axis of velocity wrt earth frame airspeed component along YB -axis component along YB -axis of velocity wrt earth frame airspeed component along ZB -axis component along ZB -axis of velocity wrt earth frame wind velocity component along XB -axis wind velocity component along YB -axis wind velocity component along ZB -axis altitude variation with respect to trim value a u˜a () = δ˜a () − 0.0095Vˆa − 0.4606 ua () = Va () cos α() cos β() ue () = ua () + uw () time-T → velocity-T time-T → velocity-T va () = Va sin β() ve () = va () + vw () time-T → velocity-T time-T → velocity-T wa () = Va () sin α() cos β() we () = wa () + ww () time-T → velocity-T time-T → velocity-T uw () = uw¯ () + uwt () + uwg () vw () = vw¯ () + vwt () + vwg () ww () = ww¯ () + wwt () + wwg () ˆ − H0 ∆H() = H() time-T → velocity-T time-T → velocity-T time-T → velocity-T natural-T → altitude-T 179 Table C.9: Auxiliary terms (continued) Symbol and expression in terms of base quantities ∆δ a () = δˆa () − δa0 Type Description natural-T → aileron defliection variation with respect to trim aileronDeflection-T value natural-T → elevator defliection variation with respect to elevatorDeflection-T trim value natural-T → rudder defliection variation with respect to trim rudderDeflection-T value natural-T → angle-T bank angle variation with respect to trim value natural-T → bankReference- bank reference variation with respect to trim T value natural-T → angle-T heading variation with respect to trim value natural-T → heading reference variation with respect to trim headingReference-T value natural-T → angle-T pitch angle variation with respect to trim value natural-T → pitchReference- pitch reference variation with respect to trim T value Auxiliary functions, operators, and predicates ∆δ e () = δˆe () − δe0 ∆δ r () = δˆr () − δr0 ˆ − φ0 ∆φ() = φ() ∆φr () = φˆr () − φr0 ˆ − ψ0 ∆ψ() = ψ() ∆ψ r () = ψˆr () − ψr0 ˆ − θ0 ∆θ() = θ() ∆θr () = θˆr () − θr0 constRef (f (), t1 , t2 ) = ∃ k ∀ t t1 ≤ t ≤ not specified predicate that evaluates true if f () is constant throughout the time interval [t1 , t2 ] (switch-T, time-T, time-T) → boolean predicate that evaluates true only if the switch SW () is engaged at t = t1 and stays engaged throughout the time interval [t1 , t2 ] (time-T, time-T) → boolean predicate that evaluates true if random and discrete turbulence wind components are not zero t2 ⇒ f (t) = k engaged(SW (), t1 , t2 ) = ∃ T ∀t ∧ ∀t 0