A Nomenclature and Acronyms A.1 List of Acronyms
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1 A Nomenclature and Acronyms A.1 List of Acronyms Acronym Definition AA Absolute Amplitude ADMIRE Aero-Data Model In Research Environment AG Action Group Airbus Airbus Deutschland GmbH AoA Angle of Attack APC Aircraft-Pilot Coupling APR Average Phase Rate BAE-S BAE SYSTEMS BU The University of Bristol CAP Control Anticipation Parameter CIRA Centro Italiano Ricerche Aerospaziali (Italian Aerospace Research Centre) c.g. Centre of Gravity CP Continuation Parameter DAv Dassault Aviation DERA Defence Evaluation and Research Agency DLR Deutsches Zentrum für Luft- und Raumfahrt (German Aerospace Centre) DSTL Defence Science & Technology Laboratories DUT Delft University of Technology EADS European Aeronautic Defence and Space Company EADS-M EADS Deutschland GmbH, Military Aircraft FBW Fly-By-Wire FC Flight Condition FCL Flight Control Laws FCS Flight Control System FE Flight Envelope FM Flight Mechanics FM(AG08) Flight Mechanics (Action Group 8) FM(AG11) Flight Mechanics (Action Group 11) FOI Totalförsvarets Forskningsinstitut (Swedish Defence Research Agency) GA Genetic Algorithms GARTEUR Group for Aeronautical Research and Technology in Europe
2 462 Nomenclature and Acronyms Acronym HIRM HIRM+ HIRM+RIDE HWEM INTA LF-PUM LFT LGM LMI LTI MCS NFE NLP NLR ODE ODE-PUM ONERA PCS PIO PS PUM QG RIDE SA SAAB SQP TMW UBOR UCAM UGM ULES UST Definition High Incidence Research Model High Incidence Research Model with parameter uncertainty HIRM+ with RIDE controller Harrier Wide Envelope Model Instituto Nacional de Técnica Aeroespacial (National Institute of Aerospace Technology) Linear Fractional - Parameter Uncertainty Model Linear Fractional Transformation Lower Gain Margin Linear Matrix Inequality Linear Time Invariant Multilevel Co-ordinate Search Number of Functional Evaluations Nonlinear Programming (problem) Nationaal Lucht- en Ruimtevaartlaboratorium (National Aerospace Laboratory) Ordinary Differential Equation Ordinary Differential Equation - Parameter Uncertainty Model Office National d Études et de Recherches Aérospatiales (National Office of Aerospace Studies and Research) Parametric Continuation Solver Pilot-in-the-loop Oscillation Pattern Search Parameter Uncertainty Model QinetiQ Group Robust Inverse Dynamics Estimation Simulated Annealing Saab AB Sequential Quadratic Programming The MathWorks Université Bordeaux University of Cambridge Upper Gain Margin University of Leicester Universität Stuttgart
3 A.2 List of Standard Symbols 463 A.2 List of Standard Symbols Symbol Definition a ij Element of the state matrix A A State matrix b ij Element of the input matrix B B Input matrix c ij Element of the output matrix C b Wingspan (m) c bar Mean aerodynamic chord (m) C Output matrix C D Coefficient of drag C L Coefficient of lift C X Coefficient of axial force C Y Coefficient of side force C Z Coefficient of normal force C l Coefficient of rolling moment C m Coefficient of pitching moment C n Coefficient of yawing moment C lβ Rolling moment coefficient derivative with respect to β C nr Yawing moment coefficient derivative with respect to r C nβ Yawing moment coefficient derivative with respect to β C mq Pitching moment coefficient derivative with respect to q C mα Pitching moment coefficient derivative with respect to α C mδt S Pitching moment coefficient derivative with respect to δ TS d ij Element of the direct transmission matrix D D Direct transmission matrix D Domain of eigenvalues in the complex plane f Frequency (Hz) F l (M, ) Lower linear fractional transformation matrix F u (M, ) Upper linear fractional transformation matrix g Acceleration due to gravity (m/s 2 ) h Altitude (feet) G long FCS gain for longitudinal stick (deg/s/mm) I x x body axis moment of inertia (kg m 2 ) I xy x-y body axis product of inertia (kg m 2 ) I xz x-z body axis product of inertia (kg m 2 ) I y y body axis moment of inertia (kg m 2 ) I yz y-z body axis product of inertia (kg m 2 ) I z z body moment of inertia (kg m 2 ) m Aircraft total mass (kg)
4 464 Nomenclature and Acronyms Symbol Definition M Mach number M LFT system matrix n x Load factor along x-axis (g) n y Load factor along y-axis (g) n z Load factor along z-axis (g) p Body-axis roll rate (deg/s) p dem Demanded roll rate (deg/s) q Body-axis pitch rate (deg/s) q bar Dynamic pressure (kn/m 2 ) q dem Demanded pitch rate (deg/s) R Real numbers space r Body-axis yaw rate (deg/s) s Complex variable in Laplace operator S Wing planform area (m 2 ) t Time (s) u Input vector V T Total velocity (m/s) x, y, z Earth axes positions (m) x State vector X cg Centre of gravity location along x-axis y Output Vector Y cg Centre of gravity location along y-axis Centre of gravity location along z-axis Z cg A.3 List of Greek Symbols Symbol α β β dem γ δ δ A δ ν δ CD δ CS δ SA δ SE δ T δ TS Definition Angle of attack (deg) Angle of sideslip (deg) Demanded sideslip (deg) Flight path angle (deg) LFT uncertainty matrix Vector of system parameters Aileron deflection (deg) ν-gap metric Differential canard deflection (deg) Symmetric canard deflection (deg) Lateral stick deflection (mm) Longitudinal stick deflection (mm) Tailplane deflection (deg) Symmetric tailplane deflection (deg)
5 A.4 List of Subscripts 465 Symbol Definition δ TD Differential tailplane deflection (deg) δ R Rudder deflection (deg) µ Structured singular value φ Bank angle (deg) Φ Phase angle (deg) ϕ i Phase offset for multi-loop stability margin (deg) λ Free parameter (continuation parameter) λ i Generic i-th eigenvalue λ i (A) Spectrum of matrix A π Point of uncertainty space Π Uncertainty set θ Pitch angle (deg) ρ Density of air (kg/m 3 ) ρ Stability degree σ Real part of complex eigenvalue σ Maximum singular value ω Reduction factor on aerodynamic uncertainty range ω Frequency (rad/sec) Ω Frequency range ψ Heading angle (deg) A.4 List of Subscripts Subscript Definition dem Demanded value (e.g., q dem is the demanded value of q) sensor Sensor measurement value (e.g., ψ sensor is the sensed value of ψ) trim Trim value of a variable (e.g., α trim is the trimmed value of α) Unc Uncertain parameter (e.g., C lβunc is the uncertainty parameter representing variation in C lβ )
6 Index ɛ-margin, 58 optimally weighted, 62 weighted, 59 ɛ scaled -margin calibration, 328 definition, 62 µ-analysis, 242 µ-analysis based clearance, 285 n z as an additional uncertainty, 274 altitude as variability, 297 analysis cycle, 309 AoA as an additional uncertainty, 271 comparison of results, 307 computational effort, 301 continuous AoA covering, 307 LFT modelling, 280 LFT validation, 293 Mach number as variability, 297 multi-loop analysis, 306 multivariable root locus method, 270 Nichols exclusion zone approximation, 289 parametric interdependency, 295 short period approximation, 270 stability margin criterion, 300 ν-gap clearance approach analysis cycle, 317 ν-gap metric, 64, 66 n z-limit exceedance criterion, 163, 421 absolute amplitude criterion, 160 aerodynamic fitting, 201, 203 AoA-limit exceedance criterion, 163, 421 average phase rate criterion, 160 baseline solution, 249 n z -limit exceedance criterion, 257 absolute amplitude criterion, 255 AoA-limit exceedance criterion, 257 average phase rate criterion, 255 stability margin criterion, 250 unstable eigenvalues criterion, 252 bifurcation analysis application, 91 background, 89, 93 bifurcation diagrams, 92 numerical continuation, 92 bifurcation and continuation method clearance analysis cycle, 98, 359 clearance implementation, 96, 355 clearance results, 376 comparison with baseline clearance, 372 computation times, 376 conventional implementation, 95 local bifurcation runs, 365 nonlinear sensitivity analysis, 365, 370, 372 selection of worst-case uncertainties, 365, 367, 370, 372, 376, 380 worst-case assumptions, 102 clearance block clearance, 17 flying quality levels, 27 frequency domain criteria, 27 handling analysis, 26 hardware tolerances, 23 limitations, 25, 32 linear handling criteria, 15, 27 linear stability margins, 15 manned simulation, 29 nonlinear analysis, 27 nonlinear simulation, 437, 444
7 468 Index off-line simulation, 28 response after failure, 29 stability analysis, 25 time domain criteria, 27 visualisation tools, 31 clearance models flying qualities, 144 HIRM+, 141 linearisation, 148 nonlinear model, 15, 141 pertubation model, 15 RIDE, 141 trimming, 142 compensation parameters, 208 conservatism, 207 D-stability, 79 robustness, 79 effect of uncertainties, 18 eigenvalue criterion, 48 flat systems, 226 generalised stability margin, 58 gridding based clearance, 108 HIRM+ actuator dynamics, 130 aerodynamics, 127 automated model code generation, 132 control surface deflections limits, 136 controls and gust inputs, 125 discontinuities in model, 317 dynamics model, 121 engine dynamics, 129 flight envelope, 152, 387, 410 linearisation, 148 load factor limits, 135 mass characteristics, 125 measurements and evaluation outputs, 124 object model, 123 sensor dynamics, 131 simulation, 133 trimming, 134 uncertain parameters, 153 industrial evaluation clearance criteria, 434 conservatism, 435 effort, 435, 439 evaluators background, 435 generality, 435 reliability, 435, 443 stability margin, 436 LFT model, 439, 440, 444 LFT modelling, 197 affine uncertainty representations, 198 gridding, 198 LFR-toolbox, 198 LFT validation, 206 min-max method, 208 order reduction, 206 partial differentiation, 206 symbolic equations of motion, 201 symbolic LFT generation, 206 LFT-based uncertainty modelling compensation parameters, 216 min-max approach, 213 nonlinearity compensation, 215 trends and bands approach, 214 linear fractional transformation (LFT), 38, 169, 237 affine dependency, 179 approximation, 189 equivalence, 183 extremal values, 190 input/output equivalence, 183 Kalman decomposition, 187 minimality, 185 Morton s approach, 179 normalization, 175 object-oriented realization, 178 operations, 178 order reduction, 183 polynomial dependency, 181 rational dependency, 181 realization, 176 similarity, 183 system similarity, 185 tree decomposition, 181 multi-loop gain/phase offsets, 62
8 Index 469 Nyquist stability theorem, 39 optimisation based clearance, 385 absolute amplitude, 402 average phase rate, 402 distance function, 385, 389, 398, 403 stability margin, 389 unstable eigenvalues, 397 optimisation methods, 112, 387 COBYLA, 115, 387 genetic algorithm, 115, 387 multilevel coordinate search, 116 pattern search, 114, 387 projected gradient quasi-newton, 114, 387 simulated annealing, 115, 387 software, 113 SQP, 113, 387 trust-region, 115 optimisation software, 387, 388 architecture, 389 COBYLA, 388 genetic algorithm, 388 pattern search, 388 projected gradient quasi-newton, 388 SQP, 388 optimisation-based clearance, 107, 415 analysis cycle, 110 distance function, 107, 416 genetic algorithm, 418 multilevel coordinate search, 418 nonlinear analysis, 415 SQP, 418 parametric uncertainty model, 201, 236 polynomial-based clearance, 76 adaptive grid generation, 82, 336 affine dependence, 78 multiaffine dependence, 80 multivariate dependence, 80 region shape computation, 336 unstable eigenvalues criterion, 333 worst case areas, 338, 348 polytopic set covering, 80 RIDE, 141 control laws, 149 linearization, 149 Riemann sphere, 66 robust inverse dynamic estimation (RIDE), 141 robust performance, 39 skew µ, 298 small gain theorem, 38 stability margin criterion, 44, 155 multi-loop analysis, 156, 314 single-loop analysis, 156 star product, 174 structured singular value (µ), 39 computation, 41 tangent plane, 69 unstable eigenvalues criterion, 158 worst case margins, 207
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