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1 Robstness Assessment of Flight Controllers for a Civil Aircraft sing,analysis. G.H.N. Looye ;2, S. Bennani 2, A. Varga, D. Moormann, G. Grbel Detsches Zentrm fr Lft- nd Ramfahrt (DLR) Institt fr Robotik nd Systemdynamik DLR-Oberpfaenhofen D Wessling, Germany Gertjan.Looye@dlr.de 2 Faclty of Aerospace Engineering Grop for Control & Simlation Delft University of Technology Klyverweg, 2629 HS Delft The Netherlands S.Bennani@lr.tdelft.nl

2 Contents Introdction Review of LFTs and Research Civil Aircraft Model (RCAM) LFT-modeling of RCAM -Analysis reslts Conclsions

3 Introdction Backgrond: GARTEUR FM(AG08): Project on robst ight control. { denition of two design benchmark for civil aircraft (3D trajectory tracking, stab. agmentation) military aircraft (Fly-by-wire, stab. agmentation); { application of modern and classical techniqes by member organizations... reslting in: 2 designs (0 methods) for Research Civil Aircraft Model (RCAM), 6 designs ( 5 methods) for High Incidence Research Model (HIRM); { evalation and comparison of design techniqes. For RCAM stability analysis by DLR-OP and TUD sing: { worst-case parameter optimization, {... and -analysis.

4 Introdction We are basically interested in:. is the controlled system stable over operating envelope? (allowed parameter ranges) 2. if so, what can we say abot a stability margin? preferably in terms of ncertain physical parameters We need a sitable manner to represent or ncertain parameters in the system model: Linear Fractional Transformations (LFTs);... and a robstness indicator in terms of ncertain parameters: strctred singlar vale.

5 LFT's Linear Fractional Transformations (LFTs): z e M M 2 M 2 M 22 w d e z M w d z e M 2 d w e = F (M; )d z = F l (M; 2 )w F = M 22 + M 2 (I, M ), M 2 F l = M + M 2 (I, 2 M 22 ), 2 M 2 Direct generalization of: y. x A C S B D x y = D + C(I, s A), s B

6 LFT's What makes LFT's so sefl? Typical algebraic operations like: cascade connections parallel connections feedback connections (inverse!) freqency response... preserve the LFT strctre. y G G2 2 G 3 y 2 3 y y M 3 Interconnections of LFT's are again LFT's Unstrctred ncertainty at component level...becomes strctred ncertainty at system level

7 LFT's The mass-spring-damper example: y x S. x S.. x m - - C k Model ncertainty in k. c and m : k = k( + 0:4 k ) c = c( + 0:3 c ) m = m(+0:5m), m ; c ; k These expressions are LFT's: F l = M + M 2 (I, M 22 ), M 2 e.g: c = c + 0:3c (, c 0 ), c M for each parameter now looks like: M c = c 0:3c 0 M k = k 0:4 k 0 M mi =,0:5 m,0:5 m w c Mc z c δ mi w M k k z k z m w m δ c δ k M mi

8 LFT's * The block-diagram, 's omitted: y x. z m x M mi S S.. x - - w m w c M c z c w k M k z k z m z c k... OR: z Gmck y w m wc w k G mck known * Now dene: = m c k A * We have a new LFT: () strctred y z G mck w G mck known Unstrctred ncertainty at component level...becomes strctred ncertainty at system level

9 M compte freqency reponse M(j ω)... is from some set with specic diagonal strctre, () We are interested in smallest for which M goes nstable... or, the smallest for which a rst pole travels over the imaginary axis, at which very moment and freqency the LFT is singlar, approach: walk along imaginary axis, and 'compte': (M (j!)) := min 2 f() : det(i, M (j!)) = 0g System is robstly stable () (M (j!)) < 8!

10 Plenty of tools available for approximating : D-G scaling (pper bond), power algorithms (lower bond), branch-and-bond schemes, freqency sweep (lower bond) [CERT-ONERA],... LMI techniqes.

11 RCAM atmosphere body6dof COG atmos RCAM engine RCAM aerodynamics engine2 RCAM Gravity Earth gravity wind The following parameters may vary: parameter nominal minimm maximm X cg 0.23c 0.5c 0.3c Z cg 0.0c 0.0c 0.2c mass kg kg kg delay 0.05 s 0.05 s 0.0 s The design speed is 80 m/s

12 LFT modeling trim data: x0, 0, p0 data fit with linear models Dymola Maple Physical RCAM system model RCAM model in Maple Parametric linear RCAM model A(p) B(p) C(p) D(p) atomatic model bilding mathematical (symbolic) system model atomatic code generation symbolic linearisation parameter transformation Matlab PUM / µ-toolbox RCAM RCAM LFT-description RCAM simlation model: * S-fnction (Simlink) * DSblock (Fortran or C) *... Example: A(7; 7) = X =, qs m We end p with: 2 V 0 (C D0, 0 C L0 ) linear nominal model, 2 states, 5 contr., 5 meas.; -block strctre: p = mi xcg I zcg I 3 A

13 LFT-modeling We complete the system: y z p RCAM (linear) w Act c 0.0 τ M τ ref K Or, in LFT form: z y tot P w tot = p c A K For K we can plg-in any of the controllers, after linearizing the Simlink strctre...

14 -Analysis Robst stability: prel. controller m(m) 0.5 pper bond lower bond freqency (rad) * Figre ot worst-case parameters: parameter :: vale mass m.092 X cg xcg.092 Z cg zcg.092 delay.092 compl. pert. c:: () = :092, and =() 0:90 * East-bond pole: 0.95 crit crit.05 crit i i i

15 -Analysis * The critical parameter vales are: m X cg = 0:23c + 0:08c :092 = 0:37c Z cg = 0:05c + 0:05c :092 = 0:220c = 0: :025 :092 = 0:02 s = :092 = kg... indeed otside the specied parameter ranges Perform nonlinear simlation with original model:... actally, sing simlation code generated from very same DYMOLA model Prel. controller 0.5 q (deg/s) *pert 0.95*pert....05*pert time (s)

16 parameters worst-case method (M ) peak freq. m xcg zcg mode remarks No. (pper bnd) (rad/s) (lon/lat) mlti-model mlti-objective lon MO-6 modal mlti-model-synth lon MM-2 eigenstrctre assignment lon EA-22 fzzy logic lon linearized FL-5 -synthesis lat di. LFT!!! MS lon 0.6 classical control lon CC-3 Lyapnov lon LY-4(2) Model following lon MF-25 eigenstrctre assignment lon EA-8 H loopshaping lon example HI-prel -synthesis lon di. LFT!!! MS-9 HI-2 H mixed sensitivity lon -Analysis

17 Conclsions -Analysis is a very promising tool for (post-design) robstness analysis Linear Fractional Transformations: { a powerfl framework for representing ncertainty;... bt, (especially for aircraft) not trivial to obtain; { atomated LFT generation is great relief... bt, better methods for redction need to be implemented. -comptation: { sed freqency gridding: tricky, { worst-case parameters, and veried in nonlin. simlations, { pperbond conservativeness {9 %, Otlook: new algorithms for atomated LFT generation from parametric nonlinear models apply improved optimization algorithms for -comptations beyond, LTI... reports on this work pblicly available via