Robustness Assessment of Flight Controllers for a Civil Aircraft using ,Analysis.
G.H.N. Looye1;2, S. Bennani2, A. Varga1, D. Moormann1, G. Grubel1
Deutsches Zentrum fur Luft- und Raumfahrt (DLR) Institut fur Robotik und Systemdynamik DLR-Oberpfa enhofen D-82234 Wessling, Germany 1
[email protected]
2
Faculty of Aerospace Engineering Group for Control & Simulation Delft University of Technology Kluyverweg 1, 2629 HS Delft The Netherlands
[email protected]
Contents
Introduction Review of LFTs and Research Civil Aircraft Model (RCAM) LFT-modeling of RCAM -Analysis results Conclusions
DLR/TUD
Introduction
DLR/TUD
Background:
GARTEUR FM(AG08): Project on robust ight control. { de nition of two design benchmark for
civil aircraft (3D trajectory tracking, stab. augmentation) military aircraft (Fly-by-wire, stab. augmentation);
{ application of modern and classical techniques by member
organizations
... resulting in:
12 designs (10 methods) for Research Civil Aircraft Model (RCAM), 6 designs ( 5 methods) for High Incidence Research Model (HIRM);
{ evaluation and comparison of design techniques.
For RCAM stability analysis by DLR-OP and TUD using: { worst-case parameter optimization, { ... and -analysis.
Introduction
DLR/TUD
We are basically interested in:
1. is the controlled system stable over operating envelope ? (allowed parameter ranges) 2. if so, what can we say about a stability margin? 3. ... preferably in terms of uncertain physical parameters
We need a suitable manner to represent our uncertain parameters in the system model:
Linear Fractional Transformations (LFTs);
... and a robustness indicator in terms of uncertain parameters:
structured singular value .
LFT's
DLR/TUD
Linear Fractional Transformations (LFTs): z e
∆1 z e
w
M 11 M 12 M 21 M 22
d
z w
M
e
∆2
d
e = Fu(M; 1)d
Direct generalization of: 1 S
y
d
z = Fl(M; 2)w
Fu = M22 + M21(I , 1M11),11M12 Fl = M11 + M12(I , 2M22),12M21
. x
w
M
A C
B D
x u
y = D + C (I , 1s A),1 s1 B
LFT's
DLR/TUD
What makes LFT's so useful? Typical algebraic operations like:
cascade connections parallel connections feedback connections (inverse!) frequency response ... preserve the LFT structure. y1
G1
G2
∆1
∆2
u
y2
G3 ∆3 ∆1
0 0 0 ∆2 0 0 0 ∆3
y1 y2
M
u
Interconnections of LFT's are again LFT's Unstructured uncertainty at component level
...becomes structured uncertainty at system level
LFT's
DLR/TUD
The mass-spring-damper example: x
y
. x
1 S
1 S
.. x
1 m
-
u
-
C
k
Model uncertainty in k. c and m1 :
k = k(1 + 0:4k ) c = c(1 + 0:3c)
1
1 = m m (1+0:5m)
,1 m; c; k 1 These expressions are LFT's:
Fl = M11 + M12 (I , M22 ),1 M21 e.g: c = c + 0:3c (1 , c 0 ),1 c 1
M for each parameter now looks like:
Mc = 1c 0:03c wc
Mc δc
k 0 : 4 k 0:5 Mk = 1 0 Mmi = , ,0:5 zc
wk
Mk δk
1 m 1 m
δmi
zk zm
M mi
wm
LFT's
DLR/TUD
* The block-diagram, 's omitted: x
y
. x
1 S
Mk
zm zc zk y
... OR:
.. x
wm
M mi -
-
u
zc
zk
wm wc wk u
Gmck
* Now de ne:
1 S
Mc
wc wk
zm
0 0 m = @ 0 c
Gmck known
1 0 0A
0 0 k
* We have a new LFT: ∆ z y
Gmck
() 1 structured w u
Gmck known
Unstructured uncertainty at component level ...becomes structured uncertainty at system level
DLR/TUD
∆
compute frequency reponse
∆ M(j ω)
M
... is from some set with speci c diagonal structure, () 1
We are interested in smallest for which M goes unstable...
or, the smallest for which a rst pole travels over the
imaginary axis, at which very moment and frequency the LFT is singular, approach: walk along imaginary axis, and 'compute': (M11(j!)) := min
1 2 f () : det(I , M11(j! )) = 0g
System is robustly stable
() (M11(j!)) < 1 8!
DLR/TUD
Plenty of tools available for approximating :
D-G scaling (upper bound), power algorithms (lower bound), branch-and-bound schemes, frequency sweep (lower bound) [CERT-ONERA], ... LMI techniques.
RCAM
DLR/TUD
atmosphere
body6DOF
COG atmos
u RCAM
engine1
aerodynamics RCAM
engine2
RCAM
Gravity
Earth
gravity
wind
The following parameters may vary: parameter nominal minimum maximum Xcg 0.23c 0.15c 0.31c Zcg 0.0c 0.0c 0.21c mass 120 000 kg 100 000 kg 150 000 kg delay 0.05 s 0.05 s 0.10 s
The design speed is 80 m/s
LFT modeling
DLR/TUD
Dymola Physical RCAM system model automatic model building mathematical (symbolic) system model automatic code generation Maple trim data: x0, u0, p0
RCAM model in Maple
data fit with linear models
Parametric linear RCAM model
symbolic linearisation
parameter transformation A(p) B(p) C(p) D(p) Matlab PUM / µ-toolbox
∆ RCAM RCAM LFT-description
RCAM simulation model: * S-function (Simulink) * DSblock (Fortran or C) * ...
2 (CD , 0CL ) Example: A(7; 7) = Xu = , qS 0 0 m V0
We end up with:
linear nominal model, 12 states, 5 contr., 15 meas.; -block structure: 0 1 mI17 0 0 p = @ 0 xcg I15 0 A 0 0 zcg I3
LFT-modeling
DLR/TUD
We complete the system: ∆p z
∆τ
w RCAM
u
(linear)
y
∆c
0.01
Act
ref
Mτ
K
Or, in LFT form:
∆ tot z y
w
P
u
0 0 0 1 p tot = @ 0 0 A 0 0 c
K For K we can plug-in any of the controllers, after linearizing the Simulink structure...
-Analysis
DLR/TUD Robust stability: prel. controller
mu(M11)
1
0.9436
upper bound 0.5
lower bound 0 −1 10
0
1
10 frequency (rad)
10
* Figure out worst-case parameters:
:: m Xcg xcg Zcg zcg delay compl. pert. c1::5
parameter mass
value 1.1092 1.1092 1.1092 1.1092 0
... () = 1:1092, and 1= () 0:90 * East-bound pole: 0.95 crit crit 1.05 crit -0.0137 0.6954i 0.0 0.6899i +0.0131 0.6844i
-Analysis
DLR/TUD
* The critical parameter values are:
m Xcg Zcg
= = = =
125 000 0:23c 0:105c 0:075
+ + + +
25 000 1:092 0:08c 1:092 0:105c 1:092 0:025 1:092
= = = =
152 300 kg 0:317c 0:220c 0:102 s
... indeed outside the speci ed parameter ranges
Perform nonlinear simulation with original model:
... actually, using simulation code generated from very same DYMOLA model Prel. controller 1
q (deg/s)
0.5
0
−0.5
−1 0
____ 1.00*pert − − − 0.95*pert ........ 1.05*pert
10
20 time (s)
30
40
method
(M11)
peak freq. (upper bnd) (rad/s) MO-16 multi-model multi-objective 0.35 6.0 MM-12 modal multi-model-synth. 0.36 8.0 EA-22 eigenstructure assignment 0.39 0.5 FL-15 fuzzy logic 0.44 5.5 MS-11 -synthesis 0.49 2.9 0.6 CC-13 classical control 0.51 0.8 LY-14(2) Lyapunov 0.57 0.8 MF-25 Model following 0.65 0.6 EA-18 eigenstructure assignment 0.83 7.0 HI-prel H1 loopshaping 0.94 0.7 MS-19 -synthesis 1.36 15.1 HI-21 H1 mixed sensitivity 1.53 1.3
No. + + + + + + + + + + + + +
+ + + + + + + + + +
+ + + + + + +
+ + + + + + + +
m xcg zcg
worst-case parameters mode (lon/lat) lon lon lon lon lat lon lon lon lon lon lon lon lon
example di . LFT!!!
linearized di . LFT!!!
remarks
-Analysis
DLR/TUD
Conclusions
DLR/TUD
-Analysis is a very promising tool for (post-design) robustness analysis
Linear Fractional Transformations:
{ a powerful framework for representing uncertainty; ... but, (especially for aircraft) not trivial to obtain;
{ automated LFT generation is great relief ... but, better methods for reduction need to be implemented.
-computation:
{ used frequency gridding: tricky, { worst-case parameters, and veri ed in nonlin. simulations, { upperbound conservativeness 1{9 %,
Outlook:
new algorithms for automated LFT generation from
parametric nonlinear models apply improved optimization algorithms for -computations beyond , LTI...
reports on this work publicly available via http://www.nlr.nl
