М.Г.Гоман. А.В.Храмцовский, Е.Н.Колесников "Методика численного анализа нелинейной динамики самолета и синтеза системы управления на основе качественных методов анализа", встреча участников проекта GARTEUR в Стокгольме, 21-22 марта 2006 года. M.Goman, A.Khramtsovsky, E.Kolesnikov "Computational Framework for Analysis of Aircraft Nonlinear Dynamics & Control Design Based on Qualitative Methods", GARTEUR meeting in Stockholm 21-22 March 2006.

1. Computational Framework for Analysis of Aircraft
Nonlinear Dynamics & Control Design Based on
Qualitative Methods
Mikhail Goman, Andrew Khramtsovsky, De Montfort University, UK
Eugene Kolesnikov, Bombardier Aerospace, Montreal, Canada
•
GARTEUR FM(AG17), Stockholm, FOI, 21-22 March 2006

2. Contents
• A bit about flight dynamics
• Computational framework for qualitative analysis of
nonlinear aircraft dynamics (methods & tools)
• Attainable equilibrium sets (AES) for ADMIRE airframe
• Closed-loop dynamics & regions of attraction (RA) for
ADMIRE+NDI
• Manoeuvre limitation based on analysis of AES & RA
• Summary
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
2

3. Kinematics of Velocity-Vector Roll Manoeuvre
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
3

4. Velocity-Vector Roll Manoeuvre Visualization
V
Fa
V
Fa
w
Fa
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
4

5. Kinematics of Velocity Vector Roll Manoeuvre
Available definitions of velocity vector roll manoeuvre like:
“ …The velocity vector roll is defined as an angular rotation of an airplane
about its instantaneous velocity vector, constrained to be performed at
constant angle-of-attack, no sideslip, and constant velocity…”
Wayne C. Durham, Frederick H. Lutze, William Mason, December 1993
are valid only at very high rates of rotation as the radius of helical
trajectory is inversely proportional to angular rate
W
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
5

6. Kinematics of Velocity Vector Roll Manoeuvre
Available definitions of velocity vector roll manoeuvre like:
“ …The velocity vector roll is defined as an angular rotation of an airplane
about its instantaneous velocity vector, constrained to be performed at
constant angle-of-attack, no sideslip, and constant velocity…”
Wayne C. Durham, Frederick H. Lutze, William Mason, December 1993
are valid only at very high rates of rotation as the radius of helical
trajectory is inversely proportional to angular rate
W
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
6

7. &
x = f ( x, d );
Kinematics of Velocity Vector Roll Manoeuvre
&
x = f ( x, d );
a
Z
Y
x = (a , b , p, q, r )T
d = (d Eleft , d Eright , d r , d c , d TVq , d TVy )T
Natural kinematic parameters for Velocity Vector Roll
X
(a,b,W)
W
b
w
V
where
r r
(w × V )
W= r
V
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
7

8. Methods & Tools for Qualitative Analysis
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
8

9. Computational Framework for Qualitative Analysis
Analysis Method
Computational Algorithms
Trimming & Linearization
Newton’s method, gradient descent methods,
predictor-corrector continuation algorithm,
direct & inverse simulation methods,
attainable equilibrium sets (ASE)
Stability local
Jacoby matrix (eigenvalues, eigenvectors),
Lyapunov function methods, computing
invariant manifolds (region attraction
boundary), 2-dim slices of region of
attraction (RA), …
Stability global - region of attraction (RA)
Multiple-attractors (critical flight regimes)
Continuation & systematic search methods
Monte Carlo method, phase portrait
investigation (iterative & incremental method),
…
rk ( B, AB, K , An -1 B) = n , convex optimization
(controllability regions for linear unstable
GARTEUR FM(AG17) FOI, systems with limited control inputs), …
Stockholm 21-22 March 2006
Controllability local and global
9

10. Simulation, Trimming & Local Stability
dx
= F ( x, d );
dt
x Î X M Ì Rn
d ÎU D Ì R m
d& Î U Ì R m
R
Direct simulation
d (t )
&
x = F ( x, d )
Inverse simulation
x(t )
x(t )
AES
Direct trimming
dE
0 = F ( x, d )
Bifurcation analysis:
det
¶F
¶x
Local stability:
Inverse trimming
xE
xE
=0
&
x = F ( x, d )
0 = F ( x, d )
Bifurcation free:
det
x E ,d E
é ¶F
det ê
ê ¶x
ë
xE ,d E
ù
- lE ú = 0
ú
û
d (t )
¶F
¶d
dE
¹0
x E ,d E
eigenvalues
eigenvectors
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
10

11. Some Formal Definitions
Attainable Equilibrium Sets (AES):
ì
ü
AES = í xE : F ( xE , d E ) = 0, d Î U D Ì R m ý
î
þ
Regions of Attraction (RA):
ì
ü
RA( xE , d E ) = í x0 Î R n : x ( x0 , d E ) ® xE , as t ® ¥ (or t £ T ) ý
î
þ
Controllability Region (CR):
ì
ü
CR ( xE , d E ) = í( x0 , u0 ) Î R n + m : $d (t , d 0 ) ® d E , s.t. x(t , x0 ) ® xE at d (t , d 0 ) Î U D , d&(t , d 0 ) ÎU R ý
î
þ
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
11

12. Attainable Equilibrium Sets (AES)
(d Eleft , d Eright , d r , d c , d LE , d TVq , d TVy )T Û (a , b , W)T
AES slice
Direct trimming
( b , W)
Inverse trimming
(d Eleft , d Eright , d r )T
(a , b , W)T
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
12

13. Connection with Bifurcation Analysis
• AES
• Control Space
bifurcation diagram
bifurcation tailoring
(d Eleft , d Eright , d r )T
(a , b , W)T
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
13

14. ADMIRE Bare Airframe Analysis
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
14

15. Attainable Equilibrium Sets and
Local Stability Maps
• Classification of eigenvalues
Iml
Rel
Iml
Rel
Iml
Iml
Rel
Im l
Rel
Iml
Rel
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
Rel
15

16. ADMIRE AES:
(a,b) and (a,W) slices
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
16

17. ADMIRE AES:
(a,b) and (W,b) slices
M=0.4
a=30 deg
a=15 deg
a=5 deg
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
17

18. ADMIRE AES:
M=0.4
(a,b) slices at different M
M=0.8
M=1.0
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
M=1.2
18

19. AES figure saves controls & eigenvalues
(b,W) AES M=0.4, a=15 deg
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
19

20. ADMIRE Closed-Loop Dynamics
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
20

21. d min £ d £ d max
d& £ d&
NDI-based control laws
CVcom
Feed-forward
transformation
NDI
BS or FL based
+ Reference Model
for HQ
Mc
Control
Allocation
max
d
(Newton-Raphson
based)
Aircraft
CV
actuators,
airframe,
sensors
Signal
conditioning
CV = [a, b, W]’ - control variables; CVcom = [a, b, W]’ - control inputs;
d = [DEL,DER,DR]’ - control effector commands
Mc – control moment demand
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
21

22. desired dynamics, stability, robustness etc.
..
.
..
a
a* - k1a(s)(a-a*) - k2a(s)(a-a*)
..
.
.
..
.
.
b = Aw + Aw + E = b* - k1b(s)(b-b*) - k2b(s)(b-b*)
.
.
W
W*
- k1W(s)(W-W*)
a*,b*,W*
NDI block-functional diagrams
J-1(-wxJw + M)
M
Feedback linearization (FL) NDI
.
Equations of motion
a
.
b = A(a,b)w + E(a,b,V,n,q,f)
W
a
b
W
.
w = J-1(-wxJw + M)
reference model for handling qualities
com
wa,b2
,
s2 + 2xa,bwa,bs + wa,b2
kW
s + kW
desired dynamics... 1st level
.
a
.
b = Aw + E =
W
.
a* - ka(s) (a-a*)
.
b* - kb(s) (b-b*)
W*
a*,b*,W*
desired dynamics…. 2nd level
.
.
w = w* - Kw(s)(w-w*)
J-1(-wxJw + M)
Backstepping (BS) NDI
M
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
22

23. Nose Pointing Manoeuvre
open-loop
closed-loop
AES
AES
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
23

24. Velocity-Vector Roll Manoeuvre
closed-loop AES
open-loop AES
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
24

25. Departure at Velocity-Vector Roll Manoeuvre
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
25

26. Regions of Attraction for NDI closed-loop system:
critical external disturbances
NDI closed-loop
system
Open-loop system
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
26

27. Convergence and Departure Dynamics
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
27

28. Effect of Rate Saturation
d& £ 50 deg/ s
d& £ 500 deg/ s
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
28

29. Regions of Attraction at
Velocity-Vector Roll Manoeuvre
Critical Boundaries
Regions of Attraction
AES of NDI closed-loop system
AES of open-loop system
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
29

30. Manoeuvre limitation based on analysis of AES & RA
Allowable NDI control inputs acom,bcom,Wcom
- allowable inputs
critical boundaries
D – safety margin selected
D
based on a size of RA
Next step - avoid rate saturation!
Why? How?
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
30

31. Summary
• Computational framework based on numerical implementation of
qualitative methods is feasible and effective approach to flight
dynamics applications.
• Matlab tools for computation of AES for velocity-vector roll manoeuvre
and regions of attraction RA for the closed-loop system allow the
effective analysis of aircraft dynamics and manoeuvrability.
• Assignment of allowable NDI control inputs can be made using analysis
of ASE and RA.
• The plan for future work: a) avoiding rate saturation in NDI control (?);
b) control allocation and inversion at the presence of saturated
control; c) …
• “… Noah’s Ark was built by amateurs and Titanic by professionals…”
GARTEUR FM(AG17) FOI, Stockholm 21-22 March 2006
31