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Transcript of 1 Adaptive, Optimal and Reconfigurable Nonlinear Control Design for Futuristic Flight Vehicles...
1
Adaptive, Optimal and Reconfigurable Nonlinear Control
Design for Futuristic Flight Vehicles
Radhakant Padhi
Assistant Professor
Dept. of Aerospace EngineeringIndian Institute of Science, Bangalore, India
Abha TripathiProject Assistant
2
Project Plan
Date of Commence: 1st October 2006 Project duration : 2.5 Years Staff members:
Shree Krishnamoorthy, Project Assistant, Oct-Dec 2006.
Kaushik Das, Ph.D. student, January-July, 2007.
Abha Tripathi, Project Assistant, Aug.2007…continuing.
Apurva chunodhkar, a B. Tech. student from IIT-Bombay and Siddharth Goyal, a B.E. student from Punjab Engineering College have worked in sporadic engagements
Jagannath Rajshekharan, Project Assistant, has also worked in sporadic engagements
3
Summary Two parallel directions have been explored in this project.
Firstly, a new dynamic inversion approach has been developed and is experimented on a low-fidelity model of a high performance aircraft (F-16). Comparatively, it leads to some potential benefits:
Elimination of non-minimum phase behavior of the closed loop response
Less oscillatory behavior
Lesser magnitude of control Robustness study was carried out for the above approach with
uncertainties in aerodynamic force and moment coefficients and inertia parameters
4
Summary Secondly, a structured neuro – adaptive control design
idea has been developed which treats the kinematics and dynamics of the problem separately.
Modeling and parameter inaccuracies are considered by using neural network which dynamically capture the unknown functions that are used to design a model-following adaptive controller.
Sigma correction was done in the weight update rule. This idea is found to be successful on a satellite attitude
problem.
5
Command Tracking in High Performance Aircrafts: A New Dynamic Inversion
Design
6
Airplane Dynamics(F-16): Six Degree-of-Freedom
A1 2 3 4 A
2 25 6 7
8 2 4 9
PQ L = c +c +c +c
= c
= c
A
A A
P QR N
Q PR c P R c M
R PQ c QR c L c N
1 sin
1 sin cos
1 cos cos
X X
Y
Z
A T
A
A
U VR WQ g F Fm
V WP VR g Fm
W UQ VP g Fm
= sin tan cos tan
= cos sin
= sin cos sec
P Q R
Q R
Q R
= sin cos sin cos cosh U V W
7
Definitions and Goal Total Velocity:
Roll Rate (about x-axis):
Roll Rate (about velocity vector):
Normal Acceleration:
Lateral Acceleration:
Goal:
P
where are pilot commands
/ 1/zz z An F m m F
/ 1/yy y An F m m F
P*, Pw*, nz*, ny*, VT*
* * * * *, , ,w w z z y y T TP P or P P n n n n V V
WP
TV
8
Control Synthesis Procedure
Define new variables:
Key observation:
Known:
* *,y y y ya n V a n V
* *,z z z za n W a n W
* * TT
z y z yn n n n * * TT
z y z ya a a a
z z
y y
z n n c
y n n c
n f g U
n f g U
T T
P P c
T V V c
P f g U
V f g U
0V W
9
Control Synthesis Procedure Longitudinal Maneuver
Pilot commands:
• Roll Rate (bank angle rate):
• Normal Acceleration:
• Lateral Acceleration:
• Total Velocity:
Lateral Maneuver Pilot commands:
• Roll Rate (bank angle rate):
• Normal Acceleration:
• Lateral Acceleration:
• Total Velocity:
*
*
0
0
z
y
T
n
n
V
* 0y
T
or P
h
n
V
10
Control Synthesis Procedure
Combined Longitudinal and Lateral Maneuver
Pilot commands:
• Roll Rate (about velocity vector):
• Normal Acceleration:
• Lateral Acceleration:
• Total Velocity:
*
* 0
w
z
y
T
P
n
n
V
11
Control Synthesis Procedure Design a controller such that
After some algebra, Finally:
ˆ ˆ 0,TT V TV K V ˆ ˆ 0T TX KX
1[ ]TT
c U UU A b T *
z y z y
T TT T T T T T
U P a a n nPA g g g K g g g
*
* *
z y z y
TT
U P a a n z n yPb f f f K P f f n f n
1 1 1, ,
z yP n n
K diag
1T
T
VV
K diag
12
Results: Longitudinal
Tracked Variables Control Variables
0 20 40 60 80-1
-0.5
0
0.5
1
Time (Sec)
(
deg)
0 20 40 60 80-1
-0.5
0
0.5
1
Time (Sec)
n y(g)
0 20 40 60 800
1
2
3
Time (Sec)
n z(g)
0 20 40 60 80500
600
700
800
Time (Sec)
VT (
ft/s
)
0 20 40 60 800
50
100
Time (Sec)
Thr
ust
(%)
0 20 40 60 80-1
-0.5
0
0.5
1
Time (Sec)
Aile
ron
defle
ctio
n (d
eg)
0 20 40 60 80-4
-2
0
2
Time (Sec)
Ele
vato
r de
flect
ion
(deg
)
0 20 40 60 80-1
-0.5
0
0.5
1
Time (Sec)
Rud
der
defle
ctio
n (d
eg)
New Method
Existing Method
13
Results: Lateral Mode
0 20 40 60-20
-10
0
10
20
Time (Sec)
P (
deg/
sec)
0 20 40 600.99
0.995
1
1.005
1.01x 10
4
Time (Sec)
Alt
(ft)
0 20 40 60-0.1
-0.05
0
0.05
0.1
Time (Sec)
n y(g)
0 20 40 60570
575
580
585
590
Time (Sec)
VT (
ft/s
)
New Method
Existing Method
0 20 40 600
20
40
60
80
Time (Sec)
Thr
ust
(%)
0 20 40 60-4
-2
0
2
Time (Sec)
Aile
ron
defle
ctio
n (d
eg)
0 20 40 60-5
0
5
10
Time (Sec)
Ele
vato
r de
flect
ion
(deg
)
0 20 40 60-4
-2
0
2
4
Time (Sec)
Rud
der
defle
ctio
n (d
eg)
Tracked Variables Control Variables
14
Results: Combined Longitudinal and Lateral
Tracked Variables Control Variables
0 20 40 60
-10
0
10
Time (Sec)
Pw
(deg/s
ec)
0 20 40 60-0.1
-0.05
0
0.05
0.1
Time (Sec)
n y (g)
0 20 40 600
0.5
1
1.5
2
2.5
Time (Sec)
n z (g)
0 20 40 60570
575
580
585
590
Time (Sec)
VT (
ft/s
)
0 20 40 600
20
40
60
Time (Sec)
Thr
ust
(%)
0 20 40 60-1
-0.5
0
0.5
1
Time (Sec)
Aile
ron
(deg
)
0 20 40 60-2.2
-2
-1.8
-1.6
-1.4
Time (Sec)
Ele
vato
r (d
eg)
0 20 40 60-0.6
-0.4
-0.2
0
0.2
Time (Sec)
Rud
der
(deg
)
15
Summary
Existing Method:
Assumption:
Need of integral control
More number of design parameters (10-12)
Works
New Method:
Assumption: No such need (No wind-up)
Less number of design parameters (5-7)
Works better...!• Lesser control magnitude
• Smoother transient response
• Better turn co-ordination
* * *
0
0
V W
0V W
16
Robustness Study
Nominal Controller given to the actual system having uncertainties
Perturbation assumed in the inertia parameters and aerodynamic force and moment coefficients
Normal distribution used for introducing randomness in the parameters with mean value as the nominal value of the parameters and standard deviation as 1/3 of maximum allowed perturbation in that parameter.
17
Robustness Study
Inertia parameters varied from 5 to 10% Aerodynamic coefficients varied from 1%
to 10%. Simulation were carried out for 50 cases
in each mode. In each simulation study, the aim was to
declare it as a success or failure
18
Longitudinal Mode
0
y
z
Pilot Command Given:
0; ; 0;
1sec; 0.9965 ;
15sec; 2 ;
Limits Imposed on the steady state error:
: 3 ; n : 0.05 ; 585 575 / sec;
n : 20%;
T T y
z
z
T
V V n
for t n g
for t n g
g V to ft
19
Longitudinal Mode
Aerody-namic
Coefficients
1% 1% 2% 2% 5% 5% 10% 10%
Inertia
Parameters
5% 10% 5% 10% 5% 10% 5% 10%
Percentage
Success
100% 100% 96% 92% 76% 70% 48% 40%
20
Lateral Mode
0 0
y
Pilot Command Given:
40 ; ; ; 0;
Limits Imposed on the steady state error:
h: 1%; n : 0.05 ; 585 575 / sec
10%;
T T y
T
V V h h n
g V to ft
21
Lateral Mode
Aerody-namic
Coefficient
1% 1% 2% 2% 5% 5% 10% 10%
Inertia
Parameter
5% 10% 5% 10% 5% 10% 5% 10%
%
Success
100 100 100 100 94 88 86 80
22
Lateral Mode
0 0
y
Pilot Command Given:
10 / sec; ; ; 0;
Limits Imposed on the steady state error:
h: 1%; n : 0.05 ; 585 575 / sec
10%;
T T y
T
P V V h h n
g V to ft
P
23
Lateral Mode
Aerody-namic
Coefficient
1% 1% 2% 2% 5% 5% 10% 10%
Inertia
Parameter
5% 10% 5% 10% 5% 10% 5% 10%
Percentage
Success
100% 100% 100% 100% 98% 94% 76% 76%
24
Lateral Mode
0 0
y
Pilot Command Given:
7 / sec; ; ; 0;
Limits Imposed on the steady state error:
h: 1%; n : 0.05 ; 585 575 / sec
10%;
w T T y
T
w
P V V h h n
g V to ft
P
25
Combined Mode
Aerody-namic
Coefficient
1% 1% 2% 2% 5% 5% 10% 10%
Inertia
Parameter
5% 10% 5% 10% 5% 10% 5% 10%
Percentage
Success
100% 100% 96% 94% 54% 42% 28% 24%
26
Conclusion
When aerodynamic coefficients are perturbed by 5% and the inertia parameters by 10%, the controller is robust
Increase in inertia parameters does not affect the percentage success
Aerodynamic coefficients are more sensitive than inertia parameters
27
Enhancement of Robustness
Augment Dynamic inversion with Neuro -Adaptive Design
28
Adaptive Approach(Lateral case)
Nominal Outputs:
Actual Outputs:
Approximate Outputs:
),( ddy
Td
yd
d
d
UXf
V
n
Q
P
)(),( XdUXf
V
n
Q
P
y
T
)()(ˆ),( aaaay
Ta
ya
a
a
XXKXdUXf
V
n
Q
P
29
Adaptive Approach
Goal: Strategy:
Steps for assuring :
Solve for adaptive controller
tasYY d
tasYYY da
da YY
0
d a d
d d d
E Y Y
E K E
30
Adaptive Approach
Steps for assuring Error
Error Dynamics
aYY
( ),a a ai i aiE Y Y e y y
aiaiii
aiiai
iT
ii
iiT
ii
aiaiiyiai
iyii
eKXdXd
yye
XWXd
XWXd
eKXdUXfy
XdUXfy
)(ˆ)(
)(ˆ)(ˆ
)()(
)(ˆ),(
)(),(
31
Adaptive Approach
Error Dynamics
NN Training
Lyapunov Function Candidate
aiaiiT
iiiT
iai eKXWXWe )(ˆ})({
2
2
)ˆ1)((
~
ˆ~
)~~
(2
1)(
2
1
aiiaiiaiiii
iaiiT
ii
iii
iT
iaiii
epkepWXepWL
WWWwhere
WWepL
32
Adaptive Approach
Weight Update Rule:
Condition For stability:
)(ˆ XepW iaiiii
0
( / )i
ai i ai
L if
e k
33
A STRUCTURED Approach for
Attitude Maneuver of Spacecrafts
34
Neuro-adaptive Control: Generic Theory
Actual plant
Total tracking error
Tracking error dynamics
( )1 1 2X = h X X
D( , ) ( , )( ) 2 1 2 1 2X = f X X g X X U+
n1X R
n2X R nU R
1 1Z X X
( , ) ( , ) ( ) ( , )X 1 2 X 1 2 1 1 2Z = F X X + G X X U + h X d X X
Assumption
Unknown function
35
Neuro-adaptive Control: Generic Theory
Objective of adaptive controller:
Approximate System:
Model-following strategy:
d da a aˆ( , ) ( , ) ( ) ( , ) ( )X 1 2 X 1 2 1 1 2Z = F X X +G X X U+h X d X X K Z Z
a (0) (0)Z ZNN Approximation
d asZ Z t
36
Universal approximation property:
Error : Error dynamics for the individual i th error
channel:
Step I: Assuring
a a1 2 1 2 1 2 i( ) ( )Ti i id X ,X = W X ,X ,X ,X +
aZ Z
a aE Z Z
Weight vectorBasis function vector
a a i i
Tj ij 1 j 1 2 1 2 i a a
1
( ) ( )
n
aj
e W h X X ,X ,X ,X + k e
i ij 1 j1
ˆwhere and ( )
n
i i ij
W W - W h X
37
Neural Network Training by Lyapunov Analysis
Lyapunov function candidate:
i i
T 1a i a i i i
1
1 1p
2 2
n
i
L e e W Γ W
i a a
i i i
T T 1a i j ij 1 j 1 2 1 2 i i i
1 1
a i i a a i1
ˆ( ) ( )
n n
i j
n
i
L e p W h X X ,X ,X ,X W Γ W
+ e p k e
38
Neural Network Training with Stability
Weight Update Rule:
Sufficient condition:
where
ji i i 1 2 a j ji 1 i i i1
ˆ ˆ( ) ( ) W
n
j
W = Γ X ,X e p h X Γ
n2
aii 1
i i
0 if ,max
( )(i 1,n)
iL < ep
2 2 2 2i i i i i i i
i ai
1 ˆ( p ( W W W ))2
1k
2
39
SATELLITE Attitude Dynamics
Attitude kinematics Angular rate dynamics
Nominal Dynamics Actual Dynamics
Objective of Control Design:
,
( )d d d d d d = B I + U
( ) =
I = I + U x
( ) =
D( ) = B I + U + x
T TT T *T 0 0(0)
( )d d d = x
40
Nominal Control : Problem Specific Formulation
Tracking error for nominal system:
Tracking error dynamics:
Solving for nominal control
d d ds
d d d d d d d, , ds f g U
d d1 d d
dd d
d d d d d d d d d
dU B
B I K
x
41
Neuro-adaptive Control : Problem Specific Formulation
Tracking error for actual plant:
Expanding the following terms as:
Tracking error dynamics:
Basis
function
selection:
s
d d, , ( ) s f g U d
1 1 12 2 21 1 1
2 2 22 2 22 2 2
3 3 32 2 23 3 3
2 2 2i a a
2 2 2
2 2 2 T
( , , , ) [
]
2 2 2
2 2 2
2 2 2
( x- ) ( x- ) ( x- )
( x- ) ( x- ) ( x- )
( x- ) ( x- ) ( x- )
e e e
e e e
e e e
d d B = B + B I = I + I
42
Simulation Results:Nominal vs. Adaptive Control for actual system
MRPs Angular rates
(I) Constant disturbances & parameter uncertainties
43
Simulation Results:Nominal vs. Adaptive Control for actual system
Control Unknown function capture
(II) Constant disturbances & parameter uncertainties
44
Publications Conference Publications
Radhakant Padhi, Narayan P. Rao, Siddharth Goyal and S.N. Balakrishnan, “Command Tracking in High Performance Aircrafts: A new Dynamic Inversion Design”, 17th IFAC Symposium on Automatic control in Aerospace, Touolose, France.
Apurva Chunodkar and Radhakant Padhi, ”Precision attitude Manoeuvers of Spacecrafts in Presence of Parameter Uncertainities and disturbances: A SMART Approach”, 17th IFAC Symposium on Automatic Control in Aerospace, Touolose, France.
Radhakant Padhi and Apurva Chunodkar, “Model-Following Neuro - adaptive Control Design for attitude maneuvers for rigid bodies in Presence of Parametric Uncertainties and disturbances", International Conference on advances in Control and Optimization of Dynamical Systems, Bangalore, India, 2007.
Abha Tripathi and Radhakant Padhi ,”Robustness Study of A Dynamic Inversion Control Law For A High Performance Aircraft”, International Conference on Aerospace Science And Technology, to be held on 26 – 28 June 2008, Bangalore, India.
45
Publications Journal Publications
Radhakant Padhi, Siddharth Goyal, Narayan P. Rao and S.N. Balakrishnan, “A Direct Approach for Nonlinear Flight Control Design of High Performance Aircrafts”, Submitted to Control Engineering Practice.
Jagannath Rajsekaran, Apurva Chunodkar and Radhakant Padhi, ” Precision Attitude Maneuver of Spacecrafts Using Structured Model-Following Neuro -Adaptive Control”, Submitted to Control Engineering Practice.
Radhakant Padhi and Apurva Chunodkar, “Precision Attitude Maneuver of Spacecrafts Using Model - Following Neuro – Adaptive Control”, To appear in Journal of Systems Science & Engineering.
46
Questions And comments