Transcript No Slide Title

Near-Optimal Missile Avoidance Trajectories
via Receding Horizon Control
Janne Karelahti, Kai Virtanen, and
Tuomas Raivio
Systems Analysis Laboratory
Helsinki University of Technology, Finland
S ystems
Analysis Laboratory
Helsinki University of Technology
Goal: avoid a closing missile
Criterion:
max capture time
min closing velocity
max miss distance
max guidance effort
max gimbal angle
max tracking rate
Controls wanted in a feedback form:
Receding Horizon Control + cost-to-go approximation
S ystems
Analysis Laboratory
Helsinki University of Technology
Select the
most suitable
criterion
Problem overview
• Some physical constraints of the missile system:
•
•
•
•
Lags in the missile guidance system dynamics
Seeker head’s gimbal angle limit
Seeker head’s tracking rate limit
Limited energy supply of the missile control system
Target aircraft
Gimbal angle
S ystems
Analysis Laboratory
Helsinki University of Technology

max
Problem overview
• Assumptions
•
•
•
•
•
The vehicles receive perfect state information about each other
The vehicles are modeled as 3-DOF point-masses
Target aircraft’s angular velocities and accelerations are limited
The missile utilizes proportional navigation
The missile has a single lag guidance system
S ystems
Analysis Laboratory
Helsinki University of Technology
Optimal Control Problem
• The target aircraft minimizes/maximizes
tf
J (u)   L(x, u, t )dt
subject to
State equations
Control/state constraints
Terminal constraint
S ystems
Analysis Laboratory
Helsinki University of Technology
t0
x  f (x, u, t ),
g(x, u)  0
h( x f )  0
x(t0 )  x 0
Receding horizon control scheme
•
•
The target makes decisions at tk  kt
0
1
N
i
t

t

t



t

t

T
Let’s define k
and
u

u
(
t
k
k
k
k
i
k)
1. Set k = 0. Set the initial state x0 and initial controls u0.
2. Solve the optimal controls over [ tk, tk+T ] by min./maximizing
t k T
JT (u)  
tk
L(x, u, t )dt  V (x(tk  T ),tk  T ) s.t.given constraints
3. Set uk  u0 and solve x k 1 by implementing
4. If the missile has reached its target set, stop.
5. Set k = k + 1 and go to step 2.

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Analysis Laboratory
Helsinki University of Technology
*
u*0 for t
Criteria & ctg. approximation
1)
2)
3)
4)
Capture time:
Closing velocity:
Miss distance:
Control effort:
JT (u)  r (tk  T )  rf / vc (tk  T )
JT (u)  vc (tk  T )
JT (u)  r (tk  T )
t k T
J T (u )   a (t )dt
tk
 a (t k  T )r (t k  T )  rf  / vc (t k  T )
5) Gimbal angle:
6) Tracking rate:
S ystems
Analysis Laboratory
Helsinki University of Technology

J T (u )   (t k  T )  e
JT (u)  (tk  T )
k a ( t k T )

1
1
Optimal controls over tk+T
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•
•
•
The direct shooting method:
Time discretization tki 1  tki  t  qit , i  0,, N 1
i 1
Explicit integration of the state x i 1 at t k by using ui
The resulting NLP problem
N 1
f (u1 ,, u N 1 )   L(xi , ui , tki )(tki 1  tki )  V (x N , tkN )
i 1
g(xi , ui )  0, i  0,, N
h(x N )  0
is solved by SNOPT SQP-solver
S ystems
Analysis Laboratory
Helsinki University of Technology


Controls
Numerical results
Capture time maximization: t  14.64 s
*
f

0
0
Missile
Aircraft
8
4
8
t[s]
12
16
T  2.4375s, t f  14.40 s
4
x[km] 2
0
-2
2.5
y[km]
Controls

0
Helsinki University of Technology
16

6
Analysis Laboratory
12

0
0
S ystems
t[s]

5
3.5
10
8
T  1.375s, t f  14.05 s
Controls
h[km]
6.5
4


0
0
4
8
t[s]
12
T  t f , t f  14.64 s
16



Controls
Numerical results
0
Miss distance maximization: r * (t *f )  8.72 m
0
Missile
8
t[s]
12
16
T  1.375s, r (t f )  1.36 m
Aircraft


5

Controls
h[km]
6.5
4
0
8
0
4
x[km] 2
-2
Helsinki University of Technology
t[s]
12
16


0
0
Analysis Laboratory
8

0
S ystems
4
T  2.4375s, r (t f )  2.24 m
6
Controls
3.5
10
2.5
y[km]
0
4
8
t[s]
12
16
T  t f , r(t f )  8.72 m
Numerical results
Capture time maximization
Miss distance maximization
90
90
120
60
28
150
120
24
30
60
25
150
20
18
12
6
5
180
10
15
20
0
[deg]
[deg]
30
15
10
5
5
180
10
15
20
r[km]
210
330
240
300
270
S ystems
Analysis Laboratory
Helsinki University of Technology
0
r
210
330
240
300
270
Conclusions
• The scheme provides near-optimal feedback controls in
virtually real-time
• Launch state maps provide a way for selecting the most
effective performance measure for the current state
• Additional research topics:
• Evaluation of the realism of the optimal trajectories by
inverse simulation
• Uncertainty about the missile establishes a need for
automatic identification of the missile parameters (e.g.
guidance law)
S ystems
Analysis Laboratory
Helsinki University of Technology