5/12/2016
Three Recent Advances in
Optimal Control of Fin Stabilized
Surface to Surface Missiles
1
5/12/2016
Technique #1 - Predictive Optimal Pulsejet Control for Symmetric Projectiles
10/7/2015
Motivation
Previous efforts (2001, 2008) required a prediction of the
uncontrolled, and controlled trajectories.
Pulsing changes angular rates leaving other states alone.
(impulse model)
Interim work (2011) found trajectory sensitivities to changes
in initial angular rates by finite differencing.
Current method uses closed-form sensitivities (2012) which
are computed six times faster than finite differences.
10/7/2015
2
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Technique
#1
Knowing the predicted
impact point and sensitivity with
respect to changes
in initial
angular rates,
correction
Predictive
Optimal
Pulse-jet
Control
for times
are
determined
by
two
methods:
Symmetric Projectiles
Perturbed trajectories
From unit δ v, δ w, δ q, δ r
Direction of correction
In target plane
{δ y, δ z, δθ, δ y}
Default trajectory
Size of correction
for 'shack'
Perturbed trajectories are not
calculated, but sensitivities
are propagated as co-states
10/7/2015
Target plane direction is mapped to angular rate correction
vector through a Jacobian matrix. This in turn maps to roll
angle at which pulse jet should be fired
Corrected trajectory
Default trajectory
Correction size is fixed by distance to go.
Firing time then determined when
remaining distance provides appropriate
size correction.
Predicted impact point
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Linear Theory Model
Ξ contains epicyclic modes
Ξ is invertible
Ξ 2 is the Magnus (=0 in this study)
10/7/2015
Linear Theory Solution
Velocity state total solution is written in matrix exponential
form for ease of differentiation.
Position state solution not so easily found since Φ matrix
singular.
Instead, treat as linear forced ODE and use the solution:
4
5/12/2016
Linear Theory Solution
The integral can be handled by a combined matrix
exponential, if:
and
Then:
Derivatives of the closed form solution are now found more
easily.
5
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Derivatives of initial angular rates at current downrange
distance:
6
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This influence must be propagated along the
trajectory:
Taylor Series Model Predictive Controller
Since pitch and yaw are unconstrained at the target,
replace the last two rows with:
7
5/12/2016
Euler-Lagrange Optimal Control
Uncontrolled Dispersion
60
Altitude (ft)
40
20
0
−20
−40
−60
−80
−60
−40
−20
0
20
40
60
80
Crossrange (ft)
8
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Controlled Dispersions
0.3
4
2
Altitude (ft)
Altitude (ft)
0.2
0.1
0
0
−2
−0.1
−4
−0.2
−0.4
−0.2
0
Crossrange (ft)
Taylor Series
0.2
0.4
−6
−6
−4
−2
0
2
Crossrange (ft)
4
6
One Step Optimal
Technique #2 - Euler-Lagrange Optimal
Control for Direct Fire
10/7/2015
9
5/12/2016
Nine State Linear Plant Model
LTI Finite Horizon Optimal Control
System matrices treated as constants such that A(p,V) = A,
etc.:
Form the finite horizon cost function:
Since Q ≥ 0, choose Q = 0, then define the Hamiltonian:
10
5/12/2016
LTI Finite Horizon Optimal Control,
Cont'd
Taking variations yields the conditions for optimality:
Thus the control law is
LTI Finite Horizon Optimal Control, Cont'd
State and co-state ODEs are collected into a single matrix
equation:
Which can be solved using state transition matrices such
that:
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LTI Optimal Control cont'd
Such that the current co-state is given by:
And the transition matrices are given by the full 18 x 18
matrix exponential:
Dispersions, uncontrolled, controlled
120
100
−3
80
x 10
Altitude (ft)
6
Altitude (ft)
4
60
40
20
0
2
−20
−40
0
−60
−100
−2
−50
0
50
Crossrange (ft)
100
Uncontrolled
−6
−4
−2
0
2
Crossrange (ft)
4
Controlled, no range correction
6
−3
x 10
12
5/12/2016
Dispersion, with range correction
−3
x 10
3
Altitude (ft)
2
1
0
−1
−2
−3
−2
−1
0
1
Crossrange (ft)
2
3
−3
x 10
Controlled, with range correction
Technique #3 - Euler-Lagrange Optimal
Control for Indirect Fire
10000
140
Ctrl
Unctrl
Target
120
8000
100
Altitude (ft)
Crossrange (ft)
6000
80
60
40
4000
2000
20
Ctrl
Unctrl
Target
Vac. Model
0
0
-20
0
0.5
1
1.5
2
Downrange (ft)
2.5
3
3.5
4
x 10
-2000
0
0.5
1
1.5
2
Downrange (ft)
2.5
3
3.5
4
x 10
10/7/2015
13
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Modified Projectile Linear Theory
Equations in Matrix Form
0
= 0
0
=
=
+
0
0
0
0 − 100 , $ =
=
%
' 0100 &
Pitch angle, roll rate, and total velocity are treated as
parameters rather than states
Pitch angle perturbation is added to the state vector to
preserve controllability
Angle of attack rate is uncontrollable but stabiliziable
state s.t. gravity is included in homogeneous solution
Point mass vacuum trajectory is contrived to
intersect early point in actual trajectory, and
target.
()) =
+ + &+,
)
1
)
+ /
&+2
&+-
0
14
5/12/2016
Time varying parameters (V, p, sinθ , cosθ )
are predicted from point mass vacuum
trajectory.
&( + ℎ) =
9
'
A
& 0( ) +
+ℎ =
=
:;< 4
B)
B C
23 506 8 23
4 7 −
/3
/3
=>?@ 8
A
−
=
:
B
B
C
C
Euler-Lagrange Optimal solution for Time
Varying Piecewise Linear systems
The control can be found from:
N(s) is the solution to the time varying Riccati eqn:
15
5/12/2016
Decompose the Riccati Eqn. into two DEs:
Set Z(s) at target plane:
D
G
E
E
=F
=H
Back propagate the solution from target plane to current
downrange arclength:
16
5/12/2016
Algorithm reviewed:
•
•
•
•
•
•
•
•
At the first time in the control sampling period, solve for and save the
coefficients for creating the vacuum model.
– The model will launch from the origin, intersect a projectile state
early in the trajectory, and hit the target.
Compute the pitch angle to get from current position to the first spot
on the vacuum trajectory. From the prediction of this angle, develop
the current value of the
state.
Recursively predict values for 9,&,' , ,andℎ using the vacuum
trajectory model while updating aerodynamic coefficients at each
segment based on new predicted velocity and altitude.
Build the corresponding matrix Hamiltonian for each segment.
Integrate backwards in time using (54) – (55).
Using Z1, compute the Riccati solution at the current state from (52).
Compute the control needed at the current state using (47).
Convert to dimensional form, limit from [-1,1] rad and rotate into roll
frame.
Downrange miss (ft)
Downrange miss (ft)
Results
Uncontrolled
Controlled
17
5/12/2016
• Nash, A. L.,†‡ and Burchett, B. T., “Euler-Lagrange
Optimal Control of Indirect Fire Symmetric Projectiles”,
Accepted for presentation at AIAA Science and
Technology Forum and Exposition 2016, San Diego,
CA, 4-8 January, 2016.
• Burchett, B. T., and Nash, A. L.,† “Euler-Lagrange
Optimal Control for Symmetric Projectiles”,
Proceedings of the AIAA Science and Technology
Forum and Exposition 2015, Kissimee, FL, 5-9
January, 2015. doi: 10.2514/6.2015-1020
• Burchett, B. T., “Predictive Optimal Pulse-jet Control for
Symmetric Projectiles”, Proceedings ofthe AIAA
Science and Technology Forum and Exposition 2014,
National Harbor, MD, AIAA paper no. 2014-0883. doi:
10.2514/6.2014-0883
0/0/2015
Backup Slides
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5/12/2016
Trade Study – Effect of Trajectory
Discretization
2
1.9
1.8
CEP (ft)
1.7
1.6
1.5
1.4
1.3
30
32
34
36
38
40
42
Number of Segments
44
46
48
50
Trade Study – Effect of Controller
Sampling Period
15
250 ms
CEP (ft)
10
5
100 ms
10 ms
25 ms
0
-2
10
50 ms
-1
10
Sampling Period (s)
0
10
19
5/12/2016
Robustness of Guidance – Convergence
to Vacuum Trajectory
14000
60 deg. launch
Ctrl
Vac
12000
10000
Altitude (ft)
8000
40 deg. launch
6000
4000
20 deg. launch
2000
0
-2000
O
0
0.5
1
1.5
Downrange (ft)
2
2.5
3
4
x 10
Limited predictions required – Accuracy
improves greatly with downrange travel.
60
7000
6000
Actual Traj.
Immed. Pred.
Half Pred.
40
Pitch angle (deg.)
5000
Altitude (ft)
4000
3000
2000
20
0
-20
1000
-40
Actual Traj.
Immed. Pred.
Half Pred.
0
-1000
-60
0
0.5
1
1.5
Downrange (ft)
2
2.5
3
2200
Actual Traj.
Immed. Pred.
Half Pred.
1
1.5
Downrange (ft)
2
2.5
3
4
x 10
Actual Traj.
Immed. Pred.
Half Pred.
-20
-30
Roll rate (rad/sec)
1800
Total velocity (fps)
0.5
-10
2000
1600
1400
1200
1000
-40
-50
-60
-70
800
600
0
4
x 10
-80
0
0.5
1
1.5
Downrange (ft)
2
2.5
3
4
x 10
-90
0
0.5
1
1.5
Downrange (ft)
2
2.5
3
4
x 10
20