Outline
•
•
•
•
Strix – Saab Bofors (BAE)
Precision Guided Mortar Munition (PGMM) – ATK
Projectile Equations of Motion
Controllability
– Bang-Bang Control
– Impulsive dynamical systems
• “Naïve” control strategy
• Simulations
• Future directions
2
Terminally Guided Mortar Munition
3
STRIX Main Parts
Package
Programming
unit
Launch unit
Projectile
Sustainer
4
Projectile, Main Parts
Impact
Sensor
Electronics &
Power
Supply
Fuze System
Control
Rockets
Assembly
Warhead Fin Assembly
Target
Seeker
5
Data
Calibre
Launch weight
Length
Range
Pressure
Muzzle velocity
120 mm
18.2 kg
0.84 m
> 7 km
< 127 MPa
180-320 m/s
Seeker
Guidance
Warhead
6
Imaging IR
Proportional navigation,
control rocket system
HEAT with ERA capability
and behind-armour effect
Sequence of Events
3. Ballistic phase
4. Guidance
phase
Forward observer
2. Launch
1. Preparations
7
Guidance Phase
Sustainer
separation
Hit
Find
Electric
arming
Guidance with
control rockets
Target seeker
activation
Target
acquisition
and
selection
Proportional
navigation
8
STRIX Target Impact
KILL
•
Initiation of warhead from impact sensor
•
Penetration of ERA and main armour
•
Behind armour effect (pressure etc.)
9
Projectile Model
Equations of Motion



Fy
Fx
Fz
u
 qw  rv v 
 ru  pw w 
 pv  qu
m
m
m
Forces:
2
2
1
v

w
Fx  Frocket  mg  sin(  )  VA2  A(C x 0  C x 2 A 2 A )
2
VA
1
v
Fy  mg  sin(  ) cos( )  VA2 A  C N A  FThruster sin(  ) K m
2
VA
1
 wA
Fz  mg  cos( ) cos( )  VA2 A  C N
 FThruster cos( ) K N
2
VA
10
Projectile Model cont’d
Equations for Rotational rates

p
L  qr ( I zz  I yy )
I xx
M  rp ( I xx  I zz )
q
I yy


r
N  pq( I yy  I xx )
I zz
Moments
L
1
1
pd
VA2 dCl 0  VA2 dClp
2
2
2VA
M
1
1
qd
VA2 dC N ( X cp  X cg )  VA2 dCmq
 Fthruster( X th  X cg ) cos( ) K m
2
2
2VA
N
1
1
qd
VA2 dC N ( X cp  X cg )   VA2 dCmq
 Fthruster( X th  X cg ) sin(  ) K n
2
2
2VA
11
Controllability
Definition: Given the system
x  Ax  Bu
y  Cx  Du
Controllability: The pair (A,B) is said to be controllable iff at
the initial time t0 there exist a control function u(t) which
will transfer the system from its initial state x(t0) to the
origin in some finite time. If this statement is true for all
time, then the system is "Completely Controllable".
12
Controllability cont’d
Is the “full-information” nonlinear model of the projectile,
with no wind, controllable such that it will land within a
terminal set T, for a given number of discrete, fixed
magnitude impulses?
Note that the control impulses have additional constraints
which include:
• each control impulse can only be fired once
• presences of a dwell-time between firings
• finite burn time
13
Controllability, cont’d
Three approaches to the nonlinear controllability problem
with finite, discrete impulses are investigated:
• Bang-Bang control
• Impulsive dynamical systems
• Naïve control design
14
Bang-Bang Control
The problem is to find a feasible bang-bang control action
that takes the system from a given initial point to a given
terminal point with time being a free parameter.
• Minimum fuel optimal control problem
Unfortunately, the theory of minimum-fuel systems is not as
well developed as the theory of minimum-time systems.
Also the design of fuel-optimal controllers is more
complex that time-optimal controllers. In fact there may
not exist a fuel-optimal control, with a finite number of
discrete thrusters, that drive the projectile from any initial
state to the origin (controllable).
15
Impulsive Dynamical Systems
Many systems exhibit both continuous- and discrete-time
behaviors which are often denoted as hybrid systems.
Impulsive dynamical systems can be viewed as a subclass
of hybrid systems and consist of three elements:
• a continuous-time differential equation, which governs
the motion of the dynamical system between impulsive or
resetting events;
• a difference equation, which governs the way the system
states are instantaneously changed when a resetting event
occurs;
• and a criterion for determining when the states of the
system are to be reset.
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Impulsive Dynamical Systems cont’d
The projectile control problem can be viewed as an
impulsive dynamical system, whose analysis can be quite
involved. In the general situation, such systems can
exhibit infinitely many switches, beating, etc.
Controllability of hybrid systems is a hot topic currently,
and despite the numerous papers on the topic efficient
numerical algorithms that provide control algorithms is
still lacking.
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Naïve Control Strategy
Projectile control algorithms are often synthesized in an ad
hoc manner. These solutions are logic based and involve
testing a performance criteria at each time step.
Consider the following control strategy to drive a projectile
from a given state to a target set:
1. If current state in the target set, STOP
2. Given current point (after apex). If an impulse is not active then
compute the corresponding impact state and the miss distances.
1. Numerically integrate EOM to determine impact location
3. If miss distance is within tolerance, NO ACTION taken.
4. If miss distance is less than target set, FIRE in positive direction
5. If miss distance is more than target set, FIRE in negative direction
Under some conditions, NCS results in optimal solution.
18
Simulations
Ideal assumptions include:
•
•
•
•
•
Restricting the problem to two dimensions (x,y)
No wind, target location, projectile position/velocity known
Each impulse can be fired more than once
Infinitely many impulses
Point mass model
Physical parameters:
•
•
•
•
•
•
weight, 33 lbs
muzzle velocity and angle: 235 m/s, 50 degs
impulse duration and magnitude: 0.015 +/- .0002s, 5.0 +/-0.3 g
sample time, 0.005s
Impact error tolerance: 0.1m
Unaided projectile path: 2772 m
19
Projectile Path
Undisturbed
+ 300 Meters
- 300 Meters
20
Point mass
model
Impact point computed exactly, 39 shots required
• Due to numerical errors, two extra shots were needed
• Chattering caused by numerical integration errors, which is typical of
NCS algorithm
21
Rigid body
model
Impact point within 0.1m, 9 shots required
• Due to numerical errors, series of extra shots were needed
22
Impact Distribution for +200m Target
200 Meters / 24 Shots / 5g Impulse @ .0015 sec
Number of Impacts
12
10
8
6
4
2
0
-0.09 -0.07 -0.05 -0.03 -0.01 0.01
0.03
0.05
0.07
0.09
Meters from Target
23
Impulse Distribution
Positive
impulse
8 shots
Negative
impulse
24
Tradeoff Accuracy
•Total impulse force constant, #shots x impulse = 24*5g
• More shots: Increased accuracy, more complex
• Less shots: Less accurate, cheaper
35
30
25
20
15
10
5
0
12 shot
24 shot
-0
.0
5
0.
05
0.
15
0.
25
0.
35
0.
45
0.
55
0.
65
0.
75
Number of Impulses
+250 Meter Simulation
24 shot / 5g vs 12 shot / 10g
Meters from Target
25
Initial Findings
It is easier to hit targets beyond the initial trajectory
• Function of the limited flight time of the projectile and
computation delay
• If the target is overshot, the projectile may not be able to react
fast enough to bring it down in time.
Current configurations allow for no more than a 225 meter
overshoot and 310 meter undershoot
26
Summary
• Interesting class of control systems for which there has been a
limited amount of theoretical results
• For the short term, focus on better understanding the naïve
control strategy
• Rigid body equations of motion
• Atmospheric disturbances
• Trajectory tracking versus end point control
• Over the long term, develop a mathematical framework for
control of nonlinear systems with a finite number of discrete,
finite duration, fixed magnitude impulses.
27
References on Projectile Control
•
B. Burchett and M. Costello, “Model Predictive Lateral Pulse Jet Control of an
Atmospheric Rocket,” Journal of Guidance, Control and Dynamics, V25, 5, 2002.
•
E. Cruck and P. Saint-Pierre, “Nonlinear Impulse Target Problems under State Constraint:
A Numerical Analysis Based on Viability Theory,” Set-Valued Analysis, 12, pp. 383-416,
2004.
•
B. Friedrich, ATK, Private Communication.
•
S.K. Lucas and C.Y. Kaya, “Switching-Time Computation for Bang-Bang Control Laws,”
Proceedings of the American Control Conference, Arlington, VA June 25-27, pp. 176180, 2001
•
C.Y. Kaya and J.L. Noakes, “Computations and time-optimal controls,” Optimal Control
Applications and Methods, 17, pp. 171--185, 1996.
•
Y. Gao, J. Lygeros, M. Quincampoix and N. Seube, “On the control of uncertain
impulsive systems: approximate stabilization and controlled invariance,” Int. J. Control,
vol. 77, 16, pp. 1393-1407, 2004.
•
E.G. Gilbert and G.A. Harasty, “A Class of Fixed-Time Fuel-Optimal Impulsive Control
Problems and an Efficient Algorithm for Their Solution,” IEEE Trans. Automatic
Control, vol. 16, 1, pp.1-11, 1971
•
Z.H. Guan, T.H. Qian and X. Yu, “On controllability and observability for a class of
impulsive systems,” Systems and Control Letters, 47, p247-257, 2002.
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References on Projectile Control
•
W.M. Haddad, V. Chellaboina and N.A. Kablar, “Non-linear impulsive dynamical
systems. Part I: Stability and dissipativity,” Int. J. Control, vol. 74, 17, pp. 1631-1658,
2001.
•
W.M. Haddad, V. Chellaboina and N.A. Kablar, “Non-linear impulsive dynamical
systems. Part II: Stability and dissipativity,” Int. J. Control, vol. 74, 17, pp. 1659-1677,
2001.
•
H. Ishii and B. A. Francis, “Stabilizing a Linear System by Switching Control with Dwell
Time,” IEEE Trans. Automatic Control, pp.1962-1973, 2002.
•
T. Jitpraphai, B. Burchett and M. Costello, “A Comparison of different guidance schemes
for a direct fire rocket with a pulse jet control mechanism,” AIAA-2001-4326, 2001.
•
R. Pytlak and R.B. Vinter, “An Algorithm for a general minimum fuel control problem,”
Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, FL,
December 1994.
•
G. N. Silva and R. B. Vinter, “Necessary conditions for optimal impulsive control
problems,” SIAM J. Control Opt., vol. 35, 6, pp. 1829-1846, 1997.
•
G. Xie and L. Wang, “Necessary and sufficient conditions for controllability and
observability of switched impulsive control systems,” IEEE Trans. Automatic Control,
vol. 49, 6, pp.960-977, 2004.
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