Title: OCP%20Flies%20The%20F-16%20!!
1Discrete Thruster Control of Precision Guided
Munitions
Gary Balas, Tessa Stranik, Chris Visker Aerospace
Engineering and Mechanics University of
Minnesota balas_at_aem.umn.edu Jozsef Bokor, Peter
Gaspar and Zoltan Szabo Computer and Automation
Research Institute Hungarian Academy of
Science Budapest, Hungary
1 March 2006
2Outline
- 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
3Terminally Guided Mortar Munition
4STRIX Main Parts
Package
Programming unit
Launch unit
Projectile
Sustainer
5Projectile, Main Parts
Fuze System
Impact Sensor
Electronics Power Supply
Control Rockets Assembly
Fin Assembly
Warhead
Target Seeker
6Data
7Sequence of Events
3. Ballistic phase
4. Guidance phase
Forward observer
2. Launch
1. Preparations
8Guidance Phase
Hit
Find
Sustainer separation
Electric arming
Guidance with control rockets
Target seeker activation
Target acquisition and selection
Proportional navigation
9STRIX Target Impact
- KILL
- Initiation of warhead from impact sensor
- Penetration of ERA and main armour
- Behind armour effect (pressure etc.)
10Projectile Model
Equations of Motion
Forces
11Projectile Model contd
Equations for Rotational rates
Moments
12Controllability
- Definition Given the system
- 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".
13Controllability contd
- 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
-
14Controllability, contd
- Three approaches to the nonlinear controllability
problem with finite, discrete impulses are
investigated - Bang-Bang control
- Impulsive dynamical systems
- Naïve control design
15Bang-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).
16Impulsive 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.
17Impulsive Dynamical Systems contd
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.
18Naïve 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 - If current state in the target set, STOP
- Given current point (after apex). If an impulse
is not active then compute the corresponding
impact state and the miss distances. - Numerically integrate EOM to determine impact
location - If miss distance is within tolerance, NO ACTION
taken. - If miss distance is less than target set, FIRE
in positive direction - If miss distance is more than target set, FIRE
in negative direction - Under some conditions, NCS results in optimal
solution.
19Simulations
- 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
20Projectile Path
Undisturbed
300 Meters
- 300 Meters
21Point 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
22Rigid body model
- Impact point within 0.1m, 9 shots required
- Due to numerical errors, series of extra shots
were needed
23Impact Distribution for 200m Target
24Impulse Distribution
Positive impulse
Negative impulse
8 shots
25Tradeoff Accuracy
- Total impulse force constant, shots x impulse
245g - More shots Increased accuracy, more complex
- Less shots Less accurate, cheaper
26Initial 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
27Summary
- 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.
28References 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. 176-180, 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.
29References 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.