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Discrete 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
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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

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Terminally Guided Mortar Munition
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STRIX Main Parts
Package
Programming unit
Launch unit
Projectile
Sustainer
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Projectile, Main Parts
Fuze System
Impact Sensor
Electronics Power Supply
Control Rockets Assembly
Fin Assembly
Warhead
Target Seeker
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Data
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Sequence of Events
3. Ballistic phase
4. Guidance phase
Forward observer
2. Launch
1. Preparations
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Guidance Phase
Hit
Find
Sustainer separation
Electric arming
Guidance with control rockets
Target seeker activation
Target acquisition and selection
Proportional navigation
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STRIX Target Impact

• KILL
• Initiation of warhead from impact sensor
• Penetration of ERA and main armour
• Behind armour effect (pressure etc.)

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Projectile Model
Equations of Motion
Forces
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Projectile Model contd
Equations for Rotational rates
Moments
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Controllability

• 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".

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Controllability 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

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Controllability, contd

• Three approaches to the nonlinear controllability
  problem with finite, discrete impulses are
  investigated
• Bang-Bang control
• Impulsive dynamical systems
• Naïve control design

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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).

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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 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.
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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
• 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.

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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

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Projectile Path
Undisturbed
300 Meters
- 300 Meters
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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

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Rigid body model

• Impact point within 0.1m, 9 shots required
• Due to numerical errors, series of extra shots
  were needed

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Impact Distribution for 200m Target
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Impulse Distribution
Positive impulse
Negative impulse
8 shots
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Tradeoff Accuracy

• Total impulse force constant, shots x impulse
  245g
• More shots Increased accuracy, more complex
• Less shots Less accurate, cheaper

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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

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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.

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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. 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.

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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.