Dynamic Adversarial Conflict with Restricted Information

1
Dynamic Adversarial Conflict with Restricted
Information

• Jason L. Speyer
• Research Asst. Ashitosh Swarup
• Mechanical and Aerospace Engineering
  DepartmentUCLA
• MURI , Review
• June 4 2002

2
Cooperative and Adversarial Static Strategies
with Restricted Information

• Static Stochastic Teams Radner
• Each team members strategy is a function of only
  its local noisy measurement of the state of the
  world.
• Minimize the expected value of a convex function
  of the team strategies and the state of the
  world.
• All a priori statistics and functions are known.
• Solution
• Stationary conditions for general convex cost
  Radner.
• Locally finiteness condition relaxed Krainak,
  Speyer, Marcus
• Solutions available for the LQG and LEG.
• Static Stochastic Nonzero-sum Games Basar
• Solutions available for LQG.

3
Dynamic Team and Game Strategies with Nonclasical
Information Patterns

• LQG and LEG team strategies with one-step
  delayed-information pattern available.
• All information except for the current
  measurement is shared.
• Solution constructed by dynamic programming where
  a static game is solved at each step in the
  backward recursion.
• Since information can not be shared, few results
  are available for game strategies Willman.
• Formal (possible) solution to LQG games of
  conflict.
• We interpret his results and discuss new
  directions.

4
Formulation of the Dynamic LQG Game With
Restricted Information

• Consider the quadratic cost criterion

• The discrete-time system dynamics are

• The measurements are

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Strategies for Games with Restricted Information

• Define the measurement history of the pursuer and
  evader as
• Define the strategies as general linear functions
  
• Since these strategies are adversarial, there is
  no cooperation and therefore, no possibility of
  cheating.
• Consider the Saddle Point Inequality as
• where ( ) denotes the saddle point strategies.
• If one player uses a linear strategy, then the
  resulting LQG problem produces the other linear
  optimal strategy.

6
Construction of the Linear Strategies

• The cost can be formed through the following
  nesting
• Assume the pursuer knows the functional form of
  the evaders strategy as
• Substitute evaders strategy into the dynamic
  system
• where is a ((i1)nie)(in(i-1)e)
  growing matrix.

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Construction (Continued)

• Knowing the evaders strategy, solve the LQG
  problem of minimizing
• where is the
  conditional mean propagated as
• The pursuers strategy given the evaders strategy
  is
• This strategy, known to the evader, must be
  reduced to the form

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Convergence to Saddle Point Strategies

• Strategies are a complex function of the
  opponents gains.
• Strategy gains are determined by an iterative
  procedure.
• Begin by assuming an adversaries strategy.
• Solve for the opponents strategy given an
  adversaries strategy.
• A sequence of LQG minimization and maximizations
  oscillate about the saddle point.
• This sequence may converge to the saddle point
  strategies.
• Require conditions for the existence of a saddle
  point of pure strategies.
• Require conditions for convergence to the saddle
  point.

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Special Cases

• Consider the full state information LQR
  differential game.
• If there exits a solution to the saddle point
  Riccati eq.,
• then the Riccati eqs. associated with sequential
  min and
• max operations converge to the saddle point
  Riccati eq.
• Convergence has been proved.
• Consider a scalar three stage dynamic game with
  restricted
• information.
• Cost criterion converges to a fixed point for
  some parameters.
• Saddle point strategies converge to a fixed
  point.
• Results indicate a hedging policy by the
  adversaries
• over their full state strategies.

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Application to SEAD

• Requires generalization of LQG strategies for
  adversarial conflict.
• Apply to suppression of enemy air defenses (SEAD)
• UCAVs allocate resources based on their
  distributed sensor information.
• SAM site allocates resources based on its sensor
  information.

11
Coordinated Flight

• Aerodynamically Coupled Formation Flight of
  Aircraft
• Chichka, Wolfe, Speyer
• Applications
• Autonomous Aerial Refueling.
• Autonomous Formation Flight for Drag Reduction.
• UCAV Clusters.