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Distributed Decision Making Jason Speyer UCLA


									Distributed Decision Making

         Jason L. Speyer
       MURI Kick Off Meeting
          May 14, 2000
        Old Results and Current

   Distributed Estimation
      Example: Target Association
   Static Teams
      Decentralized Control with Relaxed Communication
   Static Games and Mixed Strategies
   Static Team on Team
           Distributed Estimation
• A historical viewpoint
• Spatially distributed measurements, but same
  dynamic system
      Global Kalman filter estimates may be algebraically
       assembled from local Kalman filter estimates and a
       local correction term
      These local transmissions can occur anytime
      Speyer (1979)
• Distributed measurements and dynamic system
      Generalized above results for Gauss-Markov systems
      Willsky, Bello, Castanon, Levy, and Verghese (1982)
   Distributed Estimation (cont.)
• Formulated the distributed Gauss-Markov
  estimation problem using the information state
      Levy , Castanon, Verghese, and Willsky (1983)
• Distributed estimation algorithm extended to
  apply to the track association problem
      Overlap of error variances used to associate track
      New approach using fault detection ideas
      Applies to passive radar (only bearings to target
       measured at each station)
    Example of Distributed
Estimation: Target Association
       Target Association Problem
• Each Station has a “track” of the angle-only
  measurement history of each target
• Associate each track at one station to the track
  corresponding to the same target at another station
• Standard technique (Pao, et. Al):
      Each station estimates the position and corresponding error
       envelope of each target
      For each target at station 1, find the track at another station
       whose error envelope comes closet to that of station 1’s
      Each target’s position can be estimated using the Modified
       Gain Extended Kalman Filter (MGEKF)
      Problem: the estimated position using only 1 station’s
       angular measurements may not be precise
        Target Association Using
           Detection Methods
• Detection approach to track association
      To associate targets between two stations
         For each track at station 1, construct a bank of MGEKFs, each
          using data from that track and a track from station 2
         Mismatched tracks bias the residual of the MGEKF
         By contrast, matching tracks generate a good estimate with a
          small residual
      To associate targets from an additional station
         Use estimated target position from 1st 2 stations
         Parity test associates their tracks to those at additional stations
Stochastic Static Team Strategies
• Stochastic Static Teams
      Minimize                ELx, u 
        u1 ( z1 ), , u N ( z N )
       L is a convex function, x is the state of the world, zi(x)
       are local measurements, ui(zi) are the team strategies
      Radner (1962) showed that person by person optimality
       implied stationary
         Optimal strategies to the static LQG problem are affine in the
      Hard to verify condition (local finiteness) was
       circumvented by Krainak, Speyer, and Markus (1982)
         Optimal strategies to the static LEG problem are affine in the
            Simulation Results
 Detection filter residuals of   Parity test of two tracks: the
  two tracks: the diagonal         diagonal plots are correctly
plots are correctly associated               associated
       Stochastic Dynamic Team
• Decentralized stochastic control requires information
  patterns that allow a dynamic programming recursion
      In the dynamic programming formalism, a stochastic static
       team problem is solved to determine the optimal strategies
      Previous LQG and LEG solutions, using only a one-step
       delayed information sharing pattern, produced affine
      New results for the LQG problem uses a control only sharing
       information pattern
         Local controller affine only in the local measurement history
         Requires increasing the state of the world to include the
          process noise sequence
         Appears to be the minimal information pattern which retains
          affine strategies
           Static Game Problems
• Results of current work in resource allocation in an air
  campaign shows that mixed strategies are seemingly
      General approach is to construct primitives so that a
       descretization of strategies can be determined
      In the following simple games, a decomposition into
       primitives naturally occurs
• Static Game (Bryson and Ho)
    Find the saddle point strategies of
    L  u 2  3uv  2v 2 , u  1 , v  1
      Where u minimizes and v maximizes L
    Note that Luu > 0 implies a pure saddle point strategy, u=0,

     Lvv > 0 implies a mixed strategy,v   1 with p  [ p1 , p2 ]
             Static Games (cont.)
• Find the saddle point strategies of
    L  u 2  3uv  2v2 , u  1, v  1
      Reduces to a matrix game of discrete primitives
                v1  1 v2  0 v3  1
        u1  1   2      1     4
        u2  0    2       0      2
        u3  1    4       1     2

      Probabilities for mixed strategies obtained by Linear Programming
• Generalized: For measurements z1=z1(x) and z2=z2(x)
  (x state of the world) find the saddle point strategies where
  u(z1) is to minimize and v(z1) is to maximize E[L(x, u, v)]
                  Team on Team
• Stochastic Static Games
      Strategies dependent on local information
• Generalize stochastic static games to many players
  on each side
      Stochastic static team on team
      Zero-sum games
      None-zero sum games
• Solutions basis for approximation methods
• Consider implications to dynamic team on team

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