Distributed Decision Making

1. Distributed Decision Making Jason L. SpeyerUCLA MURI Kick Off Meeting May 14, 2000

2. Old Results and Current Thoughts • Distributed Estimation • Example: Target Association • Static Teams • Decentralized Control with Relaxed Communication • Static Games and Mixed Strategies • Static Team on Team

3. 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)

4. 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)

5. Example of Distributed Estimation: Target Association

6. 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 target • 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

7. 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

8. Stochastic Static Team Strategies • Stochastic Static Teams • Minimize 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 measurements • 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 measurements

9. Detection filter residuals of two tracks: the diagonal plots are correctly associated Parity test of two tracks: the diagonal plots are correctly associated Simulation Results

10. Stochastic Dynamic Team Problems • 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 strategies • 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

11. Static Game Problems • Results of current work in resource allocation in an air campaign shows that mixed strategies are seemingly generic • 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 Where u minimizes and vmaximizes L • Note that Luu > 0 implies a pure saddle point strategy, u=0, Lvv> 0 implies a mixed strategy,

12. Static Games (cont.) • Find the saddle point strategies of • Reduces to a matrix game of discrete primitives • 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)]

13. 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