APPLICATION OF GEOMETRICAL APPLICATION OF GEOMETRICAL EXTRAPOLATION METHOD BASED EXTRAPOLATION METHOD BASED

HYBRID SYSTEM CONTROLLER ON HYBRID SYSTEM CONTROLLER ON PURSUIT-AVOIDANCE DIFFERENTIAL PURSUIT-AVOIDANCE DIFFERENTIAL

GAMEGAME

Ginzburg Pavel and Slavnaya Lyudmila

Supervisory by Dr. Mark Moulin

List of contents:

Optimal control and Hamiltonian formalism Hybrid control Intuitive Geometrical Controller Circle Extrapolation Controller Second Order Approximation in Geometrical

Extrapolation Nonlinear Heading Estimator The Supper Hybrid Controller Conclusion

Hamiltonian Formalism

System representation

Solutions of Backward Reachable Set

Capture radius 5Linear velocities 5

Angular velocities 1

Capture radius 5Linear velocities 5

Angular velocity Evader 2Angular velocity Pursuer 1

Capture radius 5Linear velocities 5

Angular velocity Evader 1Angular velocity Pursuer 2

Open loop via closed loop control

Open loop:+ Safe for initial condition check - Sensitive to measurement noise - Non flexible to task changing - Slow and massive calculation (offline ) - And more…

Closed loop: + Safe control + Online and fast calculations - Non optimal

Intuitive Geometrical Controller

Plane division to provide control signal

Problematic situation for Simple Controller

Results of Intuitive Geometrical Controller

Initial conditions (-2, 2, π/2)

Initial conditions (-2, 2, π/3) Control signal output

Control signal output

Circle Extrapolation Controller

Red Cross – estimated positionBlue Star – measurement data, stored in controller memory

Assumptions:

- The opponent control signal unchanged during the sampling

Principe:

- 3 points define the only one circle

- Forth point is the opponent estimated position

-Controller chooses the optimal output depends on the estimated position

- Each step correction provided (like Kalman filter measurement correction)

Results of Circle Extrapolation Controller

Initial conditions (-2, 2, π/2) Control signal output

Initial conditions (-2, 2, π/3) Control signal output

Second Order Approximation in Geometrical Extrapolation

Results Geometrical Extrapolation Controller

Initial conditions (-2, 2, π/3)

Initial conditions (-2, 2, π/2) Control signal output

Control signal output

Geometrical Extrapolation Controller used by both players

Border point (0, 3.44, -π)Applied control signals

Applied control signalsBorder point (2, 1.42, -π/2)

(0, 3, -π)

(1.6, 1, -π/2)

Results Geometrical Extrapolation Controller with “arctan”

“Sign” function replaced by “Arctangent”

Initial conditions (-2, 2, π/2) Control signal output

Control signal output Initial conditions (-2, 2, π/3)

Noise in coordinate measurement

1 variance noiseBorder point (2, 1.42, -π/2)

0.1 variance noiseBorder point (2, 1.42, -π/2)

Nonlinear Heading Estimator Assumptions: - The opponent heading signal unchanged during the sampling Principe: - 5 points is a trade off between noise filtering and device flexibilityController provides the most fitted line (MSE)

Red points – measurement dataBlue lines – calculated slopes, the estimator output

Performances of Nonlinear Heading Estimator

Constant control input (0), noisy case (0.01 variance)

no noise exist

Constant control input (0.1), noisy case (0.01 variance).

The Supper Hybrid Controller

Idea:- fitting noisy data to estimated orbit - verify the noise level by R-condition checking - choose more successful controller

implementation

Travels between hybrids controllers (double hybridism exists)

Conclusions

Advantages of Hybrid Control Implementation The geometrical extrapolation method in

Hybrid system provides wise plane division Noise filtering using The Method More levels in Hybrid Implementation

Directions

Non linear filtering (fit the surface to measurement data)

Analytical approximated solution of HJB equations

More complex differential games General case