11/21/2006 Massachusetts Institute of Technology Motivation - PDF document

11/21/2006 Massachusetts Institute of Technology Motivation • Complex embedded systems DNNF-based Belief State Estimation Paul Elliott Brian Williams 2 Belief State Estimation Approximate Belief State Estimation Behavior Belief Behavior k-best Belief State Belief State Model State Model States Estimation Estimation (PCCA) Ŝ (PCCA) Ŝ Commands Observations Commands Observations Plant Plant S S 3 4 Compiled Approx Belief State Estimation Compiled Approx Belief State Estimation Behavior Belief k-best Model Compiler State States (PCCA) Estimation Ŝ Commands Observations Behavior k-best Behavior k-best Compiler Belief State Compiler Belief State Plant Model States Model States S (C2D) Estimation (C2D) Estimation (PCCA) Ŝ (PCCA) Ŝ Commands Observations Commands Observations Plant Plant S S Darwiche C2D 2005 v2.20 5 6 1

11/21/2006 Prior Work Contributions Behavior k-best Behavior k-best • MEXEC (Barrett ’05) Belief • New encoding of Belief Model Compiler States Model Compiler States State State (PCCA) Ŝ (PCCA) Ŝ Estimation Estimation estimation for compilation – Trajectories Commands Observations Commands Observations • Max-Prod Plant Plant – sd-DNNF S S • New use of sd-DNNF • BFBSU (Martin ’05) – Approximate Belief State • New k-best algorithm for • Max-Sum-Prod sd-DNNF – Optimal constraint-solver • Dynamic Bayesian Networks • New bounded believe – Max-Prod state estimation algorithm – Sum-Prod – Distribution on variables 7 8 Model: Probabilistic PCCA: Switch Concurrent Constraint Automata (PCCA) Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S Switch !TurnOn Constraints Off TurnOn Off : S out = Low 0.99 0.03 On : S out = High Failed 0.01 0.97 Failed : S out ∈ {Low, High} TurnOff On 0.999 0.001 !TurnOff 9 10 PCCA: Inverter PCCA Model Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S Switch Inverter !TurnOn 0.99 0.01 Constraints Inverter Off I in I out TurnOn 0.99 0.01 High Low Ok Failed Ok : 0.99 0.03 Low High Failed 0.01 Ok Failed Failed : I in ∈ {Low, High} Constraints 0.97 TurnOff On ^ Ok Low On 0.999 0.001 !TurnOff Off ^ Ok High 11 12 2

11/21/2006 Estimation Function PCCA as a Relation Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S t t+1 TurnOn Switch Inverter Output Switch Inverter Output Off Ok On Ok Low High On Ok Low On Ok Low 13 14 PCCA as a Probabilistic Relation PCCA as a Probabilistic Relation Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S ∏ τ 1 B t (S)/#M(S) 1 B t+1 (S) #M(S,O)/#M(S) Prev Obs Next State Prev Obs Next State Commands Prev State Next Obs Commands Prev State Next Obs TurnOn TurnOn Switch Inverter Output Switch Inverter Output Switch Inverter Output Switch Inverter Output On Ok On Ok Off Ok High Low Off Ok High Low 0.960 On Ok On Ok On Ok On Ok 0.989 Low Low Low Low 15 16 sd-DNNF: Relation Encoding sd-DNNF: Model Counting Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S ∏ τ ∏ τ B t (S)/#M(S) 1 B t+1 (S) #M(S,O)/#M(S) B t (S)/#M(S) 1 B t+1 (S) #M(S,O)/#M(S) 1 1 Commands Prev State Prev Obs Next State Next Obs Commands Prev State Prev Obs Next State Next Obs TurnOn TurnOn Switch Inverter Output Switch Inverter Output Switch Inverter Output Switch Inverter Output Off Ok High On Ok Low 0.960 Off Ok High On Ok Low 0.960 On On Ok On On Ok Ok Low Low 0.989 Ok Low Low 0.989 17 18 3

11/21/2006 sd-DNNF: Probabilities sd-DNNF: States Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S ∏ τ ∏ τ B t (S)/#M(S) 1 B t+1 (S) #M(S,O)/#M(S) B t (S)/#M(S) 1 B t+1 (S) #M(S,O)/#M(S) 1 1 Commands Prev State Prev Obs Next State Next Obs Commands Prev State Prev Obs Next State Next Obs TurnOn Switch Inverter Output Switch Inverter TurnOn Switch Inverter Output Switch Inverter Output Output Off Ok On Ok Low 0.960 Off Ok On Ok Low 0.960 High High On Ok Low On Ok Low 0.989 On Ok Low On Ok Low 0.989 19 20 Estimation Estimation: k-best Approx Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S #M(S,O)/#M(S) ∏ τ B t (S)/#M(S) TurnOn TurnOn Switch Inverter Output Switch Inverter Output Switch Inverter Output Switch Inverter Output Off Ok High On Ok Low 0.960 Off Ok High On Ok Low 0.960 B t+1 (S) argmax B t+1 (S) On On Ok Low On Ok Low 0.989 Ok Low On Ok Low 0.989 k On Ok On Ok 21 22 Estimation: k-best Approx Estimation: k-best Approx (Division) Behavior Belief k-best Behavior Belief k-best Model Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Commands Observations Commands Observations Switch Inverter Output Switch Inverter Output Plant Plant S S #M(S,O)/#M(S) ∏ τ #M(S,O)/#M(S) ∏ τ B t (S)/#M(S) B t (S)/#M(S) TurnOn Switch Inverter Output Switch Inverter Output TurnOn Switch Inverter Output Switch Inverter Output Off Ok On Ok Low 0.960 Off Ok On Ok Low 0.960 High High argmax B t+1 (S) argmax B t+1 (S) On Ok Low On Ok Low 0.989 On Ok Low On Ok Low 0.989 k k On Ok On Ok Max-Sum-Product-Sum Max-Sum-Product-Sum 23 24 4

11/21/2006 Estimation: k-best Approx (Division) C2D 2005 v2.20: Compilation Behavior Belief k-best Behavior Belief k-best Model Model Compiler State States Model Compiler State States (PCCA) Estimation Ŝ Ok (PCCA) Estimation Ŝ Switch Commands Observations Commands Observations Inverter On Switch Inverter Output Low High Plant Plant Failed S S #M(S,O)/#M(S) ∏ τ Output B t (S)/#M(S) Ok B t TurnOn Switch Inverter Output Switch Inverter Output Switch Off Ok High On Ok Low 0.960 argmax B t+1 (S) Inverter On Ok On Ok 0.989 Low Low 0.001 Off 0.03 Failed 1 k 0.999 On Ok On 0.97 Output Max-Sum-Product-Sum Low High 25 26 Labeled sd-DNNF Estimation Algorithm Behavior Belief k-best 1. Assign known values and Behavior Belief k-best Model Compiler State States Model Compiler State States Ok (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Switch probabilities Commands Observations Commands Observations 2. First Pass: Compute all Inverter On High Low summations Plant Plant Failed S S 3. Merge summation results Output B t Ok 4. Second Pass: Extract the k-best solutions Switch 1. Upward part: Compute the Inverter probabilities and paths of 0.03 Failed 1 0.001 Off 0.999 the k-best estimates On 0.97 Output 2. Downward part: Follow the paths to extract the k-best estimates Low High 27 28 Algorithm: Initial Assignment Algorithm: Initial Assignment Behavior Belief k-best Behavior Belief k-best Model Compiler State States Model Compiler State States Ok Ok (PCCA) Estimation Ŝ (PCCA) Estimation Ŝ Switch Switch Commands Observations Commands Observations Inverter Inverter On On Low High Low Low High Plant Plant Failed Failed S S Output Output Ok B t Ok Ok B t 1 Switch Switch Inverter Inverter 0.03 Failed 1 0.03 Failed Failed 1 0.001 Off 0.001 Off Off 0.999 0.999 On On On 0.97 0.97 Output Output Given: Given: Low High Low High B t ( ) = 1 Low B t ( ) = 1 Low On On 29 30 5