1. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Diagnosing Hybrid Systems: A Bayesian Model Selection Approach Sheila McIlraith Knowledge Systems Lab Stanford University

2. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Problem Statement Task: Diagnose continuous systems w/ embedded supervisory controllers. Given: –a hybrid representation of system behavior, –a history of executed controller actions, and –a history of observations, including an observation of aberrant behavior, Determine: what components failed and their associated parameter values. Assumptions: –discrete time observations and state estimation –hybrid system model contains no autonomous jumps, –fault occurrence is abrupt, –failure of component may be partial or full. Approach: Hybrid diagnosis as Bayesian model selection [MacKay,91] –Qualitative analysis to reduce and focus search space. –Quantitative analysis via Bayesian tracking and model selection.

3. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Illustrative Example NASA Sprint AERCam Robotic camera unit with 12 thrusters, T1-T12, that enable both linear and rotational motion. Discrete Mode Transitions thrusters turn on and off Continuous Newtonian Dynamics point mass, m, at position (x,y,z), with translational and angular velocities: V = (u,v,w) and  = (p,q,r) : d(mV)/dt = F + 2m (V x  ) V dm/dt + m dV/dt = F - 2m(  x V) For each coordinate du/dt = F x /m -2(qw - vr) - (u/m) * dm/dt dv/dt = F y /m -2(ru - pw) - (v/m) * dm/dt dw/dt = F z /m -2(pv - uq) - w/m) * dm/dt Simulated in HCC (Hybrid Concurrent Constraint language) [Alenius and Gupta ‘98]

4. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Hybrid System, M – discrete modes  M comprising: - behavior mode of system - fault mode of component [¬] ab(c), for every c in COMPS. X  R N – continuous state vector, x  X  – discrete action inputs,    that cause mode transitions,  C   – controller actions  E   – exogenous actions. V  R V – continuous inputs f – system dynamics function f: M x X x  x V x R M  M x X. ( t+1,x t+1 ) = f ( t, x t,  t, v t, w t ) Hybrid System Representation Model s = (,  ), a time-indexed sequence of modes and parameters s t =( t,  t ).

5. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Architecture Bayesian Tracker Qualitative Monitoring & Diagnosis Plant Controller observations controller actions

6. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Example Scenario point of detection point of failure desired trajectory actual trajectory Behavior modes: accelerate-x, cruise-x, decelerate-x, accelerate-y, cruise-y,... y x Task: Given controller action history,  and observation history,  determine the model s = (,  ) that best fits the data. behavior mode & (failed) components parameter values

7. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Model 1: s = (,  )  behavior (failed) components parameter values... ([accelerate-x, ab(T2), ¬ ab(T1,T3,…,T12) ], [20, 100, 100, …, 100]) ([cruise-x, ab(T2), ¬ ab(T1,T3,…,T12) ], [20, 100, 100, …, 100]) ([decelerate-x, ab(T2), ¬ ab(T1,T3,…,T12) ], [20, 100, 100, …, 100]) ([accelerate-y, ab(T2), ¬ ab(T1,T3,…,T12) ], [20, 100, 100, …, 100])... Example Scenario point of detection point of failure desired trajectory actual trajectory Behavior modes: accelerate-x, cruise-x, decelerate-x, accelerate-y, cruise-y,... y x

8. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Model 2: s = (,  )  behavior (failed) components parameter values... ([accelerate-x, ¬ ab(T1,…,T12) ], [100, 100, 100, …, 100]) ([cruise-x, ¬ ab(T1,…,T12) ], [100, 100, 100, …, 100]) ([decelerate-x, ¬ ab(T1,…,T12) ], [100, 100, 100, …,100]) ([accelerate-y, ab(T6), ¬ ab(T1,..,T5,T7…,T12) ], [100,..., 100, 33, …,100])... Example Scenario point of detection point of failure desired trajectory actual trajectory Behavior modes: accelerate-x, cruise-x, decelerate-x, accelerate-y, cruise-y,... y x

9. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Challenges: nonlinear dynamics multiple models  multimodal distribution Approach: Bayesian Tracking & Model Selection Determine the posterior probability distribution over models and model parameters, given the system observations p(model | observations)  p(observations | model) p(model) posterior likelihood prior Represent the posterior distribution as discrete samples and propagate the distribution through time using particle filtering [Gordon et al., 93], [Isard and Blake, 98]. How do we represent and propagate complex multimodal distr’ns?

10. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Bayesian Tracking Markov assumption for temporal dynamics p( s t | s t-1,…, s 0 ) = p( s t | s t-1 ) Hence, p( s t | O t ) = k p( obs t | s t ) p( s t | O t-1 ) posterior likelihood temporal prior where s t =( t,  t ) is the model at time t, obs t is the vector of observations at time t, O t = (obs t obs t-1,…,obs 0 ) is the observation history to time t.

11. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Particle Filtering [Gordon et al., 93], [Isard and Blake, 98] posterior at t-1 posterior likelihood p( s t-1 | O t-1 ) p( obs t | s t ) p( s t | O t ) temporal dynamics p( s t | s t-1 ) fair random sample of temporal prior p( s t | O t-1 )

12. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Focusing Bayesian Tracking Problem: state space is sparsely sampled large number of potential models delayed manifestation of faults fault modes are unexpected  low prior

13. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Architecture Bayesian Tracker Qualitative Monitoring & Diagnosis Plant Controller observations controller actions

14. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Focusing Bayesian Tracking Problem: state space is sparsely sampled large number of potential models delayed manifestation of faults fault modes are unexpected  low prior Solution: Exploit qualitative reasoning techniques to identify models, “candidate qualitative diagnoses” that are qualitatively consistent with the observation history Use candidate qualitative diagnoses to bias the temporal prior  reduced search space  focus sampling on consistent diagnoses p( s t | O t, oracle) = k p( obs t | s t, oracle ) p( s t | O t-1, oracle ) posterior likelihood temporal prior

15. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Qualitative Diagnosis ‘Oracle’ Qualitative (linearized) representation in terms of temporal causal graphs [Mosterman & Biswas, 99]. Qualitatively propagate aberrant behavior back through time to generate candidate failed components Qualitatively propagate candidate diagnoses forward to generate model at time t -- ( t,  t ). Output is a set of weighted candidate models at time t, s t =( t,  t ). Weights favor minimal diagnoses and have a temporal discounting.

16. Sheila McIlraith, Knowledge Systems Lab, Stanford University DX’00, 06/2000 Summary Task: Diagnose continuous systems w/ embedded supervisory controllers. Approach: Bayesian tracking and model selection Challenge: How to represent & propagate complex multimodal distributions. How to predict unlikely events (component failure). Solution: Represent the posterior distribution as discrete samples. Propagate the distribution through time using particle filtering. Exploit qualitative monitoring and diagnosis techniques to reduce search space and focus on qualitatively consistent models.