Diagnosing Hybrid Systems: A Bayesian Model Selection Approach

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


Problem StatementTask: 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.


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 = Fx /m -2(qw - vr) - (u/m) * dm/dt

dv/dt = Fy /m -2(ru - pw) - (v/m) * dm/dt

dw/dt = Fz /m -2(pv - uq) - w/m) * dm/dt

Simulated in HCC (Hybrid Concurrent Constraint language) [Alenius and Gupta ‘98]


Hybrid System < M, X, , V, f ,COMPS >,

M – discrete modes M comprising:

- behavior mode of system

- fault mode of component [¬] ab(c), for every c in COMPS.

X RN – continuous state vector, x X

– discrete action inputs, that cause mode transitions,

C – controller actions

E – exogenous actions.

V RV – continuous inputs

f – system dynamics function f: M x X x x V x RM M x X . (t+1,xt+1) = f (t, xt, t, vt , wt)

Hybrid System Representation

Model s = ( , ),

a time-indexed sequence of modes and parameters st=( t,t ).


Architecture

BayesianTracker

QualitativeMonitoring& DiagnosisPlant

Controller

observations

controller actions


Example Scenario

point of detection

point of failure

desired trajectoryactual 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


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



y x


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



y x


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?


Bayesian Tracking

Markov assumption for temporal dynamics

p(st | st-1,…, s0) = p(st | st-1)

Hence,

p(st | Ot ) = k p( obst | st ) p(st | Ot-1 )

posterior likelihood temporal priorwhere

• st=( t,t ) is the model at time t,

• obst is the vector of observations at time t,

• Ot = (obst obst-1 ,…,obs0 ) is the observation history to time t.


Particle Filtering [Gordon et al., 93], [Isard and Blake, 98]

posterior at t-1

posterior

likelihood

p(st-1 | Ot-1 )

p(obst | st )

p(st | Ot )

temporal dynamics p(st | st-1)

fair random sampleof temporal prior

p(st | Ot-1 )


Focusing Bayesian TrackingProblem:

• state space is sparsely sampled

• large number of potential models

• delayed manifestation of faults

• fault modes are unexpected low prior


Architecture

BayesianTracker

QualitativeMonitoring& DiagnosisPlant

Controller

observations

controller actions


Focusing Bayesian TrackingProblem:

• 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(st | Ot, oracle) = k p( obst | st , oracle ) p(st | Ot-1 , oracle ) posterior likelihood temporal prior


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, st=( t, t ).Weights favor minimal diagnoses and have a temporal discounting.


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.

Documents

Diagnosing Hybrid Systems: A Bayesian Model Selection Approach