Hybrid Diagnosis with Unknown Behavioral Modes Michael W. Hofbaur 1 - - PowerPoint PPT Presentation

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Hybrid Diagnosis with Unknown Behavioral Modes Michael W. Hofbaur 1 - - PowerPoint PPT Presentation

Jan 08, 2023 131 likes •602 views

Hybrid Diagnosis with Unknown Behavioral Modes Michael W. Hofbaur 1 & Brian C. Williams 2 1) Department of Automatic Control, Graz University of Technology, Austria 2) Artificial Intelligence & Space Systems Laboratories MIT, USA

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Hybrid Diagnosis with Unknown Behavioral Modes

Michael W. Hofbaur1 & Brian C. Williams2

1) Department of Automatic Control,

Graz University of Technology, Austria

2) Artificial Intelligence & Space Systems Laboratories

MIT, USA

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2

Motivation / Aim

  • Hybrid mode estimation / diagnosis of a

highly complex artifact that exhibits both, continuous and discrete behaviors

  • model-based: does the model capture

every possible situation?

  • unknown environment ....
  • how can we cope with unmodeled situations?

⇒ show how structural analysis and decomposition techniques can enable “unknown mode detection” for hybrid estimation

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3

Overview

  • Concurrent Probabilistic Hybrid Automata
  • Hybrid Estimation
  • Unknown Mode
  • Decomposition - intuitively
  • Decomposition - algorithmically
  • Example
  • Discussion & Conclusion
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4

  • Probabilistic Hybrid Automata

Probabilistic Hybrid Automata xd mode (discrete state) with domain Xd xc continuous state with domain ud discrete command with domain Ud uc continuous command with domain yc continuous output with domain F ................. discrete-time dynamics for each mode (sampling-period Ts) T ................. guarded probabilistic transitions between modes

, , , , , ,

d d s

F T X U T x w

...{ } ........

d c

x x x

  • ...

...

d c c

w u u y

  • n
  • i

m

  • m
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5

concurrent Probabilistic Hybrid Automata

  • observed variables:

internal variable + additive Gaussian noise

  • independent variables: uc, ud, noise
  • dependent variables:

xc, xd, internal variables

  • continuous

input uci

  • utput / observed

variable yci (cont.) PHA component

  • internal

variable discrete input udj

  • ,( )

( ) ,( 1) ,( 1) ,( 1) ( ) ( ) ,( ) ,( ) ,( )

, ( , )

c k k c k c k s k k k c k c k

  • k

f g x x u v y x u v

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Hybrid Mode / State Estimation

Task Overview:

Hybrid Estimation Problem: Given a cPHA model for a system, a sequence

  • f observations and the history of the control inputs generate the leading set
  • f most likely states at time-step k

PHA1 PHA2 PHA3 PHA4

continuous input uci

  • utput / observed

variable yci (cont.) discrete input ucj

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hybrid Mode Estimation

At each time step k, we evaluate for each trajectory:

new estimate x(k) = {xd,(k) , xc,(k)}, h(k) = Po h’ continuous behavior x’c,(k-1) → xc,(k) , xd,(k)= x’d,(k-1) Pt Po

  • ld estimate:

x(k-1)={xd,(k-1) , xc,(k-1)}, h(k-1) mode transition: xd,(k-1) = mi → x’d,(k-1) = mj x’(k-1) = {x’d,(k-1) , xc,(k-1)}, h’ = Pt h(k-1)

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hybrid Mode Estimation

At each time step k, we evaluate for each trajectory:

new estimate x(k) = {xd,(k) , xc,(k)}, h(k) = Po h’ continuous behavior x’c,(k-1) → xc,(k) , xd,(k)= x’d,(k-1) Po

  • ld estimate:

x(k-1)={xd,(k-1) , xc,(k-1)}, h(k-1) Pt

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Filter Calculation

Hypothesis: mode mj={m11,m21,m31}

F1(m11)={uc1=5 wc1 } F2(m21)={xc1,(k)=0.8 xc1,(k-1)+ wc1,(k-1), yc1=xc1} F3(m31)={xc2,(k)=xc3,(k-1)+ yc1,(k-1), xc3,(k)=0.4 xc2,(k-1)+ 0.5 uc1,(k-1), yc2=2 xc2+ xc3 } 1) retrieve cPHA equations for mj 2) solve equations independent vars: uc, observed vars: yc xc,(k)=f(k) (xc,(k-1),uc,(k-1))+ vs,(k-1) yc,(k)=g(k) (xc,(k),uc,(k))+ vo,(k) 3) calculate extended Kalman filter and evaluate it

  • MIMO Filter
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Unknown Mode

Hypothesis: mode mj={?,m21,m31}

F1( ? ) ={ } F2(m21) ={xc1,(k)=0.8 xc1,(k-1)+ wc1,(k-1), yc1=xc1} F3(m31) ={xc2,(k)=xc3,(k-1)+ yc1,(k-1), xc3,(k)=0.4 xc2,(k-1)+ 0.5 uc1,(k-1), yc2=2 xc2+xc3 } 1) retrieve cPHA equations for mj 2) solve equations FAILS - additional independent variable: wc1 we cannot calculate the extended Kalman Filter that is necessary for hybrid estimation

  • ?
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Unknown Mode

What about partial estimation?

  • ?

yc1 ... noisy measurement of outputsignal of PHA A2 PHA A3 fully specified → partial estimation possible

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Intuitive Decomposition

  • MIMO Filter

alternative: calculate clustered Filter

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Intuitive Decomposition

  • MIMO Filter
  • Filter 1

Filter 2 Filter Cluster

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Implications of Decomposition

  • Factorization of PO: PO = Π POj
  • eg. unknown mode in A1: PO ≤ PO2
  • Additional (virtual) noise at inputs (e.g. vo2 acting upon yc2)
  • Reduced computational complexity of the filter cluster

[ Kalman filter O(n3) < filter cluster O(n1

3+n2 3) ]

  • Filter 1

Filter 2 Filter Cluster

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Algorithmic Decomposition

  • 2) Remapping:

insert virtual inputs 1) generate the causal graph from the equations [bipartite matching based algorithm, Nayak-95]

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Algorithmic Decomposition

continuous estimation is limited to the observable part of the system - observable with respect to the measurements yci and the known input values uci

structurally observable (SO) variable: Either directly observed, or there exists at least one path in the causal graph that connects the variable to an output variable structurally determined (SD) variable: Either an input variable of the automaton,

  • r there does not exist a path in the causal

graph that links the variable with an undetermined exogenous variable

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Algorithmic Decomposition

prior structural analysis: eliminate loops by calculating the strongly connected components (SCC) [Aho-83]

  • structural analysis of variables

structural analysis of SCC

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Algorithmic Decomposition

Observability analysis algorithm and labeling/partitioning

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Example - BIO Plex

Airlock Plant Growth Chamber Crew Chamber

CO

2

tank lighting system chamber control flow regulator 2 pulse injection valves

CO2

flow regulator 1

Advanced Life Support System

  • BIO-Plex
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Example

PHA example: Flow Regulator

umax u t t Q Qmax

closed partially open

  • pen
  • perational modes:

closed mr1, partially open mr2, open mr3 fault modes: stuck closed mr4, stuck open mr5

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Example

PHA example: Flow Regulator

  • perational modes:

closed mr1, partially open mr2, open mr3 fault modes: stuck closed mr4, stuck open mr5 unknown mode mr0

example: DRIFT FAULT

  • t

Qmax

drifting

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Example

Hybrid estimation with a cPHA model of BIO-Plex and Simulation data obtained from NASA‘s simulator

t = 700: flow regulator drift t = 900: partial lighting blackout t = 1100: flow regulator repair t = 1300: light repair

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Example

cPHA model and decomposition

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Example

  • 1. Flow regulator drift detection detail

t = 700: Flow regulator 2 drifts t = 727: unknown mode classification t = 769: hME prefers stuck open as symptom explanation t = 800: FR2 becomes fully open

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Example

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Discussion / Conclusion

Summary

  • unknown mode for hybrid estimation
  • structural analysis and decomposition
  • example

Current & Future Research

  • decomposition → conflicts
  • conflict directed search to improve hybrid estimation
  • hybrid mode estimation & reconfiguration
  • fault tolerant control - autonomous automation
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SLIDE 36
  • ptional slides
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PHA1 PHA2 PHA3 PHA4

concurrent Probabilistic Hybrid Automata

Concurrent Probabilistic Hybrid Automata A ................ set of PHAs continuous and discrete command variables yc ................. observed continuous variables vs, vo ............ state disturbances and sensor noise inputs characterized by Nx, Ny

, , , , , ,

c s

  • x

y

A N N u y v v

... ...

d c

u u u

  • 1

2 1 2 ,( ) ,( 1) ,( 1) ,( 1) ,( 1) ( ) ,( ) ,( ) ,( ) ,( )

... { , ,..., } , , ' ( , , )

c c c cl d d d dl c k c k c k d k s k k c k c k d k

  • k

x x x f g x x x x x x x u x v y x u x v

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Mode / State Transition

State transition: no transition is triggered (x’d,(k) = xd,(k+1)) and time proceeds for

  • ne sampling period: t(k+1)= t(k) + Ts. . The evolution of the continuous state

x’c,(k) → xc,(k+1) is captured by the discrete-time dynamic model that holds for x’d,(k).

,( ) ,( )

' '

d k c k

x x

,( ) ,( ) d k c k

x x

Ti

,( 1) ,( 1) d k c k

x x

  • t(k)

t’(k) t(k+1)

Fj

Mode Transition State Transition Mode transition: time proceeds only infinitesimally t’(k)= t(k)+ ε so that the evolution of the continuous state xc,(k) → x’c,(k) can be neglected: x’c,(k)= xc,(k)

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Observation Probability Po

We compare the sensor signal yc(k) with its estimation for mode mj using an extended Kalman filter.

→ one extended Kalman filter for each hypothesis

1 T

  • P

e

−

−

=

r S r

  • peration performed by an (extended) Kalman filter:
  • state prediction:

xc,(k-1), P(k-1) , uc,(k-1) → x’c,(k), P’(k)

  • residual calculation:

x’c,(k), P’(k), yc(k)

→

r(k) , S(k) , Po

  • Kalman filter gain calculation:

P’(k) , S(k)

→

k(k)

  • state estimate refinement:

x’c,(k), P’(k), k(k) , r(k) → xc,(k), P(k)

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Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning
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Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning
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Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning
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Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning
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44

Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning
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Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning
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Algorithmic Decomposition

  • Observability analysis algorithm and labeling/partitioning