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
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 2
Overview • Concurrent Probabilistic Hybrid Automata • Hybrid Estimation • Unknown Mode • Decomposition - intuitively • Decomposition - algorithmically • Example • Discussion & Conclusion 3
Probabilistic Hybrid Automata x w , , F T X , , , U , T Probabilistic Hybrid Automata d d s � x x x d mode (discrete state) with domain X d ...{ x } ........ d c n x c continuous state with domain � � � w u u y ... ... u d discrete command with domain U d d c c m � u c continuous command with domain i m y c continuous output with domain � o F ................. discrete-time dynamics for each mode (sampling-period T s ) T ................. guarded probabilistic transitions between modes ��� � � � � �� � �� � �� ��� � � � � � ��� ��� ��� � � � ��� ��� ��� � �� � �� �������������� 4
concurrent Probabilistic Hybrid Automata internal PHA component variable � �� � �� � �� continuous � �� � � � � � � input u ci � �� output / observed discrete �� variable y ci (cont.) � �� input u dj • observed variables: internal variable + additive Gaussian noise • independent variables: u c , u d , noise • dependent variables: x c , x d , internal variables � � � � x f x , u v ,( ) ( ) ,( 1) ,( 1) ,( 1) c k k c k c k s k � � � � � y g ( x , u ) v ( ) k ( ) k c k ,( ) c k ,( ) o k ,( ) 5
Hybrid Mode / State Estimation Task Overview: PHA4 PHA1 PHA2 continuous PHA3 output / observed input u ci variable y ci (cont.) discrete input u cj Hybrid Estimation Problem: Given a cPHA model for a system, a sequence of observations and the history of the control inputs generate the leading set of most likely states at time-step k 6
hybrid Mode Estimation At each time step k , we evaluate for each trajectory: new estimate old estimate: x ’ (k-1) = { x ’ d,(k-1) , x c,(k-1) } , x (k) = { x d,(k) , x c,(k) } , x (k-1) = { x d,(k-1) , x c,(k-1) } , h (k-1) h’ = P t h (k-1) h (k) = P o h’ P o P t continuous behavior mode transition: x d,(k-1) = m i → x ’ d,(k-1) = m j x ’ c,(k-1) → x c,(k) , x d,(k) = x ’ d,(k-1) 7
hybrid Mode Estimation At each time step k , we evaluate for each trajectory: new estimate old estimate: P t x (k) = { x d,(k) , x c,(k) } , x (k-1) = { x d,(k-1) , x c,(k-1) } , h (k-1) h (k) = P o h’ P o continuous behavior x ’ c,(k-1) → x c,(k) , x d,(k) = x ’ d,(k-1) 8
Filter Calculation Hypothesis: mode m j = { m 11 ,m 21 ,m 31 } � �� 1) retrieve cPHA equations for m j � �� � �� � �� � � � � � � � �� F 1 ( m 11 ) = { u c1 =5 w c1 } � �� �� F 2 ( m 21 ) = { x c1,(k) =0.8 x c1,(k-1) + w c1,(k-1) , y c1 =x c1 } F 3 ( m 31 ) = { x c2,(k) =x c3,(k-1) + y c1,(k-1) , x c3,(k) =0.4 x c2,(k-1) + 0.5 u c1,(k-1) , y c2 =2 x c2 + x c3 } 2) solve equations independent vars: u c , observed vars: y c � �� � �� � �� � �� x c,(k) = f (k) ( x c,(k-1) , u c,(k-1) )+ v s,(k-1) MIMO Filter � �� y c,(k) = g (k) ( x c,(k) , u c,(k) )+ v o,(k) � �� � � 3) calculate extended Kalman filter and evaluate it 9
Unknown Mode Hypothesis: mode m j = { ? ,m 21 ,m 31 } � �� 1) retrieve cPHA equations for m j � �� � �� � �� � � � � � � � �� F 1 ( ? ) = { } � �� �� F 2 ( m 21 ) = { x c1,(k) =0.8 x c1,(k-1) + w c1,(k-1) , y c1 =x c1 } F 3 ( m 31 ) = { x c2,(k) =x c3,(k-1) + y c1,(k-1) , x c3,(k) =0.4 x c2,(k-1) + 0.5 u c1,(k-1) , y c2 =2 x c2 +x c3 } 2) solve equations � �� FAILS - additional independent variable: w c1 � �� � �� ? � �� � �� we cannot calculate the extended Kalman Filter � �� � � that is necessary for hybrid estimation 10
Unknown Mode What about partial estimation? � �� � �� � �� � �� ? � � � �� � � � �� �� y c1 ... noisy measurement of outputsignal of PHA A 2 PHA A 3 fully specified → partial estimation possible 11
Intuitive Decomposition � �� � �� � �� � �� � �� � �� � �� � �� � �� MIMO Filter � �� � �� � �� � �� � �� � �� � � � � �� � � � � � � �� �� alternative: calculate clustered Filter 12
Intuitive Decomposition � �� � �� � �� � �� � �� � �� � �� � �� � �� � �� � �� MIMO Filter � �� � �� � �� � � �� � � � �� � � � � � � �� �� � �� � �� � �� � �� Filter 1 � �� � � � � � � �� � �� � � � �� � �� � �� � �� � �� � �� � �� Filter 2 � �� � �� � � �� � � � � �� � � �� Filter Cluster 13
Implications of Decomposition � �� � �� � �� � �� Filter 1 � �� � � � � � � �� � �� � � � �� � �� � �� � �� � �� � �� � �� Filter 2 � �� � �� � � �� � � � � �� � � �� Filter Cluster • Factorization of P O : P O = Π P Oj eg. unknown mode in A 1 : P O ≤ P O2 • Additional (virtual) noise at inputs (e.g. v o2 acting upon y c2 ) • Reduced computational complexity of the filter cluster [ Kalman filter O(n 3 ) < filter cluster O(n 1 3 +n 2 3 ) ] 14
Algorithmic Decomposition 1) generate the causal graph from the equations [bipartite matching based algorithm, Nayak-95] � �� � �� � �� � �� � �� � �� � �� 2) Remapping: � �� � �� � �� � �� insert virtual inputs �� �� � �� � �� � �� 15
Algorithmic Decomposition continuous estimation is limited to the observable part of the system - observable with respect to the measurements y c i and the known input values u ci structurally observable (SO) variable: structurally determined (SD) variable: Either directly observed, or there exists Either an input variable of the automaton, at least one path in the causal graph that or there does not exist a path in the causal connects the variable to an output graph that links the variable with an variable undetermined exogenous variable � �� � �� � �� � �� � �� � �� � �� � �� 16
Algorithmic Decomposition prior structural analysis: eliminate loops by calculating the strongly connected components (SCC) [Aho-83] � �� � �� � �� � �� � �� � �� � �� � �� �� �� � �� � �� � �� �� �� � �� � � �� � �� structural analysis of variables structural analysis of SCC 17
Algorithmic Decomposition Observability analysis algorithm and labeling/partitioning � �� � � �� � �� � �� � �� � � � � �� � �� �� �� �� � �� � � �� � �� � �� � � �� � �� � �� � �� 18
Algorithmic Decomposition Observability analysis algorithm and labeling/partitioning � �� � � � �� �� � �� � �� � � �� �� �� � �� � � �� � � �� � �� 19
Algorithmic Decomposition Observability analysis algorithm and labeling/partitioning � �� � � � �� �� � �� � �� � � ��� �� �� � �� � � �� � � �� � �� 20
Algorithmic Decomposition Observability analysis algorithm and labeling/partitioning � �� � � � �� �� � �� � �� � � � �� ��� �� �� � �� � � �� � � �� � �� 21
Algorithmic Decomposition Observability analysis algorithm and labeling/partitioning � �� � � � �� �� � �� � �� ��� � ��� ��� ��� �� �� � �� � � �� � �� � � �� 22
Algorithmic Decomposition Observability analysis algorithm and labeling/partitioning � �� � � � �� �� � �� � �� ��� � ��� ��� ��� �� �� � �� � � � �� �� � �� � �� � � �� 23
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