Identifiability in dynamic network identification Harm Weerts 1 Arne - PowerPoint PPT Presentation

Identifiability in dynamic network identification Harm Weerts 1 Arne Dankers 2 Paul Van den Hof 1 1 Control Systems, Department of Electrical Engineering, Eindhoven University of Technology, The Netherlands. 2 Department of Electrical Engineering, University of Calgary, Canada. 21-10-2015

Dynamic networks appear in different domains r ( t ) w ( ζ, t ) r ( t ) m m m m w 1 ( t ) w 2 ( t ) w 3 ( t ) w 4 ( t ) e ( t ) r ( t ) u ( t ) G ( q ) y ( t ) C ( q ) 1

Illustration of the problem w 1 ◮ We have some measured node signals ◮ We want to identify the dynamics between the node signals w 2 w 3 2

Illustration of the problem w 1 ◮ We have some measured node signals ◮ We want to identify the dynamics between the node signals w 2 w 3 2

The identifiability question w 1 w 1 A A AB B w 2 w 3 w 2 w 3 Can we distinguish between the networks?? Identifiability?? 3

Outline ◮ Introduction ◮ Dynamic network setup ◮ Network predictor ◮ Global network identifiability ◮ Conclusions 4

Network definition Nodes / w j ( t ) Internal variables External variables R jk ( q ) G jl ( q ) w l ( t ) r k ( t ) . . . . . for all k ∈ N r for all l ∈ N j . j v j ( t ) Process Noise           w 1 w 1 r 1 v 1 0 G 12 · · · G 1 L . ...           . w 2 G 21 0 . w 2 r 2 v 2           = + R +           . . . . . ... ...  .   .   .   .   .  . . G L − 1 L . . .           · · · 0 G L 1 G L L − 1 w L w L r K v L � �� � � �� � � �� � � �� � � �� � G ( q ) w ( t ) w ( t ) r ( t ) v ( t ) 5

Typical assumptions in network identification literature       v 1 e 1 H 11 ( q ) 0 · · · 0 . ...       . v 2 e 2 0 H 22 ( q ) .       = H ( q ) , with H ( q ) =       . . . ... ...      .  . . . 0 . .       0 · · · 0 H LL ( q ) v L e L � �� � e ( t ) ◮ Van den Hof et. al., Automatica, 2013 ◮ Yuan et. al., Automatica, 2011 ◮ Sanandaji et. al., ACC, 2011 ◮ Materassi & Salapaka, IEEE trans AC, 2012 6

Typical assumptions in network identification literature       v 1 e 1 H 11 ( q ) 0 · · · 0 . ...       . v 2 e 2 0 H 22 ( q ) .       = H ( q ) , with H ( q ) =       . . . ... ...      .  . . . 0 . .       0 · · · 0 H LL ( q ) v L e L � �� � e ( t ) ◮ Van den Hof et. al., Automatica, 2013 ◮ Yuan et. al., Automatica, 2011 ◮ Sanandaji et. al., ACC, 2011 ◮ Materassi & Salapaka, IEEE trans AC, 2012 Consequence: identification problem can be split into MISO problems 6

Our approach       v 1 e 1 H 11 ( q ) H 12 ( q ) · · · H 1 L ( q ) . ...       . v 2 e 2 H 21 ( q ) H 22 ( q ) .       = H ( q ) , with H ( q ) =       . . . . ... ...      . .  . . . . . .       H L 1 ( q ) · · · · · · H LL ( q ) v L e L � �� � e ( t ) Noise contribution on all nodes can be correlated with each other 7

Our approach       v 1 e 1 H 11 ( q ) H 12 ( q ) · · · H 1 L ( q ) . ...       . v 2 e 2 H 21 ( q ) H 22 ( q ) .       = H ( q ) , with H ( q ) =       . . . . ... ...      . .  . . . . . .       H L 1 ( q ) · · · · · · H LL ( q ) v L e L � �� � e ( t ) Noise contribution on all nodes can be correlated with each other Consequence: In the network identification problem all nodes should be treated symmetrically 7

The problem ◮ Identify the whole network, G ( q ), H ( q ) and R ( q ), on the basis of a predictor for every node signal w j . ◮ Formulate a condition, based on the network predictor, under which the networks can be distinguished from each other. ◮ The introduced identifiability notion is related to uniqueness of dynamics instead of parameters 8

The problem ◮ Identify the whole network, G ( q ), H ( q ) and R ( q ), on the basis of a predictor for every node signal w j . ◮ Formulate a condition, based on the network predictor, under which the networks can be distinguished from each other. ◮ The introduced identifiability notion is related to uniqueness of dynamics instead of parameters 8

The problem ◮ Identify the whole network, G ( q ), H ( q ) and R ( q ), on the basis of a predictor for every node signal w j . ◮ Formulate a condition, based on the network predictor, under which the networks can be distinguished from each other. ◮ The introduced identifiability notion is related to uniqueness of dynamics instead of parameters 8

Problem statement Under which condition is there a one-to-one relation between model dynamics and network predictor? For standard configuration (open-loop, closed-loop) and for diagonal H the answer is relatively simple. For non-diagonal H the answer is nontrivial! 9

Network predictor w ( t ) = G ( q ) w ( t ) + R ( q ) r ( t ) + H ( q ) e ( t ) 10

Network predictor w ( t ) = G ( q ) w ( t ) + R ( q ) r ( t ) + ( H ( q ) − I ) e ( t ) + e ( t ) 10

Network predictor � � e ( t ) = H − 1 ( q ) ( I − G ( q )) − 1 w ( t ) − R ( q ) r ( t ) w ( t ) = G ( q ) w ( t ) + R ( q ) r ( t ) + ( H ( q ) − I ) e ( t ) + e ( t ) 10

Network predictor � � e ( t ) = H − 1 ( q ) ( I − G ( q )) − 1 w ( t ) − R ( q ) r ( t ) w ( t ) = G ( q ) w ( t ) + R ( q ) r ( t ) + ( H ( q ) − I ) e ( t ) + e ( t ) � � I − H − 1 ( q ) w ( t )+ H − 1 ( q ) G ( q ) w ( t )+ H − 1 ( q ) R ( q ) w ( t ) = r ( t )+ e ( t ) � �� � � �� � � �� � New filter Typical input filter Typical output filter Typical structure results from the network structure! 10

Network predictor � � e ( t ) = H − 1 ( q ) ( I − G ( q )) − 1 w ( t ) − R ( q ) r ( t ) w ( t ) = G ( q ) w ( t ) + R ( q ) r ( t ) + ( H ( q ) − I ) e ( t ) + e ( t ) � � I − H − 1 ( q ) w ( t )+ H − 1 ( q ) G ( q ) w ( t )+ H − 1 ( q ) R ( q ) w ( t ) = r ( t )+ e ( t ) � �� � � �� � � �� � New filter Typical input filter Typical output filter � � I − H − 1 ( q ) ( I − G ( q )) w ( t ) + H − 1 ( q ) R ( q ) r ( t ) w ( t | t − 1) = ˆ 10

Model structure Define the network model structure: M ( θ ) = { G ( q , θ ) , H ( q , θ ) , R ( q , θ ) } � � I − H − 1 ( q , θ ) ( I − G ( q , θ )) w ( t )+ H − 1 ( q , θ ) R ( q , θ ) w ( t | t − 1 , θ ) = ˆ r ( t ) � �� � � �� � W r ( q ,θ ) W w ( q ,θ ) 11

One-to-one relation w ( t | t − 1 , θ 1 ) = ˆ w ( t | t − 1 , θ 2 ) ˆ Predictor equality θ 1 = θ 2 Parameter equality 12

One-to-one relation w ( t | t − 1 , θ 1 ) = ˆ w ( t | t − 1 , θ 2 ) ˆ Predictor equality M ( θ 1 ) = M ( θ 2 ) Model equality � c Classical identifiability . [Ljung, 1999] θ 1 = θ 2 Parameter equality 12

One-to-one relation w ( t | t − 1 , θ 1 ) = ˆ w ( t | t − 1 , θ 2 ) ˆ Predictor equality Informative data [Ljung, 1999] � c Φ w , r ( ω ) > 0 . W w ( θ 1 ) = W w ( θ 2 ) Predictor filter equality W r ( θ 1 ) = W r ( θ 2 ) M ( θ 1 ) = M ( θ 2 ) Model equality 12

One-to-one relation W w ( θ 1 ) = W w ( θ 2 ) Predictor filter equality W r ( θ 1 ) = W r ( θ 2 ) M ( θ 1 ) = M ( θ 2 ) Model equality 12

One-to-one relation W w = I − H − 1 ( I − G ) W r = H − 1 R W w ( θ 1 ) = W w ( θ 2 ) Predictor filter equality W r ( θ 1 ) = W r ( θ 2 ) M ( θ 1 ) = M ( θ 2 ) Model equality 12

One-to-one relation W w = I − H − 1 ( I − G ) W r = H − 1 R W w ( θ 1 ) = W w ( θ 2 ) Predictor filter equality W r ( θ 1 ) = W r ( θ 2 ) � ??? M ( θ 1 ) = M ( θ 2 ) Model equality 12

One-to-one relation W w = I − H − 1 ( I − G ) W r = H − 1 R W w ( θ 1 ) = W w ( θ 2 ) Predictor filter equality W r ( θ 1 ) = W r ( θ 2 ) Definition: Global network identifiability � ??? M ( θ ) is globally network identifiable when the implication holds in both directions. M ( θ 1 ) = M ( θ 2 ) Model equality 12

Global network identifiability Proposition T =( I − G ) − 1 � � W w ( θ 1 ) = W w ( θ 2 ) H R W r ( θ 2 ) ⇔ T ( θ 1 ) = T ( θ 2 ) W r ( θ 1 ) = 13

Global network identifiability Proposition T =( I − G ) − 1 � � W w ( θ 1 ) = W w ( θ 2 ) H R W r ( θ 2 ) ⇔ T ( θ 1 ) = T ( θ 2 ) W r ( θ 1 ) = Theorem M ( θ ) is globally network identifiable if ∃ P ( q ) nonsingular, such that � � � � H ( q , θ ) R ( q , θ ) P ( q ) = D ( q , θ ) F ( q , θ ) with D ( q , θ ) diagonal, ∀ θ . The condition is necessary when G ( q , θ ) is fully and independently parameterized. 13