Identifiability in dynamic network identification

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 w(ζ, t) r(t) r(t) m m m m w 1 (t) w 2 (t) w 3 (t) w 4 (t) r(t) u(t) G(q) e(t) 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

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 / Internal variables w l (t) G jl (q) for all l N j... w j (t) External variables R jk (q). for all k N r j r k (t) vj(t) Process Noise w 1 0 G 12 G 1L w 1 r 1 v 1 w 2. = G 21 0... w 2........ +R r 2 + v 2 GL 1 L... w L }{{} G L1 } G L L 1 {{ 0 w } L }{{} r K }{{} v L }{{} w(t) G(q) w(t) r(t) v(t) 5

Typical assumptions in network identification literature v 1 e 1 H 11 (q) 0 0 v 2. = H(q) e 2., with H(q) = 0 H 22 (q)........... 0 v L e L }{{} e(t) 0 0 H LL (q) Van den Hof et. al., Automatica, 2013 Yuan et. al., Automatica, 2011 Sanandaji et. al., ACC, 2011 Materassi & Salapaka, IEEE trans AC, 2012 6

Our approach v 1 e 1 H 11 (q) H 12 (q) H 1L (q) v 2. = H(q) e 2., with H(q) = H 21 (q) H 22 (q)............ v L e L }{{} e(t) H L1 (q) H LL (q) 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

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) w(t) = ( I H 1 (q) ) w(t)+h 1 (q)g(q) w(t)+h 1 (q)r(q) r(t)+e(t) }{{}}{{}}{{} Typical output filter New filter Typical input 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) w(t) = ( I H 1 (q) ) w(t)+h 1 (q)g(q) w(t)+h 1 (q)r(q) r(t)+e(t) }{{}}{{}}{{} Typical output filter New filter Typical input filter ŵ(t t 1) = { I H 1 (q) (I G(q)) } w(t) + H 1 (q)r(q)r(t) 10

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

One-to-one relation ŵ(t t 1, θ 1 ) = ŵ(t t 1, θ 2 ) Predictor equality θ 1 = θ 2 Parameter equality 12

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

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

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

Global network identifiability Proposition W w (θ 1 ) = W w (θ 2 ) W r (θ 1 ) = W r (θ 2 ) T (θ 1) = T (θ 2 ) T =(I G) 1 [ H R ] 13

Global network identifiability Proposition W w (θ 1 ) = W w (θ 2 ) W r (θ 1 ) = W r (θ 2 ) T (θ 1) = T (θ 2 ) T =(I G) 1 [ H R ] 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

Global network identifiability Diagonal H(q, θ) trivially satisfies the theorem. 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

Global network identifiability Diagonal H(q, θ) trivially satisfies the theorem. Interpretation: Every node has excitation that enters only at that node 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

What if we have a particular structure in G? Flexibility of G(q, θ) can be reduced based on knowledge of the network. 0 G 12 (θ) 0 Example: G(q, θ) = G 21 (θ) 0 0 G 31 (θ) G 32 (θ) 0 14

What if we have a particular structure in G? Flexibility of G(q, θ) can be reduced based on knowledge of the network. 0 G 12 (θ) 0 Example: G(q, θ) = G 21 (θ) 0 0 G 31 (θ) G 32 (θ) 0 Column 3 not parameterized, no need to excite node 3! 14

Conclusions New identifiability concept for dynamic networks. Identifiability split into two parts, dynamics and parameters Restrictions on G(q, θ), H(q, θ) and R(q, θ) can guarantee a one-to-one relation between M(θ) and ŵ(t t 1, θ). Ability to deal with correlated noise in the network. 15

Thank you for the attention!! Comments, questions and points of discussion are appreciated :) Harm Weerts, Arne Dankers, Paul Van den Hof