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

Identifiability in dynamic network identification

Harm Weerts1 Arne Dankers2 Paul Van den Hof1

1Control Systems, Department of Electrical Engineering,

Eindhoven University of Technology, The Netherlands.

2Department 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

w1(t) w2(t) w3(t) w4(t)

G(q) e(t) C(q) y(t) u(t) r(t)

1

Illustration of the problem

w2 w3 w1

◮ We have some measured

node signals

◮ We want to identify the

dynamics between the node signals

2

Illustration of the problem

w2 w3 w1

◮ We have some measured

node signals

◮ We want to identify the

dynamics between the node signals

2

The identifiability question

w2 w3 w1 A B w2 w3 w1 A AB

Can we distinguish between the networks?? Identifiability??

3

Outline

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

4

Network definition Internal variables External variables

wj(t)

Gjl(q) Rjk(q)

wl(t) rk(t)

. . . for all k ∈ N r j for all l ∈ Nj . . .

vj(t) Process Noise

Nodes /

      w1 w2 . . . wL      

w(t)

=       G12 · · · G1L G21 ... . . . . . . ... ... GL−1 L GL1 · · · GL L−1      

• G(q)

      w1 w2 . . . wL      

w(t)

+R       r1 r2 . . . rK      

r(t)

+       v1 v2 . . . vL      

v(t)

5

Typical assumptions in network identification literature

      v1 v2 . . . vL       = H(q)       e1 e2 . . . eL      

e(t)

, with H(q) =       H11(q) · · · H22(q) ... . . . . . . ... ... · · · HLL(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

Typical assumptions in network identification literature

      v1 v2 . . . vL       = H(q)       e1 e2 . . . eL      

e(t)

, with H(q) =       H11(q) · · · H22(q) ... . . . . . . ... ... · · · HLL(q)      

◮ 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

      v1 v2 . . . vL       = H(q)       e1 e2 . . . eL      

e(t)

, with H(q) =       H11(q) H12(q) · · · H1L(q) H21(q) H22(q) ... . . . . . . ... ... . . . HL1(q) · · · · · · HLL(q)       Noise contribution on all nodes can be correlated with each other

7

Our approach

      v1 v2 . . . vL       = H(q)       e1 e2 . . . eL      

e(t)

, with H(q) =       H11(q) H12(q) · · · H1L(q) H21(q) H22(q) ... . . . . . . ... ... . . . HL1(q) · · · · · · HLL(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 wj.

◮ 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

• f 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 wj.

◮ 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

• f 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 wj.

◮ 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

• f 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))−1w(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))−1w(t) − R(q)r(t)
• Typical structure results

from the network structure! w(t) = G(q)w(t) + R(q)r(t) + (H(q) − I) e(t) + e(t) w(t) =

• I − H−1(q)
• Typical output filter

w(t)+H−1(q)G(q)

• New filter

w(t)+H−1(q)R(q)

• Typical input filter

r(t)+e(t)

10

Network predictor

e(t) = H−1(q)

• (I − G(q))−1w(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)
• Typical output filter

w(t)+H−1(q)G(q)

• New filter

w(t)+H−1(q)R(q)

• Typical input filter

r(t)+e(t) ˆ w(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)} ˆ w(t|t−1, θ) =

• I − H−1(q, θ) (I − G(q, θ))
• Ww(q,θ)

w(t)+H−1(q, θ)R(q, θ)

• Wr(q,θ)

r(t)

11

One-to-one relation

ˆ w(t|t − 1, θ1) = ˆ w(t|t − 1, θ2)

Predictor equality

θ1 = θ2

Parameter equality

12

One-to-one relation

ˆ w(t|t − 1, θ1) = ˆ w(t|t − 1, θ2)

Predictor equality

M(θ1) = M(θ2)

Model equality

c

Classical identifiability. [Ljung, 1999]

θ1 = θ2

Parameter equality

12

One-to-one relation

ˆ w(t|t − 1, θ1) = ˆ w(t|t − 1, θ2)

Predictor equality

c

Informative data [Ljung, 1999] Φw,r(ω) > 0 .

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2)

Predictor filter equality

M(θ1) = M(θ2)

Model equality

12

One-to-one relation

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2)

Predictor filter equality

M(θ1) = M(θ2)

Model equality

12

One-to-one relation

Ww = I − H−1(I − G) Wr = H−1R

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2)

Predictor filter equality

M(θ1) = M(θ2)

Model equality

12

One-to-one relation

Ww = I − H−1(I − G) Wr = H−1R

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2)

Predictor filter equality

??? M(θ1) = M(θ2)

Model equality

12

One-to-one relation

Ww = I − H−1(I − G) Wr = H−1R

Definition: Global network identifiability

M(θ) is globally network identifiable when the implication holds in both directions.

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2)

Predictor filter equality

??? M(θ1) = M(θ2)

Model equality

12

Global network identifiability

Proposition

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2) ⇔ T(θ1) = T(θ2) T =(I −G)−1 H R

• 13

Global network identifiability

Proposition

Ww(θ1) Wr(θ1) = = Ww(θ2) Wr(θ2) ⇔ T(θ1) = T(θ2) 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

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. Diagonal H(q, θ) triv- ially satisfies the theo- rem.

13

Global network identifiability

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. Diagonal H(q, θ) triv- ially satisfies the theo- rem. Interpretation: Every node has excitation that enters only at that node

13

What if we have a particular structure in G?

Flexibility of G(q, θ) can be reduced based on knowledge of the network. Example: G(q, θ) =   G12(θ) G21(θ) G31(θ) G32(θ)  

14

What if we have a particular structure in G?

Flexibility of G(q, θ) can be reduced based on knowledge of the network. Example: G(q, θ) =   G12(θ) G21(θ) G31(θ) G32(θ)   Column 3 not parame- terized, 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

• ne-to-one relation between M(θ) and ˆ

w(t|t − 1, θ).

◮ Ability to deal with correlated noise in the network.

15

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

• ne-to-one relation between M(θ) and ˆ

w(t|t − 1, θ).

◮ Ability to deal with correlated noise in the network.

15

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

• ne-to-one relation between M(θ) and ˆ

w(t|t − 1, θ).

◮ Ability to deal with correlated noise in the network.

15

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

• ne-to-one relation between M(θ) and ˆ

w(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