Dankers, AG Arne

Hof, PMJ Paul  van den

Weerts, HHM Harm

Weerts, H.H.M.

Dankers, A.G.

van den Hof, P.M.J.

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 Are model and predictor filters equivalent?Citation for published version (APA):Weerts, H. H. M., Dankers, A. G., & van den Hof, P. M. J. (2015). Are model and predictor filters equivalent?studying identifiability in a dynamic network identification setting. Poster session presented at 24th Workshop ofthe European Research Network on System Identification, ERNSI 2015, Varberg, Sweden.Document status and date:Published: 01/01/2015Document Version:Other versionPlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 17. Nov. 2023Identifiability in dynamic network identificationAre model and predictor filters equivalent?Studying identifiability in a dynamic network identification settingHarm H.M. Weerts (h.h.m.weertstue.nl), Arne G. Dankers, Paul M.J. Van den Hof/ Department of Electrical Engineering - Control SystemsMain questionIn open- and closed-loop identification there is a one-to-one relation-ship between (parameterized) predictor filters and model1, is that thecase for dynamic networks?Example: Can we distinguish the two networks below on the basisof their transfers r → w?r1w1w2w3r3Network 1r1w1w2w3r3Network 2A(q)B(q)A(q)B(q)A(q)Figure 1: OH NO!! Two models generate the same mappingfrom external variables r to internal variables w.Network, predictor, model structureInternal variables External variableswjG0jl R0jkwl rk... for all k ∈ Nrjfor all l ∈ Nj...vj Process NoiseNetworkUnknownexcitationKnownexcitationFigure 2: The network (top) is constructed by connecting all thebuilding blocks (bottom).All nodes wj are stacked into columnvector w. Then we haveNetwork: w = G0w + R0r + H0ePredictor: ŵ(t|t − 1, θ) = Ww(q, θ)w(t) + Wr(q, θ)r(t)Predictor filters: Ww = I − H−1(I − G), Wr = H−1RModel structure: M = {G(q, θ), H(q, θ), R(q, θ)}.Note that G and G0 have zeros on the diagonal.Approach of the problemPredictor filters have a one-to-one relation to the filterT (q, θ) =(I − G(q, θ))−1 [H(q, θ) R(q, θ)]which is under conditions has a one-to-one relation to the filters inM(q, θ).ŵ(θ1) = ŵ(θ2) Predictor equality➀ m Conditional: Data must be informative.Ww(θ1)Wr(θ1)==Ww(θ2)Wr(θ2) Predictor filter equality➁ m Unconditional 2T (θ1) = T (θ2) Mapping equality➂ m Conditional 2: see conditions below,M(θ1) = M(θ2) Model equality➃ m Conditional: classical identifiability.θ1 = θ2 Parameter equalityResultsDefinition: M is called a globally network identifiable model struc-ture when ➂ holds.Theorem: When G is as flexible as possible then ➂ holds when[H(q, θ) R(q, θ)]P(q) has a leading diagonal for some P(q) 2.example: Choose H(q, θ) diagonal. Plugging it into predictor filter Wwshows that M is globally network identifiable. Diagonal H correspondsto having uncorrelated noise on every node.ConclusionThe one-to-one relation between (parameterized) predictor filters andmodel is conditional in case of dynamic networks. Combinations of Hand R can lead to a network identifiable M, it is not necessary that His diagonal. When H is not diagonal networks where noises are corre-lated can be considered.References1L. Ljung. System Identification: Theory for the User. Prentice-Hall, 1999.2H.H.M. Weerts, A.G. Dankers, and P.M.J. Van den Hof. Identifiability in dynamic network identification.In 17th IFAC Symposium on System Identification, 2015.

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Are model and predictor filters equivalent?:studying identifiability in a dynamic network identification setting

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 Are model and predictor filters equivalent?Citation for published version (APA):Weerts, H. H. M., Dankers, A. G., & van den Hof, P. M. J. (2015). Are model and predictor filters equivalent?studying identifiability in a dynamic network identification setting. Poster session presented at 24th Workshop ofthe European Research Network on System Identification, ERNSI 2015, Varberg, Sweden.Document status and date:Published: 01/01/2015Document Version:Other versionPlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Identifiability in dynamic network identificationAre model and predictor filters equivalent?Studying identifiability in a dynamic network identification settingHarm H.M. Weerts (h.h.m.weertstue.nl), Arne G. Dankers, Paul M.J. Van den Hof/ Department of Electrical Engineering - Control SystemsMain questionIn open- and closed-loop identification there is a one-to-one relation-ship between (parameterized) predictor filters and model1, is that thecase for dynamic networks?Example: Can we distinguish the two networks below on the basisof their transfers r → w?r1w1w2w3r3Network 1r1w1w2w3r3Network 2A(q)B(q)A(q)B(q)A(q)Figure 1: OH NO!! Two models generate the same mappingfrom external variables r to internal variables w.Network, predictor, model structureInternal variables External variableswjG0jl R0jkwl rk... for all k ∈ Nrjfor all l ∈ Nj...vj Process NoiseNetworkUnknownexcitationKnownexcitationFigure 2: The network (top) is constructed by connecting all thebuilding blocks (bottom).All nodes wj are stacked into columnvector w. Then we haveNetwork: w = G0w + R0r + H0ePredictor: ŵ(t|t − 1, θ) = Ww(q, θ)w(t) + Wr(q, θ)r(t)Predictor filters: Ww = I − H−1(I − G), Wr = H−1RModel structure: M = {G(q, θ), H(q, θ), R(q, θ)}.Note that G and G0 have zeros on the diagonal.Approach of the problemPredictor filters have a one-to-one relation to the filterT (q, θ) =(I − G(q, θ))−1 [H(q, θ) R(q, θ)]which is under conditions has a one-to-one relation to the filters inM(q, θ).ŵ(θ1) = ŵ(θ2) Predictor equality➀ m Conditional: Data must be informative.Ww(θ1)Wr(θ1)==Ww(θ2)Wr(θ2) Predictor filter equality➁ m Unconditional 2T (θ1) = T (θ2) Mapping equality➂ m Conditional 2: see conditions below,M(θ1) = M(θ2) Model equality➃ m Conditional: classical identifiability.θ1 = θ2 Parameter equalityResultsDefinition: M is called a globally network identifiable model struc-ture when ➂ holds.Theorem: When G is as flexible as possible then ➂ holds when[H(q, θ) R(q, θ)]P(q) has a leading diagonal for some P(q) 2.example: Choose H(q, θ) diagonal. Plugging it into predictor filter Wwshows that M is globally network identifiable. Diagonal H correspondsto having uncorrelated noise on every node.ConclusionThe one-to-one relation between (parameterized) predictor filters andmodel is conditional in case of dynamic networks. Combinations of Hand R can lead to a network identifiable M, it is not necessary that His diagonal. When H is not diagonal networks where noises are corre-lated can be considered.References1L. Ljung. System Identification: Theory for the User. Prentice-Hall, 1999.2H.H.M. Weerts, A.G. Dankers, and P.M.J. Van den Hof. Identifiability in dynamic network identification.In 17th IFAC Symposium on System Identification, 2015.

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Are model and predictor filters equivalent?:studying identifiability in a dynamic network identification setting

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