Local module identification in dynamic networks with ... · Single module identification 6 w 1 w 2 w 6 w 7 w 8 G 21 G 32 w 3 G 43 w 4 G 54 w 5 G 61 G 26 G 27 G 37 G 12 G 23 G 34 G

Paul M.J. Van den Hof, Karthik Ramaswamy, Arne Dankers and Giulio Bottegal

www.sysdynet.eu58th IEEE Conf. Decision and Control (CDC 2019), Nice France www.pvandenhof.nl

[email protected]

Local module identification in dynamic networks with correlated noise –the full input case

Introduction – dynamic networks

Decentralized process control

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Autonomous driving

www.envidia.com

Smart power grid

Metabolic network Hydrocarbon reservoirs

Pierre et al. (2012)

Hillen (2012) Mansoori (2014)

Dynamic network setup

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ri external excitationvi process noisewi node signal

P.M.J. Van den Hof, A.G. Dankers, P.S.C. Heuberger and X. Bombois. Automatica, 2013.

Dynamic network setup

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Assumptions:• Total of L nodes• Network is well-posed and stable• Modules are dynamic, may be unstable• Disturbances are stationary stochastic and

can be correlated

Single module identification

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For a network with known topology:• Identify on the basis of measured signals• Which signals to measure? Preference for local measurements


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Identifiying is part of a multi-input, single-output problem


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Multi-input single-output identification problemto be addressed by a closed-loop identification method

Options:

1. Indirect identification2. Direct identificationwk

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[1] VdHof et al., Automatica 2013[2] Gevers et al., SYSID 2018; Bazanella et al., CDC 2019

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1. Indirect identification[1][2]

- sufficient number of external excitations r- estimate and consistently, and

determine

- consistent estimate, also if correlated- noise signals not used for estimation

(no minimum variance) - freedom in location of r-signals

(e.g. directly on )- we do not necessarily need all inputs to

to be included in [3]

[3] Dankers et al., IEEE-TAC, 2016

- Estimate transfer and modelthe disturbance process on the output.

- consistent estimate and ML properties- provided there is enough excitation,- and uncorrelated with other signals- input signal set can be further reduced[2]


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2. Direct identification[1]

[1] VdHof et al., Automatica 2013[2] Dankers et al., IEEE-TAC, 2016; Dankers et al., IFAC 2017

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- Estimate transfer and modelthe disturbance process on the output.

- consistent estimate and ML properties- provided there is enough excitation,- and uncorrelated with other signals- input signal set can be further reduced[2]


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2. Direct identification[1]

[1] VdHof et al., Automatica 2013[2] Dankers et al., IEEE-TAC, 2016; Dankers et al., IFAC 2017

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How to deal with correlations between signals in the direct method?

Direct identification

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Currently available results

For a consistent and minimum variance estimate(direct method) there is one additional condition: • absence of confounding variables, [1][2] i.e.

correlated disturbances on inputs and outputs

[1] J. Pearl, Stat. Surveys, 3, 96-146, 2009[2] A.G. Dankers et al., Proc. IFAC World Congress, 2017.

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Two different types of confounding variables:

• Direct-type: is correlated to any term in

• Indirect-type: is correlated to any otherthat has a path to

Direct identification of can be consistent provided that v1 and v2 are uncorrelated

Direct confounding variables

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In case of correlation between v1 and v2 (direct confounding variable): MIMO approachjoint prediction of and leads to ML results,

model and both as input and output,and model the joint disturbance process

Joint estimation of and : Joint–direct method[1,2,3,4].

[1] P.M.J. Van den Hof et al. Proc. 56th IEEE CDC, 2017 [2] H.H.M. Weerts et al., Automatica, Dec. 2018.[3] T.S. Ng, G.C. Goodwin, B.D.O. Anderson, Automatica, 1977 [4] B.D.O. Anderson and M. Gevers, Automatica 1982.

Direct confounding variables: add a predicted output and model the correlated disturbances

Indirect confounding variables

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If and are correlated, then the effect of the confounding variable can be “blocked” by

• measuring a node on each path fromto , and

• including the “blocking nodes” as predictor inputs in the model

A.G. Dankers et al., Proc. IFAC World Congress, 2017.

General algorithm philosophy1) Start with output of target module and its predictor inputs2) Handle direct confounding variables

• Add inputs to predicted outputs • Add predicted inputs for the modified outputs• Repeat step 2

3) Handle indirect confounding variables • Add predictor inputs

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Sets of signals:• Only predictor inputs • Only predicted output • Both predictor inputs and predicted outputs

Direct identification : Identification of can be consistent with ML properties

MIMO identification setup

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Target module

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Full input caseWe include all in-neighbors of the predicted outputs as predictor inputs

Maximum use of information in the signals

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Network with 𝑣𝑣1 correlated with 𝑣𝑣3 and 𝑣𝑣6.

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Handling direct confounding variable

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Network with 𝑣𝑣1 correlated with 𝑣𝑣3 and 𝑣𝑣6.

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Handling indirect confounding variable

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Network with 𝑣𝑣1 correlated with 𝑣𝑣3 and 𝑣𝑣6.Direct identification:

Generalization of result

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Conditions for consistent (and ML) estimation of :

• System in the model set• Any indirect confounding variable for

is blocked by a node in • There are no confounding variables for• There are no direct or unmeasured paths from to• There is persistence of excitation, i.e. at a sufficient number of

frequencies, withand the innovation process of

• All modules in are strictly proper or satisfy some technical delay conditions

𝐺𝐺𝑗𝑗𝑗𝑗�̅�𝐺

Summary

• Methods for consistent and minimum variance estimation of a single module

• For direct method: treatment of confounding variables / correlated disturbances

• Particular situation: full-input case. Can be generalized to other setups to create more flexibility in choice of sensors[1].

• A priori known modules can be accounted for

• Generalizing towards combining direct and indirect approach: Ramaswamy et al. (later in this session)

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[1] K..R. Ramaswamy et al., ArXiv, 2018.

Further reading • P.M.J. Van den Hof, A. Dankers, P. Heuberger and X. Bombois (2013). Identification of dynamic models in complex networks

with prediction error methods - basic methods for consistent module estimates. Automatica, Vol. 49, no. 10, pp. 2994-3006.

• A. Dankers, P.M.J. Van den Hof, P.S.C. Heuberger and X. Bombois (2016). Identification of dynamic models in complex networks with predictior error methods - predictor input selection. IEEE Trans. Autom. Contr., 61 (4), pp. 937-952, 2016.

• H.H.M. Weerts, P.M.J. Van den Hof and A.G. Dankers (2018). Identifiability of linear dynamic networks. Automatica, 89, pp. 247-258, March 2018.

• H.H.M. Weerts, P.M.J. Van den Hof and A.G. Dankers (2018). Prediction error identification of linear dynamic networks with rank-reduced noise. Automatica, 98, pp. 256-268, December 2018.

• H.H.M. Weerts, P.M.J. Van den Hof and A.G. Dankers (2018). Single module identifiability in linear dynamic networks. Proc. 57th IEEE CDC 2018, ArXiv 1803.02586.

• K.R. Ramaswamy, G. Bottegal and P.M.J. Van den Hof (2018). Local module identification in dynamic networks using regularized kernel-based methods. Proc. 57th IEEE CDC 2018.

• K.R. Ramaswamy, P.M.J. Van den Hof and A.G. Dankers(2019). Generalized sensing and actuation schemes for local module identification in dynamic networks. Proc. 58th IEEE 2019 CDC.

• K.R. Ramaswamy and P.M.J. Van den Hof (2019). A local direct method for module identification in dynamic networks with correlated noise. Submitted for publication. ArXiv:1908.00976.

21Papers available at www.pvandenhof.nl

Data-driven modeling in linear dynamic networks22

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Local module identification in dynamic networks with ... · Single module identification 6 w 1 w 2 w 6 w 7 w 8 G 21 G 32 w 3 G 43 w 4 G 54 w 5 G 61 G 26 G 27 G 37 G 12 G 23 G 34 G

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