System Identification in Dynamic Networks...Introduction –dynamic networks Decentralized process...
Transcript of System Identification in Dynamic Networks...Introduction –dynamic networks Decentralized process...
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Data-driven modeling
in linear dynamic networks
Paul M.J. Van den Hof
18th IFAC Symposium on System Identification,
10 July 2018, Stockholm, Sweden www.sysdynet.eu
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Introduction – dynamic networks
Decentralized process control
2
Smart power grid
www.envidia.com
Autonomous driving Metabolic network
Hillen (2012)
Pierre et al. (2012)
Hydrocarbon reservoirs
Mansoori (2014)
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Introduction
Overall trend:
(a) The dynamic systems to be handled become
(large-scale) interconnected systems of systems
(c) The related monitoring, control and optimization problems
become distributed, multi-agent type
(d) Data is “everywhere”, big data era
(e) Modelling problems will need to consider this
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(b) With hybrid dynamics (continuous / switching)
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Introduction
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Distributed / multi-agent control:
With both physical and communication links between
systems and controllers
How to address data-driven modelling problems in such a setting?
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G +
v
u y
Introduction
+ G +
v
C
- u yr
The classical (multivariable) identification problems[1]
:
open loop closed loop
We have to move from a fixed and known configuration
to deal with and exploit structure in the problem.
[1] Ljung (1999), Söderström and Stoica (1989), Pintelon and Schoukens (2012)
Identify a plant model on the basis of measured signals u, y (and possibly r),
focusing on continuous LTI dynamics.
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• Introduction and motivation
• How to model a dynamic network?
• Single module identification – known topology
• Network identifiability
• Extensions
• Discussion
Contents 8
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Dynamic networks for data-driven modeling
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Dynamic networks
State space representations
Module representation
11
(Goncalves, Warnick, Sandberg, Yeung, Yuan, Scherpen,…)
(Van den Hof, Dankers, Gevers, Bazanella,…)
• E.M.M. Kivits and P.M.J. Van den Hof, SYSID2018, WeA02.1.
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ri external excitation
vi process noise
wi node signal
Dynamic network setup 14
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ri external excitation
vi process noise
wi node signal
15Dynamic network setup
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ri external excitation
vi process noise
wi node signal
16Dynamic network setup
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ri external excitation
vi process noise
wi node signal
17Dynamic network setup
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Assumptions:
• Total of L nodes
• Network is well-posed and stable
• Modules may be unstable
• Disturbances are stationary stochastic and can
be correlated
(I ¡ G0)¡ 1
• P.M.J. Van den Hof, A.G. Dankers, P.S.H. Heuberger and X. Bombois. Automatica, 2013.
18Dynamic network setup
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(I ¡ G0)¡ 1
Many new identification questions
can be formulated:
19Dynamic network setup
• Identification of a local module (known toplogy)
• Identification of the full network
• Topology estimation
• Sensor and excitation selection
• Fault detection
• Experiment design
• ..........
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(I ¡ G0)¡ 1
Early literature
• Topology detection: Materassi, Innocenti, Salapaka, Yuan, Stan,
Warnick, Goncalves, Sanandaji, Vincent, Wakin, Chiuso, Pillonetto
exploring Granger causality, Bayesian networks, Wiener filters
• Subspace algorithms for spatially distributed systems with
identical modules (Fraanje, Verhaegen, Werner), or
non-identical ones (Torres, van Wingerden, Verhaegen, Sarwar, Salapaka, Haber)
Here: focus on prediction error methods and concepts for identification
in generally structured (linear) dynamic networks, including non-measured
disturbances.
20Dynamic network setup
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Single module identification - known topology
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Single module identification
w1 w2
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G12 G23 G34 G45
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For a network with known topology:
• Identify on the basis of measured signals
• Which signals to measure?
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• Identify the full MIMO system:
Options for identifying a module:
from measured and .
Global approach with “standard” tools
(I ¡ G0)¡ 1
w1 w2
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w8
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G12 G23 G34 G45
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• Identify a local (set of) module(s)
from a (sub)set of measured and
Local approach with “new” tools and structural conditions
23Single module identification
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• Identifiying is part of a 4-input, 1-output problem
24Single module identification
w1 w2
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w8
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• Identifiying is part of a 4-input, 1-output problem
25Single module identification
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
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0 0
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00
0 0
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00
0
0 0
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Identification methods
• 2-stage/projection/IV (indirect) method
Consistency; no need for noise models; no ML
Excitation through external excitation signals only
4-input 1-output problem
to be addressed by a
closed-loop identification method
P.M.J. Van den Hof, A.G. Dankers, P.S.H. Heuberger and X. Bombois. Automatica, 49, 2994-3006, 2013.
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
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0 0
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0 0
00
0 0
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• Direct PE method
ML properties
Disturbances uncorrelated over channels
Excitation through all signals
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4 input nodes to be measured:
Can we do with less?w1 w2
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w8
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• An immersed network is constructed by removing node signals, but leaving the
remaining node signals invariant
• Modules and disturbance signals are adapted
• Abstraction through variable elimination (Kron reduction[2] in network theory).
Network immersion [1]
27Single module identification
[2] F. Dörfler and F. Bullo, IEEE Trans. Circuits and Systems I (2013)
[1] A. Dankers. PhD Thesis, 2014.
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Immersion
w1 w2
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G12 G23 G34 G45
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0 0
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0 0
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0 0
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0 0
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w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
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Immersion
Immersing
When does immersion leave
invariant?
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w1 w2
w6
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G36
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v8
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r4 r5
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0 0
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0 0
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0 0
0
~
~~
~
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When does immersion leave
invariant?
Proposition
Consider an immersed network where and are retained.
Then if
• A.G. Dankers, P.M.J. Van den Hof, A.G. Dankers, X. Bombois and P.S.C. Heuberger, IEEE Trans. Automatic Control, 61, 937-952, 2016.
a) Every path other than the one through goes through
a node that is retained. (parallel paths)
w1 w2
w6 w7
w8
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G61 G37G26 G27
G12 G23 G34 G45
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v1 v2v3 v4 v5
v8
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b) Every path goes through a node that is retained.
(loops around the output)
29Immersion
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parallel paths, and loops around the output
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
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0 0
30Single module identification
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w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
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0 0
parallel paths, and loops around the output
31Single module identification
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parallel paths, and loops around the output
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
32Single module identification
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parallel paths, and loops around the output
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
33Single module identification
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Choose as an additional input (to be retained)
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
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0 0
0
0 0
00
0 0
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0 0
34Single module identification
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parallel paths, and loops around the output
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
35Single module identification
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parallel paths, and loops around the output
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
36Single module identification
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parallel paths, and loops around the output
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
37Single module identification
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Choose as an additional input, to be retained
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
38Single module identification
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w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
Conclusion:
With a 3-input, 1 output model we can
consistently identify
The immersion reasoning is sufficient but not necessary to arrive at a
consistent estimate, see e.g. Linder and Enqvist[1]
[1] J. Linder and M. Enqvist. Int. J. Control, 90(4), 729-745, 2017.
39Single module identification
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w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
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
Conclusion:
With a 3-input, 1 output model we can
consistently identify
with an indirect method.
[1] J. Pearl, Stat. Surveys, 3, 96-146, 2009; [2] A.G. Dankers et al., Proc. IFAC World Congress, 2017.
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Confounding variables
Back to the (classical) closed-loop problem:
40
Direct identification of can be consistent
provided that v1 and v2 are uncorrelated
w1 w2G G43
H2H1
G12 G G34
v2
r1
G
0
0
21
12
v1
r2
In case of correlation between v1 and v2:
joint prediction of and leads to ML results,
related to the classical joint-io method [3,4]
Joint estimation of and : Joint–direct method [1,2]
[1] P.M.J. Van den Hof, A.G. Dankers, H.H.M. Weerts, Proc. 56th IEEE CDC, 2017; [2] H.H.M. Weerts et al., Automatica, 2018, to appear.
[3] T.S. Ng, G.C. Goodwin, B.D.O. Anderson, Automatica, 1977; [4] B.D.O. Anderson and M. Gevers, Automatica 1982.
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Confounding variables
[1] A.G. Dankers, P.M.J. Van den Hof, D. Materassi and H.H.M. Weerts, Proc. IFAC World Congress, 2017.
• (not measured) now acts as a disturbance
• For minimum variance: MISO direct method
loses consistency if there are confounding variables
w1 w2
w7
G21
G27
G12 G23
v7
v1 v2
r1
0
0
0
00
0
0
w6 G26
v6
0
w3 G23
v3
0
• This requires:
uncorrelated with
and no path from to an input
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w1 w2
w7
G21
G27
G12 G23
v7
v1 v2
r1
0
0
0
00
0
0
w6 G26
v6
0
w3 G23
v3
0
Confounding variables
Solutions while restricting to MISO models:
[1] A.G. Dankers, P.M.J. Van den Hof, D. Materassi and H.H.M. Weerts, Proc. IFAC World Congress, 2017.
(a) Including the node as additional
input, or
(a) Block the paths from to inputs/
outputs by measured nodes, to be
used as additional inputs.
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Solutions:
[1] A.G. Dankers, P.M.J. Van den Hof, D. Materassi and H.H.M. Weerts, Proc. IFAC World Congress, 2017.
(b) Block the paths from to
input by measuring node
to be used as additional
inputs.w1 w2
w7
G21
G27
G12 G23G34
v7
v1 v2
r1
0
0 0
0 0
00
0
0
w6 G26
v6
0
w3 G23
v3
0
w4
v4
r4
42Confounding variables
Relation with d-separation in graphs
(Materassi & Salapaka)
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Summary single module identification
• Single module identification in a network with known topology
• Methods for consistent and minimum variance module estimation
• Excitation through excitation signals only or through all external signals
• For direct method / ML results: treatment of confounding variables /
correlated disturbances
• A priori known modules can be accounted for
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Network Identifiability
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Network identifiability
Question: Can the dynamics/topology of a network be uniquely determined
from measured signals wi ,ri ?
Required: Can different dynamic networks be distinguished from each other
from measured signals wi ,ri ?
w1 w2
w6 w7
w8
G21 w3G32 w4G43 w5G54
G61 G37G26 G27
G12 G23 G34 G45
G18 G84
G76
v6 v7
v1 v2v3 v4 v5
v8
r1
r4 r5
r8
0 0
0
0 0
00
0 0
0
00
0
0 0
blue = unknown
red = known
Starting assumption: all signals wi ,ri that are present are measured
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Network identifiability
Network:
The network is defined by:
a network model is denoted by:
rank p
and a network model set by:
dim(r) = K
represents prior knowledge on the network models:
• topology
• disturbance correlation
• known modules
• the signals used for identification
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Network identifiability
How to define identifiability?
Clasically:
In the network situation:
• Property of a model set
• Unique mapping between parameters and models
• Property of a model set
• Unique mapping between models and identified objects
Weerts et al, SYSID2015; Weerts, Van den Hof and Dankers, Automatica, 2018;
Denote:
Objects that are uniquely identified from data :
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Network identifiability
Weerts et al, SYSID2015; Weerts, Van den Hof and Dankers, Automatica, 2018;
Denote the parametrized objects:
Definition
A network model set is network identifiable at
if for all models :
is network identifiable if this holds for all models
Objects identified from data:
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Network identifiability
is network identifiable at if there exists a
nonsingular and rational such that
Proposition (full excitation case) (based on Goncalves & Warnick, 2008)
has a leading diagonal matrix,
that is full rank for all .
Let
Goncalves and Warnick, IEEE-TAC, 2008; Weerts et al, SYSID2015;
Gevers and Bazanella, 2017; Weerts, Van den Hof and Dankers, Automatica, 2018;
Note: can be fully unknown (no prior knowledge on topology)
# independent external signals # nodes
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Example 5-node network 52
There are noise-free nodes, and and are expected to be correlated
fully parametrized
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Example 5-node network
There are noise-free nodes, and and are expected to be correlated
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Example 5-node network
There are noise-free nodes, and and are expected to be correlated
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Example 5-node network
There are noise-free nodes, and and are expected to be correlated
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Example 5-node network
There are noise-free nodes, and and are expected to be correlated
We can not arrive at a diagonal structure in
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Network identifiability
Observations:
b) The condition is typically fulfilled if each node is excited by
an independent external signal (either an or a )
a) A simple test can be performed to check the condition
c) The result is rather conservative:
1. Restricted to having full row rank
2. No account of prior knowledge in
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• For each row i the transfer matrix has full row rank, with
:
Network identifiability
Theorem 3 – identifiability for general model sets
If:
H.H.M. Weerts, P.M.J. Van den Hof and A.G. Dankers, Automatica, Vol. 89, pp. 247-258, 2018.
Then is network identifiable at if and only if
• Each row of has at most K+p parametrized entries
a) Each unknown entry in covers the set of all proper rational
transfer functions
b) All unknown entries in are parametrized independently
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Example 5-node network
Number of parametrized entries in each row < K+p = 5
If we restrict the structure of :
First condition:
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Example 5-node network
Row 1: Full row rank of transfer:
Rank condition:
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Example 5-node network
Verifying the rank condition for :
w5 w1G15 w4G21 w3G34
G12 G23
G53
v1 v2 v3r4
w2
r5
Evaluate the rank of the transfer matrix
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Vertex-disjoint paths
Theorem (Van der Woude, 1991; Hendrickx et al. 2017; Weerts et al., 2018)
The generic rank of a transfer function matrix between
inputs r and nodes w
is equal to the maximum number of vertex-disjoint paths between
the sets of inputs and outputs.
62
A (path-based) check on the topology of the network can decide
whether the conditions for identifiabiltiy are satisfied generically.
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Vertex-disjoint paths
r1
r2
r3
w1
w2
w3
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Vertex-disjoint paths
r1
r2
r3
w1
w2
w3
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Vertex-disjoint paths
r1
r2
r3
w1
w2
w3
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Vertex-disjoint paths
r1
r2
r3
w1
w2
w3
Generic rank = 3
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Example 5-node network
Verifying the rank condition for :
Evaluate the rank of the transfer matrix from to
w5 w1G15 w4G21 w3G34
G12 G23
G53
v1 v2 v3r4
w2
r5
2 vertex-disjoint paths full row rank 2
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Example 5-node network
Verifying the rank condition for :
Evaluate the rank of the transfer matrix from to
w5 w1G15 w4G21 w3G34
G12 G23
G53
v1 v2 v3r4
w2
r5
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# unknown modules # external signals uncorrelated with
For each row
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Summary identifiability
Identifiability of network model sets is determined by
• Presence and location of external signals, and
• Correlation of disturbances
• Prior knowledge on modules
• Fully applicable to the situation (i.e. reduced-rank noise)
So far:
• All node signals assumed to be measured
• Identifiability of the full network model – conditions per row/output node
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Extensions:
2. When topology is known and every node is excited,
which nodes need to be measured for unique identification
of a particular module on the basis of ?
(Hendrickx, Gevers & Bazanella, CDC 2017, ArXiv 2018)
3. When all nodes are measured,
what are conditions for identifiability of a particular module
in a network model set?(Weerts, Van den Hof, Dankers, CDC 2018 submitted)
71Summary identifiability
1. Relation back to state space structures,(Hayden, Yuan, Goncalves, TAC 2017)
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Extensions
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Reduced rank noise
Question:
How to identify (parts of) a dynamic network,
when the process noise is of reduced rank (p < L)?
73
Typically considered in dynamic factor models[1]
[1] M. Deistler, W. Scherrer and B.D.O. Anderson, 2015
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Weighted LS criterion:
Properties:
• Consistent estimate under regularity conditions,
• But for minimum variance an optimal has to be chosen
Typical choice, leading to minimum variance estimator:
but in our situation is singular
Typical: multi-output situation
74Reduced rank noise
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Solution: Parametrize dependencies in innovation process, and
include them as constraints:
Identification criterion becomes a
constrained quadratic problem
with ML properties for Gaussian noise.
H.H.M. Weerts, P.M.J. Van den Hof and A.G. Dankers, IFAC World Congress, 2017; Automatica, 2018 to appear.
Some parameters can be estimated variance-free.
Reformulation of Cramer-Rao bound
75Reduced rank noise
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76Sensor noise
Identification of a single module under the influence of sensor noise:
• Typical tough problem in open-loop identification (errors-in-variables)
• In dynamic networks this may become more simple due to
the presence of multiple (correlated) node signals
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76Sensor noise
Solution strategies:
1. Apply a correlation (IV) approach to mitigate the effect of sensor noise
choosing instrument signals (based on topology)
2. Combine:
1. a correlation (IV) technique to mitigate sensor noise, and
2. BJ noise models for addressing process noise.
A.G. Dankers, P.M.J. Van den Hof et al.; Automatica, 62 (2015), 39-50.
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76Discussion
• Dynamic network identification:
intriguing research topic with many open questions
• The (centralized) LTI framework is only just the beginning
• Further move towards data-aspects related to distributed control
• Move towards including nonlinear dynamics (contribution by M. Schoukens -TuB01.1)
• Session on dynamic network identification - WeA02
• From classical PE methods to regularized (kernel-based) approaches(Everitt, Bottegal, Hjalmarsson, 2018; Ramaswamy et al, 2018)
• Algorithmic aspects for large-scale data handling
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Acknowledgements
Co-authors, contributorsand discussion partners:
Arne DankersHarm Weerts
Xavier BomboisPeter HeubergerDonatelllo MaterassiManfred DeistlerMichel GeversJonas LinderSean WarnickAlessandro ChiusoHakan HjalmarssonMiguel Galrinho
Lizan Kivits, Shengling Shi, Khartik Ramaswamy,
Tom Steentjes, Mircea Lazar, Jobert Ludlage,
Giulio Bottegal, Maarten Schoukens
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The End
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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 (2014). System identification in dynamic networks. Dr. Dissertation, TU Delft.
• A. Dankers, P.M.J. Van den Hof, X. Bombois and P.S.C. Heuberger (2015). Errors-in-variables identification in
dynamic networks - consistency results for an instrumental variable approach. Automatica, Vol. 62, pp. 39-50,
December 2015.
• 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. Automatic Control, 61 (4),
pp. 937-952, April 2016.
• P.M.J. Van den Hof, A.G. Dankers and H.H.M. Weerts (2018). System identification in dynamic networks.
Computers & Chemical Engineering, 109, pp. 23-29, 2018. ArXiv: 1710.08865.
• 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, 2018. ArXiv: 1711.06369, 2017.
• 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. To appear in Automatica. ArXiv: 1711.06369.
• H.H.M. Weerts, P.M.J. Van den Hof and A.G. Dankers (2018). Single module identifiability in linear dynamic
networks. Submitted to 57th IEEE CDC 2018, ArXiv 1803.02586.
Papers available at www.pvandenhof.nl
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