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Verified Parameter Identification of Quasi-Linear Cooperative System Models: A Combination of Branch-and-Bound as well as Contractor Techniques

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Andreas Rauh at Carl von Ossietzky University of Oldenburg
Andreas Rauh
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Abstract

Many dynamic system models in (control) engineering, especially in the frame of thermo-fluidic applications, are described after a first-principle modeling by sets of either discrete-time difference equations or by sets of ordinary differential equations that have certain monotonicity properties. The most important property that allows for a simplification of parameter identification tasks is the cooperativity of the state equations that often results from properties such as conservation of mass and energy. In this case, (quasi-)linear system models are characterized by dynamics matrices that can be bounded by interval-valued Metzler matrices. Those matrices are characterized by non-negative off-diagonal elements. In the case that all system states are additionally ensured to be non-negative, the property of cooperativity leads to a decoupling of lower and upper bounding systems that enclose all possible state trajectories with certainty. This presentation gives an overview of branch-and-bound techniques as well as contractor-based approaches for the parameter identification of cooperative systems on the basis of uncertain measurements. An overview of current research towards the GPU-based identification of nonlinear models of the thermal behavior of high-temperature fuel cells concludes this talk.
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Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Parameter Identification of Quasi-Linear Cooperative
System Models: A Combination of Branch-and-Bound as well
as Contractor Techniques
Computational Mathematics Seminar
Jagiellonian University, Krak´ow, Poland (virtual)
April 15, 2021
Andreas Rauh
Lab-STICC (Robex)
ENSTA Bretagne, Brest, France
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 1/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Contents
Different approaches for verified parameter identification
Linear example: Finite volume model of a distributed heating system
IControl-oriented modeling
IParameter identification
IUse for interval observer design and cooperativity-preserving control
Extensions to nonlinear processes
Combination of branch&bound (branch&prune) techniques with contractor approaches
Modeling and identification for a high-temperature Solid Oxide Fuel Cell System (SOFC)
INon-stationary thermal behavior during heating and reaction phases
IPhysical constraints of nonlinear models for heat capacities and reaction enthalpies
IStability constraints
Massive parallelization and combination with identification of neural network models
Conclusions with outlook to non-cooperative systems in robotics
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 2/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Contents
Different approaches for verified parameter identification
Linear example: Finite volume model of a distributed heating system
IControl-oriented modeling
IParameter identification
IUse for interval observer design and cooperativity-preserving control
Extensions to nonlinear processes
Combination of branch&bound (branch&prune) techniques with contractor approaches
Modeling and identification for a high-temperature Solid Oxide Fuel Cell System (SOFC)
INon-stationary thermal behavior during heating and reaction phases
IPhysical constraints of nonlinear models for heat capacities and reaction enthalpies
IStability constraints
Massive parallelization and combination with identification of neural network models
Conclusions with outlook to non-cooperative systems in robotics
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 2/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Contents
Different approaches for verified parameter identification
Linear example: Finite volume model of a distributed heating system
IControl-oriented modeling
IParameter identification
IUse for interval observer design and cooperativity-preserving control
Extensions to nonlinear processes
Combination of branch&bound (branch&prune) techniques with contractor approaches
Modeling and identification for a high-temperature Solid Oxide Fuel Cell System (SOFC)
INon-stationary thermal behavior during heating and reaction phases
IPhysical constraints of nonlinear models for heat capacities and reaction enthalpies
IStability constraints
Massive parallelization and combination with identification of neural network models
Conclusions with outlook to non-cooperative systems in robotics
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 2/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (1)
Observer-based approach
ym,ny(tk)
Fusion of
measurement
information
Sensor 1
Sensor 2
Sensor ny
Nonlinear measurement
model with uncertainties
Estimate in the
correction step
ym,1(tk)
ym,2(tk)Intersection of
both verified
state enclosures
Nonlinear dynamical
with uncertainties
system model
unit delay
State and parameter estimate in the prediction step
ˆ
x(tk−1)
Improved estimate after transition from tk−1to tk
Verified integration of state equations between two subsequent measurement points =⇒
Structure close to Luenberger observer/ (Extended) Kalman Filter
Exclusion of inadmissible intervals
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 3/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (1)
Observer-based approach
ym,ny(tk)
Fusion of
measurement
information
Sensor 1
Sensor 2
Sensor ny
Nonlinear measurement
model with uncertainties
Estimate in the
correction step
ym,1(tk)
ym,2(tk)Intersection of
both verified
state enclosures
Nonlinear dynamical
with uncertainties
system model
unit delay
State and parameter estimate in the prediction step
ˆ
x(tk−1)
Improved estimate after transition from tk−1to tk
Requirement for problem-specific techniques for the reduction of overestimation
However: Dependency problem and wrapping effect when using standard simulation
techniques
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 3/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (2)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
y
m
(
t
k
)∈[
y
m
](
t
k
)
Measured data are available at discrete points of time
Worst-case bounds for measurement tolerances
Necessity for information about uncertain initial states and bounds on uncertain
parameters
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 4/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (3)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
simulated output enclosure
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
Prerequisite: Correctness of model structure
Initial state/ parameter intervals are subdivided for candidates, for which no decision
about admissibility can be made
Intersection of directly measured and simulated state intervals possible
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 5/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (4)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
Search for guaranteed admissible initial state/ parameter intervals
Subdivision until undecided region is sufficiently small
Needs to be fulfilled for each available sensor if dim(ym)>1
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 6/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (5)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
Exclusion of inadmissible intervals (for at least one of the sensors)
Prerequisite: Efficient computation of the interval enclosure of all reachable states for
given control inputs
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 7/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Methods for Parameter Identification (5)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
Exclusion of inadmissible intervals (for at least one of the sensors)
Prerequisite: Efficient computation of the interval enclosure of all reachable states for
given control inputs
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 7/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Linear Example: Spatially Distributed Heating System
Early lumping: Finite volume semi-discretization
Differential equation for the rod temperature
˙
ϑi(t) = 1
ci·mi
·˙
Qλ,i
i−1(t) + ˙
Qλ,i
i+1(t) + ˙
Qα,i
B(t) + ˙
Qα,i
˜
N+i(t) + ˜ui(t)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 8/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Linear Example: Spatially Distributed Heating System
Early lumping: Finite volume semi-discretization
Differential equation for the rod temperature
˙
ϑi(t) = 1
ci·mi
·˙
Qλ,i
i−1(t) + ˙
Qλ,i
i+1(t) + ˙
Qα,i
B(t) + ˙
Qα,i
˜
N+i(t) + ˜ui(t)
with ˜ui(t) = 1
2M+ 1uξ(t), ξ =i
2M+ 1, i ∈ {1,..., ˜
N}
Heat conduction between neighboring elements
˙
Qλ,i
i−1(t) = λR·Ac
ls
·(ϑi−1(t)−ϑi(t))
Heat convection between rod and air canal
˙
Qα,i
˜
N+i(t) = α·As·ϑ˜
N+i(t)−ϑi(t)=−˙
Qα, ˜
N+i
i(t)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 8/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Cooperative and Positive Dynamic Systems
Simplified computation of state enclosures for cooperative systems
Sufficient condition for cooperativity of the dynamic system
˙
x(t) = f(x(t)) ,x∈Rn
Ji,j ≥0, i, j ∈ {1, . . . , n}, i 6=jwith J=∂f(x)
∂x
For initial conditions in the positive orthant
Rn
+={x∈Rn|xi≥0∀i∈ {1, . . . , n}} ,
positivity of all state trajectories is ensured if
˙xi(t) = fi(x1, . . . , xi−1,0, xi+1,...xn)≥0
holds for all components i∈ {1, . . . , n}
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 9/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Definition of Point-Valued Bounding Systems
Decoupled bounding systems, A(p)∈Ap;A(p): uncertain Metzler matrix
Ap·v(t) + B·u(t) = ˙
v(t)≤˙
x(t)≤˙
w(t) = A(p)·w(t) + B·u(t)
with the parameter intervals α∈[α;α],αB∈[αB;αB],αT∈[αT;αT],∆α∈∆α; ∆α,
∆ma∈∆ma; ∆ma, and λR∈λR;λR
Block-wise definition of the system matrix
A(p) = Ah11i(p)Ah12i(p)
Ah21i(p)Ah22i(p)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 10/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Uncertain Cooperative System Model (1)
Block-wise definition of the system matrix
A(p) = Ah11i(p)Ah12i(p)
Ah21i(p)Ah22i(p)
Example for the sign pattern in the system matrix A(p)for M= 0
A(p) =
−+ 0 0 + 0 0 0
+−+ 0 0 + 0 0
0 + −+ 0 0 + 0
0 0 + −0 0 0 +
+ 0 0 0 −000
0 + 0 0 0 −0 0
0 0 + 0 0 0 −0
000+000−
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 11/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Uncertain Cooperative System Model (2)
Block-wise definition of the system matrix
A(p) = Ah11i(p)Ah12i(p)
Ah21i(p)Ah22i(p)
Example for the parameter-dependent matrix entries
ah11i
i,j (p) =
p1
ci·mi<0for i=j= 1 and i=j=˜
N
p2
ci·mi>0for i=j−1, j ∈ {2,..., ˜
N}
p2
ci·mi>0for i=j+ 1 , j ∈ {1,..., ˜
N−1}
p3
ci·mi<0for i=j , j ∈ {2,..., ˜
N}
0else
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 12/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Uncertain Cooperative System Model (3)
Block-wise definition of the system matrix
A(p) = Ah11i(p)Ah12i(p)
Ah21i(p)Ah22i(p)
Parameterization of the lower bounding system
p=
−λR·Ac
ls+ (αB+α)·As
λR·Ac
ls
−2λR·Ac
ls+ (αB+α)·As
α·As
−αT+α+ ∆α·δ˜
N+i, ˜
N+1·As
ma
˜
N·1 + δ˜
N+i,2˜
N·∆ma−1
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 13/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Uncertain Cooperative System Model (4)
Block-wise definition of the system matrix
A(p) = Ah11i(p)Ah12i(p)
Ah21i(p)Ah22i(p)
Parameterization of the upper bounding system
p=
−λR·Ac
ls+ (αB+α)·As
λR·Ac
ls
−2λR·Ac
ls+ (αB+α)·As
α·As
−αT+α+ ∆α·δ˜
N+i, ˜
N+1·As
ma
˜
N·1 + δ˜
N+i,2˜
N·∆ma−1
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 14/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Stability Analysis (1)
Application of the Gershgorin circle theorem for the rows i={1,..., ˜
N}of the system
matrix A(p)
<{λi} ≤ ah11i
i,i +
2˜
N
X
j=1,j6=i
|ai,j |
=
1
ci·mi
·(p1+p2+p4) = −αB·As
ci·mi
<0
f¨
ur i∈ {1,˜
N}
1
ci·mi
·(p3+ 2p2+p4) = −αB·As
ci·mi
<0
f¨
ur i∈ {2,..., ˜
N−1}
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 15/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Stability Analysis (2)
Application of the Gershgorin circle theorem for the rows ˜
N+iwith i={1,..., ˜
N}of
the system matrix A(p)
<{λ˜
N+i} ≤ ah22i
i,i +
2˜
N
X
j=1,j6=˜
N+ia˜
N+i,j =p4+p5,i
c˜
N+i·p6,i
=−αT+ ∆α·δ˜
N+i, ˜
N+1·As
c˜
N+i·p6,i
<0
Stability-based parameter exclusion
Convexity of the eigenvalue domains
Subdivide those interval parameters which influence the eigenvalue discs with
infima/suprema of opposite sign and may cause unstable (upper) bounding systems
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 16/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification (1)
Normalization of the parameter domain, ζ∈ {1, . . . , L}
hzhζii=h˜
N
ma
Ac
ls·αhζi,˜
N
ma
Ac
ls·hαhζi
Bi,˜
N
ma
Ac
ls·hαhζi
Ti,
˜
N
ma
Ac
ls·∆αhζi,1
As·Ac
ls·h∆mhζi
ai,˜
N
ma·As·hλhζi
RiiT
Selection of the list element to be subdivided
ζ∗= arg max
ζ=1,...,L
K
X
k=1
I
X
i=1
Yhζi
i,k with
Yhζi
i,k = diam nhyhζi
ii(tk)o−diam n[ym,i] (tk)∩hyhζi
ii(tk)o
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 17/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification (2)
Approach 1: Selection of the parameter with the maximum sensitivity on the variation
of all state variables
ξ∗= arg max
ξ=1,...,6
1T·
∂[f]
∂ηη∈[ηhζ∗i]
·µξ
with the interval evaluation [f] := A(p)·[x] + B·[u]of the state equations for all possible
states and inputs with the current parameter box
hηhζ∗ii=hαhζ∗ihαhζ∗i
Bi hαhζ∗i
Ti∆αhζ∗ih∆mhζ∗i
ai hλhζ∗i
RiiT,
the weighting
µ= diag
diam nhηhζ∗i
ξio2
diam nhηh0i
ξio
,and the initial domain hηh0ii
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 18/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification (3)
Approach 2: Selection of the parameter with the maximum output sensitivity
after setting [f]to
[f] := C·(A(p)·[x] + B·[u])
(alternative: use of sensitivity of computed output trajectories as a first-order variation of the
uncertain set of ODEs)
Approach 3: Splitting into the direction of the largest (weighted) interval diameter
ξ∗= arg max
ξ=1,...,6ndiam nhzhζi
ξioo
Possible performance enhancement by GPU implementation of simulation-based exclusion test
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 19/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification (3)
Approach 2: Selection of the parameter with the maximum output sensitivity
after setting [f]to
[f] := C·(A(p)·[x] + B·[u])
(alternative: use of sensitivity of computed output trajectories as a first-order variation of the
uncertain set of ODEs)
Approach 3: Splitting into the direction of the largest (weighted) interval diameter
ξ∗= arg max
ξ=1,...,6ndiam nhzhζi
ξioo
Enhance the likelihood for parameter exclusion by
(Linear) Transformation of parameter vector: p=Tq
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 19/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification (3)
Approach 2: Selection of the parameter with the maximum output sensitivity
after setting [f]to
[f] := C·(A(p)·[x] + B·[u])
(alternative: use of sensitivity of computed output trajectories as a first-order variation of the
uncertain set of ODEs)
Approach 3: Splitting into the direction of the largest (weighted) interval diameter
ξ∗= arg max
ξ=1,...,6ndiam nhzhζi
ξioo
Enhance the likelihood for parameter exclusion by
Input optimization (Pontryagin’s Maximum Principle for parameter distinguishabilty)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 19/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification (4)
A. Rauh, J. Kersten, Julia: From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers. Acta
Cybernetica. Vol. 24, pp. 509–537. (2020) DOI:10.14232/actacyb.24.3.2020.13.
A. Rauh, J. Kersten, H. Aschemann: Interval and Linear Matrix Inequality Techniques for Reliable Control of Linear Continuous-Time Cooperative Systems
with Applications to Heat Transfer. International Journal of Control. Vol. 93, pp. 1–31. (2019) DOI:10.1080/00207179.2019.1708966.
E. Auer, A. Rauh, J. Kersten: Experiments-based parameter identification on the GPU for cooperative systems. Journal of Computational and Applied
Mathematics. Vol. 371. Paper-Id: 112657. (2019) DOI:10.1016/j.cam.2019.112657.
A. Rauh, J. Kersten, H. Aschemann: Interval methods and contractor-based branch-and-bound procedures for verified parameter identification of quasi-linear
cooperative system models. Journal of Computational and Applied Mathematics. Vol. 367. Paper-Id: 112484. (2019) DOI:10.1016/j.cam.2019.112484.
N. Cont, W. Frenkel, J. Kersten, A. Rauh, H. Aschemann: Interval-Based Modeling of High-Temperature Fuel Cells for a Real-Time Control
Implementation Under State Constraints, IFAC World Congress, Berlin, Germany (virtual) (2020).
L. Senkel, A. Rauh, H. Aschemann: Optimal input design for online state and parameter estimation using interval sliding mode observers. Proceedings of
the IEEE Conference on Decision and Control, pp. 502-507, Firenze, Italy. (2013) DOI:10.1109/CDC.2013.6759931.
A. Rauh, J. Kersten, H. Aschemann. Linear Matrix Inequality Techniques for the Optimization of Interval Observers for Spatially Distributed Heating
Systems. Proceedings of the 23rd International Conference on Methods and Models in Automation and Robotics, pp. 138-143, Mi ,
edzyzdroje, Poland.
(2018) DOI:10.1109/MMAR.2018.8486120.
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 20/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Interval Observer Design: Virtual Sensor
Bounding systems: x∈[v;w]and ˆ
x∈[ˆ
v;ˆ
w] =⇒x∈[ˆ
v;ˆ
w]
AOˆ
v+Bu +Hym≤˙
ˆ
x≤AOˆ
w+Bu +Hym
with the observer system matrices
AO=A−HC and AO=A−HC
Uncertain measurements
[ym] = ym+ [−∆ym; ∆ym]with a linear output model y=Cx
Parameterize Hsuch that [AO]is Metzer and Hurwitz
Use of state estimates as further virtual measurements
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 21/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch-and-Bound Procedure: Verified Identification
Initial parameter domain
α∈[5 ; 100] W
m2·K,αB∈[15 ; 300] W
m2·K,αT∈[5 ; 100] W
m2·K,∆α∈[1 ; 200] W
m2·K,
∆ma∈[0.01 ; 0.2] und λR∈[10 ; 200] W
m·K
State ϑ1(t)(not measured)
tin s
[ϑ1](t) in K
0 1200400200 600 800 1000
20
16
12
0
8
4
approach 2
approach 1
observer
approach 3
State ϑ2(t)(measured)
tin s
[ϑ2](t) in K
0 1200400200 600 800 1000
20
16
12
0
8
4
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 22/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Algebraic Consistency Tests for Parameter Identification (1)
A-priori knowledge of stability
Enclosure of eigenvalues by means of the Gershgorin circle criterion: nuncertain discs
[D] := h[m],[r]iwith an interval-valued midpoint [m]and radius [r]
[Di,k] = *[aii ] ([ˆ
xk],[p]) ,
n
X
j=1,j6=i
|[aij ] ([ˆ
xk],[p])|+, i ∈ {1, . . . , n}
A parameter subinterval [p]hζi⊆[p]is guaranteed to be infeasibile if
< {[Di,k]}>0,i.e., inf < {[Di,k ]}>0
for at least one eigenvalue i∈ {1, . . . , n}and at least one time instant tk
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 23/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Algebraic Consistency Tests for Parameter Identification (2)
Constraints on the temporal variation rates of measured and non-measured state
variables
A parameter subinterval [p]hζi⊆[p]which violates at least one of the following
constraints is classified as infeasible:
supfi([xk],[p],[uk])≥infh˙
ˆxi,ki or inffi([xk],[p],[uk])≤suph˙
ˆxi,ki
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 24/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Algebraic Consistency Tests for Parameter Identification (3)
Contractors for enclosing admissible parameter domains
Contraction of the j-th component of the subbox [p]hζiby a symbolic rearrangement of
the expression
fi(xk,p,uk)−˙xi,k = 0
for the parameter pj,j∈ {1, . . . , q},p∈Rq
Result of rearrangement
pj=Ci,j,k (xk,p,uk,˙xi,k)
A contraction is possible according to
p∗
jhζi= [pj]hζi∩ Ci,j,k [xk],[p]hζi,[uk],h˙
ˆxi,ki ,[pj]hζi:= p∗
jhζi
if hp∗
jihζi6=∅holds; for hp∗
jihζi=∅, the parameter box ζis infeasible
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 25/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Algebraic Consistency Tests for Parameter Identification: Example
System model
Continuous-time state equations
˙
x=−p1+p2cos x2
2 p2
p2−p1+p2cos x2
2·x=: f1
f2
Initial parameter domain p1p2T∈[0 ; 5] [0 ; 5]T
Imperfect knowledge about the system’s initial state
ˆ
x(0) ∈2 1.5T+[−0.1 ; 0.1] [−0.1 ; 0.1]T
Measurements at equidistant time instants tk=k·h,h= 0.1
x2∈[ym] = ym+ [∆] ,[∆] = [−0.1 ; 0.1]
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 26/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
State-Independent Pre-Identification Stage (1)
Fundamental feasibility requirement
Verification of asymptotic stability according to the Gershgorin circle theorem for each point at
which measured data are available =⇒exclude all subintervals ζfor which
inf n[p2]hζi−[p1]hζi−[p2]hζicos [ym]2o>0
holds
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 27/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
State-Independent Pre-Identification Stage (2)
Characterization of a bounding system for the time derivative of the measured state
(output); infeasible parameters violate one of the inequalities
supX1·[p2]hζi−[p1]hζi+ cos [ym,k ]2·[p2]hζi·[ym,k ]≥inf[ ˙ym,k ]
infX1·[p2]hζi−[p1]hζi+ cos [ym,k ]2·[p2]hζi·[ym,k ]≤sup[ ˙ym,k ]
Note
Assumption of time-invariant bounds for all non-measured states that are valid for all
t∈[0 ; tf], here: tf= 2
x1(t)∈X1;X1
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 28/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
State-Independent Pre-Identification Stage (3)
Estimation of interval bounds for the time derivative of ym, sampling time h=tk+1 −tk
(h= 0.1in the example)
Construction of the conservative derivative interval with the convex hull operator F
[ ˙ym,k] = Gninf ([ym,k+1]) −inf ([ym,k ]) ,
sup ([ym,k+1]) −sup ([ym,k ]) o·h−1+κ·[−1 ; 1]
Pre-identification stage (algebraic)
Exclusion of all subintervals which violate the inequalities on the measured output derivatives
with certainty
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 29/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
State-Independent Pre-Identification Stage (3)
Estimation of interval bounds for the time derivative of ym, sampling time h=tk+1 −tk
(h= 0.1in the example)
Construction of the conservative derivative interval with the convex hull operator F
[ ˙ym,k] = Gninf ([ym,k+1]) −inf ([ym,k ]) ,
sup ([ym,k+1]) −sup ([ym,k ]) o·h−1+κ·[−1 ; 1]
Pre-identification stage (algebraic)
Exclusion of all subintervals which violate the inequalities on the measured output derivatives
with certainty
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 29/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Branch&Bound Identification Stage
Cooperativity-based simulation of lower and upper bounding systems
Investigate all parameter intervals ζwhich are yet undecided
Exclude all intervals which violate the output constraints (including measurement
tolerances)
Application of the parameter contractor
Difference to the naive identification: No large further amount of subdivisions required
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 30/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Contractors for Each Measurement Instant
C1,1,k : [p∗
1]hζi:= [p1]hζi∩
−h˙
ˆx1,ki−[p2]hζi·[ ˆx2,k ]+[p2]hζi·[ˆx1,k ]·cos [ˆx2,k]2
[ˆx1,k ]
C1,2,k : [p∗
2]hζi:= [p2]hζi∩h˙
ˆx1,ki+ [p1]hζi·[ ˆx1,k ]
[ˆx2,k ]−[ˆx1,k ]·cos [ˆx2,k]2
C2,1,k : [p∗
1]hζi:= [p1]hζi∩
−h˙
ˆx2,ki−[p2]hζi·[ ˆx2,k ]+[p2]hζi·[ˆx2,k ]·cos [ˆx2,k]2
[ˆx2,k ]
C2,2,k : [p∗
2]hζi:= [p2]hζi∩h˙
ˆx2,ki+ [p1]hζi·[ ˆx2,k ]
[ˆx1,k ]−[ˆx2,k ]·cos [ˆx2,k]2
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 31/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Re-Evaluation of the Algebraic Parameter Constraints (1)
Construction of bounding systems for the time derivatives of all states (possibly feasible
domains)
sup[ˆx1,k ]·[p2]hζi−[p1]hζi+ cos [ˆx2,k ]2·[p2]hζi·[ˆx2,k]≥infh˙
ˆx2,ki
inf[ˆx1,k ]·[p2]hζi−[p1]hζi+ cos [ˆx2,k ]2·[p2]hζi·[ˆx2,k]≤suph˙
ˆx2,ki
Time-varying bounds for all states over the complete time horizon t∈[0 ; tf], here: tf= 2
x1(tk)∈[ˆx1,k ]⊆X1;X1
x2(tk)∈[ˆx2,k ]⊆[ym,k ]
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 32/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Re-Evaluation of the Algebraic Parameter Constraints (2)
Construction of bounding systems for the time derivatives of all states (possibly feasible
domains)
sup−[p1]hζi+ cos [ˆx2,k ]2·[p2]hζi·[ˆx1,k] + [ˆx2,k ]·[p2]hζi≥inf h˙
ˆx1,ki
inf−[p1]hζi+ cos [ˆx2,k ]2·[p2]hζi·[ˆx1,k] + [ˆx2,k ]·[p2]hζi≤suph˙
ˆx1,ki
Time-varying bounds for all states over the complete time horizon t∈[0 ; tf], here: tf= 2
x1(tk)∈[ˆx1,k ]⊆X1;X1
x2(tk)∈[ˆx2,k ]⊆[ym,k ]
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 33/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Re-Evaluation of the Algebraic Parameter Constraints (3)
Construction of bounding systems for the time derivatives of all states
η1·−[p1]hζi+ cos [ˆx2,k ]2·[p2]hζi·[ˆx1,k] + [ˆx2,k ]·[p2]hζi+
η2·[ˆx1,k ]·[p2]hζi−[p1]hζi+ cos [ˆx2,k ]2·[p2]hζi·[ˆx2,k]!
∩η1·h˙
ˆx1,ki+η2·h˙
ˆx2,ki6=∅
η1= 1,η2= 1
−[p1]hζi−[p2]hζi·cos [ˆx2,k ]2−1·([ˆx1,k ] + [ˆx2,k])∩h˙
ˆx1,ki+h˙
ˆx2,ki6=∅
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 34/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Re-Evaluation of the Algebraic Parameter Constraints (4)
Construction of bounding systems for the time derivatives of all states
η1·[f1] + η2·[f2]∩η1·h˙
ˆx1,ki+η2·h˙
ˆx2,ki6=∅
η1=p1,η2=p2
−[p1]hζi2−cos [ˆx2,k ]2·[p1]hζi·[p2]hζi+[p2]hζi2·[ˆx1,k ]
−[p2]hζi2·[ˆx2,k ]·cos [ˆx2,k ]2!
∩[p1]hζi·h˙
ˆx1,ki+ [p2]hζi·h˙
ˆx2,ki6=∅
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 35/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Re-Evaluation of the Algebraic Parameter Constraints (5)
Construction of bounding systems for the time derivatives of all states
η1·[f1] + η2·[f2]∩η1·h˙
ˆx1,ki+η2·h˙
ˆx2,ki6=∅
η1=p2,η2=−p1
−[p1]hζi·[p2]hζi−[p2]hζi·[p1]hζi+ [p2]hζi·cos [ˆx2,k ]2·[ˆx1,k ] +
[p1]hζi·[p1]hζi+ [p2]hζi·cos [ˆx2,k ]2+[p2]hζi2·[ˆx2,k ]
∩[p2]hζi·h˙
ˆx1,ki−[p1]hζi·h˙
ˆx2,ki6=∅
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 36/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Illustrative Example: Results of Parameter Identification
Option 1 Option 2 Option 3 Option 4 Option 5 Option 6
Initial grid 10 ×10 50 ×50 10 ×10 50 ×50 3 ×3 10 ×10
Adaptive refinement no no no no yes yes
Number of iterations 3 3 3 3 14 16
Parameter contractor no no yes yes yes yes
Number of system evaluations 50 654 50 662 461 687
Remaining parameter domain in % 7.00 1.76 3.72 1.72 1.22 1.11
Options 1–4: Evaluation up to the point, where no further exclusions are possible without
further parameter subdivision
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 37/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Illustrative Example: Results of Parameter Identification
p1
p2
5
2
543
4
3
1
0
012
(a) Option 1.
p1
p2
5
2
543
4
3
1
0
012
(b) Option 2.
p1
p2
5
2
543
4
3
1
0
012
(c) Option 3.
blue: infeasible domains; yellow: identification result; •: true parameter value
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 38/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Illustrative Example: Results of Parameter Identification
p1
p2
5
2
543
4
3
1
0
012
(d) Option 4.
p1
p2
5
2
543
4
3
1
0
012
(e) Option 5.
p1
p2
5
2
543
4
3
1
0
012
(f) Option 6.
blue: infeasible domains; yellow: identification result; •: true parameter value
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 39/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Control-Oriented Modeling of SOFC Systems (1)
Configuration of a SOFC test rig
Supply of fuel gas (hydrogen and/or mixture
of methane, carbon monoxide, water vapor)
Supply of air
Independent preheaters for fuel gas and air
Stack module containing fuel cells in electric
series connection
Electric load as disturbance
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 40/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature distribution
=⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
1HT: Heat transfer (heat conduction and
convection)
2G: Enthalpy flows of supplied gases
3R: Exothermic reaction enthalpy
4EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
˙
Q
R
,
˙
Q
EL
I
i
=
I
=
I
+
stack element
=(
i
,
j
,
k
)
G
˙
Q
G,
j
∑
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 41/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature distribution
=⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
1HT: Heat transfer (heat conduction and
convection)
2G: Enthalpy flows of supplied gases
3R: Exothermic reaction enthalpy
4EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
˙
Q
R
,
˙
Q
EL
I
i
=
I
=
I
+
stack element
=(
i
,
j
,
k
)
G
˙
Q
G,
j
∑
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 41/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature distribution
=⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
1HT: Heat transfer (heat conduction and
convection)
2G: Enthalpy flows of supplied gases
3R: Exothermic reaction enthalpy
4EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
I
i
=
I
=
I
+
stack element
=(
i
,
j
,
k
)
G
˙
Q
G,
j
∑
˙
Q
R
,
˙
Q
EL
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 41/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature distribution
=⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
1HT: Heat transfer (heat conduction and
convection)
2G: Enthalpy flows of supplied gases
3R: Exothermic reaction enthalpy
4EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
I
i
=
I
=
I
+
stack element
=(
i
,
j
,
k
)
G
˙
Q
G,
j
∑
˙
Q
R
,
˙
Q
EL
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 41/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Control-Oriented Modeling of SOFC Systems (3)
Modeling assumptions and model properties
Capability to represent time-varying hotspot locations
Representation of the specific heat capacities of the anode and cathode gases by
second-order temperature-dependent polynomials
Representation of the reaction enthalpy by a second-order temperature-dependent
polynomial
Availability of gas mass flows, preheater temperatures as well as inlet and outlet manifold
temperatures as measured data
Non-measured state variables (temperatures) can be reconstructed by an observer and
also be used for a parameter identification (if additive error bounds are accounted for)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 42/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Definition of Point-Valued Bounding Systems
Bounding systems: x∈[v;w]
A·v+Bu =˙
v≤˙
x≤˙
w=A·w+Bu
with the element-wise bounding matrices
A≤A(x,p)≤Aand 0≤B≤B(x,p)≤Bas well as Bu ≥0
Definition of the system and input matrices (L=N= 1,M= 3)
A(x,p) =
a11 a12 0
a21 a22 a23
0a32 a33
and B(x,p) =
b11 b12 b13 b14
b21 0 0 b24
b31 0 0 b34
with the state vector
x=ϑ(1,1,1) ϑ(1,2,1) ϑ(1,3,1)T
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 43/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Definition of Point-Valued Bounding Systems
Bounding systems: x∈[v;w]
A·v+Bu =˙
v≤˙
x≤˙
w=A·w+Bu
with the element-wise bounding matrices
A≤A(x,p)≤Aand 0≤B≤B(x,p)≤Bas well as Bu ≥0
Definition of the system and input matrices (L=N= 1,M= 3)
A(x,p) =
a11 a12 0
a21 a22 a23
0a32 a33
and B(x,p) =
b11 b12 b13 b14
b21 0 0 b24
b31 0 0 b34
with the input vector
u=ϑAϑAG,in ϑCG,in 1
3IT
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 43/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Definition of Point-Valued Bounding Systems
Bounding systems: x∈[v;w]
A·v+Bu =˙
v≤˙
x≤˙
w=A·w+Bu
with the element-wise bounding matrices
A≤A(x,p)≤Aand 0≤B≤B(x,p)≤Bas well as Bu ≥0
Example for the sign pattern of the system matrix A(x,p)for L=N= 1,M= 3
A(x,p) =
−+ 0
+−+
0 + −
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 43/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Structural Analysis of the Bounding Systems
Cooperativity
Verified by the sign pattern of the system matrix =⇒Metzler matrix for all physically
reasonable parameterizations
Open-loop stability analysis using the Gershgorin circle theorem
<{λi} ≤ aii +
nx=3
X
j=1,j6=i
|aij |=
1
cImI·−α(2lNlM+lLlN+ 2lLlM)
−CAG,(1,1,1)(ϑ(1,1,1), t)
−CCG,(1,1,1)(ϑ(1,1,1), t)<0for i= 1
−2α
cImI·(lNlM+lLlM)<0for i= 2
−α
cImI·(2lNlM+lLlN+ 2lLlM)<0for i= 3
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 44/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Verified Interval-Based Parameter Identification
Use of Verified (Global Optimization) Procedures
General Branch-and-Bound procedure excluding all unphysical parameter domains:
Comparison of simulated system outputs with interval-bounded measurements
Extension by exploiting stability and cooperativity properties and constraints
A. Rauh, T. D¨
otschel, E. Auer, H. Aschemann: Interval Methods for Control-Oriented Modeling of the Thermal Behavior of High-Temperature Fuel Cell
Stacks. Proceedings of the 16th IFAC Symposium on System Identification, Brussels, Belgium, pp. 446-451. (2012)
DOI:10.3182/20120711-3-BE-2027.00374.
A. Rauh, L. Senkel, J. Kersten, H. Aschemann: Reliable control of high-temperature fuel cell systems using interval-based sliding mode techniques. IMA
Journal of Mathematical Control and Information, Vol. 33, No. 2, pp. 457–484. (2016) DOI:10.1093/imamci/dnu051.
A. Rauh, J. Kersten, H. Aschemann: An Interval Observer Approach for the Online Temperature Estimation in Solid Oxide Fuel Cell Stacks. Proceedings of
the 17th European Control Conference, Limassol, Cyprus, pp. 1596–1601. (2018) DOI:10.23919/ECC.2018.8550158.
S. Ifqir, A. Rauh, J. Kersten, D. Ichalal, N. Ait-Oufroukh, S. Mammar: Interval Observer-Based Controller Design for Systems with State Constraints:
Application to Solid Oxide Fuel Cells Stacks. Proceedings of the 24th International Conference on Methods and Models in Automation and Robotics,
pp. 372–377, Mi,
edzyzdroje, Poland. (2019) DOI:10.1109/MMAR.2019.8864718.
E. Auer, A. Rauh, J. Kersten: Experiments-based parameter identification on the GPU for cooperative systems. Journal of Computational and Applied
Mathematics. Vol. 371. Paper-id: 112657. (2019) DOI:10.1016/j.cam.2019.112657.
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 45/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Efficient Representation of Measured Data and Inputs
Verified enclosures: Time-dependent Bernstein polynomials with interval remainder
Reduction of memory requirements (full experiment with >300,000 sampling instants)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 46/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Constraints for Parameter Identification of the SOFC
State-independent parameters (3parameters)
Heat conductivity λ > 0
Heat transfer α > 0
Ohmic cell resistance R > 0
Possibility to split up the identification into heating and reaction phases
Stability-based subdivision of the heat conductivity intervals [λ]so that the bounding systems
become stable prior to the simulation-based identification
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 47/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Constraints for Parameter Identification of the SOFC
State-dependent characteristics (15 parameters) with θ∈ {ϑ(1,1,1), ϑ(1,2,1), ϑ(1,3,1) }
Heat capacities of all gases cχ(θ) = γ0,χ +γ1,χθ+γ2,χ θ2>0
Monotonicity of the heat capacities of all gases ∂cχ(θ)
∂θ =γ1,χ + 2γ2,χ θ > 0
Reaction enthalphy HRχ(θ) = γHR,0+γHR,1θ+γHR,2θ2>0
Monotonicity of the reaction enthalphy ∂HRχ(θ)
∂θ =γHR,1+ 2γHR,2θ > 0
Implementation of a parameter pre-identification, e.g.
γHR,0+γHR,1θ+γHR,2θ2>0
(γHR,1+ 2γHR,2θ > 0if γHR,2>0
γHR,1+ 2γHR,2θ > 0if γHR,2<0
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 48/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Cooperative Bounds for Point-Valued Parameters
Floating-point parameter identification; not yet fully verified
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 49/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Quasi-Linear, Input-Affine Neural Network Structure
A(x,q)·x
b(x,q)·ϑCG,in
d(x,q)
.
.
.
.
.
.
.
.
.
P
.
.
.
P
•
.
.
.
•
P
.
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•
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•
P
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Bias Bias
ϑCG,in(tk)
q1(tk)
qm(tk)
x1(tk)
xn(tk)
H1
HL2
˙x1(tk)
˙xn(tk)
˙xR,1(tk)
˙xR,n(tk)
states
A. Rauh, J. Kersten, W. Frenkel, N. Kruse, T. Schmidt: Physically
Motivated Structuring and Optimization of Neural Networks for
Multi-Physics Modeling of Solid Oxide Fuel Cells, under review for
Mathematical and Computer Modelling of Dynamical Systems.
A. Rauh: Kalman Filter-Based Real-Time Implementable
Optimization of the Fuel Efficiency of Solid Oxide Fuel Cells. Clean
Technologies. Vol. 3, pp. 206–226. (2021)
DOI:10.3390/cleantechnol3010012.
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 50/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Interval Extension of Neural Networks for the Electric Power
Characteristics
.
.
.
.
.
.P
Bias Bias
q1,k
qm,k
H1
HL
Uk
Input
layer
Hidden
layer
Ouput
layer
•
Ik
PEL,k
A. Rauh, J. Kersten, W. Frenkel, N. Kruse,
T. Schmidt: Physically Motivated Structuring and
Optimization of Neural Networks for Multi-Physics
Modeling of Solid Oxide Fuel Cells, under review for
Mathematical and Computer Modelling of
Dynamical Systems.
A. Rauh: Kalman Filter-Based Real-Time
Implementable Optimization of the Fuel Efficiency of
Solid Oxide Fuel Cells. Clean Technologies. Vol. 3,
pp. 206–226. (2021)
DOI:10.3390/cleantechnol3010012.
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 51/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Interval Extension of Neural Networks: Electric Power Characteristics
Two-stage training procedure
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 52/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Future Work — GPU-Based Parallelization (1)
Performance enhancement
Massive parallelization of the system evaluation on the GPU
Need for the extension of existing GPU interval libraries by function evaluations
(e.g. trigonometric expressions) that are currently only available for the CPU
Basic arithmetic operations (+,−,∗, /)are defined for interval data types in CUDA
Kernel Elementwise multiplication
__g l o b a l _ _ void m ul t _e l em ( double * v1 , con s t d o u b l e * v2 )
{
in t i dx = t h re a dI d x .x ;
v1 [ i d x ] *= v 2 [ i dx ] ;
}
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 53/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Future Work — GPU-Based Parallelization (2)
Performance enhancement
Massive parallelization of the system evaluation on the GPU
Need for the extension of existing GPU interval libraries by function evaluations
(e.g. trigonometric expressions) that are currently only available for the CPU
Basic arithmetic operations (+,−,∗, /)are defined for interval data types in CUDA
Usage in MatLab
k er n el _ m ul t = p a r al l el . g p u . CU D A Ke r n el ( ’ m y _ mu l t . pt x ’ ,.. .
’ m y_ m u lt . c u ’ ,’ m u lt _ el e m ’ );
% .. . d ef i ni ti o n of A 1 , A2 a s gp u Ar ra y
% .. . d ef i nt i on of k er n el _ mu lt . T h re a dB l oc k Si z e = N;
% .. . a nd p o ss i bl y a ls o k e rn e l_ m ul t . G ri dS i ze = . .. ;
result = (f e va l ( k er n e l_ m u lt , A 1 , A 2 )) ;
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 54/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Conclusions and Outlook on Future Work
Control-oriented modeling and identification of cooperative dynamic systems
(e.g. thermo-fluidic applications, compartment models for cell maturation in biology and
medicine)
Bounding systems account for parameter uncertainty and quasi-linear state dependencies
Prerequisite for the implementation of cooperativity-preserving state observers and
controllers
Extension to further nonlinear cooperative systems with large number of parameters
(optimal design of experiment in combination with pre-identification and contractor
technique)
Extension towards non-cooperative system models (e.g. from the field of robotics)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 55/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
Conclusions and Outlook on Future Work
Control-oriented modeling and identification of cooperative dynamic systems
(e.g. thermo-fluidic applications, compartment models for cell maturation in biology and
medicine)
Bounding systems account for parameter uncertainty and quasi-linear state dependencies
Prerequisite for the implementation of cooperativity-preserving state observers and
controllers
Extension to further nonlinear cooperative systems with large number of parameters
(optimal design of experiment in combination with pre-identification and contractor
technique)
Extension towards non-cooperative system models (e.g. from the field of robotics)
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 55/56
Verified Parameter Identification Application Scenario 1 Consistency Tests Application Scenario 2 Conclusions
State, Parameter, and Disturbance Estimation for the DDBOATS
Uncertain Dubins car model
xk+1 =xk+T·
cos(x4,k)x3,k
sin(x4,k)x3,k
0
0
augmented with additive input disturbances
Future work: Estimation of input characteristics
and inertia parameters (nonlinear and
non-cooperative)
A. Rauh, A. Bourgois, L. Jaulin: Union and Intersection Operators for Thick Ellipsoid
State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design.
Algorithms. 14(3), 88. (2021). DOI:10.3390/a14030088.
A. Rauh: Verified Parameter Identification of Quasi-Linear Cooperative System Models 56/56
ResearchGate has not been able to resolve any citations for this publication.
Interval Observer-Based Controller Design for Systems with State Constraints: Application to Solid Oxide Fuel Cells Stacks
Conference Paper
Full-text available
  • Aug 2019
For many practical applications, it is necessary to design controllers that are, one the one hand, robust against bounded uncertainty in selected parameters and external disturbances. On the other hand, it is also quite common for such systems, that state constraints need to be accounted for which must not be exceeded for any possible parameter value. Although the task is quite challenging, several robust control design procedures were designed in recent years which allow for solving these problems, especially with respect to the latter aspect. In this paper, such robust control procedures are combined in a novel manner with an interval observer to additionally handle the influence of unknown but bounded measurement noise and to prevent the violation of state constraints in a provable fashion.
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An Interval Observer Approach for the Online Temperature Estimation in Solid Oxide Fuel Cell Stacks
Conference Paper
Full-text available
  • Jun 2018
Interval observers that are based on the structural property of cooperativity allow for the computation of guaranteed lower and upper bounds for all state trajectories of dynamic systems by defining two sets of bounding systems for the dynamic behavior. In such a way, they remove the disadvantage of predictor-corrector interval estimation schemes which often suffer from the fact that the dynamic system model has to be evaluated over finitely large domains of state variables and parameters (often interval boxes or zonotopes). This evaluation typically results in overestimation that includes unphysical parts of the state-space in the computed results. To counteract this so-called wrapping effect, computationally expensive, problem-specific algorithms need to be implemented. However, this drawback can be removed for systems that have certain monotonicity and stability properties. In this paper, it is shown that appropriately defined system models for the thermal behavior of high-temperature fuel cells belong to this class of systems. The corresponding interval observer design is presented methodologically and demonstrated with the help of measured data from a test rig available at the Chair of Mechatronics at the University of Rostock.
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Interval Methods for Control-Oriented Modeling of the Thermal Behavior of High-Temperature Fuel Cell Stacks
Conference Paper
Full-text available
  • Jul 2012
Solid oxide fuel cells (SOFCs) can be used as decentralized energy supply devices for providing electricity and heat directly by converting chemical energy. In such applications of SOFCs, the electric power demand is commonly varying over time. Therefore, all processes in fuel cell systems are typically instationary. For example, the heating and cooling phases for starting up and shutting down the fuel cell system as well as the response to varying electrical load demands characterize the instationarity of the operating conditions for the thermal subprocess. In contrast to most existing control approaches, which only cover stationary operating strategies, our work aims at controlling SOFC systems in instationary operating regions. This means that a mathematical system model for these regions is necessary. Such control-oriented models for the temperature distribution in a fuel cell stack module can be obtained by the method of finite volume discretization. On the basis of the first law of thermodynamics, local energy balances are derived for each volume element, which leads to a system of coupled nonlinear ordinary differential equations. In this paper, parameter identification routines are compared which are based both on classical floating point techniques and on verified interval arithmetic approaches. In particular, interval techniques are employed to deal with imperfect system knowledge expressed by bounded parameter uncertainties and to search for globally instead of locally optimal system parameterizations.
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Experiments-based parameter identification on the GPU for cooperative systems
Article
  • Dec 2019
  • J COMPUT APPL MATH
In this paper, we describe how to enhance experimental parameter identification for cooperative systems using the GPU. This kind of identification is necessary, for example, in the area of developing robust controllers and observers based on available measurement data. First, we point out which parts of the general, global optimization based parameter identification procedure can be parallelized and compare it to the experiments-based technique. After that, we illustrate the principle and its implementation with the help of a close-to-life textbook example. Finally, we demonstrate how the GPU-based experimental parameter identification works for identifying experimentally the parameters of the (reduced-order) temperature model for the distributed heating system. This is an important step towards working with uncertain models in the context of solid oxide fuel cells.
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Reliable control of high-Temperature fuel cell systems using interval-based sliding mode techniques
Article
  • Dec 2014
In previous work, it has been shown that novel interval-based sliding mode techniques can be employed effectively for the design of guaranteed stabilizing control strategies for high-temperature fuel cells. These control strategies aim at both trajectory tracking during non-stationary heating phases and at compensating disturbances at desired constant operating points. The implementation of the corresponding control strategies involves three main building blocks: (i) transformation of the state equations into a non-linear controller canonical form, (ii) application of interval arithmetic techniques in real time for the computation of stabilizing system inputs despite bounded uncertainty and (iii) an online optimization for the synthesis of the input variables, namely suitable preheater mass flows and temperatures. For practical application, the transformation into the controller canonical form based on the computation of Lie derivatives can be impeded by the complexity of the symbolic representation of the state equations. Hence, techniques for algorithmic differentiation are compared in this contribution with alternative approaches based on symbolic formula manipulation. The latter one is only efficient for low-dimensional finite volume models in the description of thermodynamic system behaviour. Since the interval-based control law involves the use of time derivatives of the measured or estimated system output up to its relative degree, suitable differentiation techniques using low-pass filtered derivatives, algebraic derivative estimation and sensitivity-based approaches conclude this contribution in a numerical comparison.
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