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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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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.
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(tk1)
Improved estimate after transition from tk1to 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(tk1)
Improved estimate after transition from tk1to 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
i1(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
i1(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
i1(t) = λR·Ac
ls
·(ϑi1(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)) ,xRn
Ji,j 0, i, j ∈ {1, . . . , n}, i 6=jwith J=f(x)
x
For initial conditions in the positive orthant
Rn
+={xRn|xi0i∈ {1, . . . , n}} ,
positivity of all state trajectories is ensured if
˙xi(t) = fi(x1, . . . , xi1,0, 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],αα; ∆α,
mama; ∆ma, and λRλR;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=j1, j ∈ {2,..., ˜
N}
p2
ci·mi>0for i=j+ 1 , j ∈ {1,..., ˜
N1}
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,..., ˜
N1}
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