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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

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 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 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 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 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 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 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 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

Linear Example: Spatially Distributed Heating System

Early lumping: Finite volume semi-discretization

inlet: air canal

outlet:

air canal

temperature

measurements in

the air canal

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

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

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

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

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

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

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

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 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 Stability Analysis (2)

Application of the Gershgorin circle theorem for the rows ˜

N+iwith i={1