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Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Veriﬁcation Techniques for Fractional Diﬀerential Equations

with Bounded Uncertainty

2e Journ´ees Virtuelles de la SAGIP, GT VS-CPS

02 Juillet 2021

Andreas Rauh1, Luc Jaulin1, Rachid Malti2

1Lab-STICC (Robex), ENSTA Bretagne, Brest, France

2IMS Laboratory, University of Bordeaux, France

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 1/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Contents

Motivation – Or why should we deal with fractional-order system models?

Generalization of an exponential enclosure technique to fractional diﬀerential equations

Alternative veriﬁed simulation methods

Consideration of temporal truncation errors

Application: Prediction of the state-of-charge for a fractional-order battery model

Observer-based quantiﬁcation of truncation errors

Conclusions and outlook on future work

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 2/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

A Time-Domain Point of View

Observations

The time response seems to be partially faster and partially slower than an exponential

The time response seems to be impossible to be explained by a linear ﬁrst- or

second-order model

Do we really need to identify a nonlinear representation?

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 4/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

A Frequency-Domain Point of View

Observations

Amplitude response does not only show slopes of ±20 dB per frequency decade

The phase response seems to have a characteristic value not only around −90◦and

−180◦, but also at ≈ −54◦

Is it necessary to expand the amplitude and phase responses into series of classical ﬁrst-

and second-order transfer function blocks?

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 6/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Resolution

Considered system model

y(2)(t)+3y(0.6)(t)+y(t) = u(t)

Fractional-order attenuation term

The system model is linear

However, not all contained derivatives have an integer order

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 7/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Resolution

Considered system model

y(2)(t)+3y(0.6)(t)+y(t) = u(t)

Fractional-order attenuation term

Representation that lies between the dependency of stress (σ(t)) and strain ((t))

according to Newton’s law

σ(t) = ηd(t)

dtwith the viscosity η

and Hooke’s law

σ(t) = E(t)with the modulus of elasticity E

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 7/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Modeling and Identiﬁcation by Means of Impedance Spectroscopy

Typical amplitude and phase response vs. integer-order system models

Amplitude variations do not correspond to integer multiples of ±20dB per frequency

decade

Phase variations do not correspond to integer multiples of ±π

2per frequency decade

Thevenin equivalent circuit consisting of series connections of resistors and RC submodels

Approximation of frequency response characteristics over a ﬁnite bandwidth

v

OC

v

T

+

i

T

C

TS

R

S

R

T S

v

T S

C

T L

v

T L

R

T L

®

¯°

5336 D. Andre et al. / Journal of Power Sources 196 (2011) 5334–5341

0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

Z’ / Ω

−Z’’ / Ω

Measurement after 0 min

Measurement after 30 min

Measurement after 50 min

Fig. 6. Effect of pause time after SOC variation (SOC =10%) on impedance spectra

at −30 ◦C and 50% SOC.

0 0.005 0.01 0.015 0.02 0.025

0

0.005

0.01

−Z´´ / Ω

Z´ / Ω

Measurement 1 (started after 70 min)

Measurement 2 (started after 90 min)

Measurement 3 (started after 115 min)

Measurement 4 (started after 170 min)

Fig. 7. Effect of pause time after temperature variation (T= 30 K) on impedance

spectra at −10 ◦C and 60% SOC.

Fig. 8. Approximation of internal cell resistance Rby seven points as a function of

temperature.

T = 50°C T = 24°C T = 4°C

02468

x 10−3

0

2

x 10−3

Z’ / Ω

−Z’’ / Ω

Fig. 9. Impedance spectra for positive temperatures at 60% SOC.

0 0.05 0.1 0.15 0.2

0

0.05

Z’ / Ω

−Z’’ / Ω

T = −9°C T = −18°C T = −24°C T = −30°C

Fig. 10. Impedance spectra for negative temperatures at 60% SOC.

Fig. 11. Bode plot of EIS measurements at 60% SOC.

−20 0 20 40

1

1.5

2

Temperature / °C

RΩ / mΩ

(a)

33.54

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1/Temperature / 1/K

log(RΩ) / mΩ

(b)

Fig. 12. Inﬂuence of temperature (a) on Rat 60% SOC and (b) Arrhenius plot of R.

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 8/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Modeling and Identiﬁcation by Means of Impedance Spectroscopy

Typical amplitude and phase response vs. fractional-order system models

Generalization of the Thevenin equivalent circuit by a replacement of capacitors (and

possibly inductors) by respective constant phase elements

Generalization of frequency response functions by expressions of the form

b0+. . . +bm(ω)mν

a0+. . . +an(ω)nν , m, n ∈N0, ν ∈(0,1)

A. Rauh & J. Kersten 11

6.2 Simpliﬁed Fractional-Order Battery Model

As a second application scenario, consider the fractional-order battery model depicted in Fig. 3. It

describes the dynamics of the charging and discharging behavior of a Lithium-Ion battery with the help

of the state of charge σ(t)as well as with the dynamics of the exchange of charge carriers in the interior

of the battery cell which is related to electrochemical double layer effects. The latter is represented with

the help of the voltage drop v1(t)across the fractional-order constant phase element Qwhich serves as

a generalization of an ideal capacitor according to [1, 10, 24]. This kind of constant phase element was

already motivated by the example in Sec. 3.

vOC(σ(t)) +

−

i(t)R0

Rv(t)

+

−

Q

v1(t)

Figure 3: Basic fractional-order equivalent circuit model of batteries.

For the following investigation of the proposed interval-based simulation approach for fractional-

order system models, assume that the state vector

x(t) = hσ(t)0D0.5

tσ(t)v1(t)iT∈R3(30)

comprises in addition to the voltage v1across the constant phase element in Fig. 3 and the state of charge

also its fractional derivative of order ν=0.5. Then, following the modeling steps described in [10],

where this dynamic system representation was employed for the derivation of a cooperativity-enforcing

interval observer design, and generalizing the charging/discharging dynamics to

0D1

tσ(t) = −η0·i(t) + η1·σ(t)·sign i(t)

3600CN

(31)

with the terminal current i(t), the commensurate-order quasi-linear state equations

0D0.5

tx(t) = A·x(t) + b·i(t)(32)

with the system and input matrices

A=

0 1 0

η1·sign(i(t))

3600CN0 0

0 0 −1

RQ

and b=

0

−η0

3600CN

1

Q

(33)

and the terminal voltage

v(t) = "4

∑

k=0

ckσk−1(t)0−1#·x(t) + −R0+d0ed1σ(t)·i(t)(34)

as the system output are obtained. The following numerical simulations are based on the system pa-

rameters summarized in Tab. 1 which were — except for η1— identiﬁed according to the exper-

imental data referenced in [10, 22] during multiple charging and discharging cycles before the oc-

currence of aging. Now, a linear state feedback controller with the structure presented in [11] for

5336 D. Andre et al. / Journal of Power Sources 196 (2011) 5334–5341

0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

Z’ / Ω

−Z’’ / Ω

Measurement after 0 min

Measurement after 30 min

Measurement after 50 min

Fig. 6. Effect of pause time after SOC variation (SOC =10%) on impedance spectra

at −30 ◦C and 50% SOC.

0 0.005 0.01 0.015 0.02 0.025

0

0.005

0.01

−Z´´ / Ω

Z´ / Ω

Measurement 1 (started after 70 min)

Measurement 2 (started after 90 min)

Measurement 3 (started after 115 min)

Measurement 4 (started after 170 min)

Fig. 7. Effect of pause time after temperature variation (T= 30 K) on impedance

spectra at −10 ◦C and 60% SOC.

Fig. 8. Approximation of internal cell resistance Rby seven points as a function of

temperature.

T = 50°C T = 24°C T = 4°C

02468

x 10−3

0

2

x 10−3

Z’ / Ω

−Z’’ / Ω

Fig. 9. Impedance spectra for positive temperatures at 60% SOC.

0 0.05 0.1 0.15 0.2

0

0.05

Z’ / Ω

−Z’’ / Ω

T = −9°C T = −18°C T = −24°C T = −30°C

Fig. 10. Impedance spectra for negative temperatures at 60% SOC.

Fig. 11. Bode plot of EIS measurements at 60% SOC.

−20 0 20 40

1

1.5

2

Temperature / °C

RΩ / mΩ

(a)

33.54

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1/Temperature / 1/K

log(RΩ) / mΩ

(b)

Fig. 12. Inﬂuence of temperature (a) on Rat 60% SOC and (b) Arrhenius plot of R.

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 9/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Initial Value Problem with Interval Uncertainty

Deﬁnition of the initial value problem (IVP)

Given set of ordinary diﬀerential equations (ODEs)

˙

x(t) = f(x(t))

with smooth right-hand sides

Uncertain initial conditions

x(0) ∈[x0] := [x(0)] = [x(0) ; x(0)]

Component-wise deﬁnition of interval vectors [x] = [x1]. . . [xn]Twith the vector

entries [xi]=[xi;xi],xi≤xi≤xi,i= 1, . . . , n

Generalization for fractional diﬀerential equations with derivative orders 0< ν ≤1?

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 10/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Exponential Enclosure Technique

Deﬁnition of the state enclosure

Representation of contracting state enclosures by using

x∗(t)∈[xe] (t) := exp ([Λ]·t)·[xe] (0)

with 06∈ [xe,i] (0),[xe] (0) = [x0]for the diagonal matrix

[Λ] := diag {[λi]}, i = 1, . . . , n

with element-wise negative real entries λi

Deﬁnition of the interval matrix exponential

exp ([Λ]·t) := diag {exp ([λ1]·t),...,exp ([λn]·t)}

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 11/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Exponential Enclosure Technique

Interval-valued iteration scheme

Picard iteration

x∗(t)∈[xe]hκ+1i(t) := [x0] +

t

Z

0

f[xe]hκi(s)ds

Resulting iteration formula

[λi]hκ+1i:=

fiexp [Λ]hκi·[0 ; T]·[xe] (0)

exp [λi]hκi·[0 ; T]·[xe,i ] (0)

, i = 1, . . . , n

with the guaranteed state enclosure at the point t=T

x∗(T)∈[xe] (T) := exp ([Λ]·T)·[xe] (0)

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 12/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Fractional Diﬀerential Equations (1)

Fractional diﬀerential equations (FDEs) of Caputo type

Generally nonlinear pseudo state equations

x(ν)(t) = f(x(t))

Initial conditions speciﬁed at t= 0 (temporally constant initialization)

Exact solution of the FDE for the linear case

Scalar, linear fractional pseudo state equation

x(ν)(t) = λ·x(t)

with the initial condition x0=x(0) that also describes the entire past of the pseudo state

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 13/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Fractional Diﬀerential Equations (2)

Exact solution of the FDE for the linear case

Scalar, linear fractional pseudo state equation

x(ν)(t) = λ·x(t)

with the initial condition x0=x(0) that also describes the entire past of the pseudo state

Explicit solution

x(t) = Eν,1(λtν)·x(0)

depending on the two-parameter Mittag-Leﬄer function

Eν,β (ζ) =

∞

X

i=0

ζi

Γ (νi +β)

with the general argument ζ∈Cand the parameters ν∈R+,β∈R

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 14/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Fractional Diﬀerential Equations (3)

Note 1

For ν= 1, the two-parameter Mittag-Leﬄer function turns into the classical exponential

function according to the solution

x(t) = E1,1λt1·x(0)

with

E1,1(ζ) =

∞

X

i=0

ζi

Γ (i+ 1) =eζ

Note 2

FDE systems are characterized by an inﬁnite memory of previous states, i.e., subdividing

the integration time horizon into short slices [tk;tk+1], where the state at t=tk+1 serves as

the initialization for the following slice, is prone to approximation errors

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 15/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Iterative Solution Procedure for Fractional System Models (1)

Generalization of exponential enclosures to the FDE case

Deﬁnition of the pseudo state enclosures

[xe]hκi(t) = Eν,1[Λ]hκi·tν·[xe] (0)

Diagonal matrix containing the evaluation of the scalar Mittag-Leﬄer function

Eν,1[Λ]hκi·tν

with

[Λ] := diag {[λi]}, i = 1, . . . , n

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 16/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Iterative Solution Procedure for Fractional System Models (2)

Element-wise notation of the ﬁxed-point iteration

diag n[λi]hκ+1io·Eν,1[Λ]hκ+1i·[t]ν·[xe] (0)

|{z }

[xe]hκ+1i([t]) ⊆[xe]hκi([t])

=: fEν,1[Λ]hκi·[t]ν·[xe] (0)

Detailed derivation

Rauh, Andreas; Kersten, Julia; Aschemann, Harald: Interval-Based Veriﬁcation Techniques for the Analysis of Uncertain Fractional-Order System Models,

2020 European Control Conference (ECC), Saint Petersburg, Russia, pp. 1853–1858, 2020.

Rauh, Andreas; Kersten, Julia: Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems, Acta

Cybernetica, 2020. DOI: 10.14232/actacyb.285660.

Rauh, Andreas; Kersten, Julia: Veriﬁcation and Reachability Analysis of Fractional-Order Diﬀerential Equations Using Interval Analysis, Proceedings 6th

International Workshop on Symbolic-Numeric Methods for Reasoning about CPS and IoT (SNR 2020), Electronic Proceedings in Theoretical Computer

Science 331, pp. 18–32. 2021. DOI: 10.4204/EPTCS.331.2

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 17/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Iterative Solution Procedure for Fractional System Models (2)

Element-wise notation of the ﬁxed-point iteration

diag n[λi]hκ+1io·Eν,1[Λ]hκ+1i·[t]ν·[xe] (0)

| {z }

[xe]hκ+1i([t]) ⊆[xe]hκi([t])

=: fEν,1[Λ]hκi·[t]ν·[xe] (0)

Necessity for an interval extension of Mittag-Leﬄer functions

Rauh, Andreas; Kersten, Julia; Aschemann, Harald: Interval-Based Veriﬁcation Techniques for the Analysis of Uncertain Fractional-Order System Models, 2020

European Control Conference (ECC), Saint Petersburg, Russia, pp. 1853–1858, 2020.

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 17/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Alternative Solution Techniques

Option 1: Interval-based Picard iteration

Multi time step Picard iteration

xhκ+1i(t) = x(0) + 1

Γ (ν)·

t

Z

0

(t−τ)ν−1·fxhκi(τ)dτ

Interval formulation

xhκ+1i(tn)∈x(0) + 1

Γ (ν)·

n−1

X

i=0

ti+1

Z

ti

(tn−τ)ν−1·fhxhκii(τ)

| {z }

[f]i+1

i

dτ

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 18/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Alternative Solution Techniques

Option 1: Interval-based Picard iteration

Interval formulation

xhκ+1i(tn)∈x(0) + 1

Γ (ν)·

n−1

X

i=0

ti+1

Z

ti

(tn−τ)ν−1·fhxhκii(τ)

| {z }

[f]i+1

i

dτ

Analytic overapproximation of the integral

xhκ+1i(tn)∈x(0) + 1

Γ (ν)·

n−1

X

i=0

[f]i+1

i·(tn−ti)ν−(tn−ti+1)ν

ν

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 18/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Alternative Solution Techniques

Series-based solution representation

Truncated series expansion of the solution (also in the complex-valued case!)

x(t)∈

L

X

i=0

[xi·ν]·ti·ν

Γ(1 + i·ν)+ [R]

Diﬀerentiation and substitution into the fractional diﬀerential equation model

Polynomial nonlinearities: Analytic (interval) expressions for the initial condition x0as

well as for the terms of order i∈ {1, . . . , L}

Finally compute guaranteed bounds [R]

Further examples (e.g. from the ﬁeld of epidemiology)

A. Rauh, L. Jaulin: Novel Techniques for a Veriﬁed Simulation of Fractional-Order Diﬀerential Equations,

Fractal Fract, 5(1), 17, 2021

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 23/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Consideration of the Inﬁnite Memory Property

Quantiﬁcation of temporal truncation errors (Podlubny)

|tkDν

tx(t)−tk+TDν

tx(t)| ≤ XT−ν

|Γ(1 −ν)|=: µ

with the component-wise deﬁned supremum of the reachable states according to

Xi= sup

t∈[tk;tk+1]

|xi(t)|, i ∈ {1, . . . , n}

and a subsequent inﬂation of the right-hand side of the system model

˜

f(x(t)) := f(x(t)) + [−µ;µ]

Note

The given bounds µare conservative, especially due to the fact that they are temporally

constant

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 24/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Visualization of the Eﬀect of Truncation Errors

Uncertain FDE model x(0.5)(t) = −x(t),x(t) = 1 for t≤0

t

x(t)

0

0

0.4

0.2

0.6

1.0

0.8

exact solution

2 4 6 8 10

∆T= 0.1

∆T= 1

t

x(t)

0

1.0

2 4 6 8 10

exact solution

0.2

0

0.4

0.6

0.8

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 25/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Illustrating Example: Mittag-Leﬄer-Type Enclosures

Uncertain FDE model x(ν)(t) = p·x3(t)

[x] (0) = [0.99 ; 1.0],[p] = [−2 ; −1.99],[ν] = [0.8 ; 0.81]

t

[x] (t)

0

0

0.4

0.2

0.6

1.0

0.8

0.2 0.4 0.6 0.8 1.0

[z] (t)

t

0

0

0.4

0.2

0.6

1.0

0.8

0.2 0.4 0.6 0.8 1.0

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 26/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Simpliﬁed Battery Model (1)

Fractional-order equivalent circuit model

A. Rauh & J. Kersten 11

6.2 Simpliﬁed Fractional-Order Battery Model

As a second application scenario, consider the fractional-order battery model depicted in Fig. 3. It

describes the dynamics of the charging and discharging behavior of a Lithium-Ion battery with the help

of the state of charge σ(t)as well as with the dynamics of the exchange of charge carriers in the interior

of the battery cell which is related to electrochemical double layer effects. The latter is represented with

the help of the voltage drop v1(t)across the fractional-order constant phase element Qwhich serves as

a generalization of an ideal capacitor according to [1, 10, 24]. This kind of constant phase element was

already motivated by the example in Sec. 3.

vOC(σ(t)) +

−

i(t)R0

Rv(t)

+

−

Q

v1(t)

Figure 3: Basic fractional-order equivalent circuit model of batteries.

For the following investigation of the proposed interval-based simulation approach for fractional-

order system models, assume that the state vector

x(t) = hσ(t)0D0.5

tσ(t)v1(t)iT∈R3(30)

comprises in addition to the voltage v1across the constant phase element in Fig. 3 and the state of charge

also its fractional derivative of order ν=0.5. Then, following the modeling steps described in [10],

where this dynamic system representation was employed for the derivation of a cooperativity-enforcing

interval observer design, and generalizing the charging/discharging dynamics to

0D1

tσ(t) = −η0·i(t) + η1·σ(t)·sign i(t)

3600CN

(31)

with the terminal current i(t), the commensurate-order quasi-linear state equations

0D0.5

tx(t) = A·x(t) + b·i(t)(32)

with the system and input matrices

A=

0 1 0

η1·sign(i(t))

3600CN0 0

0 0 −1

RQ

and b=

0

−η0

3600CN

1

Q

(33)

and the terminal voltage

v(t) = "4

∑

k=0

ckσk−1(t)0−1#·x(t) + −R0+d0ed1σ(t)·i(t)(34)

as the system output are obtained. The following numerical simulations are based on the system pa-

rameters summarized in Tab. 1 which were — except for η1— identiﬁed according to the exper-

imental data referenced in [10, 22] during multiple charging and discharging cycles before the oc-

currence of aging. Now, a linear state feedback controller with the structure presented in [11] for

Mathematical formulation

Pseudo state vector x(t) = σ(t)0D0.5

tσ(t)v1(t)T∈R3

Parameters identiﬁed experimentally

J. Reuter, E. Mank, H. Aschemann, A. Rauh: Battery state observation and condition monitoring using online

minimization, 21st International Conference on Methods and Models in Automation and Robotics (MMAR),

Miedzyzdroje, Poland, pp. 1223–1228, 2016.

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 27/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Simpliﬁed Battery Model (2)

Mathematical formulation

State equations

0D0.5

tx(t) = A·x(t) + b·i(t)

with the system and input matrices

A=

0 1 0

η1·sign(i(t))

3600CN0 0

0 0 −1

RQ

and b=

0

−η0

3600CN

1

Q

Terminal voltage

v(t) = 4

P

k=0

ckσk−1(t) 0 −1·x(t) + −R0+d0ed1σ(t)·i(t)

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 28/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Simulation Results

Prediction of state-of-charge and terminal voltage (control with pole assignment)

State-of-charge

tin s

[σ] (t)

0 2 4 6 8 10

0.85

1.00

1.05

1.10

0.90

0.95

1.15

Terminal voltage

tin s

[v] (t) in V

0 2 4 6 8 10

2.0

2.5

3.0

3.5

4.0

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 29/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Interval-Observer Technique for Improved Bounds of Truncation Errors (1)

Reference model: virtual measurements for a cooperative dynamic system model

Pseudo state enclosures

x(t)∈[v(t) ; w(t)] with 0D(ν)

tx(t) = f(x(t))

Without loss of generality: temporally constant initialization

x(t)∈[x0],˙

x(t) = 0for t≤0

Bounding systems

0D(ν)

tv(t) = fv(v(t)) ,v(t) = x0and

0D(ν)

tw(t) = fw(w(t)) ,w(t) = x0

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 30/35

Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Interval-Observer Technique for Improved Bounds of Truncation Errors (2)

Observer-based approach

Denote the truncation error bounds at t=T > 0by [−µT;µT]

Augmented observer model: z(t) = ˜

vT(t)˜

wT(t)µT

v(t)µT

w(t)T

TD(ν)

tz(t) =

fv(˜

v(t)) + µv(t)

fw(˜

w(t)) + µw(t)

0

0

+H·v(t)−˜

v(t)

w(t)−˜

w(t)with z(T) =

v(T)

w(T)

−µT

µT

Choose the observer gain Hsuch that [v(t) ; w(t)] ⊆[˜

v(t) ; ˜

w(t)]

Applicable to (periodically) reset the fractional integrator for the diﬀerential equation model

A. Rauh et al.: Veriﬁcation Techniques for Fractional Diﬀerential Equations with Bounded Uncertainty 31/35