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Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Fractional-Order System Models and

Their Veriﬁed Numerical Analysis Using Interval Methods

8th Intl. Online Seminar on

Interval Methods in Control Engineering

Virtually from Brest, France

January 15, 2021

Andreas Rauh

Lab-STICC (Robex)

ENSTA Bretagne, Brest, France

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 1/43

Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

Contents

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

Exponential enclosure technique for a veriﬁed integration of ordinary diﬀerential equations

Generalization of the iteration procedure to fractional-order diﬀerential equations

Consideration of temporal truncation errors

Which other simulation methods are possible in the frame of a veriﬁed simulation?

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

Conclusions and outlook on future work

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 2/43

Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions

A Time-Domain Point of View

Comparison of “velocity” y(1)(t)and “position” y(t)

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 3/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 4/43

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 frequency responses into series of classical

ﬁrst- and second-order transfer function blocks?

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 6/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 7/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 7/43

A Further Simple Example

Time domain representation: Fractional-order diﬀerential equation

y(0.5)(t) = −y(t) + u(t)

Frequency domain representation: Typical fractional transfer function (impedance)

F(ω) = Y(ω)

U(ω)=1

1+(ω)0.5

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 8/43

A Further Simple Example

Approximation by integer-order model

Amplitude response

ωin rad/s

magnitude in dB

0

−30

−20

−10

ω ω10−1

10−3101103

Phase response

ωin rad/s

phase in rad

0

−0.2

−0.4

−0.6

−0.8ω ω10−310−1101103

−−−fractional model; —integer order 5; —integer order 11

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 8/43

Further Appearances of Fractional-Order Models

Electrochemistry

Electrochemical impedance spectroscopy for monitoring corrosion processes

=⇒Non-destructive determination of corrosion layer thickness

Intermediate phenomena between diﬀusion (ν= 1) and wave propagation (ν= 2)

∂νu

∂tν=a·∂2u

∂x2

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 9/43

Further Appearances of Fractional-Order Models

Continued-fraction integer-order transfer function model of multi-robot formation

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 9/43

Further Appearances of Fractional-Order Models

Fractional-order representation of control laws

Generalization of linear PID-type controllers with the Laplace-domain formulation

G(s) = KP+sν1KD+s−ν2KI, ν1>0, ν2>0

=⇒Enhanced degrees for freedom for loop-shaping design methods

Replacement of integral terms in sliding mode-type control laws by fractional-order

integrals

=⇒Reduction of the danger for an integrator windup

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 9/43

Further Appearances of Fractional-Order Models

Modeling of electrochemical processes in batteries and fuel cells on the basis of measured data

from impedance spectroscopy

=⇒Low-dimensional linear and nonlinear equivalent circuit representations for modeling the

dynamics of battery charging and discharging (double-layer eﬀects, polarization, . . . ) as well as

for changes between various electric operating points of fuel cells

Theoretic Background

Stability domains of linear commensurate, fractional-order systems

(t0Dγ

tx(t) = Ax(t) + Bu(t) + Ez(t),x(t0) = x0

y(t) = Cx(t) + Du(t) + F z(t), γ ∈(0,2)

the observer (in terms of short transient operating phases)

and for achieving insensitivity of the computed estimates

with respect to noise. The latter aspect was addressed,

e.g. by Ichalal et al. (2018), where despite bounded noise,

the observer gain was optimized by an H∞methodology

dealing with random disturbances within the prespeciﬁed

error bounds.

In Sec. 2, preliminaries with respect to the system descrip-

tion, controllability, observability, and stability analysis of

fractional-order linear time-invariant models (FLTI) are

given. Sec. 3 summarizes the aforementioned procedures

for the interval observer design of FLTI models, before

numerical simulations are presented in Sec. 4 to highlight

the practical applicability in the frame of SOC estimation

for Lithium-Ion batteries. This paper is concluded with an

outlook on future work in Sec. 5.

Throughout the paper, the following set of notations is

used: The transpose of a matrix Mis denoted by MT,

its conjugate is ¯

Mand its conjugate transpose MH.

Sym{M}means M+MH. The left and right endpoints

of an interval [x] are denoted respectively by xand xsuch

that [x] = [x;x]. For any two vectors x1,x2or matrices

M1,M2, the relations x1≤x2and M1≤M2are

understood element-wise. The relation M≺0 (M0)

means that the matrix M∈Cn×nis negative (positive)

deﬁnite. The symbol Idenotes the identity matrix.

2. PRELIMINARIES

In this paper, commensurate FLTI systems are considered

permitting a state-space representation

(Dγ

tx(t) = Ax(t) + Bu(t) + Ez(t),x(t0) = x0

y(t) = Cx(t) + Du(t) + F z(t), γ ∈(0,2) (1)

with the state vector x(t)∈Rn, the input vector u∈Rp,

the output vector y(t)∈Rm, and the disturbance input

vector z(t)∈Rq. Additionally, A,B,C,D,E, and Fare

constant real matrices. Dγis the fractional diﬀerentiation

operator of order γ, whereby the presented results are valid

regardless of the deﬁnition used, e.g. Gr¨unwald-Letnikov,

Riemann-Liouville, Caputo etc. (Podlubny, 1999). The

transfer function matrix between u(t) and y(t) is H(s) =

C(sγI−A)−1B+D.

The properties of controllability and observability inﬂu-

ence the solvability of control problems signiﬁcantly. In

Monje et al. (2010), it was shown that, as with LTI

systems, a necessary and suﬃcient criterion for the con-

trollability and observability exists, mainly depending on

the structure of the analyzed system.

Theorem 1. An FLTI system of the form (1) is fully

controllable if and only if the controllability matrix

QC=B AB A2B. . . An−1B(2)

satisﬁes the condition rank{QC}=n.

Theorem 2. An FLTI system of the form (1) is fully

observable if and only if the observability matrix

QO=CT(CA)T(C A2)T. . . (CAn−1)TT(3)

satisﬁes the condition rank{QO}=n.

Throughout the paper, the dynamic system is always as-

sumed to be fully controllable and observable. Stability in

terms of the location of eigenvalues, is checked by extended

the well-known stability domain of LTI systems (the com-

plex left half-plane) to the more general commensurate

FLTI case, cf. Sabatier and Farges (2012).

Theorem 3. The system (1) is asymptotically stable if and

only if the following condition is satisﬁed

arg(eig(A))> γ π

2, γ ∈(0,2),(4)

where eig(A) represents the set of all eigenvalues of the

matrix A. A corresponding graphical interpretation of the

stability region is shown in the shaded areas in Fig. 1.

stable

region

unstable

region

γπ

2

Re

Im

(a) Order: γ∈(0,1).

stable

region

unstable

region

γπ

2

Re

Im

(b) Order: γ∈[1,2).

Fig. 1. Stability regions of FLTI systems.

In this paper, LMIs are used to verify and ensure the

global asymptotic stability of uncertain systems. Alongside

Lyapunov’s method, the bounded real lemma forms the

basis of many LMI approaches in robust control. Although

the bounded real lemma is used in both, linear and

nonlinear control engineering, the actual result is based on

the state-space representation of an LTI system (partially

after overbounding nonlinear dynamics). The worst case

performance of a stable system measured in terms of the

maximum ampliﬁcation between the input and output is

quantiﬁed by the H∞norm

kH(s)k∞= sup

Re(s)>0

δ(H(s)) = sup

ω∈R

δ(H(jω)),(5)

where δdenotes the maximum singular value. The H∞

norm for an integer-order system can be written in terms

of an LMI. The result is called the bounded real lemma,

see VanAntwerp and Braatz (2000).

Theorem 4. Let µ > 0 be a given real number and the

order of the system (1) be γ= 1. Then kH(s)k∞< µ is

equivalent to the existence of a symmetric matrix ∃P=

PT∈Rn×nsatisfying the LMIs

P A +ATP P B C T

BTP−µI DT

C D −µI

≺0,P0.(6)

Moreover, LMIs can be used to test whether the eigenval-

ues of a matrix belong to a speciﬁc area in the complex

plane. Thus, existing approaches do not try to extend the

bounded real lemma to FLTI systems of the form (1), but

rather to ﬁnd conditions describing the stability region in

terms of performance requirements such as damping ratios,

cf. Chilali et al. (1999), in Fig. 1.

Theorem 5. Let µ > 0 be a given real number and the

fractional order of the system (1) be in the range of γ∈

[1,2). Then, kH(s)k∞< µ is equivalent to the existence

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 10/43

Theoretic Background

Stability domains of linear commensurate, fractional-order systems

(t0Dγ

tx(t) = Ax(t) + Bu(t) + Ez(t),x(t0) = x0

y(t) = Cx(t) + Du(t) + F z(t), γ ∈(0,2)

Possible use: Design of interval observers

System matrices of the observer need to become stable Metzler matrices

Decoupled lower and upper bounding systems for veriﬁed enclosure of states despite

bounded parameter uncertainty and noise

T. Ra¨ıssi, A. Mohamed: On Robust Pseudo State Estimation of Fractional Order Systems, Positive Systems. POSTA. Lecture Notes in Control and

Information Sciences, pp. 97–111. Springer, Cham, 2017.

E. Hildebrandt, J. Kersten, A. Rauh, H. Aschemann: Robust Interval Observer Design for Fractional-Order Models with Applications to State Estimation

of Batteries, In Proc. of IFAC World Congress, Berlin, Germany, 2020.

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 10/43

Selected References

H. Delavari, P. Lanusse, J. Sabatier: Fractional Order Controller Design for A Flexible Link Manipulator Robot. Asian Journal of Control. Vol. 15, No. 3,

pp. 783–795, 2013.

B. Goodwine: Modeling a Multi-Robot System with Fractional-Order Diﬀerential Equations, 2014 IEEE International Conference on Robotics and

Automation (ICRA), Hong Kong, pp. 1763–1768, 2014.

S. Kempﬂe, I. Sch¨

afer and H. Beyer: Fractional Diﬀerential Equations and Viscoelastic Damping, 2001 European Control Conference (ECC), Porto,

pp. 1738–1743, 2001.

P. Lanusse, J. Sabatier, and A. Oustaloup: Extension of PID to Fractional Orders Controllers: A Frequency-Domain Tutorial Presentation. IFAC

Proceedings Volumes (IFAC-PapersOnline). Vol. 19, pp. 7436–7442, 2014.

M. Lazarevic: Biologically Inspired Control and Modeling of (Bio)Robotic Systems and Some Applications of Fractional Calculus in Mechanics, Theoretical

and Applied Mechanics, Vol. 40, pp. 163–187, 2013.

C. Monje, B. Vinagre, V. Feliu, Y, Chen: Tuning and Auto-Tuning of Fractional Order Controllers for Industry Applications. Control Engineering Practice,

Vol. 16, pp. 798–812, 2008.

M. Troparevsky, S. Seminara, M. Fabio: A Review on Fractional Diﬀerential Equations and a Numerical Method to Solve Some Boundary Value Problems.

IntechOpen, 2019, DOI:10.5772/intechopen.86273.

B. Zhou and W. Gu, Numerical Study of Some Intelligent Robot Systems Governed by the Fractional Diﬀerential Equations, IEEE Access, Vol. 7,

pp. 138548–138555, 2019.

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 11/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 12/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 12/43

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

References

1A. Rauh, et al.: Nonlinear State Observers and Extended Kalman Filters for Battery

Systems, AMCS, 23(3), 539–556, 2013.

2D. Andre, et al.: Characterization of High-Power Lithium-Ion Batteries by Electrochemical

Impedance Spectroscopy: I. Experimental Investigation, Journal of Power Sources,

196(12), 5334–5341, 2011.

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 12/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 13/43

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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 13/43

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)

References

1C. Zou, et al.: A Review of Fractional-Order Techniques Applied to Lithium-Ion Batteries,

Lead-Acid Batteries, and Supercapacitors Journal of Power Sources, 390, 286–296, 2018.

2D. Andre, et al.: Characterization of High-Power Lithium-Ion Batteries by Electrochemical

Impedance Spectroscopy: I. Experimental Investigation, Journal of Power Sources,

196(12), 5334–5341, 2011.

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 13/43

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

Goal: Analogously for fractional-order systems, restriction to the derivative orders 0< ν ≤1

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 14/43

Exponential Enclosure Technique

Desired properties

Avoidance of time-consuming gridding and Monte-Carlo simulations

Computation of guaranteed outer enclosures for all reachable states

Preservation of asymptotic stability by non-diverging interval bounds =⇒Counteracting

the well-known wrapping eﬀect

Alternative to Taylor series-based techniques such as VNODE-LP by N.S. Nedialkov

Reference

A. Rauh, R. Westphal, E. Auer, H. Aschemann: Exponential Enclosure Techniques for the Computation

of Guaranteed State Enclosures in ValEncIA-IVP, Reliable Computing: Special volume devoted to

material presented at SCAN 2012, Novosibirsk, Russia, Vol. 19, Issue 1, pp. 66-90, 2013.

A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 15/43