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# Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods

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## Abstract and Figures

Verified integration of initial value problems for sets of ordinary differential equations were investigated in many research projects and resulted in numerous software implementations over the last decades. However, ordinary differential equations with integer orders of the derivatives are not the only relevant dynamic system models in modeling dynamic systems in engineering as well as computational physics, biology and other disciplines. One of the emerging modeling assumptions studied over the recent years is the use of fractional-order models that are characterized by non-integer order temporal derivatives. This presentation briefly touches upon their use in modeling applications from the areas of robotics and energy systems as well as for describing dynamic elements that are useful for so-called loop shaping techniques in control design. Novel, interval-based simulation routines for fractional-order systems are presented with an illustrating example from the field of modeling Lithium-ion batteries.
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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
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 4/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
A Frequency-Domain Point of View
Amplitude and phase response
A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 5/43
Why? Battery Models Problem Statement 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 90and
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
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 7/43
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 7/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
A Further Simple Example
Approximation by integer-order model
Amplitude response
magnitude in dB
0
30
20
10
ω ω101
103101103
Phase response
0
0.2
0.4
0.6
0.8ω ω103101101103
−−−fractional model; integer order 5; integer order 11
A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 8/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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+sν2KI, ν1>0, ν2>0
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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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
A. Rauh: Fractional-Order System Models and Their Veriﬁed Numerical Analysis Using Interval Methods 9/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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 Hmethodology
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 x1x2and M1M2are
understood element-wise. The relation M0 (M0)
means that the matrix MCn×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 uRp,
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(sγIA)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. . . An1B(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. . . (CAn1)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 Hnorm
kH(s)k= sup
Re(s)>0
δ(H(s)) = sup
ωR
δ(H()),(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=
PTRn×nsatisfying the LMIs
P A +ATP P B C T
BTPµI 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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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
Why? Battery Models Problem Statement 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
Phase variations do not correspond to integer multiples of ±π
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
Why? Battery Models Problem Statement 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
Phase variations do not correspond to integer multiples of ±π
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
Why? Battery Models Problem Statement 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
Phase variations do not correspond to integer multiples of ±π
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
Why? Battery Models Problem Statement 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)
a0+. . . +an(ω), 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)iTR3(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
ckσk1(t)01#·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
Why? Battery Models Problem Statement 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)
a0+. . . +an(ω), 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)iTR3(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
ckσk1(t)01#·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
Why? Battery Models Problem Statement 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)
a0+. . . +an(ω), 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
Why? Battery Models Problem Statement 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],xixixi,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
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
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