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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
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
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
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
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+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
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 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
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
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
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
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
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
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
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)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
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)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
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)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
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],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
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
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 16/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Exponential Enclosure Technique
Derivation of the iteration scheme
Picard iteration
x∗(t)∈[xe]hκ+1i(t) := [x0] +
t
Z
0
f[xe]hκi(s)ds
Differentiation with respect to time and evaluation for t∈[0 ; T]
˙
x∗([0 ; T]) ∈diag n[λi]hκ+1io·exp[Λ]hκ+1i·[0 ; T]·[xe] (0)
⊆fexp[Λ]hκi·[0 ; T]·[xe] (0)
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 17/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Exponential Enclosure Technique
Derivation of the 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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 17/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Fractional-Order System Models (1)
Fractional-order differential equations (FDEs) of Caputo type
Quasi-linear state equations
x(ν)(t) = f(x(t)) = A(x(t)) ·x(t)
Initial conditions specified at t= 0
Exact solution of the FDE for the linear case
Scalar, linear fractional-order state equation
x(ν)(t) = λ·x(t)
with the initial condition x0=x(0)
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 18/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Fractional-Order System Models (2)
Exact solution of the FDE for the linear case
Scalar, linear fractional-order state equation
x(ν)(t) = λ·x(t)
with the initial condition x0=x(0)
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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 19/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Fractional-Order System Models (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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 20/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Derivation of an Iterative Solution Procedure (1)
Generalization of exponential enclosures to the FDE case
Redefinition of the assumption of 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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 21/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Derivation of an Iterative Solution Procedure (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)
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 22/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Derivation of an Iterative Solution Procedure (3)
Resulting iteration scheme
[λi]hκ+1i:= aii [xe]hκi([t])+
n
X
j=1
i6=j
aij [xe]hκi([t])·
Eν,1[λj]hκi·[t]ν
Eν,1[λi]hκi·[t]ν·[xe,j ] (0)
[xe,i] (0)
Advantageous to perform a state-space transformation into a new coordinate frame zin which
the state equations are decoupled as far as possible with the equilibrium in the origin
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 23/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.10
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.15
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.20
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.25
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.30
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.35
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.40
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.45
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.50
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.55
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.50
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.65
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.70
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.75
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.80
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.85
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.90
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.95
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 1.00
Real part
<{Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
Imaginary part
={Eν,1(z)}
<{z}
={z}
−10
0
−10 −50
−1
0
1
10
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 24/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.10
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.15
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.20
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.25
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.30
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.35
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.40
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.45
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.50
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.55
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.50
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.65
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.70
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.75
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.80
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.85
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.90
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 0.95
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval Evaluation of the Mittag-Leffler Function
Eν,1(z),z∈C,ν= 1.00
Real part
<{Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
Imaginary part
={Eν,1(z)}
={z}
<{z}
−10
0010
−10 −5
−1
0
1
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 25/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Consideration of the Infinite Memory Property
Quantification of temporal truncation errors
|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]
Useful contractor
[˜
λi] = ˜
fi([xe] ([tk+T;t∗])) ∩˜
fi([xref ] ([tk+T;t∗]))
[xe,i] ([tk+T;t∗]) ∩[xi,ref ] ([tk+T;t∗])
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 26/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
So far (obvious)
Use the Mittag-Leffler-type enclosures for a longer time span, i.e., typically enclosures with
start of the derivative operator at an earlier point of time
Other options (1)
Gr¨
unwald-Letnikov formulation of the derivative operator (step size: ∆T)
x(tk+1) = νI·x(tk)+∆T·f(x(tk)) −
∞
X
i=2
(−1)i·ν
i·x(tk+1−i)
Necessity for interval evaluation of the infinitely long sum up to some i < ∞and
bounding the remainder in guaranteed way
Drawback: Slow convergence of this expansion
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 27/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Other options (2)
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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 28/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Other options (2)
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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 28/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) = 1,ti+1 −ti= 4 ·10−4,κ∈ {2,...,5}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 29/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) = 1,ti+1 −ti= 4 ·10−4,κ∈ {6,...,10}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 30/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) = 1,ti+1 −ti= 4 ·10−4,κ∈ {11,...,15}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 31/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) = 1,ti+1 −ti= 4 ·10−4,κ∈ {46,...,50}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 32/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) ∈[0.9 ; 1.1],ti+1 −ti= 2 ·10−4,κ∈ {2,...,5}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 33/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) ∈[0.9 ; 1.1],ti+1 −ti= 2 ·10−4,κ∈ {6,...,10}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 34/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) ∈[0.9 ; 1.1],ti+1 −ti= 2 ·10−4,κ∈ {11,...,15}
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 35/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Other options (3)
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, substituting into the (nonlinear) fractional-order 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]
These options (1)–(3) can be used if 0∈[xi] (t)for some i∈ {1, . . . , n}, where the
Mittag-Leffer enclosures are not applicable
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 36/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Other options (3)
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, substituting into the (nonlinear) fractional-order 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 reading: Picard iteration technique (not interval-based in the sense of this talk)
R. Lyons, V.S. Vatsala, R.A. Chiquet: Picard’s Iterative Method for Caputo Fractional Differential Equations
with Numerical Results. Mathematics. 2017; 5(4):65.
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 36/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x,x(0) = 1,t= 0.1,E0.5,1(−0.10.5)≈0.723578438477616
L= 1,x(t)∈[0.51585122229592 ; 0.77049913124297]
L= 2,x(t)∈[0.70749269444638 ; 0.77885765909251]
L= 3,x(t)∈[0.71089859158917 ; 0.72787511885232]
L= 4,x(t)∈[0.72260273110458 ; 0.72617097933690]
L= 5,x(t)∈[0.72309579181353 ; 0.72377485290406]
L= 10,x(t)∈[0.72357841848901 ; 0.72357847795982]
L= 15,x(t)∈[0.72357843847658 ; 0.72357843847820]
L= 20,x(t)∈[0.72357843847761 ; 0.72357843847762]
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 37/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Computation of the Reference Solutions [xi,ref ] ([tk+T;t∗])
Example: x(0.5) =−x3,x(0) = 1,t= 0.02
L= 1,x(t)∈[0.76402779810064 ; 0.91681837757821]
L= 2,x(t)∈[0.85950797443780 ; 0.94133820124105]
L= 3,x(t)∈[0.85027277294607 ; 0.89602057301508]
L= 4,x(t)∈[0.87354773949736 ; 0.89969321255168]
L= 5,x(t)∈[0.87207743027399 ; 0.88721927623248]
L= 6,x(t)∈[0.87894132726984 ; 0.88778811489730]
L= 7,x(t)∈[0.87874194996238 ; 0.88394324648305]
L= 8,x(t)∈[0.88092344966538 ; 0.88399577967748]
L= 9,x(t)∈[0.88092471319144 ; 0.88274607728973]
L= 10,x(t)∈[0.88164599262517 ; 0.88272887011466]
Note
The intersection of intervals for different orders is possible
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 38/43
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 39/43
Why? Battery Models Problem Statement 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
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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 40/43
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 41/43
Why? Battery Models Problem Statement 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: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 42/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Conclusions and Outlook on Future Work
Computationally efficient verified simulation of fractional-order system models
Readily applicable for use in predictive control strategies, system identification and state
estimation
Handling of uncertainty in initial conditions and parameters
Further work on the case of complex eigenvalues required
Refinement of error bounds for temporal truncation errors
Further development of optimized step-size control procedures
Point out further similarities between series-based solutions for classical integer-order and
fractional-order differential equations
Solution representation by further fundamental shapes such as ellipsoids
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 43/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Conclusions and Outlook on Future Work
Computationally efficient verified simulation of fractional-order system models
Readily applicable for use in predictive control strategies, system identification and state
estimation
Handling of uncertainty in initial conditions and parameters
Further work on the case of complex eigenvalues required
Refinement of error bounds for temporal truncation errors
Further development of optimized step-size control procedures
Point out further similarities between series-based solutions for classical integer-order and
fractional-order differential equations
Solution representation by further fundamental shapes such as ellipsoids
A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods 43/43
Why? Battery Models Problem Statement Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Related publications
Rauh, Andreas; Westphal, Ramona; Aschemann, Harald: Verified Simulation of Control Systems with Interval Parameters Using an Exponential State
Enclosure Technique, Proc. of IEEE Intl. Conference on Methods and Models in Automation and Robotics MMAR 2013, Miedzyzdroje, Poland, 2013
Rauh, Andreas; Westphal, Ramona; Aschemann, Harald; Auer, Ekaterina: Exponential Enclosure Techniques for Initial Value Problems with Multiple
Conjugate Complex Eigenvalues, 16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics
SCAN2014, W˜
A¼rzburg, Germany, 2014. Vol. 9553 of Lecture Notes in Computer Science, pp. 87–122
Rauh, Andreas; Kersten, Julia; Aschemann, Harald: Techniques for Verified Reachability Analysis of Quasi-Linear Continuous-Time Systems, Proc. of IEEE
Intl. Conference on Methods and Models in Automation and Robotics MMAR 2019, Miedzyzdroje, Poland, 2019
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
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A. Rauh: Fractional-Order System Models and Their Verified Numerical Analysis Using Interval Methods
