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Verification Techniques for Fractional Differential Equations with Bounded Uncertainty

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Abstract

Fractional differential equations (FDEs) are powerful for modeling many applications with non-standard dynamics due to memory effects concerning an infinite horizon of previous state evolutions. Practical scenarios, in which such phenomena can be observed, include the charging and discharging dynamics of batteries, rheological spring-damper elements with a behavior "between" Newton's and Hooke's laws, or models from the field of epidemiology. This presentation gives an overview of current developments toward a verified simulation of FDEs, which make use of interval extensions of Mittag-Leffler functions, Picard iteration schemes, and appropriate temporal series expansions. To correctly account for the initialization of pseudo-states that are non-constant over a bounded past time window and to allow - among others - for the reinitialization of fractional integrators in state estimation of continuous-time systems with discrete-time measurements, a novel observer-based scheme for the quantification of time-domain truncation errors is proposed and visualized in simulations.
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Verification Techniques for Fractional Differential Equations
with Bounded Uncertainty
2e Journ´ees Virtuelles de la SAGIP, GT VS-CPS
02 Juillet 2021
Andreas Rauh1, Luc Jaulin1, Rachid Malti2
1Lab-STICC (Robex), ENSTA Bretagne, Brest, France
2IMS Laboratory, University of Bordeaux, France
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 1/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Contents
Motivation – Or why should we deal with fractional-order system models?
Generalization of an exponential enclosure technique to fractional differential equations
Alternative verified simulation methods
Consideration of temporal truncation errors
Application: Prediction of the state-of-charge for a fractional-order battery model
Observer-based quantification of truncation errors
Conclusions and outlook on future work
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 2/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
A Time-Domain Point of View
Comparison of “velocity” y(1)(t)and “position” y(t)
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 3/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
A Time-Domain Point of View
Observations
The time response seems to be partially faster and partially slower than an exponential
The time response seems to be impossible to be explained by a linear first- or
second-order model
Do we really need to identify a nonlinear representation?
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 4/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
A Frequency-Domain Point of View
Amplitude and phase response
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 5/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
A Frequency-Domain Point of View
Observations
Amplitude response does not only show slopes of ±20 dB per frequency decade
The phase response seems to have a characteristic value not only around 90and
180, but also at ≈ −54
Is it necessary to expand the amplitude and phase responses into series of classical first-
and second-order transfer function blocks?
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 6/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Resolution
Considered system model
y(2)(t)+3y(0.6)(t)+y(t) = u(t)
Fractional-order attenuation term
The system model is linear
However, not all contained derivatives have an integer order
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 7/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Resolution
Considered system model
y(2)(t)+3y(0.6)(t)+y(t) = u(t)
Fractional-order attenuation term
Representation that lies between the dependency of stress (σ(t)) and strain ((t))
according to Newton’s law
σ(t) = ηd(t)
dtwith the viscosity η
and Hooke’s law
σ(t) = E(t)with the modulus of elasticity E
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 7/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Modeling and Identification by Means of Impedance Spectroscopy
Typical amplitude and phase response vs. integer-order system models
Amplitude variations do not correspond to integer multiples of ±20dB per frequency
decade
Phase variations do not correspond to integer multiples of ±π
2per frequency decade
Thevenin equivalent circuit consisting of series connections of resistors and RC submodels
Approximation of frequency response characteristics over a finite bandwidth
v
OC
v
T
+
i
T
C
TS
R
S
R
T S
v
T S
C
T L
v
T L
R
T L
®
¯°
5336 D. Andre et al. / Journal of Power Sources 196 (2011) 5334–5341
0 0.05 0.1 0.15 0.2
0
0.05
0.1
0.15
Z’ / Ω
−Z’’ / Ω
Measurement after 0 min
Measurement after 30 min
Measurement after 50 min
Fig. 6. Effect of pause time after SOC variation (SOC =10%) on impedance spectra
at 30 C and 50% SOC.
0 0.005 0.01 0.015 0.02 0.025
0
0.005
0.01
−Z´´ / Ω
Z´ / Ω
Measurement 1 (started after 70 min)
Measurement 2 (started after 90 min)
Measurement 3 (started after 115 min)
Measurement 4 (started after 170 min)
Fig. 7. Effect of pause time after temperature variation (T= 30 K) on impedance
spectra at 10 C and 60% SOC.
Fig. 8. Approximation of internal cell resistance Rby seven points as a function of
temperature.
T = 50°C T = 24°C T = 4°C
02468
x 10−3
0
2
x 10−3
Z’ / Ω
−Z’’ / Ω
Fig. 9. Impedance spectra for positive temperatures at 60% SOC.
0 0.05 0.1 0.15 0.2
0
0.05
Z’ / Ω
−Z’’ / Ω
T = −9°C T = −18°C T = −24°C T = −30°C
Fig. 10. Impedance spectra for negative temperatures at 60% SOC.
Fig. 11. Bode plot of EIS measurements at 60% SOC.
−20 0 20 40
1
1.5
2
Temperature / °C
RΩ / mΩ
(a)
33.54
x 10−3
−0.2
0
0.2
0.4
0.6
0.8
1/Temperature / 1/K
log(RΩ) / mΩ
(b)
Fig. 12. Influence of temperature (a) on Rat 60% SOC and (b) Arrhenius plot of R.
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 8/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Modeling and Identification by Means of Impedance Spectroscopy
Typical amplitude and phase response vs. fractional-order system models
Generalization of the Thevenin equivalent circuit by a replacement of capacitors (and
possibly inductors) by respective constant phase elements
Generalization of frequency response functions by expressions of the form
b0+. . . +bm(ω)
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 et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 9/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Initial Value Problem with Interval Uncertainty
Definition of the initial value problem (IVP)
Given set of ordinary differential equations (ODEs)
˙
x(t) = f(x(t))
with smooth right-hand sides
Uncertain initial conditions
x(0) [x0] := [x(0)] = [x(0) ; x(0)]
Component-wise definition of interval vectors [x] = [x1]. . . [xn]Twith the vector
entries [xi]=[xi;xi],xixixi,i= 1, . . . , n
Generalization for fractional differential equations with derivative orders 0< ν 1?
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 10/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Exponential Enclosure Technique
Definition of the state enclosure
Representation of contracting state enclosures by using
x(t)[xe] (t) := exp ([Λ]·t)·[xe] (0)
with 06∈ [xe,i] (0),[xe] (0) = [x0]for the diagonal matrix
[Λ] := diag {[λi]}, i = 1, . . . , n
with element-wise negative real entries λi
Definition of the interval matrix exponential
exp ([Λ]·t) := diag {exp ([λ1]·t),...,exp ([λn]·t)}
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 11/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Exponential Enclosure Technique
Interval-valued iteration scheme
Picard iteration
x(t)[xe]hκ+1i(t) := [x0] +
t
Z
0
f[xe]hκi(s)ds
Resulting iteration formula
[λi]hκ+1i:=
fiexp [Λ]hκi·[0 ; T]·[xe] (0)
exp [λi]hκi·[0 ; T]·[xe,i ] (0)
, i = 1, . . . , n
with the guaranteed state enclosure at the point t=T
x(T)[xe] (T) := exp ([Λ]·T)·[xe] (0)
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 12/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Fractional Differential Equations (1)
Fractional differential equations (FDEs) of Caputo type
Generally nonlinear pseudo state equations
x(ν)(t) = f(x(t))
Initial conditions specified at t= 0 (temporally constant initialization)
Exact solution of the FDE for the linear case
Scalar, linear fractional pseudo state equation
x(ν)(t) = λ·x(t)
with the initial condition x0=x(0) that also describes the entire past of the pseudo state
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 13/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Fractional Differential Equations (2)
Exact solution of the FDE for the linear case
Scalar, linear fractional pseudo state equation
x(ν)(t) = λ·x(t)
with the initial condition x0=x(0) that also describes the entire past of the pseudo state
Explicit solution
x(t) = Eν,1(λtν)·x(0)
depending on the two-parameter Mittag-Leffler function
Eν,β (ζ) =
X
i=0
ζi
Γ (νi +β)
with the general argument ζCand the parameters νR+,βR
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 14/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Fractional Differential Equations (3)
Note 1
For ν= 1, the two-parameter Mittag-Leffler function turns into the classical exponential
function according to the solution
x(t) = E1,1λt1·x(0)
with
E1,1(ζ) =
X
i=0
ζi
Γ (i+ 1) =eζ
Note 2
FDE systems are characterized by an infinite memory of previous states, i.e., subdividing
the integration time horizon into short slices [tk;tk+1], where the state at t=tk+1 serves as
the initialization for the following slice, is prone to approximation errors
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 15/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Iterative Solution Procedure for Fractional System Models (1)
Generalization of exponential enclosures to the FDE case
Definition of the pseudo state enclosures
[xe]hκi(t) = Eν,1[Λ]hκi·tν·[xe] (0)
Diagonal matrix containing the evaluation of the scalar Mittag-Leffler function
Eν,1[Λ]hκi·tν
with
[Λ] := diag {[λi]}, i = 1, . . . , n
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 16/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Iterative Solution Procedure for Fractional System Models (2)
Element-wise notation of the fixed-point iteration
diag n[λi]hκ+1io·Eν,1[Λ]hκ+1i·[t]ν·[xe] (0)
|{z }
[xe]hκ+1i([t]) [xe]hκi([t])
=: fEν,1[Λ]hκi·[t]ν·[xe] (0)
Detailed derivation
Rauh, Andreas; Kersten, Julia; Aschemann, Harald: Interval-Based Verification Techniques for the Analysis of Uncertain Fractional-Order System Models,
2020 European Control Conference (ECC), Saint Petersburg, Russia, pp. 1853–1858, 2020.
Rauh, Andreas; Kersten, Julia: Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems, Acta
Cybernetica, 2020. DOI: 10.14232/actacyb.285660.
Rauh, Andreas; Kersten, Julia: Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis, Proceedings 6th
International Workshop on Symbolic-Numeric Methods for Reasoning about CPS and IoT (SNR 2020), Electronic Proceedings in Theoretical Computer
Science 331, pp. 18–32. 2021. DOI: 10.4204/EPTCS.331.2
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 17/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Iterative Solution Procedure for Fractional System Models (2)
Element-wise notation of the fixed-point iteration
diag n[λi]hκ+1io·Eν,1[Λ]hκ+1i·[t]ν·[xe] (0)
| {z }
[xe]hκ+1i([t]) [xe]hκi([t])
=: fEν,1[Λ]hκi·[t]ν·[xe] (0)
Necessity for an interval extension of Mittag-Leffler functions
Rauh, Andreas; Kersten, Julia; Aschemann, Harald: Interval-Based Verification Techniques for the Analysis of Uncertain Fractional-Order System Models, 2020
European Control Conference (ECC), Saint Petersburg, Russia, pp. 1853–1858, 2020.
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 17/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Option 1: Interval-based Picard iteration
Multi time step Picard iteration
xhκ+1i(t) = x(0) + 1
Γ (ν)·
t
Z
0
(tτ)ν1·fxhκi(τ)dτ
Interval formulation
xhκ+1i(tn)x(0) + 1
Γ (ν)·
n1
X
i=0
ti+1
Z
ti
(tnτ)ν1·fhxhκii(τ)
| {z }
[f]i+1
i
dτ
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 18/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Option 1: Interval-based Picard iteration
Interval formulation
xhκ+1i(tn)x(0) + 1
Γ (ν)·
n1
X
i=0
ti+1
Z
ti
(tnτ)ν1·fhxhκii(τ)
| {z }
[f]i+1
i
dτ
Analytic overapproximation of the integral
xhκ+1i(tn)x(0) + 1
Γ (ν)·
n1
X
i=0
[f]i+1
i·(tnti)ν(tnti+1)ν
ν
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 18/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Example: x(0.5) =x3,x(0) = 1,ti+1 ti= 4 ·104,κ∈ {2,...,5}
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 19/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Example: x(0.5) =x3,x(0) = 1,ti+1 ti= 4 ·104,κ∈ {6,...,10}
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 20/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Example: x(0.5) =x3,x(0) = 1,ti+1 ti= 4 ·104,κ∈ {11,...,15}
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 21/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Example: x(0.5) =x3,x(0) = 1,ti+1 ti= 4 ·104,κ∈ {46,...,50}
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 22/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Alternative Solution Techniques
Series-based solution representation
Truncated series expansion of the solution (also in the complex-valued case!)
x(t)
L
X
i=0
[xi·ν]·ti·ν
Γ(1 + i·ν)+ [R]
Differentiation and substitution into the fractional differential equation model
Polynomial nonlinearities: Analytic (interval) expressions for the initial condition x0as
well as for the terms of order i∈ {1, . . . , L}
Finally compute guaranteed bounds [R]
Further examples (e.g. from the field of epidemiology)
A. Rauh, L. Jaulin: Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations,
Fractal Fract, 5(1), 17, 2021
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 23/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Consideration of the Infinite Memory Property
Quantification of temporal truncation errors (Podlubny)
|tkDν
tx(t)tk+TDν
tx(t)| ≤ XTν
|Γ(1 ν)|=: µ
with the component-wise defined supremum of the reachable states according to
Xi= sup
t[tk;tk+1]
|xi(t)|, i ∈ {1, . . . , n}
and a subsequent inflation of the right-hand side of the system model
˜
f(x(t)) := f(x(t)) + [µ;µ]
Note
The given bounds µare conservative, especially due to the fact that they are temporally
constant
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 24/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Visualization of the Effect of Truncation Errors
Uncertain FDE model x(0.5)(t) = x(t),x(t) = 1 for t0
t
x(t)
0
0
0.4
0.2
0.6
1.0
0.8
exact solution
2 4 6 8 10
T= 0.1
T= 1
t
x(t)
0
1.0
2 4 6 8 10
exact solution
0.2
0
0.4
0.6
0.8
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 25/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Illustrating Example: Mittag-Leffler-Type Enclosures
Uncertain FDE model x(ν)(t) = p·x3(t)
[x] (0) = [0.99 ; 1.0],[p] = [2 ; 1.99],[ν] = [0.8 ; 0.81]
t
[x] (t)
0
0
0.4
0.2
0.6
1.0
0.8
0.2 0.4 0.6 0.8 1.0
[z] (t)
t
0
0
0.4
0.2
0.6
1.0
0.8
0.2 0.4 0.6 0.8 1.0
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 26/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Simplified Battery Model (1)
Fractional-order equivalent circuit model
A. Rauh & J. Kersten 11
6.2 Simplified Fractional-Order Battery Model
As a second application scenario, consider the fractional-order battery model depicted in Fig. 3. It
describes the dynamics of the charging and discharging behavior of a Lithium-Ion battery with the help
of the state of charge σ(t)as well as with the dynamics of the exchange of charge carriers in the interior
of the battery cell which is related to electrochemical double layer effects. The latter is represented with
the help of the voltage drop v1(t)across the fractional-order constant phase element Qwhich serves as
a generalization of an ideal capacitor according to [1, 10, 24]. This kind of constant phase element was
already motivated by the example in Sec. 3.
vOC(σ(t)) +
i(t)R0
Rv(t)
+
Q
v1(t)
Figure 3: Basic fractional-order equivalent circuit model of batteries.
For the following investigation of the proposed interval-based simulation approach for fractional-
order system models, assume that the state vector
x(t) = hσ(t)0D0.5
tσ(t)v1(t)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
Mathematical formulation
Pseudo state vector x(t) = σ(t)0D0.5
tσ(t)v1(t)TR3
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.
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Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Simplified Battery Model (2)
Mathematical formulation
State equations
0D0.5
tx(t) = A·x(t) + b·i(t)
with the system and input matrices
A=
0 1 0
η1·sign(i(t))
3600CN0 0
0 0 1
RQ
and b=
0
η0
3600CN
1
Q
Terminal voltage
v(t) = 4
P
k=0
ckσk1(t) 0 1·x(t) + R0+d0ed1σ(t)·i(t)
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 28/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Simulation Results
Prediction of state-of-charge and terminal voltage (control with pole assignment)
State-of-charge
tin s
[σ] (t)
0 2 4 6 8 10
0.85
1.00
1.05
1.10
0.90
0.95
1.15
Terminal voltage
tin s
[v] (t) in V
0 2 4 6 8 10
2.0
2.5
3.0
3.5
4.0
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 29/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval-Observer Technique for Improved Bounds of Truncation Errors (1)
Reference model: virtual measurements for a cooperative dynamic system model
Pseudo state enclosures
x(t)[v(t) ; w(t)] with 0D(ν)
tx(t) = f(x(t))
Without loss of generality: temporally constant initialization
x(t)[x0],˙
x(t) = 0for t0
Bounding systems
0D(ν)
tv(t) = fv(v(t)) ,v(t) = x0and
0D(ν)
tw(t) = fw(w(t)) ,w(t) = x0
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 30/35
Motivation Exponential Enclosures Fractional Systems Series & Truncation Simulations Conclusions
Interval-Observer Technique for Improved Bounds of Truncation Errors (2)
Observer-based approach
Denote the truncation error bounds at t=T > 0by [µT;µT]
Augmented observer model: z(t) = ˜
vT(t)˜
wT(t)µT
v(t)µT
w(t)T
TD(ν)
tz(t) =
fv(˜
v(t)) + µv(t)
fw(˜
w(t)) + µw(t)
0
0
+H·v(t)˜
v(t)
w(t)˜
w(t)with z(T) =
v(T)
w(T)
µT
µT
Choose the observer gain Hsuch that [v(t) ; w(t)] [˜
v(t) ; ˜
w(t)]
Applicable to (periodically) reset the fractional integrator for the differential equation model
A. Rauh et al.: Verification Techniques for Fractional Differential Equations with Bounded Uncertainty 31/35