Robust Interval Observer Design for Fractional-Order Models with Applications to State Estimation of Batteries

Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis

“…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 v 1 (t) across the fractional-order constant phase element Q which 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 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…”

“…The following numerical simulations are based on the system parameters summarized in Tab. 1 which were -except for η 1 -identified according to the experimental data referenced in [10,22] during multiple charging and discharging cycles before the occurrence of aging. Now, a linear state feedback controller with the structure presented in [11] for the discharging phase (i(t) > 0) is parameterized by assigning the asymptotically stable eigenvalues λ ∈ {−0.0001; −0.0002; −0.4832} to the closed-loop dynamics, where the last value corresponds to the fastest asymptotically stable dynamics of the open-loop system.…”

“…Such applications can be found, for example, in simulation, control design, its validation and verification, the design of state-of-charge estimators and model-based aging detection for battery systems. The solution of these tasks relies on 1. deriving state equations for an electric equivalent circuit model in which mostly capacitive elements are generalized by corresponding fractional-order constant phase elements, 2. identifying suitable system parameters by means of impedance spectroscopy, and finally 3. evaluating the resulting system models numerically for a certain finite time horizon [10,23,24].…”

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

“…Fractional differential equations (FDEs) are powerful modeling tools in many engineering applications in which non-standard dynamics, characterized by infinite horizon states, can be observed [21,23,37,40]. Examples for such applications are modeling the charging and discharging dynamics of batteries [11], the identification of dynamic system models by means of impedance spectroscopy [2] if amplitude and phase variations do not correspond to integer multiples of ±20 dB and ± π 2 per frequency decade, respectively, modeling of multi-robot systems [9], control design for flexible manipulators [4], and generally for the representation of dynamic systems with long-term memory effects. Moreover, advanced models of visco-elastic damping [13] can be described with the help of FDEs.…”