*4.1. Simplified Fractional-Order Battery Model*

Figure 4 illustrates a fractional-order equivalent circuit model of the charging/discharging behavior of a Lithium-ion battery, where the state of charge *σ*(*t*), its fractional derivative <sup>0</sup>D0.5 *<sup>t</sup> σ*(*t*) of order *ν* = 0.5, and the voltage drop *v*1(*t*) across the fractional-order constant phase element *Q* (as a generalization of an ideal capacitor [28–30]) are employed as pseudo-state variables.

**Figure 4.** Basic fractional-order equivalent circuit model of batteries according to [25].

These pseudo-state variables are summarized in the vector

$$\mathbf{x}(t) = \begin{bmatrix} \sigma(t) & \_0\boldsymbol{\otimes}\!\!/\_t^{0.5}\sigma(t) & v\_1(t) \end{bmatrix}^T \in \mathbb{R}^3 \text{ .}\tag{37}$$

Using the modeling steps described in [28] and generalizing the charging/discharging dynamics to

$$\mathbb{E}\_0 \mathbb{\hat{o}}\_t^1 \sigma(t) = -\frac{\eta\_0 \cdot i(t) + \eta\_1 \cdot \sigma(t) \cdot \text{sign}\left(i(t)\right)}{3600 \text{C}\_\text{N}} \tag{38}$$

as described in [25] with the terminal current *i*(*t*) as the system input, the commensurate fractional-order quasi-linear state equations

$$\mathbf{a}\_0 \boxtimes\_t^{0.5} \mathbf{x}(t) = \mathbf{a}\mathbf{f} \cdot \mathbf{x}(t) + \mathbf{f} \cdot i(t) \tag{39}$$

with the system and input matrices

$$\mathbf{M} = \begin{bmatrix} 0 & 1 & 0 \\ \frac{\eta\_1 \cdot \text{sign}\left(i(t)\right)}{3600 \text{C}\_N} & 0 & 0 \\ 0 & 0 & -\frac{1}{\mathbb{K}Q} \end{bmatrix} \quad \text{and} \quad \mathfrak{E} = \begin{bmatrix} 0 \\ -\frac{\eta\_0}{3600 \text{C}\_N} \\ \frac{1}{\mathbb{Q}} \end{bmatrix} \tag{40}$$

are obtained.

The simulations discussed in the following two subsections consider the parameters listed in Table 1 which are a subset of those used in [25]. To make the evaluation based on the novel exponential enclosures for the Mittag-Leffler function according to Theorem 3 comparable with the previous work reported in [25], we employ the same linear state feedback controller for the discharging phase (*i*(*t*) > 0) that was designed in terms of an assignment of the asymptotically stable eigenvalues *λ* ∈ {−0.0001; −0.0002; −0.4832} to the closed-loop dynamics. This leads to the closed-loop dynamic model

$$
\boldsymbol{\Omega}\_t^{0.5} \mathbf{x}(t) = \left(\boldsymbol{\mathcal{A}} - \boldsymbol{\mathcal{J}} \boldsymbol{\mathcal{A}}^T\right) \cdot \mathbf{x}(t) =: \boldsymbol{\mathcal{A}}\_\mathbb{C} \cdot \mathbf{x}(t) \; , \tag{41}
$$

where the matrix entries AC,2,1, AC,2,2, AC,2,3 are inflated to interval parameters by symmetric bounds of a 1% radius of the respective nominal quantity and AC,3,3 to 10%, respectively.

**Table 1.** Parameters of the Lithium-ion battery model.


To investigate the effect of the novel exponential enclosure approach for Mittag-Leffler functions, the following simulations also make use of the intersection of the solution to the system model (41) with an alternative representation resulting from a time-invariant similarity transformation of the system matrix **A**<sup>C</sup> into an interval-valued diagonally dominant representation. This transformation, as detailed in [25], employs the matrix of eigenvectors for a matrix containing the elementwise-defined interval midpoints of **A**C. As discussed in [25], this transformation reduces the overestimation due to the wrapping effect of interval analysis [23,31] when evaluating both iteration Formulas (31) and (34).

For the rest of this paper, consider the two sets of initial pseudo-state vectors

$$\mathbf{x}(0) \in [\mathbf{x}]\_1(0) = \begin{bmatrix} [&0.9000 \ \vdots & 1.1000] \\ [-0.0011 \ \vdots & -0.0009] \\ [& 0.0900 \ \vdots & 0.1100] \end{bmatrix} \tag{42}$$

and

$$\mathbf{x}(0) \in [\mathbf{x}]\_2(0) = \begin{bmatrix} [&0.99000 \ \vdots & 1.01000] \\ [-0.00101 \ \vdots & -0.00099] \\ [& 0.09900 \ \vdots & 0.10100] \end{bmatrix} \tag{43}$$

in the sense of a Caputo-type initialization of the controlled battery model (41), i.e., with **x**(0) corresponding to **x**(*t*) for all *t* < 0.

In all simulations summarized in the following two subsections, the final points in time *T* for the evaluation of the iteration Formulas (31) and (34) are chosen as either of the ten values

$$T \in \left\{ 0.5, 0.5 + \frac{10 - 0.5}{9}, 0.5 + 2 \cdot \frac{10 - 0.5}{9}, \dots, 10 \right\} \tag{44}$$

with *<sup>t</sup><sup>ξ</sup>* − *<sup>t</sup>ξ*−<sup>1</sup> = 0.005. Note that this latter choice for the interval subdivision is not identical to the one used in [25], where for all the different values *T* the considered time ranges were always subdivided into 100 equally spaced slices.
