# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging
## 5 Design of the Fast Charging Robust Controller
### CRONE Control Methodology for Robust Controller Design  
The controller \(C(s)\) in Figure 6 was designed using the CRONE control methodology. It uses fractional differentiation to define high-level design parameters [26] that are tuned in the frequency domain. The third generation CRONE control was used in this work. It is based on the optimization of the transfer function \(\beta_{nom}(s)\) parameters (\(\beta_{nom}(s)\) being computed for the nominal plant denoted \(G_{nom}\)) defined by  
\[\beta_{nom}(s)=\beta_{l}(s)\beta_{m}(s)\beta_{h}(s) \tag{27}\]  
in which \(\beta_{l}(s)\) is an integer proportional integrator of order \(n_{l}\) defined by  
\[\beta_{l}(s)=C_{l}\bigg{(}\frac{\omega_{N^{-}}}{s}+1\bigg{)}^{n_{l}} \tag{28}\]  
\(\beta_{m}(s)\) groups several band-limited generalized templates  
\[\beta_{m}(s)=\prod_{k=-N^{-}}^{N^{+}}\beta_{mk}(s) \tag{29}\]  
with  
\[\beta_{mk}(s)=C_{k}\bigg{(}\alpha_{k}\frac{1+\frac{s}{\omega_{k+1}}}{1+\frac{ s}{\omega_{k}}}\bigg{)}^{a_{k}}\bigg{(}\Re_{k}\Bigg{\{}\Bigg{(}\alpha_{k}\frac{1+ \frac{s}{\omega_{k+1}}}{1+\frac{s}{\omega_{k}}}\Bigg{\}}^{\beta_{k}}\Bigg{)} \Bigg{)}^{-q_{k}sign(b_{k})} \tag{30}\]  
where \(\alpha_{k}=\big{(}\frac{\omega_{k+1}}{\omega_{k}}\big{)}^{0.5}\) for \(k\
eq 0\), and \(\alpha_{0}= \begin{pmatrix}1+\big{(}\frac{\omega_{r}}{\omega_{0}}\big{)}^{2}\\ 1+\big{(}\frac{\omega_{r}}{\omega_{1}}\big{)}^{2}\end{pmatrix} ^{0.5}\) and \(\beta_{h}(s)\) is a low-pass filter of order \(n_{h}\):  
\[\beta_{h}(s)=C_{h}\bigg{(}\frac{s}{\omega_{N^{+}}}+1\bigg{)}^{-n_{h}} \tag{31}\]  
Order \(n_{l}\) must be adjusted to get the desired controller accuracy. The constants \(C_{l}\), \(C_{k}\) and \(C_{h}\) are tuned to ensure a given resonance frequency \(\omega_{r}\) of the closed loop transfer function \(T_{nom}(s)=\beta_{nom}(s)\)/(1 + \(\beta_{nom}(s)\)). Order \(n_{h}\) was chosen to ensure a bi-proper or strictly proper controller. The open-loop parameters are tuned \(M_{T}\), where \(M_{T}\) denotes the resonance peak of the complementary sensitivity function \(T(s)\). Such an objective is attained by minimizing the objective function  
\[J=sup_{G}M_{T}-M_{Tnom} \tag{32}\]  
Figure 6: Fast charging control closed-loop. in which \(M_{Thom}\) denotes a whised value for the closed-loop resonance peak computed with the nominal plant denoted \(G_{nom}\). The minimisation of Relation (32) is achieved under the constraints on the four closed loop sensitivity functions that follows:  
\[\underset{G}{inf}[T(j\omega)]\geq T_{I}(\omega),\underset{G}{sup}[T(j\omega)] \leq T_{u}(\omega)\underset{G}{sup}[S(j\omega)]\leq S_{u}(\omega),\underset{G }{sup}[CS(j\omega)]\leq CS_{u}(\omega)\underset{G}{sup}[GS(j\omega)]\leq GS_{u }(\omega) \tag{33}\]  
with  
\[\left\{ \begin{array}{c}T(s)=\frac{C(s)C(s)}{1+G(s)C(s)}\ S(s)=\frac{1}{1+G( s)C(s)}\\ CS(s)=\frac{C(s)}{1+G(s)C(s)}\ SG(s)=\frac{G(s)}{1+G(s)C(s)}\end{array} \right. \tag{34}\]  
Using the Nichols plot, by minimizing criterion \(J\) in Relation (32), the optimal parameters move the frequency uncertainty domains so that they cover areas of low stability margin as little as possible around the (\(-180\), \(0\) dB) critical point on the Nichols chart (as the open-loop frequency response uncertainty domains are those of the plant). Only a nonlinear optimization method can be used to compute the optimal parameters as uncertainties are not norm-bounded and taken into account by the least conservative method. If \(N^{+}=N^{-}=0\) in Relation (29), only the independent four parameters \(\omega_{0}\), \(\omega_{1}\), \(\omega_{r}\) and \(Y_{r}\)=\(|\beta(j\omega)|_{\text{dB}}\) need to be optimized in Relation (27). In the Nichols chart, the tangency of the frequency response \(\beta_{nom}(j\omega)\) to the desired \(M_{Thom}\) circle can be ensured by computing the other parameters of Relation (27), such as \(a_{0}\) and \(b_{0}\).