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Table 2 Mathematical expressions of fractional order models

From: A Comparative Study of Fractional Order Models on State of Charge Estimation for Lithium Ion Batteries

FOM Parameters Discretised model equations State space for SOC estimation
R(RQ) [41,42,43,44,45,46] [Uoc, Ri, R1, Q1, α1] \(\left\{ {\begin{array}{*{20}l} {U_{\text{c1}} \left( k \right) = \frac{{\tau_{1} }}{{\tau_{1} + 1}}\left[ {\frac{1}{{Q_{1} }}I\left( k \right) - {\text{D}}^{{\left( {\alpha_{1} } \right)}} U_{\text{c1}} \left( k \right)} \right],} \hfill \\ {U_{\text{t}} \left( k \right) = U_{\text{oc}} + I\left( k \right)R_{\text{i}} + U_{\text{c1}} \left( k \right).} \hfill \\ \end{array} } \right.\) \(\left\{ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\rm{D}}^{{\alpha _1}}}{U_{{\rm{c1}}}}\left( {k + 1} \right)}\\ {{{\rm{D}}^1}z\left( {k + 1} \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - \frac{1}{{{\tau _1}}}}&0\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{U_{{\rm{c1}}}}\left( k \right)}\\ {z\left( k \right)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{Q_1}}}}\\ {\frac{1}{{{C_{\max }}}}} \end{array}} \right]I\left( k \right),\\ {U_{\rm{t}}}\left( k \right) = {U_{{\rm{oc}}}}\left( {z\left( k \right)} \right) + I\left( k \right){R_{\rm{i}}} + {U_{{\rm{c1}}}}\left( k \right). \end{array} \right.\)
R(RQ)W [32,33,34] [Uoc, Ri, R1, Q1, α1, W1, β1] \(\left\{ {\begin{array}{*{20}l} \begin{aligned} U_{\text{c1}} \left( k \right) = \frac{{\tau_{1} }}{{\tau_{1} + 1}}\left[ {\frac{1}{{Q_{1} }}I\left( k \right) - {\text{D}}^{{\left( {\alpha_{1} } \right)}} U_{\text{c1}} \left( k \right)} \right], \hfill \\ U_{\text{W}} \left( k \right) = \frac{1}{{W_{1} }}I\left( k \right) - {\text{D}}^{{\left( {\beta_{1} } \right)}} U_{\text{W}} \left( k \right), \hfill \\ \end{aligned} \hfill \\ {U_{\text{t}} \left( k \right) = U_{\text{oc}} + I\left( k \right)R_{\text{i}} + U_{\text{c1}} \left( k \right).} \hfill \\ \end{array} } \right.\) \(\left\{ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\rm{D}}^{{\alpha _1}}}{U_{{\rm{c1}}}}\left( k \right)}\\ {{{\rm{D}}^{{\beta _1}}}{U_{\rm{W}}}\left( k \right)}\\ {{{\rm{D}}^1}z\left( k \right)} \end{array}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}} { - \frac{1}{{{\tau _1}}}}&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{U_{{\rm{c1}}}}\left( k \right)}\\ {{U_{\rm{W}}}\left( k \right)}\\ {z\left( k \right)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{Q_1}}}}\\ {\frac{1}{{{W_1}}}}\\ {\frac{1}{{{C_{\max }}}}} \end{array}} \right]I\left( k \right),\\ {U_{\rm{t}}}\left( k \right) = {U_{{\rm{oc}}}}\left( {z\left( k \right)} \right) + I\left( k \right){R_{\rm{i}}} + {U_{{\rm{c1}}}}\left( k \right) + {U_{\rm{W}}}\left( k \right). \end{array} \right.\)
R(RWQ) [35, 36] [Uoc, Ri, R1, Q1, α1, W1, β1] \(\left\{ {\begin{array}{*{20}l} {U_{\text{c}} \left( k \right) = \frac{{\left[ \begin{aligned} I\left( k \right) + W_{1} R_{1} {\text{D}}^{{\beta_{{_{1} }} }} I\left( k \right) - W_{1} {\text{D}}^{{\left( {\beta_{{_{1} }} } \right)}} U_{\text{c1}} \left( k \right) \hfill \\ - W_{1} \tau_{1} {\text{D}}^{{\left( {\alpha_{1} + \beta_{{_{1} }} } \right)}} U_{\text{c1}} \left( k \right) - Q_{1} {\text{D}}^{{\left( {\alpha_{1} } \right)}} U_{\text{c1}} \left( k \right) \hfill \\ \end{aligned} \right]}}{{W_{1} + W_{1} \tau_{1} + Q_{1} }},} \hfill \\ {U_{\text{t}} \left( k \right) = U_{\text{oc}} + I\left( k \right)R_{\text{i}} + U_{\text{c1}} \left( k \right).} \hfill \\ \end{array} } \right.\) \(\left\{ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\rm{D}}^{{\alpha _1}}}{U_{{\rm{c1}}}}\left( k \right)}\\ {{{\rm{D}}^{ - {\beta _1}}}{I_1}\left( k \right)}\\ {{{\rm{D}}^1}z\left( k \right)} \end{array}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}} { - \frac{1}{{{\tau _1}}}}&0&0\\ W&0&0\\ 0&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{U_{{\rm{c1}}}}\left( k \right)}\\ {{I_1}\left( k \right)}\\ {z\left( k \right)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{Q_1}}}}\\ {W{R_1}}\\ {\frac{1}{{{C_{\max }}}}} \end{array}} \right]I\left( k \right)\\ {U_{\rm{t}}}\left( k \right) = {U_{{\rm{oc}}}}\left( k \right) + I\left( k \right){R_{\rm{i}}} + {U_{{\rm{c1}}}}\left( k \right) \end{array} \right.\)
R(RQ)(RQ) [37,38,39,40] [Uoc, Ri, R1, Q1, α1, R2, Q2, α2] \(\left\{ {\begin{array}{*{20}l} \begin{aligned} U_{\text{c1}} \left( k \right) = \frac{{\tau_{1} }}{{\tau_{1} + 1}}\left[ {\frac{1}{{Q_{1} }}I\left( k \right) - {\text{D}}^{{\left( {\alpha_{1} } \right)}} U_{\text{c1}} \left( k \right)} \right], \hfill \\ U_{\text{c2}} \left( k \right) = \frac{{\tau_{2} }}{{\tau_{2} + 1}}\left[ {\frac{1}{{Q_{2} }}I\left( k \right) - {\text{D}}^{{\left( {\alpha_{2} } \right)}} U_{\text{c2}} \left( k \right)} \right], \hfill \\ \end{aligned} \hfill \\ {U_{\text{t}} \left( k \right) = U_{\text{oc}} + I\left( k \right)R_{\text{i}} + U_{\text{c1}} \left( k \right) + U_{\text{c2}} \left( k \right).} \hfill \\ \end{array} } \right.\) \(\begin{aligned} \left[ {\begin{array}{*{20}c} {{\text{D}}^{{\alpha_{1} }} U_{\text{c1}} \left( k \right)} \\ {{\text{D}}^{{\alpha_{2} }} U_{\text{c2}} \left( k \right)} \\ {{\text{D}}^{1} z\left( k \right)} \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} { - \frac{1}{{\tau_{1} }}} & 0 & 0 \\ 0 & { - \frac{1}{{\tau_{2} }}} & 0 \\ 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {U_{\text{c1}} \left( k \right)} \\ {U_{\text{c2}} \left( k \right)} \\ {z\left( k \right)} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\frac{1}{{Q_{1} }}} \\ {\frac{1}{{Q_{2} }}} \\ {\frac{1}{{C_{\text{max} } }}} \\ \end{array} } \right]I\left( k \right), \hfill \\ U_{\text{t}} \left( k \right) = U_{\text{oc}} \left( {z\left( k \right)} \right) + I\left( k \right)R_{\text{i}} + U_{\text{c1}} \left( k \right) + U_{\text{c2}} \left( k \right). \hfill \\ \end{aligned}\)
R(RQ)(RQ)W [29,30,31] [Uoc, Ri, R1, Q1, α1, R2, Q2, α2, W1, β1] \(\left\{ {\begin{array}{*{20}l} \begin{aligned} U_{\text{c1}} \left( k \right) = \frac{{\tau_{1} }}{{\tau_{1} + 1}}\left[ {\frac{1}{{Q_{1} }}I\left( k \right) - {\text{D}}^{{\left( {\alpha_{1} } \right)}} U_{\text{c1}} \left( k \right)} \right], \hfill \\ U_{\text{c2}} \left( k \right) = \frac{{\tau_{2} }}{{\tau_{2} + 1}}\left[ {\frac{1}{{Q_{2} }}I\left( k \right) - {\text{D}}^{{\left( {\alpha_{2} } \right)}} U_{\text{c2}} \left( k \right)} \right], \hfill \\ U_{\text{W}} \left( k \right) = \frac{1}{{W_{1} }}I\left( k \right) - {\text{D}}^{{\left( {\beta_{1} } \right)}} U_{\text{W}} \left( k \right), \hfill \\ \end{aligned} \hfill \\ {U_{\text{t}} \left( k \right) = U_{\text{oc}} + I\left( k \right)R_{\text{i}} + U_{\text{c1}} \left( k \right) + U_{\text{c2}} \left( k \right) + U_{\text{W}} \left( k \right).} \hfill \\ \end{array} } \right.\) \(\left\{ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{{\rm{D}}^{{\alpha _1}}}{U_{{\rm{c1}}}}\left( k \right)}\\ {{{\rm{D}}^{{\alpha _2}}}{U_{{\rm{c2}}}}\left( k \right)}\\ {{{\rm{D}}^{{\beta _1}}}{U_{\rm{W}}}\left( k \right)}\\ {{{\rm{D}}^1}z\left( k \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} { - \frac{1}{{{\tau _1}}}}&0&0&0\\ 0&{ - \frac{1}{{{\tau _2}}}}&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{U_{{\rm{c1}}}}\left( k \right)}\\ {{U_{{\rm{c2}}}}\left( k \right)}\\ {{U_{\rm{W}}}\left( k \right)}\\ {z\left( k \right)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{Q_1}}}}\\ {\frac{1}{{{Q_2}}}}\\ {\frac{1}{{{W_1}}}}\\ {\frac{1}{{{C_{\max }}}}} \end{array}} \right]I\left( k \right),\\ {U_{\rm{t}}}\left( k \right) = {U_{{\rm{oc}}}}\left( k \right) + I\left( k \right){R_{\rm{i}}} + {U_{{\rm{c1}}}}\left( k \right) + {U_{{\rm{c2}}}}\left( k \right) + {U_{\rm{W}}}\left( k \right). \end{array} \right.\)