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\begin{document}

\begin{equation}
\begin{split}
\begin{array}{l}
{P_k} = \left( {I - K(k)H'} \right){{\tilde P}_k}{\left( {I - K(k)H'} \right)^T} - M(k){K^T}(k) - K(k){M^T}(k)\\
\begin{array}{*{20}{c}}
{}&{}
\end{array} + K(k)\left( {H'M(k) + {M^T}(k){{H'}^T} + H\left( {\sum\limits_{j = 1}^k {{C_j}} {{\tilde P}_{k + 1 - j}}C_j^T} \right){H^T} + R'(k)} \right){K^T}(k)\\
\begin{array}{*{20}{c}}
{}& =
\end{array}{{\tilde P}_k} - K(k)H'{{\tilde P}_k} - {{\tilde P}_k}{{H'}^T}{K^T}(k) - M(k){K^T}(k) - K(k){M^T}(k)\\
\begin{array}{*{20}{c}}
{}&{}
\end{array} + K(k)\left( {H'{{\tilde P}_k}H{'^T} + H'M(k) + {M^T}(k){{H'}^T} + H\left( {\sum\limits_{j = 1}^k {{C_j}} {{\tilde P}_{k + 1 - j}}C_j^T} \right){H^T} + R'(k)} \right){K^T}(k)\\
\begin{array}{*{20}{c}}
{}& =
\end{array}{{\tilde P}_k} - K(k)\left( {H'{{\tilde P}_k} + {M^T}(k)} \right) - \left( {{{\tilde P}_k}{{H'}^T} + M(k)} \right){K^T}(k)\\
\begin{array}{*{20}{c}}
{}&{}
\end{array} + \left( {{{\tilde P}_k}{{H'}^T} + M(k)} \right){\left( {H'{{\tilde P}_k}{{H'}^T} + H'M(k) + {M^T}(k){{H'}^T} + H\left( {\sum\limits_{j = 1}^k {{C_j}} {{\tilde P}_{k + 1 - j}}C_j^T} \right){H^T} + R'(k)} \right)^{ - 1}}\\
\begin{array}{*{20}{c}}
{}&{}
\end{array}\left( {H'{{\tilde P}_k} + {M^T}(k)} \right)\\
\begin{array}{*{20}{c}}
{}& =
\end{array}{{\tilde P}_k} - K(k)\left( {H'{{\tilde P}_k} + {M^T}(k)} \right) - \left( {{{\tilde P}_k}{{H'}^T} + M(k)} \right){K^T}(k) + \left( {{{\tilde P}_k}{{H'}^T} + M(k)} \right){K^T}(k)\\
\begin{array}{*{20}{c}}
{}& =
\end{array}{{\tilde P}_k} - K(k)\left( {H'{{\tilde P}_k} + {M^T}(k)} \right)
\end{array}
\end{split}
\end{equation}
\end{document}