Algorithm 1: Solution steps for $P_{k|k-\tau }^{(m)}$

Initialize:  $A_1=I_n,C_0=I_n.$

Iterate:

for  $m=1:\tau$

$A_{m+1}=\Phi A_m-\Phi K_{k-\tau +m}^{}H{A}_m$

end

if $\tau =1,\mathit{B}=0$

else $\tau =2,m=2,\mathit{B}=-{\Phi }^{\tau -m}\Gamma Q_{k-\tau +m+1}{\Gamma }^{\mathrm{T}}{I}_n^{\mathrm{T}}{H}^{\mathrm{T}}{K}_{k-\tau +m}^{\mathrm{T}}{({\Phi }^{\tau -m})}^{\mathrm{T}}$

else ${\mathit{B}}_m^{}=0$

for $m=3:\tau$

$h=m-2, p=m-2$

for $\xi =1:m-2$

for $\partial =0:h-1$

$C_{\partial +1}=\Phi C_{\partial }-\Phi K_{k-\tau +m-h+\partial }^{}{H}^{}{C}_{\partial }$

end

${\mathit{B}}_m^{\psi }=-{\Phi }^{\tau -m+p}\Gamma Q_{k-\tau +m-p-1}{\Gamma }^{\mathrm{T}}{C}_h^{\mathrm{T}}{H}^{\mathrm{T}}{K}_{k-\tau +m}^{\mathrm{T}}{({\Phi }^{\tau -m})}^{\mathrm{T}}$

$h=h-1, p=p-1$

end

${\mathit{B}}_m^{}={\mathit{B}}_m^{}+{\mathit{B}}_m^{\psi }$

end

$\mathit{B}={\mathit{B}}_m-{\Phi }^{\tau -m}\Gamma Q_{k-\tau +m+1}{\Gamma }^{\mathrm{T}}{I}_n^{\mathrm{T}}{H}^{\mathrm{T}}{K}_{k-\tau +m}^{\mathrm{T}}{({\Phi }^{\tau -m})}^{\mathrm{T}}$

end

$P_{k|k-\tau }^{(m)}=-{\Phi }^{\tau -1}{P}_{k-\tau +1|k-\tau }{(A_m)}^{\mathrm{T}}{H}^{\mathrm{T}}{K}_{k-\tau +m}^{\mathrm{T}}{({\Phi }^{\tau -m})}^{\mathrm{T}}+\mathit{B}$