Suboptimal filtering for a singular discrete-time stochastic linear system (Q915705)

Suboptimal filtering for the singular discrete-time stochastic linear system \(\psi_ 1(k+1) = \phi_{11}\psi_ 1(k)+\phi_{13}\psi_ 3(k)+\Gamma_{11}\xi_ 1(k)+\Gamma_{12}\xi_ 2(k)\), 0 \(=\psi_ 2(k)+\Gamma_{22}\xi_ 2(k),\) 0 \(=\phi_{31}\psi_ 1(k)+\Gamma_{33}\xi_ 3(k),\) \(Z_ 1(k)\) \(=\) \(H_{11}\psi_ 1(k)+\eta_ 1(k),\) \(Z_ 2(k)\) \(=\) \(H_{21}\psi_ 1(k)+H_{22}\psi_ 2(k)+\eta_ 2(k),\) \(Z_ 3(k)\) \(=\) \(H_{33}\psi_ 3(k)+\eta_ 3(k)\) is investigated, where \(\{\xi_ i(k)\}\) and \(\{\eta_ i(k)\}\) are white noise sequences. The suboptimal estimation of \(\psi_ i(k)\) are given as follows: \({\hat \psi}{}_ 1(k| k)\) \(=\) \({\hat \psi}\)(k\(| k-1)+K(k)[Z(k)- M{\hat \psi}_ 1(k| k-1)],\) \({\hat \psi}{}_ 1(k| k-1)\) \(=\phi_{11}{\hat \psi}_ 1(k| k- 1)+\psi_{13}H^{-1}_{33}Z_ 3(k),\) K(k) \(=\) \(P_ 1(k| k-1)M^ T[MP_ 1(k| k-1)M^ T+\Theta (k)]^ 1,\) \(P_ 1(k| k-1)\) \(=\phi_{11}P(k-1| k-1)\phi^ T_{11}+\Omega (k-1),\) \(P_ 1(k| k)\) \(=\) \([I-K(k)M]P_ 1(k| k-1),\) where \({\hat \phi}{}_ 1(0| 0)\) and P(0\(| 0)\) is given, \({\hat \psi}{}_ 2(k| k)\) \(=\Gamma_{22}\Sigma_ 2(k)\Gamma^ T_{22}H^ T_{22}(H_{22}\Gamma_{22}\Sigma_ 2(k)\Gamma^ T_{22}H^ T_{22}+\pi_ 2(k)\) \(+H_{21}P_ 1(k| k)H^ T_{21})^{-1}[Z_ 2(k)-H_{21}{\hat \psi}_ 1(k| k)],\) \({\hat \psi}{}_ 2(k| k)\) \(=\) \(H^{-1}_{33}Z_ 3(k).\) where Q257.