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Robust \(H_\infty\) filtering of stationary continuous-time linear systems with stochastic uncertainties. (English) Zbl 1016.93067

The authors consider the following linear mean-square stable system \[ \begin{aligned} dx &= (Ax+ B_1w) dt+ Dx d\beta,\\ dy &= (Cx+ B_2w) dt+ Fx d\zeta,\\ z &= Lx,\end{aligned} \] where \(x\in \mathbb{R}^n\) is the system state vector, \(y\in\mathbb{R}^r\) is the measurement, \(z\in\mathbb{R}^m\) is the state combination to be estimated, \(\beta\) and \(\zeta\) are Wiener processes, \(w\) is the disturbance signal satisfying \[ \int^\infty_0 E\|w(t)\|^2 dt<\infty,\quad w(t)\in\mathbb{R}^q, \] and all the matrices are constants and of the appropriate dimensions. They consider the following filter for the estimation of \(z(t)\): \[ d\widehat x= A_f\widehat x dt+ B_f dy,\quad\widehat z= C_f\widehat x, \] and invstigate the stochastic \(H_\infty\) filtering problem: given \(\gamma> 0\), find an asymptotically stable linear filter of the above form that leads to an estimation such that \[ J:= \int^\infty_0 E\|z(t)-\widehat z(t)\|^2 dt-\gamma^2 \int^\infty_0 E\|w(t)\|^2 dt \] is negative for all nonzero \(w\).

MSC:

93E11 Filtering in stochastic control theory
93C73 Perturbations in control/observation systems
93B36 \(H^\infty\)-control
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