## 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

### Keywords:

stochastic $$H_\infty$$ filtering
Full Text: