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Robust $H_2/H_{\infty}$ filtering for linear systems with error variance constraints. (English) Zbl 1003.93051
The authors deal with the following class of linear uncertain continuous time systems: $$\dot x(t)=(A+\Delta A)x(t)+D_1w(t);\quad y(t)=(C+\Delta C)x(t)+D_2w(t),$$ where $x\in{\Bbb R}^n$, $y\in{\Bbb R}^p$, $w(t)$ is a Gaussian white noise and $A,C,D_1,D_2$ are known constant matrices. The linear full-order filter under consideration is determined by $\dot{\widehat{x}}(t)=G\widehat{x}(t)+Ky(t)$, where $\widehat{x}(t)$ denotes the state estimate and $G,K$ are filter parameters to be determined. The corresponding discrete time system is considered, too. The problem is to find a linear filtering procedure that does not depend on the parameter perturbations such that: the filtering process is asymptotically stable; the variance of the estimation is not greater than the prescribed value; the transfer function from noise inputs to error state outputs is not greater than the prescribed $H_{\infty}$ norm upper bound. The authors show that in both continuous and discrete time cases the considered filtering problem can be solved in terms of solutions of the corresponding algebraic Riccati type equations/inequalities. A numerical example demonstrates the performance of the proposed algorithms.

MSC:
93E11Filtering in stochastic control
93E10Estimation and detection in stochastic control
93B35Sensitivity (robustness) of control systems
62M20Prediction; filtering (statistics)
93C73Perturbations in control systems
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