# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 93E11 Filtering in stochastic control 93E10 Estimation and detection in stochastic control 93B35 Sensitivity (robustness) of control systems 62M20 Prediction; filtering (statistics) 93C73 Perturbations in control systems
##### Keywords:
filtering problem; Riccati type equation; uncertain system
Full Text: