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Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters. (English) Zbl 0986.93066

The authors consider the problem of Kalman filtering for a class of uncertain linear continuous-time systems with Markovian jumping parameters described by

$\stackrel{˙}{x}\left(t\right)=\left[A\left(r\left(t\right)\right)+{\Delta }A\left(t,r\left(t\right)\right)\right]x\left(t\right)+w\left(t\right),\phantom{\rule{1.em}{0ex}}x\left(0\right)={x}_{0},\phantom{\rule{1.em}{0ex}}{r}_{0}=i,$
$y\left(t\right)=\left[C\left(r\left(t\right)\right)+{\Delta }C\left(t,r\left(t\right)\right)\right]x\left(t\right)+v\left(t\right),$

where $x\in {ℝ}^{n}$ and $y\in {ℝ}^{m}$ are the state and measurement vectors, respectively; $w\in {ℝ}^{n}$ and $v\in {ℝ}^{m}$ are the state and measurement noises, respectively; $A\left(r\left(t\right)\right)$, ${\Delta }A\left(t,r\left(t\right)\right)$, $C\left(r\left(t\right)\right)$ and ${\Delta }C\left(t,r\left(t\right)\right)$ are matrices of appropriate dimensions; $\left\{r\left(t\right),t\ge 0\right\}$ represents a homogeneous continuous-time discrete-state Markov process taking values in a finite set $S=\left\{1,2,\cdots ,s\right\}$ with stationary transition probabilities. For each $r\left(t\right)\in S$, ${\Delta }A\left(t,r\left(t\right)\right)$ and ${\Delta }C\left(t,r\left(t\right)\right)$ represent the system’s uncertainties.

The authors design a stochastic quadratic estimator that guarantees both the stability and boundedness of the estimation error dynamics.

##### MSC:
 9.3e+12 Filtering in stochastic control 9.3e+16 Stochastic stability
##### Keywords:
Kalman filtering; Markovian jumping parameters