DISCRETE TIME SYSTEMS Edited by Mario A. Jordán and Jorge L. by Mario A. Jordán, Jorge L. Bustamante
By Mario A. Jordán, Jorge L. Bustamante
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Extra resources for DISCRETE TIME SYSTEMS Edited by Mario A. Jordán and Jorge L. Bustamante
Kq ), Σ, Υ are deﬁned in (49), (53) and (55) respectively. Using the Schur Lemma and the notation R = L T P, the inequality (48) is equivalent to ⎡ ⎤ Γ11 Γ12 Γ13 Γ14 ⎢( ) Γ22 Γ23 Γ24 ⎥ ⎢ ⎥ ⎣( ) ( ) Γ33 Γ34 ⎦ < 0. ( ) ( ) ( ) Γ44 33 (77) Consequently, we deduce that under the condition (48), the estimation error converges asymptotically towards zero. 2. 3. 3 Numerical example Now, we present a numerical example to show the performances of the proposed method. We consider the modiﬁed chaotic system introduced in Cherrier et al.
In this case the prediction estimate xˆ opt t+Δ and its error opt covariance Pt+Δ are determined by the combining of time update and observation update, ⎧ xˆ opt ˆ opt ˆ opt ˆ opt s =Fs x s , t k ≤ s ≤ t+Δ , x s=t k =x t k , ⎪ ⎨ opt opt opt opt opt T ⎪⎩ Ps =FsPs +Ps Fs +Q s , Ps=t k =Ptk , (5) and its error where the initial conditions represent filtering estimate of the state xˆ opt tk covariance Ptopt which are given by the continuous-discrete Kalman filter equations (Lewis, k 1986, Gelb, 1974): Time update between observations : ⎧ ˆ opt , t k-1 ≤ τ ≤ t k , xˆ opt τ=t k-1 =x t k-1 , ⎪ ⎨ opt ⎪ Pτopt =Fτ Pτopt +Pτopt FτT +Q τ , Pτ=t =Ptopt , k-1 k-1 ⎩ xˆ opt =Fτ xˆ opt τ τ (6a) Observation update at time t k : ) ( ⎧ xˆ opt =xˆ opt - +Lopt Y -H xˆ opt - , tk tk tk tk tk ⎪ tk ⎪ ⎪ opt opt - T opt - T ⎨ L tk =Pt k H t k H tk Pt k Ht k +R t k ⎪ ⎪ P opt = I -Lopt H P opt - .
E. (1960). A new approach to linear ﬁltering and prediction problems, Trans. ASME, Journal of Basic Engineering, Vol. 82, pp. 35-45. S. (1960). New Results in Linear Filtering and Prediction Theory, Trans. ASME, Journal of Basic Engineering, Series D, Vol. 83, pp. W. (1994). Investigation of trellis based ﬁlters for tracking, IEE Proceedings Radar, Sonar and Navigation, Vol. 1 pp. 9-18. Julier S. ; Uhlmann J. K. (2004). Unscented Filtering and Nonlinear Estimation Proceedings of the IEEE, 92, pp.