Filtering, smoothing and prediction for wide-band noise driven linear systems

Bashirov, Agamirza E.; Etikan, Hüseyin; Semi, Nidai

doi:10.1016/s0016-0032(96)00052-x

Cited by 11 publications

(8 citation statements)

References 7 publications

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“…In the following reasoning we will omit (for simplification) the time dependance notation. From the form C s,r (15) and the fact, that the covariance matrix of random noise v(t) is positive, we can conclude that C is positive definite and hence invertable. The assumption concerning the observability of the system (12) implies G T G > 0 and hence G T C −1 G. That justifies the existence of (G T C −1 G) −1 .…”

Section: Theorem 2 For the Vector State X(t) Given By The Formulamentioning

confidence: 93%

“…If the system parameters are measured directly the most popular (but not the best) method takes the arithmetic mean of few observations symmetrically: before and after the estimating moment. The early papers [3], [4], [15] present the recursive methods and the continuous time solutions using the linear least squares estimator. The smoothers applied in the discrete time systems based on the Kalman theory are described in [16].…”

Section: Introductionmentioning

confidence: 99%

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Off-line estimation of trajectory in discrete state space using the minimal-covariance adaptive FIR smoothing with extended output vector

Podsedkowski

Podsȩdkowska

2015

2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR)

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The new tool for off-line estimation of the state of discrete linear systems is presented. The algorithm of Finite Impulse Response (FIR) smoother is described and its optimality is proven. The optimality of this smoother means that error covariance matrix of the estimation is minimal. It places this method in the Least Square Estimation (LSE) methods group, which are much better than frequently used Least Mean Square (LMS) methods. This method can also be used as FIR filter or predictor. In the paper the simple simulation, showing the mode of actions of described smoother, is presented. It is compared with others presented in the literature.

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Section: Theorem 2 For the Vector State X(t) Given By The Formulamentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Off-line estimation of trajectory in discrete state space using the minimal-covariance adaptive FIR smoothing with extended output vector

Podsedkowski

Podsȩdkowska

2015

2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR)

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show abstract

“…On the other hand,

((), \frac{\partial}{∂t} MathClass-bin+ \frac{\partial}{∂θ}) ψ (t MathClass-punc, θ) MathClass-rel= Ψ^{MathClass-rel'} (θ) w (t) MathClass-punc.

This equality can be verified straightforwardly by calculation of the partial derivatives of ψ , but its strong justification needs deeper mathematical considerations because the integral in is a stochastic integral defined as a limit in the mean square sense (not in the pointwise sense). We refer to (see also ) for mathematics of this issue, noticing that in the partial differentiation − ∂ / ∂θ can be interpreted as a differential operator − d / d θ on L 2 , which has the domain D ( − d / d θ ) as a collection of all h ∈ L 2 such that h ′ ∈ L 2 and h ( − ϵ ) = 0. The operator − d / d θ generates the strongly continuous semigroup T defined by

([], T (t) h) (θ) MathClass-rel= ({, \begin{matrix} falsenonefalsearray \\ arrayleft h MathClass-open(θ - t MathClass-close), & arrayleft θ - t ≥ - ϵ, \\ arrayleft 0 & arrayleft θ - t < - ϵ, \end{matrix}) MathClass-punc, MathClass-bin- ϵ ≤ θ ≤ 0 MathClass-punc, t ≥ 0 MathClass-punc, h MathClass-rel\in L_{2} MathClass-punc,

which is called a semigroup of right translation.…”

Section: White and Wideband Noisesmentioning

confidence: 99%

Delay structure of wideband noises with application to filtering problems

Bashirov

Mazhar

Etikan

et al. 2012

Optim Control Appl Methods

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SUMMARYThe paper presents a mathematical theory of handling and working with wideband noises. We demonstrate that a wideband noise can be represented as a distributed delay of a white noise. From this, we deduce that the behavior of a wideband noise is the same as the behavior of an infinite dimensional colored noise along the boundary line. All these are used to deduce a complete set of formulae for the Kalman-type optimal filter and also to derive nonlinear filtering equation for wideband-noise-driven linear and nonlinear systems.

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“…Another approach to wide band noise based on representation in a certain integral form was suggested in [3] and its applications to space engineering and gravimetry was discussed in [4,5]. Filtering, smoothing, and prediction results for wide band noise driven linear systems are obtained in [3,6]. The proofs in [3, 6] are given through the duality principle and, technically, they are routine, making further developments in the theory difficult.…”

Section: Motivation and History Of The Problemmentioning

confidence: 99%

Problem 2.1 On error of estimation and minimum of cost for wide band noise driven systems

Bashirov¹

2009

Unsolved Problems in Mathematical Systems and Control Theory

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Filtering, smoothing and prediction for wide-band noise driven linear systems

Cited by 11 publications

References 7 publications

Off-line estimation of trajectory in discrete state space using the minimal-covariance adaptive FIR smoothing with extended output vector

Off-line estimation of trajectory in discrete state space using the minimal-covariance adaptive FIR smoothing with extended output vector

Delay structure of wideband noises with application to filtering problems

Problem 2.1 On error of estimation and minimum of cost for wide band noise driven systems

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