State space realization of bilinear continuous-time input–output equations

Kotta, Ülle; Mullari, Tanel; Zinober, A.S.I.; Kotta, P.

doi:10.1080/00207170701447106

Cited by 6 publications

(5 citation statements)

References 9 publications

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“…Note that the results, similar to those of [12], were obtained earlier for discrete-time bilinear i/o equations [8]. Though both papers provide the necessary and sufficient realizability conditions for low-order bilinear systems and suggest a few realizable subclasses together with the corresponding state equations for the arbitrary-order bilinear systems, there is no formal similarity in the realizability conditions for the discrete-and continuous-time cases.…”

Section: Introductionsupporting

confidence: 55%

See 1 more Smart Citation

Removing the input derivatives in the generalized bilinear state equations

Mullari¹,

Kotta²,

Kotta³

et al. 2009

Proc. Estonian Acad. Sci.

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The paper suggests constraints on the coefficients a i , b i , c i j of the bilinear continuous-time input-output model that yield generalized state equations with input derivative order lower than that in the input-output equations. In the limiting case when one removes the input derivatives altogether, these conditions provide a solution of the realizability problem. The new state coordinates are found step by step. We first find a coordinate transformation allowing the reduction of the maximal order of the input time derivatives by one and write the corresponding state equations. At the second step we find the next coordinate transformation to lower the maximal order of input time derivative in the new state equations, etc. At each step we check, what condition the coefficients should satisfy to make the next step possible.Preliminary results of this paper were presented at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS) 2007, Pretoria, South Africa.

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Section: Introductionsupporting

confidence: 55%

“…When deriving the conditions in [12] shown for the case n = 4 in Fig. 1, we assumed s = n − 1 and for the case n = 4 three possible realizable structures resulted.…”

Section: Theorem 1 Using a Generalized State Transformation (4) Thementioning

confidence: 99%

Removing the input derivatives in the generalized bilinear state equations

Mullari¹,

Kotta²,

Kotta³

et al. 2009

Proc. Estonian Acad. Sci.

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“…Moreover, the results of Kotta et al (2003) were extended to include the complete list of realizable structures for the case of the 4th order bilinear systems and one additional realizable subclass of an arbitrary order. Finally, note that the results for discrete-time bilinear systems, given in this paper, are very much different from the continuous case presented in Kotta et al (2007).…”

Section: Discussionmentioning

confidence: 73%

Realizability of bilinear input/output difference equations: corrections and extensions

Belikov

Kotta

2010

UKACC International Conference on CONTROL 2010

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This article studies the realizability property of discrete-time bilinear input-output (i/o) equations in the classical state-space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. Using new formulas for computing basis vectors of certain vector spaces of differential one-forms, we present in this paper the complete list of the third-and fourth-order realizable input-output bilinear models. Moreover, we provide a new realizable subclass of an arbitrary order.

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“…Application of these models have been broadly developed in statistics, electrical engineering, and signal processing (Sarkka, 2007;Grewal et al, 2001;Kotta et al, 2007). State estimation of time varying systems is a main purpose when only noisy observation is available.…”

Section: Introductionmentioning

confidence: 99%

“…A large number of physical phenomena can be modeled as stochastic differential equations (SDEs) which can be observed at discrete instance of time and considered as a dynamic model in the state space models. Application of these models have been broadly developed in statistics, electrical engineering, and signal processing (Sarkka, 2007;Grewal et al, 2001;Kotta et al, 2007). State estimation of time varying systems is a main purpose when only noisy observation is available.…”

Section: Introductionmentioning

confidence: 99%

Estimation of parameters in the state space model of stochastic RL electrical circuit

Farnoosh

Hajrajabi²

2013

COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

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Purpose -The purpose of this paper is to consider the stochastic differential equation of the RL electrical circuit as the dynamic model of a state space system when the current in the circuit is hidden and corrupted by the measurement noise. Estimation of the corrupted current and the values of missing or unknown parameters (resistance, the observed current variance in the measurement model, the mean and variance of the current prior distribution) which are the main concern in electrical engineering is considered. Design/methodology/approach -Optimal filtering is proposed for estimation of the hidden current from the noisy observations. Also, the problem of analyzing this model based on estimation of the unknown parameters is addressed from the likelihood-based and Bayesian perspective. Findings -Computational techniques for parameter estimation are carried out by the Maximum likelihood (ML) approach using Expectation-Maximization type optimization and Bayesian Monte Carlo perspective using Metropolis-Hastings scheme. The explicit formulas for the ML estimator are obtained and it is shown that the smoothers, the filters and the predictions for the current have the best confidence intervals, respectively. Some numerical simulation examples which are performed by R programming software are considered to show the efficiency and applicability of the proposed approaches. Results show an excellent estimation of the parameters based on these approaches. Practical implications -Due to the fact that in an empirical situation of electrical engineering, observing the current in the circuit regardless of the measurement noise and knowing the exact value of the parameters are unrealistic assumptions, this paper can be used in various types of real time projects. Originality/value -To the best of the authors' information, the problem of analyzing the state space model of RL electrical circuit has not been studied before. Furthermore, the estimation of the hidden current as the state of the system and estimation of the unknown parameters of the model via both ML and Bayesian approaches have been investigated for the first time in the present study.

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State space realization of bilinear continuous-time input–output equations

Cited by 6 publications

References 9 publications

Removing the input derivatives in the generalized bilinear state equations

Removing the input derivatives in the generalized bilinear state equations

Realizability of bilinear input/output difference equations: corrections and extensions

Estimation of parameters in the state space model of stochastic RL electrical circuit

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