Achieving robustness at diagnosis of nonlinear systems

Zhirabok, Alexey; Kucher, D. N.; Filaretov, Vladimir

doi:10.1134/s000511791001011x

Cited by 20 publications

(16 citation statements)

References 5 publications

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“…We consider two ways to obtain such a transformation. The first one is based on the mathematical approach called the algebra of functions; the second one on the logic‐dynamic approach …”

Section: Basic Modelsmentioning

confidence: 99%

“…It gives the general solution where the transformation of the initial model into the form is nonlinear generally. If a class of transformations is limited by linear functions, then a solution can be obtained by methods of linear algebra on the basis of the logic‐dynamic approach . To apply this approach, the initial system should be represented in the form

\begin{matrix} right x (t + 1) & left = F x (t) + G u (t) + C (\begin{matrix} ψ_{1} false(A_{1} x false(t false), u false(t false) false) \\ . . . \\ ψ_{q} false(A_{q} x false(t false), u false(t false) false) \end{matrix}) + ∑_{i = 1}^{s} D_{i} d_{i} (t), & right \\ right y (t) & left = H x (t), \end{matrix}

where F , G , and C are constant matrices under nominal values of the parameters; F and G describe the linear dynamics; H , D 1 ,…, D s are constant matrices; d 1 ( t ),…, d s ( t ) are scalar functions describing faults; if the i th parameter has nominal value, ie, the fault ρ i is absent, then d i ( t ) = 0; otherwise, d i ( t ) becomes an unknown function of time, i = 1,2,…, s ; ψ 1 , …, ψ q are arbitrary nonlinearities, A 1 , …, A q are row matrices.…”

Section: Logic‐dynamic Approach–based Solutionmentioning

confidence: 99%

“…The logic‐dynamic approach has three steps . On the first step, one removes the nonlinear term from , and then the problem is solved for linear system with additional linear restriction; on the third step, the obtained linear solution is supplemented by the transformed nonlinear term.…”

Section: Logic‐dynamic Approach–based Solutionmentioning

confidence: 99%

“…Now, we describe the solution on the second step where the linear model is presented in the canonical form, which is feedback‐free and is given by

\begin{matrix} right x_{* i} (t + 1) & left = x_{* i + 1} (t) + J_{i} y (t) + G_{* i} u (t), i = 1, 2, ⋯, k - 1, & right \\ right x_{* k} (t + 1) & left = J_{k} y (t) + G_{* k} u (t), \\ right y_{*} (t) & left = x_{* 1} (t), \end{matrix}

where x ∗ ∈ R k is the state vector and x ∗ = Φ x , y ∗ = R y for some matrices Φ and R . It is known that matrices describing the linear part of and the model satisfy the following equations :

R H = Φ_{1}, Φ_{i} F = Φ_{i + 1} + J_{i} H, i = 1, ⋯, k - 1, Φ_{k} F = J_{k} H,

where Φ i and J i are the i th row of the matrices Φ and J , i = 1,…, k . The additional restriction on the matrix Φ is caused by the nonlinear term in the form

A = A_{*} {((), \begin{matrix} center \\ array {normalΦ}^{T} & array H^{T} \end{matrix})}^{T}

for some matrix A ∗ , where

A = {false(\begin{matrix} center \\ array A_{1}^{T} & array . . . & array A_{q}^{T} \end{matrix} false)}^{T}

.…”

Section: Logic‐dynamic Approach–based Solutionmentioning

confidence: 99%

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Nonparametric methods for fault diagnosis in dynamic systems

Zhirabok

Shumsky

Scherbatyuk

2018

Intl J Robust & Nonlinear

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Summary The problem of fault diagnosis in engineering systems with uncertainties described by nonlinear discrete‐time models is studied within the scope of analytical redundancy concept. Solution of the problem assumes the checking redundancy relations existing among system inputs and outputs measured over a finite time window. The nonparametric method is used to construct redundancy relations involving transformation of the initial system model into canonical form with special properties. The obtained results are illustrated by the general electric servoactuator of manipulation robots.

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“…We consider two ways to obtain such a transformation. The first one is based on the mathematical approach called the algebra of functions; the second one on the logic‐dynamic approach …”

Section: Basic Modelsmentioning

confidence: 99%

\begin{matrix} right x (t + 1) & left = F x (t) + G u (t) + C (\begin{matrix} ψ_{1} false(A_{1} x false(t false), u false(t false) false) \\ . . . \\ ψ_{q} false(A_{q} x false(t false), u false(t false) false) \end{matrix}) + ∑_{i = 1}^{s} D_{i} d_{i} (t), & right \\ right y (t) & left = H x (t), \end{matrix}

Section: Logic‐dynamic Approach–based Solutionmentioning

confidence: 99%

Section: Logic‐dynamic Approach–based Solutionmentioning

confidence: 99%

“…Now, we describe the solution on the second step where the linear model is presented in the canonical form, which is feedback‐free and is given by

\begin{matrix} right x_{* i} (t + 1) & left = x_{* i + 1} (t) + J_{i} y (t) + G_{* i} u (t), i = 1, 2, ⋯, k - 1, & right \\ right x_{* k} (t + 1) & left = J_{k} y (t) + G_{* k} u (t), \\ right y_{*} (t) & left = x_{* 1} (t), \end{matrix}

R H = Φ_{1}, Φ_{i} F = Φ_{i + 1} + J_{i} H, i = 1, ⋯, k - 1, Φ_{k} F = J_{k} H,

where Φ i and J i are the i th row of the matrices Φ and J , i = 1,…, k . The additional restriction on the matrix Φ is caused by the nonlinear term in the form

A = A_{*} {((), \begin{matrix} center \\ array {normalΦ}^{T} & array H^{T} \end{matrix})}^{T}

for some matrix A ∗ , where

A = {false(\begin{matrix} center \\ array A_{1}^{T} & array . . . & array A_{q}^{T} \end{matrix} false)}^{T}

.…”

Section: Logic‐dynamic Approach–based Solutionmentioning

confidence: 99%

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Nonparametric methods for fault diagnosis in dynamic systems

Zhirabok

Shumsky

Scherbatyuk

2018

Intl J Robust & Nonlinear

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“…It is based on so-called logic-dynamic approach. The feature of this approach is that the problems for nonlinear systems are solved by methods of linear algebra [22,23] but it does not guarantee the minimal dimension of the system designed.…”

Section: Introductionmentioning

confidence: 99%

Fault Isolation in Technical Systems Based on Non-parametric Method

Zhirabok¹,

Filaretov²,

Shumsky³

2017

DAAAM Proceedings

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The paper is devoted to the fault isolation problem in technical systems described by nonlinear dynamic models. So-called non-parametric method is used to solve the problem. The feature of this method is that parameters of the system under consideration may be unknown. To apply this method, the system is transformed into the input-out form. The methods to obtain such a transformation and to isolate faults are suggested. They are based on so-called logic-dynamic approach. The feature of this approach is that the system transformation and fault isolation problems for nonlinear systems are solved by methods of linear algebra. The adaptive threshold is used to avoid a false alarm. The theoretical results are illustrated by practical example of diagnosis of general electric servoactuator.

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Non-Parametric Method for Fault Isolation in Nonlinear Dynamic Systems

Zhirabok

Shumsky

2017

IFAC-PapersOnLine

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Achieving robustness at diagnosis of nonlinear systems

Cited by 20 publications

References 5 publications

Nonparametric methods for fault diagnosis in dynamic systems

Nonparametric methods for fault diagnosis in dynamic systems

Fault Isolation in Technical Systems Based on Non-parametric Method

Non-Parametric Method for Fault Isolation in Nonlinear Dynamic Systems

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