Novel Direct and Accurate Identification of Kalman Filter for General Systems Described by a Box-Jenkins Model

Doraiswami, R.; Cheded, Lahouari

doi:10.5772/intechopen.81793

Cited by 4 publications

(36 citation statements)

References 15 publications

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“…The identification is based on the above‐mentioned key necessary and sufficient properties of the KF. They were fully stated and proved in [7, 8 ] and are restated herein for clarity and completeness.…”

Section: Identification Of the System Signal And Disturbance Modelsmentioning

confidence: 99%

“…Proof The signal model is derived from the residual model of the KF, as proved in [8, 15 ]. The signal model becomes

{\hat{G}}_{s} ()(, z) = \frac{\overset{false¯}{N} false(z false)}{\overset{false¯}{D} false(z false)} = \frac{N_{s} (z) D_{w} (z)}{D_{s} (z) D_{w} (z)} = \frac{N_{s} false(z false)}{D_{s} false(z false)} ◻

…”

Section: Identification Of the System Signal And Disturbance Modelsmentioning

confidence: 99%

“…The augmented state‐space model

()(, A, B, {bold-italicE}_{w}, C)

termed as Box–Jenkins model [8, 14 ] is given by

\begin{matrix} right leftthickmathspace.5em \\ bold-italicx (k + 1) & = bold-italicA bold-italicx (k) + bold-italicB u_{normalest} (k) + E_{w} u_{w} (k) \\ y (k) & = bold-italicC bold-italicx (k) + v (k) \end{matrix}

where

x = (][, \begin{matrix} 1em4pt \\ {bold-italicx}_{normals} \\ {bold-italicx}_{w} \end{matrix}) \in R^{n}

A = (][, \begin{matrix} 1em4pt \\ {bold-italicA}_{normals} & 0 \\ 0 & {bold-italicA}_{w} \end{matrix}) \in R^{n \times n}

is a transition matrix;

B = (][, \begin{matrix} 1em4pt \\ {bold-italicB}_{s} \\ 0 \end{matrix}) \in R^{n x 1}

is the input vector,

{bold-italicE}_{w} = (][, \begin{matrix} 1em4pt \\ 0 \\ {bold-italicB}_{w} \end{matrix}) \in R^{n \times 1}

is the disturbance entry vector and

C = (][, \begin{matrix} 1em4pt \\ {bold-italicC}_{s} & bold-i... \end{matrix})

…”

Section: Disturbance Modelmentioning

confidence: 99%

“…The residual model relating the residual

e false(z false)

to the input

u_{est} ()(, z)

and the output

y false(z false)

e false(z false) = \frac{\overset{false¯}{D} false(z false)}{F false(z false)} y (z) - \frac{\bar{N} (z)}{F (z)} u_{est} false(z false)

where

()(, \overset{false¯}{D} ()(, z) / F false(z false)) = I - C {(z bold-italicI - bold-italicA bold+ bold-italicK bold-italicC)}^{- 1} K

()(, \overset{false¯}{N} ()(, z) / F false(z false)) = C {(z bold-italicI - bold-italicA bold+ bold-italicK bold-italicC)}^{- 1} B

and are transfer functions relating, respectively, the output y to the residual e ; and the input u to the residual e ;

F false(z false)

is the scalar characteristic polynomial given by

F false(z false) = (||, z I - A + K C)

. It is shown in [5, 7, 8 ] that the expressions for

\overset{false¯}{D} false(z false)

and

\overset{false¯}{N} false(z false)

are …”

Section: Disturbance Modelmentioning

confidence: 99%

“…The proposed scheme is an extension of the model‐order selection schemes [1–6 ] when the inputs and outputs are both corrupted by disturbances and measurement noise, and is based on the key properties of the Kalman filter (KF) [7 ]. The system and the associated KFs are identified for various selected orders by minimising the residual of the KF without any a‐priori knowledge of the covariance of the disturbance and/or of measurement noise [8 ]. The signal model and its associated KF are identified similarly to the system model.…”

Section: Introductionmentioning

confidence: 99%

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New approach to model‐order selection

Doraiswami

Cheded

2020

Electron. lett.

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A new, direct and practical scheme is proposed for determining the model orders of a system, its signal and disturbance using key properties of Kalman filter (KF). Unlike conventional methods, it enjoys the unique property of being both necessary and sufficient. The system is described by the Box–Jenkins model, whose accessible input and output are corrupted by unknown zero‐mean white Gaussian‐distributed disturbances and measurement noise. The signal and disturbance are outputs of asymptotically‐stable linear time‐invariant systems driven by an inaccessible input and a zero‐mean white Gaussian noise process, respectively. Predictive analytics is used to estimate the input by exploiting its smoothness and the randomness of the noisy input. The system, signal, and disturbance models and their associated KFs are identified for various selected model orders by minimising the KF residuals so that these become zero‐mean white noise processes. The selected model‐order corresponds to the minimum‐variance residual. Equivalently, the minimum order is selected when the number of poles or the output estimates of the identified models are all identical for all orders equal to, or exceeding the minimal order. The scheme is successfully evaluated and shown to outperform the commonly‐used but only sufficient Akaike Information Criterion.

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Section: Identification Of the System Signal And Disturbance Modelsmentioning

confidence: 99%

“…Proof The signal model is derived from the residual model of the KF, as proved in [8, 15 ]. The signal model becomes

{\hat{G}}_{s} ()(, z) = \frac{\overset{false¯}{N} false(z false)}{\overset{false¯}{D} false(z false)} = \frac{N_{s} (z) D_{w} (z)}{D_{s} (z) D_{w} (z)} = \frac{N_{s} false(z false)}{D_{s} false(z false)} ◻

…”

Section: Identification Of the System Signal And Disturbance Modelsmentioning

confidence: 99%

“…The augmented state‐space model

()(, A, B, {bold-italicE}_{w}, C)

termed as Box–Jenkins model [8, 14 ] is given by

\begin{matrix} right leftthickmathspace.5em \\ bold-italicx (k + 1) & = bold-italicA bold-italicx (k) + bold-italicB u_{normalest} (k) + E_{w} u_{w} (k) \\ y (k) & = bold-italicC bold-italicx (k) + v (k) \end{matrix}

where

x = (][, \begin{matrix} 1em4pt \\ {bold-italicx}_{normals} \\ {bold-italicx}_{w} \end{matrix}) \in R^{n}

A = (][, \begin{matrix} 1em4pt \\ {bold-italicA}_{normals} & 0 \\ 0 & {bold-italicA}_{w} \end{matrix}) \in R^{n \times n}

is a transition matrix;

B = (][, \begin{matrix} 1em4pt \\ {bold-italicB}_{s} \\ 0 \end{matrix}) \in R^{n x 1}

is the input vector,

{bold-italicE}_{w} = (][, \begin{matrix} 1em4pt \\ 0 \\ {bold-italicB}_{w} \end{matrix}) \in R^{n \times 1}

is the disturbance entry vector and

C = (][, \begin{matrix} 1em4pt \\ {bold-italicC}_{s} & bold-i... \end{matrix})

…”

Section: Disturbance Modelmentioning

confidence: 99%

“…The residual model relating the residual

e false(z false)

to the input

u_{est} ()(, z)

and the output

y false(z false)

e false(z false) = \frac{\overset{false¯}{D} false(z false)}{F false(z false)} y (z) - \frac{\bar{N} (z)}{F (z)} u_{est} false(z false)

where

()(, \overset{false¯}{D} ()(, z) / F false(z false)) = I - C {(z bold-italicI - bold-italicA bold+ bold-italicK bold-italicC)}^{- 1} K

()(, \overset{false¯}{N} ()(, z) / F false(z false)) = C {(z bold-italicI - bold-italicA bold+ bold-italicK bold-italicC)}^{- 1} B

and are transfer functions relating, respectively, the output y to the residual e ; and the input u to the residual e ;

F false(z false)

is the scalar characteristic polynomial given by

F false(z false) = (||, z I - A + K C)

. It is shown in [5, 7, 8 ] that the expressions for

\overset{false¯}{D} false(z false)

and

\overset{false¯}{N} false(z false)

are …”

Section: Disturbance Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New approach to model‐order selection

Doraiswami

Cheded

2020

Electron. lett.

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Fault tolerance in non‐linear systems: A model‐based approach with a robust soft sensor design

Doraiswami

Cheded²

2021

IET Control Theory & Appl

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A novel multiple Kalman filtre (KF)-based scheme is proposed, as a generalisation of conventional gain-scheduling techniques, for fault diagnosis and tolerance in a large class of multiple-input and multiple-output non-linear systems. The outputs are corrupted by unknown stochastic disturbance and measurement noise. A reliable and computationally efficient, two-stage identification of a piecewise linear parameter-varying Box-Jenkins dynamic model that better approximates the non-linear system, at each operating point, and the design of the associated KFs are proposed. Novel emulators, whose induced parameter changes mimic likely and predictive operating scenarios, are used to provide an accurate model identification and robustness to noise, disturbance, non-linearity errors and model perturbations. These crucial emulators generate missing representative data, aid predictive analytics, and improve the reliability and accuracy of the identified KF model. A novel formulation of the KF is used for fault isolation, and the Bayes strategy is used to isolate difficult-to-detect incipient faults in noisy environments. The proposed scheme leads to the design of a novel robust soft sensor aimed at replacing the maintenance-prone hardware sensor in practical applications including product quality assessment, performance monitoring, condition-based maintenance, fault diagnosis and fault-tolerant control. The proposed soft sensor was successfully evaluated on simulated and laboratory-scale physical control systems.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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New Kalman Filter Residue-Based Identification and Soft Sensor Design for Accurate Trajectory Tracking with a Fault-tolerant Robot

Doraiswami¹,

Cheded²,

Mamani³

et al. 2021

International Conference of Control, Dynamic Systems, and Robotics

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A Kalman filter(KF)-based identification, internal model-based controller for accurate tracking a specified trajectory despite the sensor errors, and fault tolerance is proposed. This study was mainly motivated by the need for precision, resolution and accuracy required in robotic applications such as robotic surgery. The computed torque approach is used to map a nonlinear model into a linear one. The sensor errors of the orientation input and the position corrupted by unknown input and output stochastic disturbance and measurement noise. Predictive analytics is used to estimate the true input by exploiting its smoothness and the randomness of the noisy input. The system is described using the Box-Jenkins(BJ) model, which is an augmented model of the true output, termed signal and the disturbance. The BJ model and the associated KF are identified without the a priori knowledge of the statistics of the disturbance and measurement noise. Using the key properties of KF the signal, the output error, the signal model, and the disturbance models, the KF associated with the signal model is accurately identified. An internal model-based state-feedback and feedforward controller is designed to accurately track the desired trajectory. The hardware sensors are replaced by KF-based sensors. The KF ensures fault tolerance. The proposed scheme was successfully evaluated on a physical robot.

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Novel Direct and Accurate Identification of Kalman Filter for General Systems Described by a Box-Jenkins Model

Cited by 4 publications

References 15 publications

New approach to model‐order selection

New approach to model‐order selection

Fault tolerance in non‐linear systems: A model‐based approach with a robust soft sensor design

New Kalman Filter Residue-Based Identification and Soft Sensor Design for Accurate Trajectory Tracking with a Fault-tolerant Robot

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