Kalman filters in non-uniformly sampled multirate systems: For FDI and beyond

doi:10.1016/j.automatica.2007.05.009

Automatica

Volume 44, Issue 1, January 2008, Pages 199-208

https://doi.org/10.1016/j.automatica.2007.05.009 Get rights and content

Abstract

The first part of the paper is the development of a data-driven Kalman filter for a non-uniformly sampled multirate (NUSM) system. Algorithms for both one-step predictor and filtering are developed and analysis of stability and convergence is conducted in the NUSM framework. The second part of the paper investigates a Kalman filter-based methodology for unified detection and isolation of sensor, actuator, and process faults in the NUSM system with analysis on fault detectability and isolability. Case studies using data respectively collected from a pilot experimental plant and a simulated system are conducted to justify the practicality of the proposed theory.

Introduction

In many industrial processes, variables are sampled at multiple rates. Therein the manipulated variables can be adjusted at relatively fast rates, while the measurements of quality variables are typically obtained from laboratory analysis after several minutes of analysis. Furthermore, the sampling is termed as non-uniform, if the sampling intervals for each variable are non-equally spaced.

First, this paper attempts to develop the Kalman filter for a non-uniformly sampled multirate (NUSM) system. The development is conducted in a generic framework: each variable in a physical system is sampled at non-uniform rates. Non-uniform sampling has advantages over uniform sampling, such as always preserving controllability and observability in discretization (Sheng, Chen, & Shah, 2002). Suppose that one describes a NUSM system by a lifted state space model. In the development of the Kalman filter, no knowledge regarding the state space model and the covariance matrices of process and measurement noise in the NUSM system is assumed. Alternatively, a subspace method of identification (SMI) for the state space model and the covariances is proposed.

The second part of this paper is concerned with the development of a novel Kalman filter-based approach towards unified detection and isolation of sensor, actuator, and process faults in NUSM systems. For single rate systems, Mehra and Peschon (1971) did the pioneering work in applying Kalman filters for fault detection. Kalman filter-based fault detection and isolation (FDI) methods proposed before the 1990s have been surveyed in Willsky (1976) and Frank (1990), and Keller's (1999) work represented the state of art in this area. However, these existing FDI schemes are applicable only for actuator and process faults and have difficulty in isolating sensor faults (White & Speyer, 1987).

Recently, several authors (Fadali and Liu, 1998, Fadali and Emara-Shabaik, 2002, Zhang et al., 2002) have considered FDI issues in uniformly sampled multirate systems. In addition, Li, Han, and Shah (2006) have developed detection and isolation methodology for sensor and actuator faults in NUSM systems. This paper investigates a novel Kalman filter-based FDI scheme that works for detection and isolation of faults in actuators, sensors, and process components in NUSM systems. Analysis of fault detectability and isolability is also given.

Section snippets

Problem formulation

Consider a multivariate system represented by the following continuous-time (CT) state space model: $\dot{x} (t) = Ax (t) + B \tilde{u} (t) + φ (t), \tilde{y} (t) = Cx (t) + D \tilde{u} (t),$ where: (i) $\tilde{u} (t) \in R^{l}$ and $\tilde{y} (t) \in R^{m}$ are noise-free inputs and outputs, respectively; (ii) $x (t) \in R^{n}$ is the state; (iii) $φ (t) \in R^{n}$ is a stationary Gaussian white noise vector with covariance $R_{φ} \in R^{n \times n}$ , e.g. $φ (t) \sim ℵ (0, R_{φ})$ , to represent process disturbances; and (iv) $A$ , $B$ , $C$ and $D$ are unknown system matrices with compatible dimensions. In the sequel, a Gaussian white

Subspace identification of the NUSM system

In the past few years, SMI algorithms for uniformly sampled multirate systems (Li, Shah, & Chen, 2001), and NUSM systems for residual models to perform FDI (Li et al., 2006) have been reported. This paper proposes an SMI, which will have a more generic applicability in contrast to the existing work, for Eq. (5).

With $i > n$ , ${\underset{̲}{Γ}}_{1} = \underset{̲}{C}$ , and ${\underset{̲}{H}}_{1} = \underset{̲}{D}$ , define ${\underset{̲}{Γ}}_{i} \equiv [\begin{matrix} \underset{̲}{C} \\ {\underset{̲}{Γ}}_{i - 1} \underset{̲}{A} \end{matrix}], {\underset{̲}{H}}_{i} \equiv [\begin{matrix} \underset{̲}{D} & 0 \\ {\underset{̲}{Γ}}_{i - 1} \underset{̲}{B} & {\underset{̲}{H}}_{i - 1} \end{matrix}],$ where $^{'}$ stands for the transpose of an argument. Since the pair $(\underset{̲}{C}, \underset{̲}{A})$ is observable, ${\underset{̲}{Γ}}_{i}$ has rank n. Define a

Kalman filter for the NUSM system

Suppose that ${\underset{̲}{A}, \underset{̲}{B}, \underset{̲}{C}, \underset{̲}{D}}$ , ${\underset{̲}{R}}_{ω}$ , ${\underset{̲}{R}}_{ɛ}$ , and ${\underset{̲}{R}}_{ɛ, ω}$ have been identified. Denote $\hat{x} (k | j)$ as the estimate of $x (k)$ based on data ${\underset{̲}{u} (1), \underset{̲}{y} (1), \dots, \underset{̲}{u} (j), \underset{̲}{y} (j)}$ , where $j = k - 1$ or $j = k$ . The Kalman filter for Eq. (5) should enable the estimation of $x (k)$ such that $x (k) - \hat{x} (k | j)$ has minimum covariance. Specifically, the Kalman filter functions as a one step predictor if $j = k - 1$ , or a filtering algorithm if $j = k$ .

Mathematical description of the NUSM system with faults

A multivariate system can be represented by $\dot{x} (t) = Ax (t) + B \tilde{u} (t) + f_{p} (t) + φ (t), \tilde{y} (t) = Cx (t) + D \tilde{u} (t),$ in the presence of process faults (Ge & Fang, 1988), where $f_{p} (t) \in R^{n}$ is the fault magnitude vector with zero or non-zero elements.

We apply the NUSM sampling approach to discretize Eq. (18), where $f_{p} (t)$ and $φ (t)$ are “sampled” at the same rate. Similarly, it can be shown that the resulting lifted model is $x (k + 1) = \underset{̲}{A} x (k) + \underset{̲}{B} \tilde{\underset{̲}{u}} (k) + \underset{̲}{W} {\underset{̲}{f}}_{p} (k) + \underset{̲}{W} \underset{̲}{φ} (k), \tilde{\underset{̲}{y}} (k) = \underset{̲}{C} x (k) + \underset{̲}{D} \tilde{\underset{̲}{u}} (k),$ where ${\underset{̲}{f}}_{p} (k) \in R^{ng}$ is the lifted vector of $f$

A numerical example

This example demonstrates the power of the Kalman filter in estimating physical variables from NUSM data. A quadruple tank system (Ge & Fang, 1988) is used as a test bed, where four tanks with the same height and same cross section are serially connected by outlets that have an identical cross section.

The tank system can be modelled by a nonlinear differential equation. After linearizing the equation at a steady operating point, one can get the following state space model (Ge & Fang, 1988): $\dot{x} (t$

Conclusion

Data-driven Kalman filters for NUSM systems have been proposed. A numerical example to illustrate estimation of the process variables from a simulated quadruple tank system with NUSM data is provided. Moreover, a novel Kalman filter-based FDI methodology has been investigated. The developed FDI scheme has been applied to an experimental CSTHS system, and its practicality and utility have been demonstrated.

Acknowledgements

Financial aid from the Natural Science and Engineering Research Council, Matrikon Inc., Suncor Energy Inc., and iCORE of Canada is gratefully acknowledged.

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Weihua Li was born in October, 1963 in WuXue City (former GuangJi County), Hubei Province, the People's Republic of China and was entirely educated in China. He got his B.Sc. in Control Engineering from South China University of Technology, in 1982 and Ph.D. in Control Engineering from Tsinghua University in 1994. He was a Research Manager for the NSERC-MATRIKON-SUNCOR Senior Industrial Research Chair Project in Computer Process Control in the Department of Chemical and Materials Engineering, University of Alberta, Canada for few years. Currently, he is working on development of process monitoring technologies in Shell Global Solutions. His research interests are process monitoring, fault detection & diagnosis, subspace methods of identification for diagnosis, multivariate statistical analysis, and computational linear algebra.

Sirish Shah received his B.Sc. degree in Control Engineering from Leeds University in 1971, a M.Sc. degree in Automatic Control from UMIST, Manchester in 1972, and a Ph.D. degree in Process Control (Chemical Engineering) from the University of Alberta in 1976. During 1977 he worked as a Computer Applications Engineer at Esso Chemicals in Sarnia, Ontario. Since 1978 he has been with the University of Alberta, where he currently holds the NSERC-Matrikon-Suncor-iCORE Senior Industrial Research Chair in Computer Process Control.

In 1989, he was the recipient of the Albright & Wilson Americas Award of the Canadian Society for Chemical Engineering in recognition of distinguished contributions to chemical engineering. He has held visiting appointments at Oxford University and Balliol College as a SERC fellow in 1985–1986, Kumamoto University, Japan as a senior research fellow of the Japan Society for the Promotion of Science (JSPS) in 1994, the University of Newcastle, Australia in 2004, IIT-Madras, India in 2006 and the National University of Singapore in 2007. The main area of his current research is process and performance monitoring, system identification and design and implementation of softsensors. He is the co-author of a book titled, Performance Assessment of Control Loops: Theory and Applications. He has held consulting appointments with a wide variety of process Industries and has also taught many industrial courses.

Deyun Xiao was born in 1945 and graduated from Tsinghua University, Beijing, China in 1970. Since 1996 he is Professor of Control Science and Engineering in the Department of Automation, Tsinghua University. His main research interests include modeling and system identification, fault diagnosis, control of hybrid dynamical systems, multi-sensor fusion, and intelligent transportation systems.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hitay Ozbay under the direction of Editor Ian Petersen.

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Brief paperKalman filters in non-uniformly sampled multirate systems: For FDI and beyond☆

Abstract

Introduction

Section snippets

Problem formulation

Subspace identification of the NUSM system

Kalman filter for the NUSM system

Mathematical description of the NUSM system with faults

A numerical example

Conclusion

Acknowledgements

Automatica

Automatica

Automatica

Automatica

Journal of Process Control

Automatica

Automatica

Automatica

Introduction to stochastic control theory

Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrice

IEEE Transactions on Automatic Control

Brief paper
Kalman filters in non-uniformly sampled multirate systems: For FDI and beyond☆