Fault detection and identification for uncertain linear time-delay systems

doi:10.1016/j.compchemeng.2005.08.012

Computers & Chemical Engineering

Volume 30, Issue 2, 15 December 2005, Pages 228-242

https://doi.org/10.1016/j.compchemeng.2005.08.012 Get rights and content

Abstract

In this paper, a robust fault detection and identification approach based on an adaptive observer is developed for uncertain continuous linear time-invariant systems with multiple discrete time-delays in both states and outputs. State and output faults of bias type that may evolve slowly or abruptly are considered, and the delay system is disturbed by unstructured bounded unknown inputs. Based on the scheme of [Trunov, A. B., & Polycarpou, M. M. (2000). Automated fault diagnosis in nonlinear multivariable systems using a learning methodology. IEEE Transactions on Neural Networks, 11, 91–101], a novel adaptive observer for detecting and estimating faults in the considered system is constructed, and robustness with respect to unknown inputs and sensitivity to faults of the detecting scheme are rigorously analyzed. The fault estimate and the state estimation error are then proved to be uniformly bounded. Finally, simulations of a heating process demonstrate that the proposed approach can detect the faults shortly after the occurrences without any false alarm and can approximate the faults with desired accuracy.

Introduction

With the rising demands of product quality, effectiveness and safety in modern industries, the research on fault diagnosis for dynamic systems has received more and more attention and has developed quickly in the past three decades. On the contrary to the intensive investigation of robust fault diagnosis for uncertain systems and fault diagnosis for nonlinear systems, which have achieved much progress in recent years (Frank & Ding, 1997; Venkatasubramanian, Rengaswamy, Yin, & Kavuri, 2003), the works on fault diagnosis for time-delay systems are very few. Time-delay exists widely in practice (Hale & Verduyn Lunel, 1993; Richard, 2003). Large delays in some reaction processes of chemical industries or time-delays induced by long-distance transportation and communication might cause the closed-loop systems unstable and deteriorate the control performance. So the development of fault diagnosis methods for time-delay systems, which are infinite dimensional systems in essence, is very important.

On the research of fault diagnosis for linear time-delay systems, Yang and Saif (1998) first proposed a scheme of actuator and sensor fault diagnosis using an unknown input observer and a technique of input estimation for systems with time-delays only in the state. In this work, modeling uncertainties were not considered and some assumptions on the system's structure decomposition were unreasonable. For systems with state and input time-delays, Ding, Zhong, Tang, and Zhang (2001) designed a robust fault detection filter that guaranteed both sensitivity to faults and insensitivity to disturbances. In the scheme of Zhong, Ding, Lam, and Zhang, 2003, the influence of disturbances on the residual was further decreased using the idea of integrated design of $H_{\infty}$ filter and unknown input observer. In the works of Ding et al. (2001) and Zhong et al. (2003) structured description of disturbances and faults (i.e. the form of time-variant function multiplied by a known constant matrix from the left) was adopted, which had some limitations in practice. Based on an adaptive observer, Jiang, Staroswiecki, and Cocquempot, 2002 developed a scheme to estimate abrupt state fault for linear (nonlinear) systems with only state time-delays, and no uncertainties were considered. For systems with constant time-delays in inputs and outputs only, Zhang, Ding, Wang, and Zhou, 2002 presented a state fault detection method based on parity space. The time-delays investigated above are either in the state or in the input/output, not in both of them. In practice, a system's dynamics may involve time-delays in states, inputs and outputs, and the influences of modeling uncertainties, noises and disturbances are perhaps not negligible. Furthermore, all components of a controlled system may deteriorate after long time running.

In the fault diagnosis scheme based on an adaptive observer, a fault cannot only be detected but also be approximated, and the fault estimate can be further used in fault-tolerant control. On the basis of an adaptive observer, Wang and Daley (1996) and Jiang et al. (2002) presented schemes that could deal with state fault only. Using the augmented error technique, Wang, Huang, and Daley, 1997 proposed another scheme that could estimate actuator and sensor gain faults occurring simultaneously or sequentially. In their approach, measurement noises were not considered and the transform needed for multi-input multi-output systems was complex. Combining adaptive observer with neural networks used as nonlinear function approximators, Polycarpou and Helmicki (1995) and Vemuri and Polycarpou (1997) developed an on-line learning scheme to detect and estimate faults for nonlinear systems. Trunov and Polycarpou (2000) generalized this scheme to detect and estimate both state and output faults in the presence of modeling uncertainties. Compared with the schemes of Wang and Daley (1996); Wang et al. (1997) and Jiang et al. (2002), the one of Trunov and Polycarpou (2000) is more general.

In short, for general linear time-invariant systems with time-delays in the states, inputs and outputs, the problem of state and output fault diagnosis in the presence of modeling uncertainties and unknown disturbances has not been solved completely up to the present. In this paper, we extend the scheme of Trunov and Polycarpou (2000) to robust fault detection and estimation for a class of uncertain continuous linear time-invariant systems with multiple discrete time-delays in both states and outputs. Bias type faults, incipient (slowly developing) or abrupt, are considered which include state and output faults. Modeling uncertainties, disturbances and noises are represented as unstructured unknown inputs appearing in the state and output equations with their magnitudes bounded. Considering the fact that time-delay systems are essentially infinite dimensional, this extension is not trivial. We demonstrate that the proposed fault detection and identification scheme is robust to unknown disturbances, and prove that both the fault estimate and the state estimation error are uniformly bounded. The sensitivity to faults is also analyzed.

The organization of this paper is as follows. The problem formulation, the robust fault detection and identification scheme based on an adaptive observer are presented in Sections 2 Problem formulation, 3 Fault detection and identification scheme. Robustness to the unknown disturbance, sensitivity to faults and the performance of fault estimates are analyzed in Sections 4 Robustness to unknown disturbances, 5 Sensitivity to faults, 6 Performance analysis of fault estimation, respectively. Simulations with three fault cases for a simulated heating process are presented in Section 7. Finally, Section 8 draws the conclusions.

Throughout this paper, $ℝ^{n}$ denotes the n dimensional Euclidean space with vector norm $| \cdot |$ , $ℝ^{n \times m}$ is the set of all real matrices with induced matrix norm $| \cdot |, C ([- d_{q}, 0], ℝ^{n})$ denotes the space of continuous functions $ϕ : [- d_{q}, 0] \to ℝ^{n}$ with norm ${∥ϕ (θ)∥}_{d_{q}} ≔ \sup_{θ \in [- d_{q}, 0]} |ϕ (θ)|$ , $L_{\infty}$ denotes the space of functions with the supremum norm ${∥\cdot∥}_{\infty}, L_{2}$ denotes the space of square integral functions, ${∥\cdot∥}_{\infty}$ also denotes the $H_{\infty}$ norm for transfer functions, $| \cdot |_{F}$ denotes the Frobenius matrix norm defined by ${|M|}_{F}^{2} ≔ \sum_{i j} {|m_{i j}|}^{2} = trace (M M^{T})$ , I_n denotes the n × n dimensional identity matrix, and the notation P > 0 for $P \in ℝ^{n \times m}$ means that P is symmetric and positive definite.

Section snippets

Problem formulation

Consider a continuous linear system with multiple discrete time-delays in both states and outputs described by $\{\begin{cases} \dot{x} (t) = \sum_{i = 0}^{q} A_{i} x (t - d_{i}) + B u (t) + B_{x} (t - T_{x}) f_{x} + η_{x} (t) \\ y (t) = \sum_{i = 0}^{q} C_{i} x (t - d_{i}) + D u (t) + B_{y} (t - T_{y}) f_{y} + η_{y} (t) \end{cases}$ where $x (t) \in ℝ^{n}, y (t) \in ℝ^{m}, u (t) \in ℝ^{l}$ are the state, measurable output and control input of the system, respectively. A_i, B, C_i, D, i = 0, …, q are known matrices with proper dimensions. $0 = d_{0} < d_{1} < \dots < d_{q} < \infty$ are a group of discrete constant time-delays. In model (1) time-delays in the state and output are assumed to

Fault detection and identification scheme

Based on the scheme of Trunov and Polycarpou (2000), for the uncertain linear time-delay system (1) an adaptive observer is constructed in this section, which will be used to detect and estimate the faults.

The adaptive observer for fault detection and identification is constructed as (Bastin & Gevers, 1988) $\{\begin{array}{l} \dot{\hat{x}} (t) = \sum_{i = 0}^{q} A_{i} \hat{x} (t - d_{i}) + B u (t) + {\hat{f}}_{x} (t) + K [y (t) - \hat{y} (t)] - Ω_{x} (t) {\dot{\hat{f}}}_{x} (t) - Ω_{y} (t) {\dot{\hat{f}}}_{y} (t) - \sum_{i = 1}^{q} A_{i} ξ_{i} (t) \\ \hat{y} (t) = \sum_{i = 0}^{q} C_{i} \hat{x} (t - d_{i}) + D u (t) + {\hat{f}}_{y} (t) - \sum_{i = 1}^{q} C_{i} ξ_{i} (t) \\ ξ_{i} (t) = Ω_{x} (t - d_{i}) [{\hat{f}}_{x} (t) - {\hat{f}}_{x} (t - d_{i})] + Ω_{y} (t - d_{i}) [{\hat{f}}_{y} (t) - {\hat{f}}_{y} (t - d_{i})] \end{array}$ $\{\begin{cases} \dot{Ω} \end{cases}$

Robustness to unknown disturbances

In the presence of unknown disturbances, robustness with respect to disturbances is required for fault detection, and this performance is guaranteed in the fault detection algorithm designed above by the dead-zone operator (11). An appropriate threshold ɛ selected can reduce the ratio of missing alarm of the detection algorithm on the premise that no false alarms exist. The following theorem indicates that, before the occurrence of any fault, i.e. when the system is only driven by the

Sensitivity to faults

In addition to robustness, another performance index of fault detection algorithm is fault sensitivity. In the following theorem, some sufficient conditions are presented which characterize the faults that can be detected by the proposed algorithm.

Theorem 2

Consider the robust fault detection and identification scheme (6), (7) and (10).

(1)
If there is a time instant t_x > 0 such that the state fault $B_{x} (t - T_{x}) f_{x}$ satisfies $|\sum_{i = 0}^{q} C_{i} \int_{0}^{T_{x} + t_{x} - d_{i}} X (T_{x} + t_{x} - d_{i} - τ) B_{x} (τ - T_{x}) f_{x} d τ| \geq 2 ε$ then the state fault will be detected.
(2)
If there

Performance analysis of fault estimation

In the previous sections, robustness to disturbances and sensitivity to faults are analyzed rigorously. In the sequel, stability of the estimation algorithm is derived. Before the theorem is presented, some additional definitions are needed $e_{x} (t - d_{i}) ≔ x (t - d_{i}) - \hat{x} (t - d_{i}), {\tilde{f}}_{k} (t - d_{i}) ≔ f_{k} - {\hat{f}}_{k} (t - d_{i}), Φ_{k} (t) ≔ I - B_{k} (t - T_{k}), k = x, y and i = 0, \dots, q$ So it can be derived from (3) and (4) that ${\dot{Φ}}_{x} (t) = - Λ_{x} Φ_{x} (t), t \geq T_{x}$ ${\dot{Φ}}_{y} (t) = - Λ_{y} Φ_{y} (t), t \geq T_{y}$ where $Φ_{x} (T_{x} + θ) = I$ , $Φ_{y} (T_{y} + θ) = I$ for $θ \in [- d_{q}, 0]$ , and $Λ_{x} = diag {α_{x 1}, \dots, α_{x n}}, Λ_{y} = diag {α_{y 1}, \dots, α_{y m}}$ .

Theorem 3

Simulations

In this section, the effectiveness of the proposed fault detection and identification method is illustrated with a simulated heating process.

This process consists of a water-water heat exchanger A, an air–water heat exchanger B with a cooling fan, long connected pipes, a three way valve, and some other elements. The schematic diagram of the plant is shown in Fig. 1, where the dash-dot line represents the primary (hot water) circuit and the dashed line represents the secondary (cold water)

Conclusions

Based on an adaptive observer, a robust fault detection and identification scheme is proposed for a class of uncertain continuous linear time-invariant systems with multiple discrete time-delays in both states and outputs. State and output faults of biased type, either incipient or abrupt, are considered. Theoretical analyses and simulation results show that the fault detection and identification scheme is robust to unknown bounded disturbances and is capable of estimating the faults with

Acknowledgements

This work was mainly supported by NSFC (Grant No., 60234010), partially supported by the Field Bus Technology and Automation Key Lab of Beijing at North China and the national 973 program (Grant No. 2002CB312200) of China.

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Fault detection and identification for uncertain linear time-delay systems

Abstract

Introduction

Section snippets

Problem formulation

Fault detection and identification scheme

Robustness to unknown disturbances

Sensitivity to faults

Performance analysis of fault estimation

Simulations

Conclusions

Acknowledgements

Automatica

Journal of Process Control

Automatica

Computers and Chemical Engineering

Automatica

Automatica

Control Engineering Practice

Stable adaptive observers for nonlinear time-varying systems

IEEE Transactions on Automatic Control

Incipient fault diagnosis of dynamical systems using online approximators

IEEE Transactions on Automatic Control

A unified approach to the optimization of fault detection systems

International Journal of Adaptive Control and Signal Processing