Sensor fault detection in a class of nonlinear systems using modal Kalman filter

doi:10.1016/j.isatra.2020.08.008

ISA Transactions

Volume 107, December 2020, Pages 214-223

https://doi.org/10.1016/j.isatra.2020.08.008 Get rights and content

Highlights

•
A novel model-based fault detection approach, based on a new modification of Kalman filter is proposed.
•
This approach, called Modal Kalman filter, is based on modal series.
•
Applied to nonlinear systems for sensor fault detection.
•
Both bias fault and loss of effectiveness fault are taken into consideration.
•
The results approve the efficiency of the proposed filter.

Abstract

Kalman filter and its different variants are commonly used as optimal methods for fault detection in various types of system components. In this paper, a newly introduced type of aforementioned filters, called modal Kalman filter, is extended and utilized in order to estimate the states of nonlinear systems, for sensor fault detection purposes, in a class of nonlinear certain systems. This method, in contrast to the extended Kalman filter, which employs only the linear term of Taylor expansion, retains higher-order terms; as a result, the estimation error will reduce accordingly. Practicality and effectivity of this method, and its superiority over Kalman filter, in terms of accuracy and promptness of sensor fault detection, are also verified with simulation results.

Introduction

The growing need for safe, optimal and efficient operation of many complicated and expensive industrial systems is one of the main incentives to find the best solution for fault detection and identification in such systems. Fault occurrence is inevitable; hence, a fast and reliable fault detection method, in order to prevent disaster event, is an essential need.

Regarding the fault detection methods used, two general categories can be identified: data-driven methods and model-based methods. Data-driven methods, utilize the information collected from the system for fault detection purposes. These approaches mostly rely on real-time or historical data and use artificial intelligence algorithms, statistically process data or consist of expert systems. The main advantage of these methods is that no accurate model of the system is required and they have been successfully applied for fault detection purposes [1]. However, the use of recorded data streams for fault detection requires a large database, and the enormous computational effort makes these methods impractical for online applications.

On the other hand, model-based methods assume that the accurate model of the system and its parameters are precisely known. These methods are based on the difference between the process measurement and the model measurement, through the generation of “residual” signal. By evaluating the generated residual, fault occurrence could be detected [2]. When a precise mathematical model of the system is available, model-based methods are more powerful, in comparison to data-based ones, and provide better real-time performance. Different model-based techniques, such as parity space [3], parameter estimation [4] and state estimation [5], have been used in literature, in order to detect fault occurrence. Among the existing model-based methods which use state estimation to generate “residual” signal, Kalman filter (KF) variants have found widespread applications.

Kalman filter has been a powerful tool, which provides optimal state estimation for linear systems with Gaussian noise, since the 1960s [6]. The algorithm consists of two steps: the state update which produces the a priori estimate of the state, which does not include observation information from the current timestep, and the measurement update which provides the a posteriori estimate of the state, by combining the current a priori prediction with current observation information to decreases state error covariance and produce the a posteriori state estimate. Its recursive nature makes it a suitable option for implementations. Optimality of linear Kalman filter (LKF) for linear systems state estimation encourages other modifications of this filter to be developed, such as extended Kalman filter (EKF), for nonlinear systems state estimation. The main idea of EKF is the linearization of the nonlinear system, and then applying LKF to obtain the state estimation. This linearization takes place at any time step around the last estimated state. To overcome the issue of linearization errors, another modification of Kalman filter, called Unscented Kalman Filter (UKF) is presented [7]. In this algorithm, the state probability distribution is represented by using a set of samples, called sigma points.

In the 1970s, the use of this method for fault detection was presented by Mehra and Peschon [8]. Since then, it has been used widely as a residual generator for fault detection. EKF and UKF are also successfully used for nonlinear systems fault detection [9], [10]. Faults can occur in process components, actuators or sensors. In this work, sensor fault detection is addressed, with the assumption of non-faulty performance of other components of the system.

Sensors are used in almost every industrial process and they are of great importance to ensure a safe and reliable operation of the systems. The information, which the sensors provide, is essential to have an accurate assessment of the condition of the monitored system and take proper actions in case of any abnormality. Faults in sensors can cause degradation in system performance and even make the control system unstable. In addition, the defective output of the sensor can lead to incorrect decisions and component replacements. Therefore, constant monitoring of sensors performance and sensors fault detection is essential and various methods are used to detect sensor faults in nonlinear systems. [11], [12], [13], [14]. Kalman filter and its modifications, as well-known optimal filters, are widely used to detect sensor faults for linear and nonlinear systems in the presence of measurement noise and disturbances [15], [16], [17]. In this paper, the ability of a new modification of KF, called modal Kalman filter (MKF) for sensors fault detection in a class of certain nonlinear systems will be investigated.

In [18] and [19] a method, called modal series, is introduced for investigation and modeling of autonomous nonlinear systems. In this proposed approach the modal series are used to transform the state estimation of a nonlinear system into an infinite number of state estimations of linear systems. In this method, system modeling is more accurate than the linearization of nonlinear systems. In [20] the approach is extended to find the solution of nonautonomous nonlinear differential equations. In [21] an extended form of modal series is used to present a nonlinear model predictive control approach.

Modal Kalman filter is introduced in [22]. In this method, in order to estimate the states of the nonlinear system, instead of using the first linear term of the Taylor series extension, higher-order terms are used. This filter, based on modal series, converts the state estimation of a nonlinear system into a series of linear systems state estimation. Since Kalman filter is an optimal and effective state estimator for linear systems, it is used to estimate the state of each linear system in this series. In other words, Kalman filter is used more than once at each step in nonlinear system state estimation. As it has been discussed in [22], the estimation error in this case is much lower than one by using KF, and in its extended modification, the error is lower than EKF. The main contribution of this paper is to extend the formulas of MKF, which are presented in [22], to make it possible to use this filter for state estimation of physical nonlinear plants with inputs, and to use this filter for the purpose of sensors fault detection in nonlinear systems.

The remainder of this paper is organized as follows: modal Kalman filter and its mathematical formulation are introduced in Section 2. In Section 3 the MKF-based sensor fault detection method is explained. In order to evaluate the proposed method and to verify its effectiveness, simulation results are given in Section 4, and the performance of MKF in sensor fault detection is compared with KF and EKF. Finally, in Section 5 conclusions are presented.

Section snippets

Problem formulation

The following nonlinear differential equation is considered: $\dot{X} (t) = f (X (t)), X (t_{0}) = X_{0}$ where $X (t) \in R^{n}$ , $f : R^{n} \times R^{m} \to R^{n}$ and $X_{0}$ are states vector, a nonlinear smooth function and initial states condition, respectively. Assuming $X_{e} = 0$ as a stable operating point for the nonlinear system (1), it is shown in [23] that the solution of the nonlinear system (1), the modal series of $X (t)$ , can be obtained as follows: $X (t) = \sum_{j = 0}^{\infty} s^{j} (t)$ where $s^{j} (t) \in R^{n}, j = 1, 2, 3, \dots$ are continuous-time vectors that satisfy the following linear

Sensor fault detection

Sensor faults classification, based on the impact on sensor readings, can be defined as follows:

Bias: The constant value added to the sensor’s output signal.

Drift: A variable with time, $ϕ (t)$ , that is added to the sensor’s signal. This value usually changes linearly with time.

Loss of effectiveness: In this case, the signal is multiplied by a coefficient $α (t)$

Hard Failure: the sensor output stays constant at zero or a non-zero value.

Bias and hard failure are abrupt faults, which occur

Mixing tank

A mixing tank process example, as shown in Fig. 3, is used in this work to evaluate the performance of the proposed methodology in sensor fault detection. Nonlinear equations of the system are according (47), (48), where $x (t) = [\begin{matrix} h (t) \\ T_{T} (t) \end{matrix}]$ is thestate vector. The elements of $x (t)$ indicate the height and the temperature of the liquid in the tank, respectively. $u (t) = [\begin{matrix} q_{C} (t) \\ q_{H} (t) \end{matrix}]$ is the input vector, represents the flow of two liquid inlets, with temperatures $T_{C}$ and $T_{H}$ .

$\dot{h} (t) = \frac{1}{A_{T}} (q_{C} (t) + q_{H} (t) - c_{D} A_{0} \sqrt{2 g h (t)})$ $\dot{T_{T}} (t) = \frac{1}{h (t) A_{T}}$

Conclusion

This paper presents a model-based sensor fault detection scheme, for a class of certain nonlinear systems, using modal Kalman filter. This method offers a great enhancement in accurate state estimation, which can make the residuals smoother in the non-faulty mode. Generation of smoother and lower residuals results in more sensitivity to fault occurrence; as a result, the false alarm rate and the detection delay are noticeably reduced. Two simulation examples, with different fault types,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Research articleSensor fault detection in a class of nonlinear systems using modal Kalman filter

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Sensor fault detection

Mixing tank

Conclusion

Declaration of Competing Interest

Inf Sci

Control Eng Pract

J Process Control

Automatica

Mech Syst Signal Process

J Process Control

Measurement

Inf Fusion

ISA Trans

Robust Model-Based Fault Diagnosis for Dynamic Systems