Observer-based fault detection and diagnosis strategy for industrial processes
Introduction
Nowadays, the flexibility introduced due to the use of computer algorithms, to perform control systems strategies, offer operators a wide variety of possibilities to achieve the desired aims, in terms of efficiency and reliability (e.g. optimizing costs and control efforts). Unfortunately, all those advantages and features are, to a greater or lesser degree, subject to the proper behaviour of all components involved in the system. In other words, the design of a modern controller for complex systems, in case of malfunctions on actuators, sensors or other components, can result in unsatisfactory performance, or even lead the system to instability.
Accordingly, the problem of achieving a suitable performance and stability, especially for safety-critical processes, requires a different strategy rather than just having a robust controller. Thereby, this challenge has motivated the study of what is now commonly known as Active Fault Tolerant Control Systems (AFTCS) [1], [2], [3], [4]. Specifically, this strategy responds to the system component failures in an active way by reconfiguration, so that the stability and acceptable performance of the entire system can be maintained.
The appropriate behaviour of an AFTCS highly depends on a solid Fault Detection and Diagnosis (FDD) module, to provide the accurate fault information before reconfiguration must be undertaken. The FDD module is perhaps the most difficult aspect involved in the design of an AFTCS, and its development has a highly important role, because an early detection and maintenance of faults can avoid system shutdown, breakdowns, and even worse, the occurrence of potential catastrophes (especially for safety-critical systems). In that sense, previous works show that the modular approach (FDD and AFTCS designed separately) presents its benefits, being more flexible for practical applications and, therefore, easier to test and implement [5], [6].
As a consequence, a wide variety of FDD methods is available in literature, such as: a deeply review of process fault detection and diagnosis is presented in the series of works [7], [8], [9]; a brief survey about fault detection methods is tackled in [10]; and a recent survey of fault diagnosis and fault-tolerant techniques is exposed in [11], [12]. In this way, several contributions and theoretical approaches about its use on linear systems can be found in [6], [13], [14], [15], [16], and the references therein. Besides, there is a considerable amount of articles that involve the development of observers applied to a non-linear system. For instance, fuzzy techniques, adaptive approaches or sliding-mode observer-based technology have been recently reported [17], [18], [19], [20], [21], [22], even some newer papers have applied the Moving Horizon Estimation (MHE) framework to develop FDD schemes [23], [24]. However, there are few works, in literature, about observer-based fault detection, isolation and estimation of multiple actuators and sensors upon non-linear systems that present its approaches over continuous-time as much as discrete-time, which motivates our current study in this paper. In short, this problem is still a challenge due to the troubles of dealing with non-linearities.
Hence, because of the difficulty to design a non-linear observer, many authors prefer to represent these systems by a Linear Parameter Varying (LPV) approach [25], [26], [27], even for the observers’ design [17], [28], [29], among others. The idea of this is to represent the system as an interpolation of i-th affine local models, which depend on scheduling parameters. These parameter values depend on endogenous variables, such as the states and inputs (typically available in real-time). That is, the interpolation technique presents a good method to schedule a set of linear models by a convex weighting function. Thus, the LPV modelling framework is powerful since it allows the application of well-known linear control and estimation design tools to a wide range of non-linear models.
For the above mentioned reasons, this paper presents the design of a novel FDD scheme, composed from a bank of two types of observers, applied to LPV systems. The first one uses a combination of LPV Reduced-order Unknown Input Observers (LPV-RUIO) to detect, isolate and estimate actuators faults, and the second one consists of a set of full-order LPV Output Observers with Unknown Input (LPV-UIOO) to detect, isolate and estimate sensors faults. The observers’ design and its stability conditions are based on the resolution of a Linear Matrix Inequalities (LMI) problem that was solved using the numerically efficient interior-point methods [30]. In addition, the continuous-time and discrete-time FDD module design is addressed.
The main motivation for the use of two type of observers, to deal with actuators and sensors faults, is the possibility to exploit each of its specific goodness. That is, the use of LPV-RUIOs allows an easy way to compute the actuator fault estimation (without the use of extended observer, or the computation of the pseudo-inverse of bad conditioned matrices), enabling a design with more degree of freedom, relieving the designer work to select the desired actuator input as unknown input without resulting unfeasible. On the other hand, the LPV-UIOOs are the simplest way to deal with sensors faults, with a better trade-off about performance and off-line design complexity, which is ideal to improve the efficiency of the proposed strategy. In such a way, the observers’ design is mainly doing off-line, and only a slightly use of computational resources is needed at each sample time to perform on-line the FDD tasks, enabling its application as a part of an AFTCS.
Lastly, with the purpose of highlighting the behaviour of the proposed FDD strategy, two numerical simulations of typical chemical industrial processes are given. These are a Heat Exchanger (HE) and a highly non-linear Continuous Stirred Tank Reactor (CSTR). Remark 1 Note that the goal of this work was not to compare the LPV observed-based methodology with other approaches, but to verify that this is a good option for FDD applied to industrial processes.
This work is organized as follows. In Section 2, the polytopic representation of an affine LPV system is provided. Sections 3 and 4 present the design proposal and stability conditions of the LPV-RUIO and LPV-UIOO observers. In Section 5, an explanation about the detection, isolation and estimation methodology is introduced. Subsequently, in Sections 6 and 7, two numerical simulations are provided to show the performance of the proposed approach. Finally, concluding remarks are made in the last section.
Notation Through this paper, with the purpose of simplifying its structure, the duality between continuous-time and discrete-time is assumed equivalent, except in cases indicated explicitly. For that, the differential operator ρ is used, which denotes the time derivative for continuous-time models and the step forward shift operator for discrete-time models. That is, ρx(ι) represents and for continuous-time and discrete-time, respectively. In addition, ι represents the time variable ( and ) for continuous-time and discrete-time cases. A similar notation was used in [31].
Section snippets
LPV system representation
By definition the dynamic of a real (non-linear) industrial process can be represented by the following general state-space form:where are the state vector, the input vector, the fault input vector, the output vector and the fault output vector, respectively. Additionally, x0 is the initial state vector of the system. Assumption 1 Considering that the state and output functions f(x(ι), u(ι), fu(ι)) and g(x(ι),
Design of the LPV-RUIO
Based on the observer’s development for linear systems with unknown input [33], recently reformulated as a reduced-order observer applied to an LPV system with unknown input [34]. If the term is considered (no sensors faults occur in the system, simultaneously), and under the assumption that the rank of (being Fi a specific column of Bi, that corresponds with the faulty input to diagnose), then a set of non-singular matrices is selected as,
Thereby, the system
Design of LPV-UIOO
Considering that the matrix H corresponds with a row of C, which is non-monitored output (faulty sensor), and choosing a transformation matrix T2, such that where only contains the C rows corresponding with the monitored outputs (non-faulty sensors). Therefore, based on the development of the unknown input observer for linear systems [37], the output equation of the system model from Eq. (4) is transformed to,
Thereby, an unknown input observer with the purpose of
Fault detection and diagnosis
As commonly used in literature [6], [13], [15], the residual pattern should be customized to follow a certain structure (generalized residuals), enabling an extra degree of freedom in the observer’s design. Therefore, the structured residuals are characterized by selective fault responses. That is, any residual is design to be sensitive to one group of faults and insensitive to others. Remark 6 The proposed FDD scheme was designed to isolate a single fault in either a sensor or an actuator, at the time.
Illustrative example I
Consider the model of a heat exchanger, represented in Fig. 2, its physical and operational parameter values appear in Table 1.
The non-linear model equations of the heat exchanger were given by Adam [39, pp. 67–72] as:Finally, bearing in mind the Fig. 2, the state variable is controlled by the cold process flow rate q1 and the hot
Illustrative example II
Consider the model of a CSTR process, represented in Fig. 7, its physical and operational parameter values appear in Table 2. This model is a modified version of the CSTR example presented by Morningred et al. [40]. In the original model, the system operates to constant volume.
Therefore, the CSTR process consists of an irreversible, exothermic reaction, A → B, in a variable volume reactor cooled by a single coolant stream which can be modeled by the following equations:
Conclusion
This paper presents the design of an FDD scheme formed by a bank of two types of observers, applied to LPV systems. The proposed observers and its stability conditions are based on the solving of an LMI problem, that had been performed using software elements.
It is important to note that the main result of this work is to address a novel design of observers (LPV-RUIO and LPV-UIOO) that allow multiple actuator and sensor fault detection and diagnosis, from the linear-like design tools and the
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.
Acknowledgments
The first author would like to thank Universidad Tecnológica Nacional for the grant support through its Doctoral Scholarship Program.
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which helped to improve the quality of the paper.
References (40)
- et al.
A review of fault tolerant control systems: advancements and applications
Measur.: J. Int. Measur. Confederation
(2019) - et al.
A review of process fault detection and diagnosis part i: quantitative model-based methods
Comput. Chem. Eng.
(2003) - et al.
A review of process fault detection and diagnosis part ii : qualitative models and search strategies
Comput. Chem. Eng.
(2003) - et al.
A review of process fault detection and diagnosis part iii: process history based methods
Comput. Chem. Eng.
(2003) - et al.
Fault analysis, detection and estimation for a microgrid via H2/H_inf LPV Observers
Int. J. Electr. Power Energy Syst.
(2019) - et al.
Actuator multiplicative fault estimation in discrete-time LPV systems using switched observers
J. Frankl. Inst.
(2016) - et al.
Actuator and sensor faults estimation based on proportional integral observer for TS fuzzy model
J. Frankl. Inst.
(2017) - et al.
Observer-based fault detection for uncertain nonlinear systems
J. Frankl. Inst.
(2018) - et al.
Robust fault detection and isolation based on zonotopic unknown input observers for discrete-time descriptor systems
J. Frankl. Inst.
(2019) - et al.
Moving horizon estimation of faults in renewable microgrids
IFAC-PapersOnLine
(2019)