Characterisation of interval-observer fault detection and isolation properties using the set-invariance approach

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Abstract

The aim of this paper is to provide a robust Fault Detection and Isolation (FDI) approach that combines the set-invariance approach with a zonotopic interval observer. The effect of the uncertainty is taken into account considering zonotopic-set representations in both the transient and steady states. The set-invariance approach is used to characterize the fault detectability and isolability properties in the steady-state operation of the system. In particular, the Minimum Detectable Fault (MDF) and the Minimum Isolable Fault (MIF) are characterized for several type of faults in separate formulations utilizing the integration of classical sensitivity analysis and set-invariance approaches. Finally, a simulation example based on a two-tanks system is employed to both illustrate and discuss the effectiveness of the proposed approach.

Introduction

In order to produce efficient and high-quality products, modern processes and systems must satisfy their function correctly. Consequently, there are some important issues as safety operation, cost efficiency or environmental protection that must be considered in engineering applications [1]. In this regard, Fault Detection and Isolation (FDI) is an increasingly important area in automatic control since the fault can be known as one of the reasons of unsatisfactory performance or even instability of a dynamic system. Generally speaking, the goal of FDI theory is referred to find the root causes of the fault occurrence. After detecting the fault, maintaining the overall system stability with an acceptable performance will be the next target to be achieved by including some fault-tolerant mechanisms [2].

Model-based FDI is nowadays a well-established approach that is becoming increasingly important in the field of automatic control. Basically, model-based FDI relies on the use of a mathematical model to describe the system behavior [3], [4]. The first step of FDI is Fault Detection (FD). In order to detect the fault, the real system behaviour obtained from sensors and the estimated one using the mathematical model are compared [3], [5]. The fault is detected when an inconsistency is found between the real and modelled behaviours.

In model-based FDI, the performance of FD relies on improving the quality of the mathematical model [6], [7]. But, due to the effect of model uncertainty, unknown disturbances and noises, the mismatch between the actual and estimated process behaviour is non-negligible even if there are no process faults [8]. Thus, consideration of the uncertainties is an important issue, and plays a key role in model-based FD framework [9], [10]. In recent years, several methods have been proposed and developed to explicitly consider model uncertainty in FD. In particular, there exist two different paradigms for considering the uncertainty in the model. In the stochastic approaches, uncertainties are represented using random variables, while in the deterministic approaches (also called set-membership approach), uncertainties are assumed unknown but bounded by means of different type of sets, e.g., interval boxes, polytopes, ellipsoids and zonotopes [5], [11], [12], [13], [14]. According to [15], polytopes provide tighter enclosures than interval boxes. However, the main drawback of using general polytopes is related to the complexity of vertices enumeration with respect to the space dimension. But using zonotopes, basic set operations can be reduced to simple matrix calculations. This fact has recently motivated the use of zonotopes1 for modeling the effect of uncertainties [19].

Any significant inconsistency between the predicted value(s) of output(s) from the model and the real measured value(s) of output(s) given by the sensor, called residual, is known as a fault occurrence [3], [6]. Therefore, detecting the existence of a fault relies on the comparison of the evaluated residual with a threshold value that takes into account the uncertainties [5], [20]. In practice, the fault will be detected if the residual is larger than such a threshold. There are several approaches associated with generating the residual [3], [6], [21]. So far, one of the most widely used paradigms for generating the residual is the observer-based approach [22]. Observer-based approaches provide state and output estimations from the measurements and the model either stochastic (e.g., Kalman filters) or deterministic approaches (e.g., Luenberger observers) uncertainties. Then, the FD test is based on generating the residual using the output estimation error [6], [9], [16], [22].

One major issue in model-based FD framework is how to consider the effect of state disturbances, measurement noise and different faults [9], [16], [23], [24], [25]. Classical methods provide only an estimation based on the nominal system model. However, they do not provide a reliable characterization of the uncertainty effect in the model prediction [26]. Moreover, in the case that the residual is evaluated using statistical methods, the uncertainty is assumed to have known distribution. But in many cases, it is difficult to validate this assumption, where the priori knowledge of the distributions of disturbances and noises is not available [27]. Therefore, assuming that the uncertainty is unknown but bounded can be more adequate. This allows to use the set-theoretical approaches for the state estimation. Recently, there has been an increasing interest in using set-theoretical approaches in FDI, e.g., interval observer, set-membership and set-invariance approaches [9], [28], [29]. Among them, set-invariance approach allows to determine the residual invariant sets that can be computed in each healthy or faulty operation mode of the system [30], [31], [32], [33]. As long as both healthy and faulty sets are separated, FDI can be performed [32], [33].

One major drawback of the set-invariance approach is related to the limitation of computing the finite description of its boundary in all cases. There is a large number of published studies describing the computation of invariant sets.

The Robust Positively Invariant (RPI) set defined as a bounded region in state-space where the system state can be confined, despite of considering the bounded system uncertainties [34], [35]. Furthermore, the minimal Robust Positively Invariant (mRPI) set is a unique and compact RPI set that contained in any closed RPI set [32], [36]. In recent years, researchers have investigated a variety of approaches to construct the RPI set. So far, the proposed approaches can be classified into two main categories: (i) explicit approaches, where the RPI set is computed using the explicit formulation of the set boundary [31], (ii) iterative approaches, where the recursive iteration of the approximation of the system dynamics can be used to reach the RPI set [23], [37]. When applied to FDI, the set-invariance approach is useful to check the separation of healthy and faulty residual set in the steady state.

Recently, most of the reported research about the field has highlighted the interest of using the capability of the set-invariance approach in FDI framework during the transient operation of the system using the set-theoretical approaches. In [35], the relationship between the classical observer-based and the set-invariance approaches in FD is proposed. Then, in [38], [39], [40], the characterization of the minimum magnitude of the fault that can be detected is computed by using both the observer-based and set-invariance approaches. However, there has been few discussion about the combination of the mentioned approaches. Then, both approaches are still considered as two different techniques into the FDI framework.

So far, the most serious weakness of the set-invariance approach in comparison with the interval observer approach is related to its limited use to detect faults in transient state. On the other hand, one important feature of the set-invariance approach is the ability of assessing both fault detectability and isolability properties by means of the off-line computation of the invariant sets for the residual that characterize the healthy and faulty operation modes of the system. In the set-invariance approach, the fault isolability can be obtained by guaranteeing the separability of faulty residual sets, not being possible through the use of the interval observer approach only. Therefore, the main contribution of this paper is to integrate the observer-based and set-invariance approaches to develop an FDI scheme such that it can be used in both transient and steady states of a system evolution. Furthermore, the Minimum Detectable Fault (MDF) and the Minimum Isolable Fault (MIF) are characterized based on the combination of the classical sensitivity analysis and the set-invariance approach. Moreover, the zonotopic representation of a set is considered for propagating the effect of uncertainties since its related operations can be reduced to simple matrix calculation in comparison with the huge number of vertices of the equivalent polytopes. Finally, a well-known benchmark based on the two-tank system is used as a case study for both illustrating and analyzing the effectiveness of the proposed approach in the paper.

The structure of the paper is the following: the problem formulation is presented in Section 2. Both the zonotopic interval observer structure and its application to FD are discussed in Section 3.1. The general framework of set-invariance approach is discussed in Section 3.2. On-line propagation of the residual set and the FDI design integrating the observer-based and set-invariance approaches are presented in Section 4. In Section 5, the application of the proposed approach to a two-tank system is used in order to illustrate its effectiveness. Finally, the conclusions are drawn in Section 6. For completeness, relevant definitions and properties of zonotopes are recalled in the Appendix.

Throughout this paper, Rn denotes the set of n-dimensional real numbers and ⊕ denotes the Minkowski sum. The matrices are written using capital letter, e.g., A, the calligraphic notation is used for denoting sets, e.g., X, the transfer functions are highlighted using script font e.g., H, ‖.‖s denotes the s-norm, [x̲,x¯] is an interval with lower bound x and upper bound x¯. The notations io and si in the mathematical formulations denote the interval observer and set-invariance approaches, respectively.

Section snippets

Problem set-up

In this paper, discrete-time linear uncertain system to be monitored is described in state space asxk+1=Axk+Buk+Eωωk,yk=Cxk+Eυυk, where uRnu, yRny and xRnx are the input, the output and the state vectors, respectively. Moreover, ARnx×nx, BRnx×nu and CRny×nx are the state-space matrices. Both state disturbance and process noise vectors are defined by ωRnx and υRny, respectively. Moreover, Eω and Eυ are the associated distribution matrices with appropriate dimensions while kN indicates

Zonotopic interval-observer approach

Using the zonotopic-set representation of the uncertainties, i.e., W and V, the state bounding observer for the dynamical model (4) can be obtained as a zonotope X^=c,R using the Luenberger observer (5) and Proposition 3.1.

Proposition 3.1 Zonotopic observer structure

Considering Assumptions 2.1 and 2.2 and the observer (5), the center c and the shape matrix R of X^ can be recursively computed asc+=(ALC)c+Bu+Ly, R+=[(ALC)R¯Ed],

where Ed=[Eω,LEυ] and R¯=q{R}. Moreover, the state inclusion property x ∈ ⟨c, Rin Properties 3 and 4 holds

Characterization of Detectability and Isolability Properties

The main objective of this section is to combine IOA and SIA in order to exploit their benefits and overcome their drawbacks. In particular, IOA will provide an on-line test that can be applied in both transient and steady states response and SIA will allow to characterize detectability and isolability properties in the steady state.

System description

The proposed FDI scheme will be tested using a two-tank system based on the well-known benchmark proposed in [45]. A schematic of the system can be seen in Fig. 2.

The input of the two-tank system is the pump flow rate that is determined when applying voltage v of the pump. Therefore, the action of the pump is to pour the tanks by extracting the water from the basin. Moreover, Tank 1 is placed below Tank 2. Furthermore, the outputs of the process are the water levels in both upper and lower

Conclusions

This paper has proposed a zonotopic interval observer-based Fault Detection and Isolation (FDI) algorithm integrated with the set-invariance approach. As a novelty, in the proposed FDI design, fault detectability and fault isolability can be guaranteed in both transient and steady states. The influences of all possible state disturbance and measurement noise are addressed using the zonotopic-set representation of a set. Furthermore, Minimum Detectable Fault (MDF) and Minimum Isolable Fault

Acknowledgments

This work has been partially funded by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERFD) through the projects DEOCS (ref. MINECO DPI2016-76493) and SCAV (ref. MINECO DPI2017-88403-R). This work has also been partially funded by AGAUR of Generalitat de Catalunya through the Advanced Control Systems (SAC) group grant (2017 SGR 482) and by Agència de Gestió d’Ajuts Universitaris i de Recerca.

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