Intermittent Fault Detection for Uncertain Networked Systems

Abstract

This paper investigates intermittent fault detection problem for a class of networked systems with multiple state delays and unknown input. Polytopic-type parameter uncertainty in the state-space model matrices is considered. A novel measurement model is employed to account for both the random measurement delays and the stochastic data missing (package dropout) phenomenon, which are typically resulted from the limited capacity of the communication networks. We aim to design an uncertainty-dependent fault detection filter such that, for all unknown input, all possible parameter uncertainties, and all incomplete measurements, the error between residual and weighted fault is made as small as possible. By converting the addressed robust fault detection problem into an alternative robust 𝐻_∞ filtering problem of a certain Markovian jumping system (MJS), a sufficient condition for the existence of the desired robust fault detection filter is derived. A residual evaluation within an incremental form is brought forward to make the whole method suitable for intermittent fault detection. A numerical example is utilized to demonstrate the effectiveness of the proposed approach.

1. Introduction

For traditional control systems with point-to-point data transmission, a variety of methods have been proposed to deal with the modeling, identification, estimation, and control problems [1–3]. During the past decades, the rapid developments in network technologies have led to more and more feedback control systems with control loops closed via digital communication channels. Compared with the traditional point-to-point wiring, in networked systems, serial communication networks are used to exchange information (reference input, plant output, control input, etc.) among control system components (sensors, controller, actuators, etc.) [4]. The use of the communication channels can reduce the costs of cables and power, simplify the installation and maintenance of the whole system, and increase the reliability, so network-based analysis and designs have many industrial applications such as in automobiles, manufacturing plants, aircrafts, and HVAC systems. However, the insertion of the communication channels raises new interesting and challenging problems such as network-induced delays or packets dropout, see [4–6] for some representative works.

With the increasing demand for higher performance, higher safety, and reliability standards, fault detection and isolation (FDI) has been an active field of research over the past decades [7, 8]. The main purpose of fault detection is to construct a residual signal which can then be compared with a predefined threshold. When the residual exceeds the threshold, the fault is detected and an alarm is generated. Among different approaches for residual generation, the model-based approaches to FDI problems for dynamic systems have received more attention. For example, in [9], the𝐻_∞norm of transfer function matrix from unknown input to residual has been designed to be small, while the𝐻_∞norm (or the smallest nonzero singular value) of transfer function matrix from fault to residual has been guaranteed to be large. In [10], the error between residual and weighted fault has been made as small as possible, and then the FDI problem can be solved by using the𝐻_∞filtering approach.

Due to the popularization of the using of network cables, it is necessary and interesting to consider the FDI problem for networked systems with network-induced delays or data missing, see [11, 12] and the references therein. Since network-induced delays and data missing (dropout) phenomenon are inherently random and time-varying [13], they have been modeled in various probabilistic ways [14]. One of the attractive approaches is to use binary switching sequence viewed as a Bernoulli distributed white sequence taking on values of 0 and 1, since such a representation is very effective to describe network-induced delays [15] or data missing [16]. Very recently, in [17], the network-induced delay and data dropout problems have been investigated in an integrated way within a unified framework and the robust filtering problem with polytopic uncertainties has been thoroughly studied. Note that in all the aforementioned results, it has been assumed that the delay or missing characteristics are statistically mutually independent from transfer to transfer. In [18], the fault detection problem for systems with missing measurements has been discussed by characterizing the residual dynamics by a discrete-time MJS. In [19], the diagnosis of intermittent faults in dynamic systems modeled as discrete event systems has been considered. So far, to the best of the authors’ knowledge, the robust intermittent fault detection problems in the presence of parameter uncertainty for networked systems with simultaneous measurement delays and data missing have not been fully investigated, which constitutes the main focus of this paper.

In this paper, intermittent fault detection problem for a class of uncertain networked systems with multiple state delays and incomplete measurement is investigated. A sequence varying in a Markov fashion is employed in the measurement model so that both the measurement delays and data missing can be simultaneously represented. Polytopic-type parameter uncertainty in state-space model matrices is considered. After augmenting the state, the addressed robust fault detection problem is converted to an equivalent robust𝐻_∞filtering problem for a certain Markovian jumping system (MJS), and a sufficient condition for the existence of the desired robust fault detection filter is brought forward. By introducing the new residual evaluation function within an incremental form, the proposed method can detect the possible intermittent fault.

Notation. The notations used throughout the paper are fairly standard.ℝ^𝑛andℝ^𝑛×𝑚denote, respectively, the𝑛-dimensional Euclidean space and the set of all𝑛×𝑚real matrices.𝑃>0means that𝑃is real symmetric and positive definite. The subscript “𝑇” denotes the matrix transpose.Pr{⋅}represents the occurrence probability of the event “⋅”, and when𝑥and𝑦are both stochastic variables,𝔼{𝑥}stands for the mathematical expectation of𝑥.𝑙₂[0,∞)is the space of all square-summable vector functions over[0,∞), with‖𝑥‖being the standard𝑙₂norm of𝑥, that is,‖𝑥‖=(𝑥^𝑇𝑥)^1/2.𝐑𝐇_∞is the set of proper and stable rational functions with real coefficients. In symmetric block matrices, we use “∗” to represent a term that is induced by symmetry, anddiag{⋯}stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations and, sometimes, the arguments of a function will be omitted in the analysis when no confusion can arise.

2. Problem Formulation and Preliminaries

Consider the following class of discrete-time linear networked systems with multiple delays in the state: 𝑥_𝑘+1=𝐴₀𝑥_𝑘+^𝑞_𝑖=1𝐴_𝑖𝑥_𝑘−𝑖+𝐵_𝑤𝑤_𝑘+𝐵_𝑓𝑓_𝑘,𝑦_𝑘=𝛿𝜏_𝑘,0𝐶₀𝑥_𝑘+^𝑞_𝑖=1𝛿𝜏_𝑘,𝑖𝐶_𝑖𝑥_𝑘−𝑖+𝐷𝑤_𝑘,𝑥_𝑘=𝜑_𝑘,𝑘=−𝑞,−𝑞+1,…,0, (1) where𝑥_𝑘∈ℝ^𝑛stands for the state vector,𝑤_𝑘∈ℝ^𝑝is the unknown input belonging to𝑙₂[0,∞), and𝑓_𝑘∈ℝ^𝑙is the fault to be detected.1≤𝑖≤𝑞(𝑞≥1) are integer time delays.𝑦_𝑘∈ℝ^𝑚is the measured output vector, which may contain random communication delays and stochastic data missing induced by the limited capacity of the communication networks. All system matrices in (1) are assumed to have appropriate dimensions.𝜑_𝑘is a given real initial sequence on[−𝑞,0], and𝛿(⋅,⋅)stands for the Kronecker delta; that is, 𝛿𝑗,𝑙=0,if𝑗≠𝑙,1,if𝑗=𝑙. (2) Furthermore,𝜏_𝑘is a stochastic variable whose role is to determine, at time𝑘, the size of the occurred delay as well as the possibility of data missing. In this paper,{𝜏_𝑘}is assumed to be a discrete-time homogeneous Markov chain taking values in the following finite state space: Ξ=−1,0,…,𝑞 (3) and stationary transition probability matrixΛ=[𝜆_𝑖𝑗], where 𝜆_𝑖𝑗=Pr𝜏_𝑘+1=𝑗∣𝜏_𝑘=𝑖. (4)

Remark 1. The assumption that the switching between difference modes abides by a Markovian chain seems realistic since, in network-based signal transmissions, time delays and data dropouts typically occur in a batch mode, and the status of the network varies slower than the sampling period and the characteristics of the network at a certain time is usually dependent on the superior time instant. The transition from one mode to another may obey certain probability distribution, and our “Markov jumping” assumption of {𝜏_𝑘} describes this phenomenon properly.

Remark 2. In our measurement model, the event {𝜏_𝑘=0} means that the measurements are ideally transmitted over the network without any delays or data missing, the event{𝜏_𝑘=𝑖}(1≤𝑖≤𝑞) corresponds to the case that the𝑖-step measurement delay occurs. Without loss of generality and for the convenience in denoting the transition probability matrix, we set the state in the state space of the Markov chain corresponding to the measurement missing situation to−1, so{𝜏_𝑘=−1}means that the measurements are missing and𝑦_𝑘consists of pure noise only.
In this paper, we are interested in the problem of robust fault detection for uncertain system described by (1) with incomplete measurements. The system matrices𝐴₀,𝐴₁,…,𝐴_𝑞,𝐵_𝑤,𝐵_𝑓,𝐶₀,𝐶₁,…,𝐶_𝑞,𝐷are assumed to be uncertain but belong to a known convex compact set of polytopic type,that is, Ω∶=𝐴₀,𝐴₁,…,𝐴_𝑞,𝐵_𝑤,𝐵_𝑓,𝐶₀,𝐶₁,…,𝐶_𝑞,𝐷∈Θ, (5) whereΘis a given convex bounded polyhedral domain described by𝑣vertices as follows: Θ∶=Ω∣Ω=^𝑣_𝑟=1𝛼_𝑟Ω_𝑟;^𝑣_𝑟=1𝛼_𝑟=1,𝛼_𝑟≥0, (6) whereΩ_𝑟∶=(𝐴_0𝑟,𝐴_1𝑟,…,𝐴_𝑞𝑟,𝐵_𝑤𝑟,𝐵_𝑓𝑟,𝐶_0𝑟,𝐶_1𝑟,…,𝐶_𝑞𝑟,𝐷_𝑟),𝑟=1,…,𝑣, denotes the𝑟th vertex of the polytope.
Consider a full-order fault detection filter of the following form: 𝑥_𝑘+1=𝐺𝜏_𝑘𝑥_𝑘+𝐾𝜏_𝑘𝑦_𝑘,𝑟_𝑘=𝐿𝜏_𝑘𝑥_𝑘, (7) where𝑥_𝑘∈ℝ^𝑛is the filter state vector and𝑟_𝑘∈ℝ^𝑙is the so-called residual that is compatible with the fault vector𝑓_𝑘. For𝜏_𝑘=𝑖∈Ξ, we denote matrices𝐺(𝜏_𝑘),𝐾(𝜏_𝑘), and𝐿(𝜏_𝑘)as𝐺_𝑖=𝐺(𝜏_𝑘=𝑖),𝐾_𝑖=𝐾(𝜏_𝑘=𝑖), and𝐿_𝑖=𝐿(𝜏_𝑘=𝑖). Our main aim is to make the error between residual𝑟_𝑘and fault signal𝑓_𝑘as small as possible.
For the purpose of fault detection, it is not necessary to estimate the fault𝑓_𝑘. Sometimes one is more interested in the fault signal of a certain frequency interval, which can be formulated as the weighted fault as follows: 𝑓(𝑧)=𝑇_𝑓(𝑧)𝑓(𝑧), (8) where𝑇_𝑓(𝑧)∈𝐑𝐇_∞is a prescribed weighting matrix.

Remark 3. Similar to [10], the introduction of a suitable weighting matrix𝑇_𝑓(𝑧)can limit the frequency interval of interest, and the system performance can then be improved. In fact, the use of weighted fault𝑓(𝑧)is more general than using the original fault𝑓(𝑧), because if we impose𝑇_𝑓(𝑧)=𝐼, we can obtain𝑓(𝑧)=𝑓(𝑧).
Suppose a minimal realization of𝑓(𝑧)=𝑇_𝑓(𝑧)𝑓(𝑧)is 𝑥_𝑘+1=𝐴_𝑡𝑥_𝑘+𝐵_𝑡𝑓_𝑘,𝑓_𝑘=𝐶_𝑡𝑥_𝑘+𝐷_𝑡𝑓_𝑘, (9) where𝑥_𝑘∈ℝ𝑛is the weighted fault state,𝑓_𝑘∈ℝ^𝑙is the original fault, and𝑓_𝑘∈ℝ^𝑙is the weighted fault.𝐴_𝑡,𝐵_𝑡,𝐶_𝑡, and𝐷_𝑡are assumed to be known real constant matrices with appropriate dimensions.
By defining 𝜁_𝑘=𝑤^𝑇_𝑘𝑓^𝑇_𝑘^𝑇,𝑟_𝑘̃=𝑟_𝑘−𝑓_𝑘,𝑥_𝑘=𝑥^𝑇_𝑘−1⋯𝑥^𝑇_𝑘−𝑞^𝑇,𝜂_𝑘=𝑥^𝑇_𝑘𝑥^𝑇_𝑘𝑥^𝑇_𝑘𝑥^𝑇_𝑘^𝑇 (10) and again, denoting matrices𝐴(𝜏_𝑘),𝐵(𝜏_𝑘),𝐶(𝜏_𝑘)and𝐷(𝜏_𝑘)as𝐴_𝑖=𝐴(𝜏_𝑘=𝑖),𝐵_𝑖=𝐵(𝜏_𝑘=𝑖),𝐶_𝑖=𝐶(𝜏_𝑘=𝑖), and𝐷_𝑖=𝐷(𝜏_𝑘=𝑖), we have the overall fault detection dynamics governed by the following system: 𝜂_𝑘+1=𝐴_𝑖𝜂_𝑘+𝐵_𝑖𝜁_𝑘,𝑟_𝑘̃=𝐶_𝑖𝜂_𝑘+𝐷_𝑖𝜁_𝑘, (11) where 𝐴_𝑖=𝐴_𝑖00𝐴_𝑡,𝐵_𝑖=𝐵_𝑖𝐵_𝑡,𝐶_𝑖=𝐶_𝑖−𝐶_𝑡,𝐷_𝑖=0−𝐷_𝑡,𝐴_𝑖=𝐴₀𝐴_𝑑0𝐴₂₁𝐴₂₂0𝛿(𝑖,0)𝐾₀𝐶₀𝐾_𝑖𝐶_𝑖𝑒_𝑖𝐺_𝑖,𝐵_𝑖=𝐵_𝑤𝐵_𝑓0_{𝑞𝑛×𝑝}0_{𝑞𝑛×𝑙}𝐾_𝑖,𝐵𝐷0^𝑡=0𝐵_𝑡,𝐴₂₁=𝐼_𝑛0_{(𝑞−1)𝑛×𝑛},𝐴₂₂=𝐼00_{(𝑞−1)𝑛×(𝑞−1)𝑛}0,𝐴_𝑑=𝐴₁⋯𝐴_𝑞,𝑒_𝑖=𝛿(𝑖,1)𝐼_𝑛×𝑛⋯𝛿𝑖,𝑞𝐼_𝑛×𝑛,𝐶_𝑖=0_𝑙×𝑛0_{𝑙×𝑞𝑛}𝐿_𝑖,𝐷_𝑖∶=𝐷_𝑖. (12)
After the above manipulations, the admissible sensor delays and data missing can be reformulated as the jumping parameters of a Markovian jumping system (11) with the same transition probability matrixΛ.
The matrices𝐴_𝑖,𝐵_𝑖,𝐶_𝑖, and𝐷_𝑖,𝑖∈Ξ, are uncertain, but they belong to prescribed matrix polytopesΩ_𝑖∶=(𝐴_𝑖,𝐵_𝑖,𝐶_𝑖,𝐷_𝑖,)∈Θ_𝑖, whereΘ_𝑖are given convex bounded polyhedral domain described by𝑣vertices as follows: Θ_𝑖∶=Ω_𝑖∣Ω_𝑖=^𝑣_𝑟=1𝛼_𝑟Ω_𝑖𝑟;^𝑣_𝑟=1𝛼_𝑟=1,𝛼_𝑟≥0, (13) andΩ_𝑖𝑟=(𝐴_𝑖𝑟,𝐵_𝑖𝑟,𝐶_𝑖𝑟,𝐷_𝑖𝑟)denotes the𝑟th vertices of the polytopes, where 𝐴_𝑖𝑟=𝐴_𝑖𝑟00𝐴_𝑡,𝐵_𝑖𝑟=𝐵_𝑖𝑟𝐵_𝑡,𝐶_𝑖𝑟=𝐶_𝑖−𝐶_𝑡=𝐶_𝑖,𝐷_𝑖𝑟=0−𝐷_𝑡=𝐷_𝑖𝑟,𝐴_𝑖𝑟=𝐴_0𝑟𝐴_𝑑𝑟0𝐴₂₁𝐴₂₂0𝛿(𝑖,0)𝐾₀𝐶_0𝑟𝐾_𝑖𝐶_𝑖𝑟𝑒_𝑖𝐺_𝑖,𝐵_𝑖𝑟=𝐵_𝑤𝑟𝐵_𝑓𝑟0_{𝑞𝑛×𝑝}0_{𝑞𝑛×𝑙}𝐾_𝑖𝐷_𝑟0,𝐴_𝑑𝑟=𝐴_1𝑟⋯𝐴_𝑞𝑟. (14) Matrices𝐴₂₁,𝐴₂₂,𝑒_𝑖,𝐵_𝑡,𝐶_𝑖,𝐶_𝑖,𝐷_𝑖, and𝐷_𝑖are the same as defined in (12). Note that from (14), matrices𝐴_𝑖𝑟,𝐵_𝑖𝑟,𝐶_𝑖𝑟, and𝐷_𝑖𝑟are affinely dependent on the matrices𝐴_0𝑟,𝐴_1𝑟,…,𝐴_𝑞𝑟,𝐵_𝑤𝑟,𝐵_𝑓𝑟,𝐶_0𝑟,𝐶_1𝑟,…,𝐶_𝑞𝑟,𝐷_𝑟, then for any𝑖∈Ξ, the uncertainty polytopes (13) have the same number of vertices𝑣, as well as the same combination coefficients𝛼_𝑟with (6), but different vertices for different Markovian model𝑖.
Recall the following definition of mean square stability for MJSs.

Definition 4 (see [20]). System (11) with𝜁_𝑘=0is said to be mean square stable if 𝔼𝜂_𝑘²⟶0,as𝑘⟶∞ (15) for any initial condition𝜂₀and initial distribution𝜏₀∈Ξ.

We further introduce the following definition.

Definition 5. System (11) with uncertainΩ_𝑖∈Θ_𝑖,𝑖∈Ξ(resp.,Λ∈Π) is robust mean square stable if (11) is mean square stable for everyΩ_𝑖∈Θ_𝑖,𝑖∈Ξ(resp.,Λ∈Π).

Assumption 6. System (1) with𝑤_𝑘=0and𝑓_𝑘=0is assumed to be robust mean square stable.
With Definition 5, we can transform the robust fault detection filter design problem of system (1) to a robust𝐻_∞filtering problem for MJS (11). What we need to do here is to find the filter parameters𝐺_𝑖,𝐾_𝑖and𝐿_𝑖(𝑖∈Ξ) such that the augmented fault detection dynamics (11) is robust mean square stable and the infimum of𝛾is made small in the feasibility of the following: sup_𝜁𝑘≠0𝔼𝑟̃_𝑘²𝜁_𝑘²<𝛾²,𝛾>0. (16)
We further adopt a residual evaluation stage including an incremental evaluation function𝐽(𝑘)and a threshold𝐽_thof the following form: 𝐽^𝐿(𝑘)=^𝑘_{ℎ=𝑘−𝐿}𝑟^𝑇_ℎ𝑟_ℎ^1/2,𝐽^𝐿_th=sup_{𝑤𝑘∈𝑙2,𝑓𝑘=0}𝔼𝐽^𝐿(𝑘), (17)(18) where𝑟_1−𝐿=𝑟_2−𝐿=⋯=𝑟₋₁=𝑟₀=0and𝐿denotes the length of time window for evaluation function.
Based on (17), the occurrence of faults can be detected by comparing𝐽(𝑘)with𝐽_thaccording to the following rule: 𝐽^𝐿(𝑘)>𝐽^𝐿_th𝐽⟹withfaults⟹alarm,_𝐿(𝑘)≤𝐽^𝐿_th⟹nofaults⟹donothing. (19)

Remark 7. By introducing the residual evaluation function with an incremental form (17), one can detect the possible intermittent fault for an uncertain networked system by analyzing the residual signal once it is generated by the fault detection filter (7). The reason is that the residual evaluation signal (11) is decreased to a small value over time once the fault disappears. This nature makes the proposed method in this paper be used for intermittent fault detection.

3. Fault Detection Filter Design

In this section, we shall discuss the robust fault detection filter design problem of system (1), under the existence of parameter uncertainty (13). We introduce the following lemma which is useful in deriving our main results in the sequel.

Lemma 8. Consider system (1) with system uncertainty (5), for a given fault detection filter of the form (7), the augmented dynamic (11) is robust mean square stable and satisfies the constraint (16), if there exist matrice 𝑃_𝑖𝑟∈ℝ^(𝑞+2)𝑛,𝑖∈Ξ,𝑟=1,…,𝑣,𝑃_𝑡∈ℝ𝑛, and𝑄_𝑖∈ℝ^(𝑞+2)𝑛such that the following LMIs −𝑃_𝑖𝑟𝐴^𝑇_𝑖𝑟𝑄^𝑇_𝑖0𝐶^𝑇_𝑖00∗−𝑄_𝑖−𝑄^𝑇_𝑖+𝒫^𝑇_𝑟𝒮_𝑖𝑄_𝑖𝐵_𝑖𝑟000∗∗−𝛾²𝐼𝐷^𝑇_𝑖0𝐵^𝑇_𝑡𝑃_𝑡∗∗∗−𝐼−𝐶_𝑡0∗∗∗∗−𝑃_𝑡𝐴^𝑇_𝑡𝑃_𝑡∗∗∗∗∗−𝑃_𝑡<0 (20) hold for all𝑖∈Ξ,𝑟=1,…,𝑣, where𝐴_𝑖𝑟,𝐵_𝑖𝑟,𝐶_𝑖, and𝐷_𝑖are defined in (14),𝐵_𝑡is defined in (12),𝐴_𝑡,𝐵_𝑡,𝐶_𝑡,𝐷_𝑡are defined in (9) and, 𝒫_𝑟=𝑃_(−1)𝑟⋯𝑃_𝑞𝑟^𝑇,𝒮_𝑖=𝜆_𝑖(−1)𝐼_(𝑞+2)𝑛⋯𝜆_𝑖𝑞𝐼_(𝑞+2)𝑛^𝑇. (21)

Proof. Considering the structure of𝐴_𝑖,𝐵_𝑖,𝐶_𝑖, and𝐷_𝑖, and imposing 𝑃_𝑖=𝑃_𝑖00𝑃_𝑡 (22) for all𝑖∈Ξ, it can be concluded from the bounded real Lemma in [21] that a sufficient condition is that there exist matrices𝑃_𝑖∈ℝ^(𝑞+2)𝑛,𝑖∈Ξ,𝑃_𝑡∈ℝ𝑛such that the following LMIs −𝑃_𝑖𝐴^𝑇_𝑖𝒫^𝑇𝒮_𝑖0𝐶^𝑇_𝑖00∗−𝒫^𝑇𝒮_𝑖𝒫^𝑇𝒮_𝑖𝐵_𝑖000∗∗−𝛾²𝐼𝐷^𝑇_𝑖0𝐵^𝑇_𝑡𝑃_𝑡∗∗∗−𝐼−𝐶_𝑡0∗∗∗∗−𝑃_𝑡𝐴^𝑇_𝑡𝑃_𝑡∗∗∗∗∗−𝑃_𝑡<0 (23) hold for any𝑖∈Ξ, where𝐴_𝑖,𝐵_𝑖,𝐶_𝑖,𝐷_𝑖, and𝐵_𝑡are defined in (12),𝐴_𝑡,𝐵_𝑡,𝐶_𝑡, and𝐷_𝑡are defined in (9),𝒮_𝑖is the same defined in (21) and 𝒫=𝑃₍₋₁₎⋯𝑃_𝑞^𝑇. (24) Following the steps as the proof of Theorem  1 in [22], it can be shown that LMIs (23) are feasible if and only if there exist matrices𝑃_𝑖∈ℝ^(𝑞+2)𝑛,𝑄_𝑖∈ℝ^(𝑞+2)𝑛,𝑖∈Ξ, and𝑃_𝑡∈ℝ𝑛satisfying −𝑃_𝑖𝐴^𝑇_𝑖𝑄^𝑇_𝑖0𝐶^𝑇_𝑖00∗−𝑄_𝑖−𝑄^𝑇_𝑖+𝒫^𝑇𝒮_𝑖𝑄_𝑖𝐵_𝑖000∗∗−𝛾²𝐼𝐷^𝑇_𝑖0𝐵^𝑇_𝑡𝑃_𝑡∗∗∗−𝐼−𝐶_𝑡0∗∗∗∗−𝑃_𝑡𝐴^𝑇_𝑡𝑃_𝑡∗∗∗∗∗−𝑃_𝑡<0. (25)
We are now in the position to prove that for system (1) with uncertainty (13), (20) ensures the robust mean square stable as well as the constraint (16) of the augmented dynamic (11). For an arbitrary fixed uncertain system with system matricesΩ, one can always find a set of coefficients𝛼_𝑟≥0, 𝑟=1,…,𝑣, such that both (6) and (13) hold. Note that LMIs (20) are affine in the matrices𝑃_𝑖𝑟,𝐴_𝑖𝑟,𝐵_𝑖𝑟,𝐶_𝑖, and𝐷_𝑖, multiplying suitable inequalities of (20) by appropriate scalars𝛼_𝑟and summing up, it can be readily shown that (25) holds for everyΩ_𝑖with a matrix𝑃_𝑖(𝛼)=∑^𝑣_𝑟=1𝛼_𝑟𝑃_𝑖𝑟,𝑖=−1,…,𝑞. By using the Bounded Real Lemma in [21], it follows that system (1) is robust mean square stable and (16) is satisfied. This concludes the proof.

Next, we give the robust fault detection filter design result for system (1) with system uncertainty (13).

Theorem 9. Consider system (1) with uncertain matrices (13), let𝛾>0be a given scalar, there exists an admissible full-order robust fault detection filter of the form (7) ensuring that the overall augmented dynamics (11) is robust mean square stable and the constraint (16) is satisfied, if there exist matrices0<𝑋^𝑇_𝑖𝑟=𝑋_𝑖𝑟∈ℝ^{(𝑞+2)𝑛×(𝑞+2)𝑛},𝑆_𝑖∈ℝ^𝑛×𝑛,𝑍_𝑖∈ℝ^𝑛×𝑛,𝑌_𝑖∈ℝ^𝑛×𝑛,𝐺_𝑖∈ℝ^𝑛×𝑛,𝐾_𝑖∈ℝ^𝑛×𝑚,𝐿_𝑖∈ℝ^𝑙×𝑛,𝑀_𝑖∈ℝ^{𝑞𝑛×𝑞𝑛},𝑖=−1,…,𝑞,𝑟=1,…,𝑣, and0<𝑃^𝑇_𝑡=𝑃_𝑡∈ℝ𝑛_×𝑛, such that the following LMIs −𝑋_𝑖𝑟Φ₁₂0Φ₁₄00∗Φ₂₂Φ₂₃000∗∗−𝛾²𝐼𝐷^𝑇_𝑖𝑟0𝐵^𝑇_𝑡𝑃_𝑡∗∗∗−𝐼−𝐶_𝑡0∗∗∗∗−𝑃_𝑡𝐴^𝑇_𝑡𝑃_𝑡∗∗∗∗∗−𝑃_𝑡<0 (26) hold for𝑖=−1,…,𝑞and𝑟=1,…,𝑣, whereΦ₁₂=𝐴^𝑇_0𝑟𝑍^𝑇_𝑖𝐼_𝑛×𝑛0_{𝑛×(𝑞−1)𝑛}𝑀^𝑇_𝑖𝐴^𝑇_0𝑟𝑌^𝑇_𝑖+𝛿(𝑖,0)𝐶^𝑇_0𝑟𝐾^𝑇₀+𝐺^𝑇_𝑖𝐴^𝑇_𝑑𝑟𝑍^𝑇_𝑖0_{(𝑞−1)𝑛×𝑛}𝐼_{(𝑞−1)𝑛×(𝑞−1)𝑛}0_𝑛×𝑛0_{𝑛×(𝑞−1)𝑛}𝑀^𝑇_𝑖𝐴^𝑇_𝑑𝑟𝑌^𝑇_𝑖+𝑒^𝑇_𝑖𝐶^𝑇_𝑖𝑟𝐾^𝑇_𝑖𝐴^𝑇_0𝑟𝑍^𝑇_𝑖𝐼_𝑛×𝑛0_{𝑛×(𝑞−1)𝑛}𝑀^𝑇_𝑖𝐴^𝑇_0𝑟𝑌^𝑇_𝑖+𝛿(𝑖,0)𝐶^𝑇_0𝑟𝐾^𝑇₀,Φ₁₄=𝐿^𝑇_𝑖00,Φ₂₂𝑍∶=−^𝑖+𝑍^𝑇_𝑖0𝑍_𝑖+𝑌^𝑇_𝑖+𝑆^𝑇_𝑖∗𝑀_𝑖+𝑀^𝑇_𝑖0∗∗𝑌_𝑖+𝑌^𝑇_𝑖+𝒳^𝑇_𝑟𝒮_𝑖,Φ₂₃=𝑍_𝑖𝐵_𝑤𝑟𝑍_𝑖𝐵_𝑓𝑟𝑌00_𝑖𝐵_𝑤𝑟+𝐾_𝑖𝐷_𝑟𝑌_𝑖𝐵_𝑓𝑟,𝒳_𝑟=𝑋_(−1)𝑟⋯𝑋_𝑞𝑟^𝑇, (27)𝐷_𝑖𝑟is defined in (14) and𝒮_𝑖is the same in (21). Moreover, if (26) is feasible, the parameters of the desired robust fault detection filter can be given by 𝐺_𝑖=𝑉_𝑖⁻¹𝐺_𝑖𝑆_𝑖⁻¹𝑉_𝑖,𝐾_𝑖=𝑉_𝑖⁻¹𝐾_𝑖,𝐿_𝑖=𝐿_𝑖𝑆_𝑖⁻¹𝑉_𝑖, (28) where𝑉_𝑖∈ℝ^𝑛×𝑛is any invertible matrix (e.g.,𝑉_𝑖could be set as𝐼).

Proof. Considering (20), Let𝑈_𝑖∈ℝ^𝑛×𝑛and𝑉_𝑖∈ℝ^𝑛×𝑛be two nonsingular matrices, and set 𝑄^𝑇_𝑖=𝑌^𝑇_𝑖0𝑄^𝑇_31𝑖0𝑀^𝑇_𝑖0𝑉^𝑇_𝑖0𝑄^𝑇_33𝑖. (29) Introducing the followind new matrices: 𝑄^𝑇_𝑖=𝑍_𝑖^−𝑇0𝑄^𝑇_31𝑖𝑈0𝐼0_𝑇𝑖0𝑄^𝑇_33𝑖, (30) where the entries𝑄^𝑇_31𝑖,𝑄^𝑇_33𝑖,𝑄^𝑇_31𝑖, and𝑄^𝑇_33𝑖are uniquely determined from the following relation: 𝑌^𝑇_𝑖𝑄^𝑇_31𝑖𝑉^𝑇_𝑖𝑄^𝑇_33𝑖𝑍_𝑖^−𝑇𝑄^𝑇_31𝑖𝑈^𝑇_𝑖𝑄^𝑇_33𝑖=𝑍_𝑖^−𝑇𝑄^𝑇_31𝑖𝑈^𝑇_𝑖𝑄^𝑇_33𝑖𝑌^𝑇_𝑖𝑄^𝑇_31𝑖𝑉^𝑇_𝑖𝑄^𝑇_33𝑖=𝐼, (31) we further have the following relation 𝑄^𝑇_𝑖𝑄^𝑇_𝑖=𝑄^𝑇_𝑖𝑄^𝑇_𝑖=𝐼000𝑀^𝑇_𝑖000𝐼. (32)
By defining 𝑇_𝑖=𝑍^𝑇_𝑖0𝑌^𝑇_𝑖0𝐼000𝑉^𝑇_𝑖, (33) we obtain𝑇^𝑇_𝑖𝑄_𝑖𝐴^𝑇_𝑖𝑟𝑄^𝑇_𝑖𝑄^𝑇_𝑖𝑇_𝑖=𝐴^𝑇_0𝑟𝑍^𝑇_𝑖𝐼_𝑛×𝑛0_{𝑛×(𝑞−1)𝑛}𝑀^𝑇_𝑖𝐴^𝑇_0𝑟𝑌^𝑇_𝑖+𝛿(𝑖,0)𝐶^𝑇_0𝑟𝐾^𝑇₀𝑉^𝑇₀+𝑍_𝑖𝑈_𝑖𝐺^𝑇_𝑖𝑉^𝑇_𝑖𝐴^𝑇_𝑑𝑟𝑍^𝑇_𝑖0_{(𝑞−1)𝑛×𝑛}𝐼_{(𝑞−1)𝑛×(𝑞−1)𝑛}0_𝑛×𝑛0_{𝑛×(𝑞−1)𝑛}𝑀^𝑇_𝑖𝐴^𝑇_𝑑𝑟𝑌^𝑇_𝑖+𝑒^𝑇_𝑖𝐶^𝑇_𝑖𝑟𝐾^𝑇_𝑖𝑉^𝑇_𝑖𝐴^𝑇_0𝑟𝑍^𝑇_𝑖𝐼_𝑛×𝑛0_{𝑛×(𝑞−1)𝑛}𝑀^𝑇_𝑖𝐴^𝑇_0𝑟𝑌^𝑇_𝑖+𝛿(𝑖,0)𝐶^𝑇_0𝑟𝐾^𝑇₀𝑉^𝑇₀,𝑇^𝑇_𝑖𝑄_𝑖𝐶^𝑇_𝑖=𝑍_𝑖𝑈_𝑖𝐿^𝑇_𝑖00,𝑇^𝑇_𝑖𝑄_𝑖𝑄_𝑖+𝑄^𝑇_𝑖𝑄^𝑇_𝑖𝑇_𝑖=𝑍_𝑖+𝑍^𝑇_𝑖0𝑍_𝑖+𝑌^𝑇_𝑖+𝑆^𝑇_𝑖∗𝑀_𝑖+𝑀^𝑇_𝑖0∗∗𝑌_𝑖+𝑌^𝑇_𝑖,𝑇^𝑇_𝑖𝑄_𝑖𝑄_𝑖𝐵_𝑖𝑟=𝑍_𝑖𝐵_𝑤𝑟𝑍_𝑖𝐵_𝑓𝑟𝑌00_𝑖𝐵_𝑤𝑟+𝑉_𝑖𝐾_𝑖𝐷_𝑟𝑌_𝑖𝐵_𝑓𝑟. (34)
Performing congruence transformations to (20) bydiag{𝑄^𝑇_𝑖𝑇_𝑖,𝑄^𝑇_𝑖𝑇_𝑖,𝐼,𝐼,𝐼,𝐼}, define 𝑋_𝑖𝑟=𝑇^𝑇_𝑖𝑄_𝑖𝑃_𝑖𝑟𝑄^𝑇_𝑖𝑇_𝑖,𝐺_𝑖=𝑉_𝑖𝐺_𝑖𝑈^𝑇_𝑖𝑍^𝑇_𝑖,𝐾_𝑖=𝑉_𝑖𝐾_𝑖,𝐿_𝑖=𝐿_𝑖𝑈^𝑇_𝑖𝑍^𝑇_𝑖,𝑆_𝑖=𝑉_𝑖𝑈^𝑇_𝑖𝑍^𝑇_𝑖, (35) then, it can be easily shown that LMIs (26) together with the additional constraints (29) and (30) are equivalent to LMIs in (20). Hence, if there exist matrices𝑋_𝑖𝑟>0,𝑆_𝑖,𝑍_𝑖,𝑌_𝑖,𝐺_𝑖,𝐾_𝑖,𝐿_𝑖,𝑀_𝑖,𝑖=−1,…,𝑞, and𝑃_𝑡>0such that LMIs (20) are feasible, the overall fault detection dynamic (11) is robust mean square stable and the constraint (16) is satisfied.
Furthermore, from LMIs (26), we have for𝑖=−1,…,𝑞, 𝑍_𝑖+𝑍^𝑇_𝑖0𝑍_𝑖+𝑌^𝑇_𝑖+𝑆^𝑇_𝑖∗𝑀_𝑖+𝑀^𝑇_𝑖0∗∗𝑌_𝑖+𝑌^𝑇_𝑖>0. (36) This indicates that𝑍_𝑖and𝑌_𝑖are nonsingular and 𝑍𝐼0−𝐼^𝑖+𝑍^𝑇_𝑖0𝑍_𝑖+𝑌^𝑇_𝑖+𝑆^𝑇_𝑖∗𝑀_𝑖+𝑀^𝑇_𝑖0∗∗𝑌_𝑖+𝑌^𝑇_𝑖𝐼0−𝐼=−𝑆_𝑖−𝑆^𝑇_𝑖>0, (37) which implies that𝑆_𝑖is nonsingular and also ensures the existence of parameter matrices𝐺_𝑖,𝐾_𝑖, and𝐿_𝑖in (28). The proof is completed.

Remark 10. In Theorem 9, uncertainty-dependent robust fault detection filter design result is provided, which reduces the conservatism than the uncertainty-independent results. If we impose 𝑋_𝑖𝑟=𝑋_𝑖,𝑖=−1,…,𝑞,𝑟=1,…,𝑣, (38) to (26), the uncertainty-independent result can be recovered.

Remark 11. In most cases, we can know the size of the measurement delay or whether the data is missing at a certain time by using the time-stamp at the system node [5], and therefore the jumping parameters of the transformed MJS are accessible. In this sense, Theorem 9 provides us with network-status-dependent fault detection filter design methods. On the other hand, if the network status is not accessible; that is, the jumping parameters of the transformed MJS are unavailable, a network-status-independent result can be easily obtained by imposing 𝑆_𝑖=𝑆,𝐺_𝑖=𝐺,𝐾_𝑖=𝐾,𝐿_𝑖=𝐿,𝑖=−1,…,𝑞, (39) in Theorem 9.
Note that (26) are LMIs over both the matrix variables and the prescribed scalar𝛾². This implies that (i) the robust full-order fault detection filter can be obtained from the solution of convex optimization problems in terms of LMIs, which can be solved via efficient interior-point algorithms [23]; (ii) the scalar𝛾²can be included as one of the optimization variables for LMIs (20), which makes it possible to obtain the minimum noise attenuation level bound for the fault detection dynamics (11). Then, the uncertainty-dependent suboptimal robust fault detection filter can be readily found by solving the following convex optimization problem.

Problem 12. Consider the parameter uncertainty (5), the sub-optimal robust fault detection filter for networked systems (1) with multiple state-delays and unknown inputs based on the idea of uncertainty dependence and network status dependence can be brought forward as follows: min_{𝑋𝑖𝑟>0,𝑆𝑖,𝑍𝑖,𝑌𝑖,𝑀𝑖,𝐺𝑖,𝐾𝑖,𝐿𝑖,𝑃𝑡>0,𝑖=−1,…,𝑞,𝑟=1,…,𝑣}𝛾²,s.t.(26). (40)
For the problems mentioned above, the parameters of the sub-optimal robust fault detection filter can be determined by (28), and the sub-optimal robust𝐻_∞attenuation level for fault detection dynamics is given by𝛾^∗=𝛾²_opt, where𝛾²_optare the sub-optimal solution of the corresponding convex optimization problems.
Here is a summary of the whole fault detection method for system (1).

Step 1. Determine the vertex of uncertain parameters of (1).

Step 2. Calculate the parameters of fault detection filter using Theorem 9.

Step 3. Get a appropriate threshold from experiments for a specific noise type.

Step 4. Generate a real-time residual signal from the fault detection filter designed in Step 2.

Step 5. Compare the evaluation function with the threshold and use the logic (19) to alarm a fault.

Remark 13. In the present work, a model-based approach is considered since there is a mathematical model for the system plant. However, when there is no such a model and only input and output data can be obtained in many complex systems, data-driven methods may work better than the model based ones since there is a leakage of prior information of system dynamics. Please see [24, 25] for typical data-driven fault detection methods.

4. A Numerical Example

To illustrate the effectiveness of the proposed method, we provide a numerical examples in this section. Consider system (1) with the following uncertain system parameters: 𝐴₀=00.50.2𝜃,𝐴₁=,𝐵0.200.70.1_𝑤=0.50.3,𝐵_𝑓=2,𝐶−1₀=𝐶₁=,0.2000.5,𝐷=0.2−0.1 (41) where𝜃is an uncertain real parameter satisfying0.1≤𝜃≤0.3. The initial state values𝜑_𝑘are set to be𝜑₋₁=𝜑₀=0. Let𝑞=1, so the state-space of the Markov chain{𝜏_𝑘}isΞ={−1,0,1}. The transition probability matrix is given by 𝜆Λ∶=^−1,−1𝜆_−1,0𝜆_−1,1𝜆_0,−1𝜆_0,0𝜆_0,1𝜆_1,−1𝜆_1,0𝜆_1,1=0.70.20.10.30.40.30.20.30.5, (42) and the initial mode is set to be𝜏₀=0. For𝑘=0,1,…,300, the unknown input𝑤_𝑘is supposed to be a random noise uniformly distributed over[−0.5,0.5], and the fault signal𝑓_𝑘is of the following intermittent form: 𝑓_𝑘=1,for𝑘=200𝑠+101,…,200𝑠+200(𝑠=0,1,2),0,others. (43)

The weighting matrix is supposed to be𝑇_𝑓(𝑧)=(0.5𝑧)/(𝑧−0.5), with the following state space realization: 𝑥_𝑘+1=0.5𝑥_𝑘+0.25𝑓_𝑘,𝑓_𝑘=𝑥_𝑘+0.5𝑓_𝑘,𝑥₀=0, (44) where𝑓_𝑘and𝑓_𝑘are shown in Figure 1.

Figure 1 

Fault𝑓𝑘and weighting fault𝑓𝑘(𝑓𝑤).

With the predefined parameters, from Theorem 9, Problem 12 can be solved by using the Matlab LMI toolbox [23]. As a result, the minimum noise attenuation level bound of the fault detection dynamic is𝛾_opt=1.0012, and the parameters of the sub-optimal fault detection filter in different modes are given by 𝐺₋₁=0.13111.34020.16200.6100,𝐾₋₁=,𝐿4.30898.622221.792943.5879₋₁=−0.70290.0724,𝐺₀=,𝐾0.03150.4950−0.03730.2187₀=−0.0002−0.0025−0.0005−0.0006,𝐿₀=,𝐺−0.1088−0.0396₁=−0.13430.8207−0.08070.3832,𝐾₁=,𝐿−0.0014−0.0001−0.0018−0.0002₁=−0.60860.2436. (45)

If we impose the uncertainty-independent and network-status-independent method indicated in Remarks 10 and 11, a fault detection filter with𝛾_opt=1.0082can be obtained.

Next, we consider the time-domain simulation using the obtained fault detection filter. we arbitrarily choose𝜃=0.15. Figure 2 shows the measurement mode with random delays and stochastic missing phenomenon.𝜏_𝑘=−1,0,1,means that the measurement is missing, transmitted over the network ideally, and with one-step delay, respectively.

Figure 2 

Measurement mode over network.

Figure 3 shows the generated residual signal𝑟_𝑘, and the evolution of𝐽^𝐿(𝑘)is presented in Figure 4, where we choose the length of time window𝐿=8. After 400 times of simulations, we get a threshold𝐽^𝐿_th=1.5215×10⁻⁶. Figure 4 shows the real-time fault detection result: 4.9199×10⁻⁸=𝐽(102)<𝐽^𝐿_th<𝐽(103)=3.1041×10⁻⁶,1.9392×10⁻⁶=𝐽(220)<𝐽^𝐿_th<𝐽(221)=1.4006×10⁻⁶,9.1655×10⁻⁸=𝐽(304)<𝐽^𝐿_th<𝐽(305)=3.4154×10⁻⁶,2.5346×10⁻⁶=𝐽(424)<𝐽^𝐿_th<𝐽(425)=6.0104×10⁻⁷,8.1835×10⁻⁸=𝐽(502)<𝐽^𝐿_th<𝐽(503)=3.2139×10⁻⁶,1.9241×10⁻⁶=𝐽(617)<𝐽^𝐿_th<𝐽(618)=1.1692×10⁻⁶. (46)

Figure 3 

Residual signal𝑟𝑘.

(a)

(b)

Figure 4 

Evolution of𝐽𝐿(𝑘).

From above inequations, we can observe the fault occurrence can be detected after 3, 5, 3 steps, while the disappearance of fault can be detected after 21, 25, 18 steps, respectively.

Remark 14. After introducing a novel residual evaluation function of the incremental type (17), one can detect not only the occurrence of a fault but also its disappearance. It can be observed from Figure 4 that the disappearance of the fault usually needs longer time to detect than the occurrence of faults. This is because the mathematical model of the system is based on the plant without the faults. This is of engineering significance and the proposed method in this paper can be applied to detect intermittent faults in many practical industrial processes.

Remark 15. The length of time window for evaluation function𝐿in (17) is also a factor that can affect the fault detection performance. Longer𝐿can reduce the missing alarm rate and false alarm rate; however, it costs longer detection time. Short𝐿can provide a rapid fault detection but this results in larger missing alarm rate and false alarm rate.

5. Conclusions

In this paper, the robust intermittent fault detection problem has been investigated for a class of discrete-time uncertain networked systems with state delays and incomplete measurements. The random delay and stochastic missing phenomenon in the measurements have been simultaneously investigated. Polytopic-type parameter uncertainty in the state-space model matrices has been considered. By augmenting the states, the addressed robust fault detection problem has been converted to an auxiliary robust𝐻_∞filtering problem for a certain Markovian jumping system (MJS). A sufficient condition for the design of the desired robust fault detection filter has been established. A numerical example has been introduced to illustrate the effectiveness of the proposed methodology.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61074084 and 61074085 and an open Grant of the Key Laboratory of Hydro-Science and Engineering in Chongqing Jiaotong University under Grant SLK2010.