Robust Fault Detection and Estimation in Nonlinear Systems with Unknown Constant Time-Delays

This paper studies the problem of fault detection and estimation in nonlinear time-delayed systems with unknown inputs, where the time-delays are supposed to be constant but unknown. A new fault detection filter, which can estimate online the time-delays, is first introduced. Then, a reference residual model is proposed to formulate the robust fault detection filter design problem as an H∞ model-matching problem. Furthermore, by a novel robust adaptive fault estimation algorithm, the classical assumption that the time derivative of the output error should be known is removed. In addition, applying a robust H∞ optimization control technique, sufficient conditions for the existence of the fault detection filter (FDF) are derived in terms of linear matrix inequality (LMI). Finally, simulation results are presented to illustrate the effectiveness of the proposed algorithm.


Introduction
Modern industrial control systems become more and more complex, with an increasing number of interconnected physical components, such as actuators and sensors [1].In the system life, faults inevitably occur in these elements, which leads to the drop of systems performances, or even worse to system damage, with dramatic consequences on the environment.Consequently, in parallel with the development of high-performance technological systems, industrials express a growing demand for reliability, maintainability, and survivability [2].Fault Tolerant Control (FTC) is an effective way of maintaining system performances under faulty conditions.FTC can be achieved either by a passive way or using an active method.Passive FTC uses feedback control laws that are robust with respect to an a priori fixed set of possible system faults [3,4].On the other hand, active FTC uses a Fault Detection and Isolation (FDI) module that provides online fault information.Active FTC may consist of a fault accommodation scheme [5,6], which is in fact an adaptive control with respect to the fault information, or may be a control system reconfiguration scheme, which consists of a switching control with a supervision layer that selects the most suitable control law for the identified faulty situation [7][8][9][10].One difficulty is that FDI module and active FTC algorithm have to be designed jointly because of their mutual interactions.
In distributed or large-scale processes, the dynamic contains time-delays which may cause instability and degrade the system performances [11].These time-delays may generally be considered as constant ones for given operating conditions.However these constant values are generally not known.The existence of such time-delays renders the control design problem much more difficult [12].Increasing attention has recently been devoted to stability control and fault diagnosis of linear/nonlinear time-delayed systems, see, for example, [4][5][6][11][12][13][14][15][16].In [14], an observer-based fuzzy control scheme with adaptation to the time-delay was proposed.In [15], fault detection and identification for uncertain linear timedelay systems were investigated.In [16], by using  ∞ control theory, a robust fault detection scheme was proposed for a class of discrete time-delay systems with parameter uncertainty.However, the sensitivity of the residual signals to the faults was not studied.In [17], the fault detection filter design problem for linear time-delay systems was studied by introducing an idealized reference residual model.Based on [16,17], the robust fault detection filter (RFDF) design problem was studied in [18] for nonlinear time-delay systems with unknown inputs.In [19],  ∞ fault detection filter design for linear discrete time systems with multiple time-delays was discussed.The fault detection filter design problem was investigated in [20] for LTI (Linear Time-Invariant) system with time-delays.In [21,22], the fault estimation problem for linear systems with state time-delays was studied, and an adaptive fault detection observer was designed.Many observers that are proposed in the literature, as in [16][17][18][19][20][21][22][23], use explicitly the time-delay value that is supposed to be accessible.However, in practical applications, this time-delay is generally not available, which makes these techniques not applicable.Another drawback of many diagnosis techniques that are proposed in the literature is that they need the derivative of the output signal for the fault estimation algorithm.These derivatives are generally not directly measured and are difficult to compute in a noisy environment.
In this paper, we investigate the problem of fault detection and estimation of nonlinear time-delayed systems.Five main contributions are worth emphasis.
(2) An adaptive fault estimation algorithm is proposed where the classical assumption that the time derivative of the output errors has to be known (see [22], for instance) and is removed.Moreover, our scheme is robust to bounded disturbances.
(3) Differing from numerous FDI schemes in the literature, the bounds of the time derivatives of the faults have not been known in our proposed adaptive fault estimation algorithm.
(4) In this paper, in contrast with [18], not only the fault detection filter design problem is discussed, but also an adaptive fault estimation algorithm is proposed.
(5) Sufficient conditions for the existence of the adaptive fault observer are expressed using the Lyapunov stability theory.
The paper is organized as follows.The problem formulation is presented in Section 2. In Section 3, the main technical results of this paper are given, which include the choice of the reference residual model, the design of a robust fault detection observer/filter, and the fault estimation algorithm.Simulations are presented in Section 4. Finally, Section 5 draws the conclusion.

Problem Formulation
Consider the following class of nonlinear time-delay systems [18]: where () ∈   is the unknown state vector, () ∈   is the control input vector, () ∈   is the measured output, () ∈   that belongs to   2 [0, +∞] is an exogenous disturbance input vector that regroups all the model uncertainties, () ∈   is the fault vector to be detected, ℎ ≥ 0 is an unknown but constant time-delay that satisfies ℎ ≤ ℎ, where ℎ is a known constant, (⋅, ⋅) :   ×   →    is a known nonlinear function, and ,   , ,   ,   , , ,   ,   ,  are all known matrices with appropriate dimensions.
Throughout this paper, the following assumptions are made.
To detect the fault, the following so-called fault detection filter (FDF) is proposed: where x() ∈   and ŷ() ∈   denote the state and the output of the filter, respectively; ĥ is the estimation of the unknown constant time-delay ℎ; () is the so-called residual signal.The observer gain matrix  and the residual weighting matrix  will be defined later.
For the above observer design, the following assumption is made.Assumption 3.There exists a known constant  > 0 ∈  such that ‖() − x()‖ ≤ .Remark 4. Assumption 3 is not restrictive.In fact, this is a common assumption in the literature concerning filter/observer design.In practical application, it may be difficult to know accurately the upper bound of the observer error.Therefore,  can be chosen large in practical applications.It must be pointed out that  is only needed for stability analysis of the closed-loop system.Choosing a large  does not degrade the performance of the closed-loop system.Define The error dynamics can be computed as where Ψ = ((), ( − ℎ)) − ( x(), x( − ĥ)).
From Newton-Leibniz formula, we have Thus, error dynamics (4) can further be described as From (6), it is seen that the dynamics of the residual signal depends not only on fault () and uncertainty (), but also on the nonlinear part Ψ. Motivated by the method presented in [18], we propose using a reference residual model that describes the desired behavior of the residual vector (), to formulate the RFDF design problem as an  ∞ modelmatching problem.In other words, the objectives are finding an idealized reference residual and minimizing the distance between the generated residual and the idealized reference residual.In the idealized case, the observed state x() and the time-delayed state x( − ĥ) should be equal, respectively, to () and ( − ℎ); thus we have () − x() = 0 and ( − ℎ) − x( − ĥ) = 0. Consequently, Ψ = ((), ( − ℎ)) − ( x(), x( − ĥ)) = 0. Therefore, according to (6), and assuming Ψ = 0, the corresponding reference residual error model is given by where   () ∈   is the reference model error state vector,   () is the reference model residual signal, and  and  are the parameters of the reference residual model to be designed.Thus the overall system can be described by where From ( 8), the design of the robust fault detection filter, which is the main objective of this work, may be formulated as an  ∞ model-matching design problem.Applying a robust  ∞ optimization control technique, for all exogenous disturbance inputs and nonlinear parts, the generated residual () will be designed as closely as possible to the reference model residual   (), independently of the unknown time-delay ℎ.
Thus, the problem of designing an observer-based RFDF can be described as designing the observer gain matrix  and the residual weighting matrix  such that (i) systems ( 8) are robustly asymptotically stable; (ii) under zero initial condition, for given constant  > 0 and any nonzero () ∈  2 [0, ∞), systems (8) satisfy the following inequality: After designing the RFDF, the remaining important task is to decide, from the generated residual, if an alarm has to be Mathematical Problems in Engineering generated.One of the widely adopted approaches is to choose a so-called threshold  th > 0 and, based on this, use the following logical relationship for fault detection: where The threshold  th > 0 can be chosen as  th =  + , where  > 0 ∈  and  > 0 ∈  denote, respectively, the upper bound of the norm of fault  and exogenous disturbance , that is, ‖‖ 2 ≤ , ‖‖ 2 ≤ .
Remark 5.There are many results on fault detection observer/filter design for time-delayed systems in the literature.In general, the observer, as the one in [16,18], is designed in the following form: However, just as said in [14], the shortcoming of the abovementioned observer is that the constant state time-delay ℎ must be exactly known.If the time-delay ℎ is unknown, the above observer will not work in practical applications.

Main Results and Proof
As mentioned in [24,25], the design of a RFDF for the nonlinear time-delay system (1) can be formulated as an  ∞ model-matching problem.In this section, we first design the reference residual model and then express the RFDF design problem under an LMI formulation.

Choice of the Reference Residual Model.
As pointed out in [18], the selection of a suitable reference residual model is one of the key steps to design an RFDF for nonlinear timedelay systems.If the reference residual model is not selected suitably, miss alarms or false alarms may occur.In order to select a suitable reference residual model, we consider the following performance index   : Therefore, according to the performance index   (13), the reference residual model can be designed, which takes into account not only the robustness of the reference model residual against disturbance but also the sensitivity to faults.
For the sake of simplicity, we assume that  = .
Consider the following transfer function: where the matrices  ∈  × ,  ∈  2× select the appropriate input/output channels or channels combinations and  satisfies the following equalities: where The following theorem provides a sufficient condition to ensure that, for a given  > 0, the reference RFDF satisfies (16).
The proof is completed.

Design of Robust Fault Detection Filter.
In this section, we propose a theorem that gives a sufficient condition to guarantee that the RFDF system is stable and has a prescribed  ∞ performance, independently of the time-delay.Before presenting the theorem, the following lemmas are introduced.

Lemma 7. Given constant matrices 𝜒
Lemma 8. Let  and  be real matrices of appropriate dimensions.For any scalar  > 0 and vectors ,  ∈   , the following inequality holds: Recall the nonlinear time-delay system (8).
If () = 0, from the above analysis, one has where ].
The proof is completed.The last step of fault detection is to evaluate the residual.This is a decision making process that always comes down to a threshold logic of a decision function.From Assumption 2 ( ∈  2 ), we can further have ‖()‖ 2 = , where  > 0 is a constant.By using Theorem 9, we obtain   = sup ∈ 2 ‖‖ 2 /‖‖ 2 .In the fault-free case, the generated residual () is only affected by the disturbance input ().Therefore, the threshold  th can be determined by The fault detection decision logic can be designed as Remark 10.In [18], Ψ was described as Ψ = ((), ( − ℎ)) − ( x(), x( − ℎ)).From Assumption 2 in [18], one has However, if time-delay ℎ is an unknown constant, ‖Ψ‖ and ‖Ψ‖ 2 are not available and can not be used in the design procedure.In this paper, an improved observer is proposed, in which ℎ is replaced by the estimation value ĥ.Then, Ψ is presented in the following form: (51)

Fault Estimation.
After detecting the fault, fault isolation and estimation are considered.Similar to [21], by a bank of so-called fault isolation observers, fault pattern and location can be obtained.In this paper, it is assumed that faults have been isolated and fault isolation is not studied further.In the following, we will investigate the fault estimation problem.

Incipient Fault Case.
As pointed out in [18], in many real physical systems, the fault evolves slowly.In this case, the fault () is assumed to be an incipient fault, which is shown in Figure 4.The generated residual signals () (including  1 () and  2 ()) and the evolution of residual evaluation function are illustrated in Figures 5 and 6, respectively.

Conclusions
In this paper, the problem of fault detection and estimation of nonlinear time-delayed systems with constant but unknown state time-delay is studied.A new fault detection filter with   adaptation to the time-delay is proposed.Then, a reference residual model is introduced to formulate the robust fault detection filter design problem as an  ∞ model-matching problem.A novel robust adaptive fault estimation algorithm is proposed where the time derivative of the output errors has not been computed.In addition, applying a robust  ∞ optimization control technique, sufficient conditions for the existence of the fault detection filter are derived in terms of LMI.In future, we will consider the fault diagnosis and fault tolerant control of actuators and sensors when both the input and the state are time-delay.