Active Fault Tolerant Control of Faulty Uncertain Neutral Time-Delay Systems

: The present paper attempts to investigate the problem of Fault Tolerant Control for a class of uncertain neutral time delay systems. In the first time, we consider an additive control that is based on adding a term to the nominal law when the fault occurs. This approach will be designed in three steps. The first step is fault detection while the second one is fault estimation. For these two steps, we consider the adaptive observer to guarantee the detection and estimation of the fault. The third step is the fault compensation. Lyapunov method and Linear Matrix Inequality (LMI) techniques were considered to improve the main method. Second, we propose a Pseudo Inverse Method "PIM" and determine the error between the closed loop and the nominal system. Finally, simulation results are presented to prove the theoretical development for an example of an uncertain neutral time delay system.


Introduction
In recent years, there has been a burst of research activities in the area of time-delay control systems [1][2][3].
Time delay systems can be classified into two different categories: the first one expressed by retarded functional differential equations (RFDE) and the second one described by neutral functional differential equations (NFDE).
The presence of delays more precisely for neutral time-delay system induces poor performance and instability in closed-loop systems. Furthermore, some faults can be occur during the system operation.
Fault is understood as any sort of malfunction that occurs in actuator, sensor or process level and that leads to any deviation from the nominal condition.
The system treated in this work has the particularity of the uncertain elements. Uncertain RFDE systems have been studied in [1,3,4,9,10]. Their objective was to determine the stability and control of this type of system.
Uncertain NFDE systems have been studied in [8,11] to achieve the adaptive robust control.
Benjemaa, R.; and al. in [1] studied the stability and active fault tolerant control for a class of certain neutral time delay systems; this work is a continuity of [1] such that, we will consider the neutral time delay systems with parametric uncertainty and we will compare two methods of active fault tolerant control.
The presence of parametric uncertainty, which can cause important error in system parameters identification, guide to more complexity of the system model. Thus, neutral variable time-delay and parameter uncertainty are sources of the oscillation generation, instability and poor performance. Benjemaa Rabeb et al.

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An adaptive robust control scheme was proposed in [7] to stabilize uncertain neutral time-delay systems, as well as in [8]. The authors proposed an adaptive Robust control of uncertain neutral time-delay systems.
However, the main drawback of this work is that it assumes the delayed state of the system in the adaptive process.
In this article, the uncertain neutral time-delay systems are studied, and a new active fault-tolerant control is proposed to this type of system.
The comparison between additive and PIM control for uncertain neutral time delay systems is the principal contribution of this paper.
First, the additive control approach (AFTC) will be presented. The main idea of this control is based on adding a term to the nominal control law when the fault occurs, to compensate the fault effect.
Second, the Pseudo-Inverse Method (PIM) will be designed for this type of system based on the error determination between the closed loop system and the nominal one.
The remainder of the paper is organized as follows. The problems statement is given in section 2. In section 3, the main results of the AFTC are presented; the method of additive control (fault detection, estimation and fault compensation) and the PIM, for uncertain neutral time-delay systems.
Simulation results are presented in section 4 to demonstrate the efficacy of the proposed method and finally, some concluding remarks are drawn in section 5.
Notation: Let us consider the following notations used in this paper. n R is the n-dimensional Euclidean space nm R  is the space of all real matrices, and the superscript "T" denotes the transpose and LMI denote Linear Matrix Inequality. I is the identity matrix of appropriate dimension. "*" denote the transposed elements in the symmetric position.

Problems Statement
Consider a class of linear uncertain neutral time delay systems described by the following state space equation: the output vector. h>0 and d>0 are respectively state and its derivative delay.  Benjemaa Rabeb et al.

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Consider the uncertain neutral delay system affected by actuator fault: Where F is a constant matrix with appropriate dimensions and f (t) is actuator fault. The presence of fault causes system damage. However, two fault tolerant control (FTC) approaches are proposed for uncertain neutral time delay systems possessing the ability to maintain nominal targets despite the presence of a fault.

Main results
In this section, the two methods, as detailed in introduction are considered to design the FTC for the system (2).

Fault detection and estimation
Considering the system (2). As a first step, the following adaptive observer is used to detect the actuator fault: To calculate L and H; we consider the following theorem: Benjemaa Rabeb et al.

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Given scalar h>0, considering the following Linear Matrix Inequality such that, if there exist symmetric positive definite matrices P, Q, W, 1 Where:  T  T  T  T   T  T  T  T  T   TT   TT   T  T  T   T  T  T   T  T  T In step 2, we use the same observer to estimate the fault with the following form: Benjemaa Rabeb et al.

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Where: Fr t (9) To find  the following theorem is considered:

Fault compensation
To ensure a zero-static error between the system output and the desired one, we add an integrator to the control law. Therefore, we can design the following state feedback controller to stabilize the closed-loop system: Where: Where K the feedback matrix gain obtained by LMI resolution in the following theorem.

Lemma 1. the system (1) is asymptotically stable if it exists
The Leibniz-Newton formula provides: The following expression is verified: Where: Then, by adding the null terms (16) and (17)    , the matrix  can be chosen as: For theorem 3 based on Lemma 1, we will determine the return gain K.
To determine the return gain K, it suffices to replace the matrix Where By developing the computation, we end up with a nonlinear LMI. To solve this problem, we recourse to the dual system. A system is stable. Then, its dual is also stable, since solution of: Therefore, is equivalent to the following system: To determine the return gain we replace Then: Where:

B B B = + 
In addition, B + is the pseudo-inverse of matrix B .

PIM for uncertain neutral time-delay system
In this part, the Pseudo-Inverse-Method is based on the error determination between the closed loop system and the nominal one.
The main idea of this method is to modify the constant feedback gains of nominal system, then, synthesized controller gains that can stabilize the faulty systems.
The reference model is defined as: Where 1 X , 2 X and 3 X are given.
Considering the state feedback, the set of admissible controls is: When the fault is detected, the system (1) can be presented as:

Simulation Results
In this section, a numerical example is presented to demonstrate the effectiveness of the proposed theoretical development. Considering the linear neutral time-delay system (2), with the system matrices described by:    By the use of additive control law method. Figure 1 and Figure 2 show that this control is able to compensate the fault (for y1add (t) and y2add (t)).
The PIM method can minimize the error between the faulty system and the reference one, Figure 1 (y1PIM (t)), but it is not the case in Figure 2 (y2PIM (t)) where the performance is minimized.

Conclusion
In this paper, two methods of active fault tolerant control are presented for an uncertain neutral time delay system.
With an additive control law added to the nominal control, the proposed AFTC achieves the minimization of fault effects.
By using a Pseudo-Inverse Method (PIM), the performance satisfaction is guaranteed for y1PIM (t) but for y2PIM (t) is not totally guaranteed. A numerical example has been given to show the effectiveness and limitation of the proposed methods.