Fault diagnosis and fault-tolerant control design for neutral time delay system

This paper presents a new approach of fault-tolerant control (FTC) for the transmission line as a neutral variable time-delay system. The main goal of this work guarantees faulty neutral variable time delay system stabilization using the state feedback control design based on Lyapunov function and the Linear Matrix Inequality resolution. The use of the FTC method is to achieve actuator and sensor fault compensation. This method is based on two steps. The first one is the synthesis of a nominal control, which remains to maintain the closed-loop system stability. The second step is based on adding a new control law to the nominal one to compensate the fault effect on system behaviour and maintain the desired performance in the closed loop system. Then, a conception of an adaptive observer is used to detect and estimate the fault. Finally, the developed approach is applied for the transmission line. The given results are presented to prove the effectiveness of this approach.

In this work, we are interested in the transmission line as a neutral variable time-delay system. The presence of variable time delay induces poor performance and instability in control systems. In addition, the fault can occur during the system operation that can be a source of degradation of its performance. Therefore, it is necessary to introduce a fault-tolerant control (FTC) strategy to maintain the system behaviour.
Benjemaa et al. [1], interested in the FTC of transmission line system with variable delay and actuator fault based on robust approach and PIM method, proved that the robust control is able to reduce the fault effect but it fails to make system output converged to desired one. The PIM method was used. It was found that the error between the closed loop system output and the nominal one minimized.
This method requires a reference model to modify it and design a control law for the fault-free system such that the closed-loop behaviour follows the reference model.
This work compensates the fault effect on system behaviour applied for two types of faults, sensor and actuator fault using a new type of FTC (additive control). This FTC is designed by adding a new control law to the nominal one. The results of FTC application for transmission line system are compared with results given in [1].
The simulation results show that the actuator fault compensation is better than the methods previously used, because it allows to minimize the fault effect on system behaviour more than the two other approaches. In addition, this approach is not used only for actuator fault compensation but also for sensor fault.
This paper is organized as follows: Section 2 presents the problems statement where the transmission line is modelled as a neutral variable time delay system. In Section 3, the main results of the proposed method; a synthesis of FTC law for the case of sensor fault and actuator fault, and a conception of an adaptive observer are given. Section 4 gives the simulation results of control application for neutral variable time delay transmission line system. The last part is the conclusion.
Notation: Consider the following notations: R n and R n×m are respectively the n-dimensional Euclidean space and the space of all real matrices, the transpose is denoted by the superscript "T" and LMI denote Linear Matrix Inequality. I is the identity matrix of appropriate dimension and " * is used to denote the transposed elements in the symmetric position.

PROBLEMS STATEMENT
Consider the transmission line shown in Figure 1, at one end (x = 0) there is a constant voltage source E and at the other end (x = l), a capacitor is connected in parallel with a tunnel diode. The current i(., .) and the voltage v(., .) are functions of t and x, which satisfy the telegraph equation, a partial derivative equation of the hyperbolic type: where l is the length of the line, L and C are the inductance and capacitance of the conductor per unit length, R is the resistance at the input and C d is the capacitance in parallel with the tunnel diode. The conditions cited in [1] give the following neutral linear system: m and R 0 are respectively a positive constant and an equilibrium resistance. The neutral variable delay system can be written as follows: where x(t) ∈ R n ,u(t) ∈ R m and y(t) ∈ R p are respectively the state, the input and the output vector.
The variation in delay is the consequence of the variation in line length.

Main results
The following lemmas are very useful for the main results of this work.

Lemma 3.2: Increase Lemma:
Given scalar X and Y two matrices of appropriate dimensions, the following inequality is always true for any matrix Q > 0:

Synthesis of an FTC, sensor fault case
Consider the following neutral variable time delay system (3) with sensor fault: where f s (t) is the sensor fault and F s is a constant matrix, and it is assumed that the pair (A i , C) is observable, i = 1, 2, 3.

Synthesis of a nominal control
The nominal control law remains to maintain stability of closed-loop system. A nominal control in case of sensor fault noted u nom s (t) which is represented as follows: where K is the feedback matrix gain. The pole placement method is used to determine K.

A strategy of additive control
For the sensor free fault system (3), the nominal control law remains to maintain stability of the closed-loop system which satisfies Equation (6). When the fault occurs, an additive control becomes necessary to compensate the fault effect.
For this reason, we suggest adding a new term u add s (t) to the nominal law to compensate the fault effect on closed loop system behaviour.
The control, to be effectively applied to the neutral variable time delay system can be written as The closed-loop state equation can be rewritten as follows:ẋ where u add s (t) satisfies the following equation: or the adaptive observer (21) is trying to estimate the fault such that lim t→∞fs = f s . wheref s is the estimate of sensor fault f s . Also, the error fault equation is described as follows: And, we knew lim t→∞fs = f s , so: Therefore, we can replace the sensor fault by its estimation in the equation (10), which gives

Synthesis of an FTC, actuator fault case
Consider the following neutral variable time delay system (3) with actuator fault: where f a (t) is actuator fault and F a is a constant matrix.

Nominal control
The following state feedback controller proposed by (2) is designed to stabilize the closed loop system:

Additive control
For the case of actuator fault, the proposed control is also written as follows: where u add a (t) satisfies the following equation: where B + is the pseudo-inverse of matrix B.
In the same way, we have f a (t) is the estimate actuator fault, and we knew lim t→∞fa = f a , sof So, we find the following equation: The FTC law synthesis needs the determination of sensor and actuator fault estimates respectivelyf s (t) and f a (t). For this reason, an adaptive observer will be used in the following paragraph to detect and estimate these faults.

Conception of an adaptive observer
The following adaptive observer generates residuals in order to detect and estimate the fault:

Adaptive observer in the case of sensor fault
Consider the adaptive observer in the case of sensor fault as follows: Based on [3], consider the estimate sensor fault as follows:f wherex(t) andŷ(t) represent respectively the estimate state, the output vector and r(t) is the residual vector.
, V 1 are respectively a positive definite and an arbitrary given matrices with appropriate dimension, σ is a given scalar.
A theorem is proposed in (2) to determine the gains of the observer L and H. Theorem 3.1: Given scalar h > 0 and if there exist matrices P 1 , Q, W,Z 1 , Z 2 , Z 3 ,G∈ R n * n , and Y 1 ,Y 2 ∈ R n * p ,M i , N i ∈ R n * n , i = 1,.., 4, such that:

Adaptive observer in the case of actuator fault
Consider the adaptive observer in the case of actuator fault to detect and estimate the fault as follows: where δ a is a positive definite matrix, it is the learning rate.
The following theorem is considered to find δ a : Proof: The error dynamic is described by where The reference model is The Lyapunov functional is chosen as where

Simulation results
In this section, the transmission line is considered with the form (3). This system is described by the following matrices: We obtained the feedback matrix gains by pole statement as follows: Figures 2 and 3 show that the residual is sensible to the fault (in the case of actuator and sensor fault). Figures 4 and 5 show that the adaptive observer can enhance the performance of fault estimation.
In conclusion, the proposed observer, based on fault tolerant additive control strategy, is considered as a good fault compensator for neutral time-delay systems. The robust and PIM methods in [1] achieve the stability and the performance system in closed loop. But the robust controller, Figure 6, cannot achieve the final value ( 0.6). Using the PIM method, Figure 7, the error between the faulty system and the reference one        is achieved, so the error can be minimized and the convergence to the final value is obtained.
Unlike the PIM method, the additive control does not require a reference model. It also allows the fault to be detected and estimated online.
The effectiveness of the proposed additive control strategy is thus noted inspite of the presence of an actuator and sensor fault. It emphasizes the ability of this strategy to maintain closed-loop performance (Figures 8 and 9). In addition, it minimizes the fault amplitude better than the PIM method (amplitude in PIM 1) and with additive control in Figure 8, (0.6 < amplitude < 0.8).
The strategy developed, therefore, shows its ability to maintain performance in terms of compensation for the considered fault.

Conclusion
In this work, the transmission line is considered as a neutral variable delay system. The FTC is composed of two terms as presented in Section 3.
This type of control is better than the PIM and the robust control. In fact, it is applicable for the sensor fault and it guarantees the desired performance of the neutral variable time-delay case systems.
Simulation results demonstrate the effectiveness of the control strategies developed for the neutral variable delay transmission line system.

Disclosure statement
No potential conflict of interest was reported by the author(s).