Fault diagnosis and fault-tolerant control for system with fast time-varying delay

This paper proposes a fault diagnosis and fault-tolerant control method for a system with a fast time-varying delay and time-varying parameters. A fault observer is designed to estimate faults, and an improved fast adaptive fault estimation (FAFE) algorithm is developed to reduce the relevant constraints in the general form of this algorithm. With newly introduced relaxation matrices, this study estimates faults in a system exhibiting a fast time-varying delay. Based on the estimated faults, an output feedback controller is designed to accommodate the faults. The fault-tolerant control is realized using the introduced relaxation matrices. An algorithm is derived to solve for the observer and controller. Finally, the theory and method are validated using a real example of a helicopter system.


Introduction
With the increasing complexity of control systems in modern industrial processes and the increasing requirements of reliable systems [1,2], the diagnosis and accommodation of faults for dynamic systems have become more important.
The diagnosis and accommodation of faults for systems with a time delay is an important and challenging task [3][4][5] because many realistic systems exhibit a time delay [6]. Furthermore, because of the uncertain parameters in most practical systems, the design becomes more difficult [7,8]. Only few studies have been conducted in the area of fault estimation for time-delay systems. A sliding mode observer has been designed to estimate faults; however, in this method, the system demand is so high that its applicability is poor [9]. Moreover, it is only suitable for systems with a constant or a slow time-varying delay. In another study [10], a fault estimation filter method was used to estimate the faults in a near spacecraft. However, this method is not suitable for all types of faults because of its strict constraints. An observer-based iterative learning method was used to estimate the fault in a spacecraft; however, the design process of this method is so complicated that its generality is poor, and this method must also satisfy difficult conditions [11]. An adaptive observer was designed to estimate faults in a system with a time-varying delay using the common adaptive fault estimation (CAFE) algorithm. This method is simple and effective while being less restrictive on the system and faults. However, the method is unsuitable for systems with a fast time-varying delay [12]. A fast adaptive fault estimation (FAFE) algorithm has been developed to estimate faults in a system with a fast time-varying delay by introducing relaxation matrices. However, in the FAFE algorithm, the corresponding constraints need to be calculated via an approximate method. Moreover, no method has been provided for fault-tolerant control of the system [13].
Studies on the fault-tolerant control for systems with a time-varying delay are limited [14]. A fault-tolerant controller for a linear system without time delay has been designed based on online fault estimates [15]. In another study, a fault-tolerant controller for a linear system with a slow time-varying delay has been designed via the adaptive control method based on online fault estimates [16]. A fault accommodation method for a system with a slow time-varying delay has been proposed based on adaptive sliding mode control [17]. The above methods can achieve satisfactory results for systems with slow time-varying delays or without any time delay but are unsuitable for systems with a fast timevarying delay. Studies on the fault-tolerant control for systems with a fast time-varying delay are lacking [18].
The main objective of this study was to develop a method for the fault diagnosis and fault-tolerant control of systems with a fast time-varying delay. Estimating and accommodating faults for such systems with loose constraints and improved applicability is a challenging task.
In this study, a fault observer is designed based on the improved FAFE algorithm for systems with a fast time-varying delay and time-varying parameters. The faults are estimated by introducing relaxation matrices. Based on this, an output feedback fault-tolerant controller is designed. The relaxation matrices are introduced to realize fault-tolerant control. Because the observer and controller cannot be solved directly, this paper derives a corresponding iterative algorithm to solve the problem. Finally, the accuracy of the method is proven through numerical and real examples.
The contributions of this study are as follows. (1) The system reported in this paper has a fast time-varying delay and time-varying parameters; we estimate the faults in this system, despite the challenging nature of the task. (2) After the fault estimation, we propose a method for fault accommodation.

System description and preliminaries
A system with a fast time-varying delay can be represented as where x(t) ∈ R n , u(t) ∈ R m , and y(t) ∈ R p are the state, control input, and output vectors, respectively. f (t) ∈ R q indicates the fault in the actuator of the system, satisfying the conditionḟ (t) ∈ L 2 [0, ∞); d(t) is the time-varying delay in the system, which satisfies 0 < d(t) ≤ h andḋ(t) ≤ τ ; and w(t) ∈ R s is the noise in the system, which satisfies w(t) ∈ L 2 [0, ∞). In addition, A, A d , B, G, W, C, D 1 , E 1 , E d , E 2 , E 3 , and E 4 are known real constant matrices with appropriate dimensions, F(t) represents the matrix associated with the time-varying parameters of the system, satisfying the condition F T (t)F(t) ≤ I, and I is a unit matrix.
Prior to diagnosing the faults and developing a faulttolerant control for the system represented in (1), we assume the following: fault estimation and accommodation. The fault studied in this paper occurs in actuators. According to a previous study [19], assumption 2.3 is reasonable and is equivalent to considering a matrix G * ∈ R m×n , implying that BG * + G = 0 and E 2 G * + E 3 = 0 hold. Lemma 2.1: [20]: Y is assumed to be a symmetric matrix, where D and E are matrices of appropriate dimensions. Accordingly, the inequality will hold for all matrices satisfying F T F ≤ I, if and only if the inequality holds for a positive scalar ε.

Design of a fault estimation observer
The following observer is designed to estimate faults: (2) wherex(t) ∈Rn is the observer state vector,ŷ(t) ∈ R p is the observer output vector,f (t) ∈ R q is the fault estimates, and L and H are the gain matrices of the observer to be designed.
We used the improved FAFE algorithm to estimate the fault:ḟ where = T > 0 is the adaptive learning factor, and K 1 and K 2 are matrices with unknown parameters to be solved. From the improved FAFE algorithm (3), the fault estimation signal can be obtained.

Remark 3.1:
The CAFE algorithm is expressed aṡ and the FAFE algorithm is expressed aṡ Equations (5) and (6) show that the FAFE is the improved version of the CAFE. Because the FAFE includes an additionalẏ(t) term, the performance of the FAFE is better than that of the CAFE. From (4) and (6), we see that in the general FAFE algorithm, K = K 1 = K 2 . It needs to satisfy the equation Because of the uncertain parameter matrix F(t), this equation is difficult to solve. The improved FAFE reported in this paper does not require the constraints of this equation; thus, we compensate for the shortcomings of the general FAFE algorithm. In (3), is an adaptive learning factor; its role is only to improve the fault estimation performance and has nothing to do with the performance indicators, namely the robustness and stability, of the observer. Therefore, we do not consider (we take = I at first, where I is a unit matrix) when the observer is designed using the H ∞ performance indicators. Once the parameters affecting the performance indicators are determined, we assign a specific value to improve the fault estimation performance.
We definẽ wherex is the state estimation error,ỹ is the output estimation error, andf is the fault estimation error. Furthermore, we define Thus, we can obtain the following extension system: where To reduce the steady state error in the system, we define the following H ∞ performance indicator:  (4) can make the state and fault estimation errors robust and stable, and the extension system (7) will satisfy the H ∞ performance ||ỹ(t)|| 2 < γ ||μ(t)|| 2 .

Proof:
The following Lyapunov function is selected: where where P = P 1 0 0 P 2 , Q = Q 1 0 0 0 , and Z = Z 1 0 0 0 . Taking the derivatives of V 1 (t), V 2 (t), andV 3 (t) with respect to time, we obtain: where Based on (17), we can solve (16); thus, the relaxation matrices N 1 and N 2 are introduced. If the relaxation matrices are not introduced, ϕ 4 of Theorem 3.1 will be −(1 − τ )Q 1 > 0. In this case, the matrix inequality of Theorem 3.1 will be unsolved when τ > 1. Because the relaxation matrices N 2 and N T 2 appear in ϕ 4 , the matrix inequality can be solved. Thus, faults can be estimated.
It is difficult to solve (9) of Theorem 3.1; therefore, we transformed it into a matrix inequality, which can be solved using the iterative method.
Note that According to lemma 1, (21) is equivalent to considering ε > 0. Therefore, the following inequality holds: According to the Schur complements and lemma 1, Therefore, we can write the following Theorem 3.2.

Theorem 3.2:
For given H ∞ performance indicators γ > 0 and constants h > 0, if there exist symmetric positive definite matrices P 1 , P 2 , Z 1 , Q 1 , 1 , and 2 ∈ R n×n ; matrices Y H ∈ R n×p , P K1 , and P K2 ∈ R n×n ; and arithmetic number ε 1 such that (23) holds, then the improved FAFE algorithm (4) makes the state and fault estimation errors robust and stable. The extension system (7) satisfies the The inequality in (23) is not linear; nevertheless, it can be solved using the iterative method. We define

and according to Schur complements, we have
We define ε −1 1 = δ 1 . Therefore, the conditions of Theorem 3.2 can be transformed into the following optimization problem: The steps involved in obtaining the solution are as follows. First, we set the error threshold ∈ and the maximum number of iterations k * .
Step 1: Solve (23) Step 2: With k = k + 1, obtain the optimal solutions late the minimized value Tr k and determine Tr k = |Tr k−1 − Tr k |.
Step 3: If Tr <∈, terminate the process. Otherwise, proceed to the next step.
Step 4: If k > k * , terminate the process. Otherwise, proceed to step 2.

Design of H ∞ fault-tolerant controller
We design the following feedback fault-tolerant controller based on the fault estimates: where ξ(t) ∈ R n is the controller state vector, r(t) ∈ R m is the reference input, and A c , B c , C c , and D c are the parameter matrices of the controller to be designed. Taking r(t) = 0, we obtain the following system based on (1).

Proof:
We define the Lyapunov function of the system represented in (32) as follows: where Taking the derivatives of V 1 (t), V 2 (t), and V 3 (t) with respect to time, we havė where

Remark 4.2:
Based on (41), we can solve (40). Thus, the relaxation matrices N c1 and N c2 are introduced to realize a faulttolerant control for the system with a fast time-varying delay with a rate of change τ > 1. If the relaxation matrices are not introduced, ϕ c3 of Theorem 4.1 will be −(1 − τ )Q c1 > 0, with no solution to the matrix inequality in Theorem 4.1.
We select the following performance indicator and obtain According to the Schur complements, we can conclude that The inequality in (43) can be expressed as According to lemma 1, (46) will hold if and only if there is a constant ε 2 > 0 that can hold (47). ⎛ According to the Schur complements, the inequality will hold when (33) holds. Equation (48) can be expressed as Hence, according to [21], the state and fault estimation errors are robust and stable. Using (49) and considering 0 the initial state, we find that the extension system (32) satisfies the H ∞ performance ||y(t)|| 2 < γ 2 ||v(t)|| 2 . The matrix inequality (33) in Theorem 4.1 is not linear, but can be transformed into one and then solved. We take matrix P c and its inverse matrix P −1 c in the following forms: where Y 1 and Y 2 ∈ R n×n are symmetric positive definite matrices, and R c1 , R c2 , T 1 , and T 2 are matrices with appropriate dimensions satisfying We can obtain We multiply both sides of (33) with diag(F T 1 , I, ..I) on the left and with diag (F 1 , I, ..I) on the right. Now, according to the Schur complements, (33) can be transformed into a matrix inequality that can be solved to prove Theorem 4.2. .I) on the right, the matrices Y 2 (N T c2 − N c1 ) and hY 2 N c1 , along with their transpose matrices, will appear in the transformed matrix inequality. According to the Schur complements and (51), we can obtain (52). ⎛ where c1 is a symmetric positive definite matrix.
The parameter matrices of the fault-tolerant controller are D c =D, The matrix inequality (51) in Theorem 4.1 is not linear but can be solved using the iterative method. We take , and ε −1 2 = δ 2 . Theorem 4.2 can now be transformed into the following optimization problem.
The following are the steps involved in obtaining the solution: First, we set the error threshold ∈ c and the maximum number of iterations k * c .
Step 2: With k = k + 1, calculate the optimal solutions Z c1 k , S c1 k , c1 k , c2 k , Q c1 k , U c1 k , N c1 k , M c1 k , ε 2 k , and δ 2 k , which satisfy (52) and (53), and minimize the objective function Calculate the minimized value Tr c k and take Tr c k = |Tr c k−1 − Tr c k |.
Step 3: If Tr c < ∈ c , terminate the process. Otherwise, proceed to the next step.
Step 4: If k c > k * c , terminate the process. Otherwise, proceed to step 2.

Simulation results
We consider a system with the following fast timevarying delay and uncertain parameters [22]:  By solving the conditions in Theorem 4.2, we find that the minimized H ∞ performance indicator is 0.4716, and the minimum objective function is 9.0507. We can obtain R c1 and R c2 by the singular value decomposition of I − Y 1 Y 2 . Accordingly, the relevant parameters of the fault-tolerant controller are as follows:   In the simulation, the input to the system is r(t) = 5, and we assume that the actuator has the following fault f 1 (t): 6,10] . Figure 1 shows the simulation results of fault f 1 (t) for = 30. Figure 1(a) shows the simulation of the fault observer. Figure 1(b) shows the fault estimation error. Figure 1(c,d) show the output curves under normal and fault-tolerance controls of f 1 (t), respectively. Figure 1(e,f) show the corresponding simulation results of CAFE.

Remark 5.1:
The higher the value of the adaptive learning factor , the better. However, if it is too high, it is difficult to implement in practice. Thus, we select it in the same order of magnitude as given elsewhere [23].
In Figure 1(a), the blue curve represents the actual value of the fault, and the red one represents the   Figure 1(a,b) show that the fault observer designed in this study can estimate the fault f 1 (t) with low error. The fault f 1 (t) occurs during 3-10 s. Figure 1(c) shows that the system will be in an abnormal state during this time under normal control. Figure 1(d) shows that the fault-tolerance controller designed in this study effectively accommodates the fault f 1 (t). Figure 1(e,f) show that the proposed algorithm outperforms the CAFE.
We assume that the actuator exhibits the following periodic time-varying fault f 2 (t): [5,30] . Figure 2 shows the simulation results of fault f 2 (t) for = 30. Figure 2(a) shows the simulation of the fault observer. Figure 2(b) shows the error in estimating the fault. Figure 2(c,d) show the output curves under normal and fault-tolerance controls of the fault f 2 (t), respectively. Figure 2(e,f) show the corresponding simulation result of CAFE. Figure 2(a,b) show that the fault observer designed in this study can accurately estimate the periodic time-varying fault f 2 (t). Furthermore, by comparing Figure 2(c,d), we find that the fault-tolerance controller designed in this study effectively accommodates the periodic time-varying fault f 2 (t).
Compared with the CAFE algorithm, the improved FAFE algorithm has better performance in fault estimation and compensation for the periodic time-varying fault.
We assume that the actuator has the following fault f 3 (t): [20,40] .    Figure 3(a) shows the fault observer simulation. Figure 3(b) shows the error in estimating the fault. Figure 3(c,d) show the output curves under normal and fault-tolerance controls of fault f 3 (t), respectively. Figure 3(e,f) show the corresponding simulation result of CAFE. Figure 3(a,b) show that the fault observer designed in this study can estimate the fault f 3 (t) with low error. When the fault f 3 (t) occurs during 10-40 s, the system responds abnormally under normal control. From Figure 3(d), we find that the proposed fault-tolerance controller effectively accommodates the fault f 3 (t). Figure 3(e,f) show that the proposed algorithm has better performance in fault estimation and compensation for the fault f 3 (t).

Real example
In this section, we show a real example of an unmanned helicopter experimental platform. Figure 4 shows the structure of the experimental platform. A control board (PC104) is used to receive the control and fault signals collected by the sensors and send them to the helicopter. A wireless network is used to realize the communication between the ground control station and the helicopter system.
The ground control station can receive the status signals of the helicopter through the PC104 and can send the control signals to the helicopter system. Moreover, the ground control station can send the fault signals to simulate the actuator fault.
The linearization equation for the helicopter at the equilibrium point is Here x 6 (t)) T , where x 1 , x 2 , and x 3 denote the lifting, convergence, and path angles, respectively, and x 4 , x 5 , and x 6 denote their angular velocities, respectively. The time-varying parameters are F(t) = 0.05 sin t, d(t) = 0.2 + 0.1 sin 9t, and w(t) = 0.1 sin 90t. u(t) = (u 1 (t) u 2 (t)) T , where u 1 and u 2 denote the voltages of the front and rear thrusters, respectively, and y(t) = (x 1 (t) x 2 (t) x 3 (t)) T .
The detailed parameters of the system are as follows:  The parameter matrices of the controller are as follows:  Figure 5(b,c) show the output curves under normal and faulttolerance controls, respectively.
The fault occurs in 20-40 s. From Figure 5(a), we can confirm that the observer designed in this study accurately estimates the fault. By comparing Figure 5(b,c), we find that the helicopter does not work properly in the presence of the fault under normal control and that the proposed fault-tolerance controller maintains the helicopter in a normal state.

Conclusions
This paper reports on the fault estimation and faulttolerant control method for a system with fast timevarying delay and time-varying parameters. The FAFE algorithm is improved, and relaxation matrices are introduced to design a fault observer. The role of the improved FAFE algorithm is to reduce the constraints, and the introduction of the relaxation matrices helps estimate faults for a system with a fast time-varying delay. Based on online fault estimates, an output feedback controller is designed to accommodate the faults. The relaxation matrices help realize a fault-tolerant control for a system with a fast time-varying delay. The solutions to the observer and controller are realized using an iterative algorithm derived in the study.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by the National Science Foundation of China [grant number 61273190].