Experimental Study of Sensor Fault-Tolerant Control for an Electro-Hydraulic Actuator Based on a Robust Nonlinear Observer

: Electro-hydraulic actuators (EHAs) have been widely used in modern industries. However, sensor faults and actuator faults in EHA systems can arise due to aging during operation, making the system unstable and unsafe. To solve these issues, fault-tolerant control (FTC) techniques for EHA systems have been studied intensively. In this paper, an FTC is proposed and developed for the mini motion package (MMP) EHA system. First, a mathematical model of the MMP system is formulated and improved to provide position tracking control using a well-known proportional-integral-derivative (PID) controller. Second, an unknown input observer (UIO) reconstruction is performed to estimate the states, disturbances, and sensor faults so that an asymptotically stable control error can be obtained by a linear matrix inequality (LMI) optimization algorithm through Lyapunov’s stability condition. Third, the FTC designed for the nonlinear discrete-time system is formed from fault compensation based on a residual logic signal to implement the fault compensation process and ensure stability and tracking performance with respect to minimizing impacts of disturbances and sensor faults. Here, residual is deﬁned by the di ﬀ erence between state response and state estimation. Finally, numerical simulations and experiments of the MMP system are presented to illustrate the e ﬃ ciency of the proposed FTC technique.


Introduction
Electro-hydraulic actuators have been applied extensively and have become commonplace for controlling the position of systems in modern industries. There have been many applications in a variety of fields, such as position control of cutting tools, aircraft wing control, and control of aircraft landing gears. This is because electro-hydraulic actuators provide stability, precise operation and control of nonlinear systems under heavy load conditions. However, accurate position control is an extremely difficult and challenging issue for the operator when faults appear in the system. In the past few decades, EHA systems have been applied to control the force or the piston position using linearized control techniques [1] and nonlinear adaptive control techniques based on back-stepping control [2,3] or sliding mode control [4]. Nonlinear adaptive control problems for the piston-cylinder position have been addressed. Nevertheless, control will become more difficult if faults or failures occur in EHA systems. These faults can arise from the components of the sensor due to aging or broken cables, or in the components of the actuator such as failures of the electrical machinery in the pump, or leakage, or friction in the pump and the cylinder. Dirty oil also causes disturbances in the

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The mathematical modeling of the MMP system which is compared with [3] to apply to the UIO reconstruction.

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Constructing an inequality under matrix is performed to determine observer gain by LMI optimization algorithm. • A procedure for evaluating the tracking performance of the MMP system under disturbances and sensor faults is proposed. Based on this evaluation process, the performance level achieved during simulations and experiments can be easily obtained.

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Our major contribution in this paper shows that the proposed SFTC technique is successfully applied to reduce minimum impacts of faults and disturbances aimed at stability and safety insurance for the system. This paper is organized as follows. In Section 2, the mathematical model of the MMP EHA system with disturbances and sensor faults is presented. In Section 3, a UIO reconstruction formulated for the nonlinear discrete-time system is performed to satisfy the discrete-time Lipschitz condition and ensure asymptotic stability of the state observer under the LMI optimization algorithm. In Section 4, an SFTC technique is proposed for the fault compensation process based on the sensor fault estimation and the residual signals. In Section 5, numerical simulations and experimental results are presented, and these results are evaluated. Section 6 contains a discussion of the results. Finally, conclusions are presented in Section 7.

Modeling of the EHA System
Considering the dynamics equations of the MMP EHA system, the model in Figure 1 can be derived using Newton's Second Law to describe the object M p [3]: Here, m p is the equivalent mass, while x p , .
x p , and ..
x p are the position, velocity, and acceleration, respectively; A 1 and A 2 are the area in the two chambers; F sp , F f rc , B v , and d are the external load force of the spring, friction force, viscous damping coefficient, and bounded UID, respectively; and P 1 and P 2 are the pressures in the two chambers.
The spring force F sp can be represented as where K sp is the spring stiffness. The F f rc friction force can be represented as [29] where F brk and F C are the breakaway friction and the Coulomb friction, respectively; and v p and v st are the velocity and Stribek velocity threshold, respectively.

Modeling of the EHA System
Considering the dynamics equations of the MMP EHA system, the model in Figure 1 can be derived using Newton's Second Law to describe the object Mp [3]: Here, p m is the equivalent mass, while p x , p x  , and p x  are the position, velocity, and acceleration, respectively; 1 A and 2 A are the area in the two chambers; sp F , frc F , v B , and d are the external load force of the spring, friction force, viscous damping coefficient, and bounded UID, respectively; and 1 P and 2 P are the pressures in the two chambers.
The spring force sp F can be represented as where sp K is the spring stiffness.
The frc F friction force can be represented as [29] ( ) 2 2 t a n h where brk F and C F are the breakaway friction and the Coulomb friction, respectively; and p v and st v are the velocity and Stribek velocity threshold, respectively. The hydraulic continuity equations for the EHA system can be described as [3] ( ) ( ) The hydraulic continuity equations for the EHA system can be described as [3] .
. where β e and Q i are the effective bulk modulus in each chamber and the internal leakage flow rate of the cylinder, respectively; and V 01 and V 02 are the initial control volumes of the first and the second chamber, respectively [3].
Here, Q pump = D p ω. Q 1v and Q 2v are the flow rate through the pilot-operated check valve on the left and on the right, respectively. Q 3v , Q 4v and Q pump are the flow rates through the pressure relief valve on the left, on the right, and the pump flow rate, respectively. D p and ω are the displacement and the speed of the servo pump, as shown in Figure 1.
Based on the system's dynamics equations shown in Equations (1)-(6), the dynamic equation for the system can be represented by a state vector The system of equations expressed in (7) can be represented as where Equation (8) shows that the matrix A is a constant matrix. The system of equations shown above can be rewritten as a nonlinear discrete-time state space model as shown below: where T s is a sampling time Obviously, system states are performed from changing the speed of the bi-directional pump, which is controlled by a DC motor. The system is controlled via the input speed ω of the DC motor so that the output position x 1k (or the measurement output y k ) tracks as closely as possible to a reference Energies 2019, 12, 4337 6 of 22 position y rk . Considering the influences and reducing the impacts of sensor fault on the measurement output signal are the objective of this paper, and these will be described in the following section.

UIO for a Nonlinear Discrete-Time System
A nonlinear discrete-time system subjected to an unknown input disturbance can be represented as follows: where x k ∈ R n is the state vector, y k ∈ R p is the output vector, d k ∈ R r is the unknown input or disturbance vector, and s k ∈ R q is the sensor fault vector.A k , B k , C k , D d , and F s are known constant matrices with suitable dimensions.
If it is assumed that the Lipschitz condition applies to the discrete-time nonlinear function vector φ x k ,u k , then there exists a constant τ > 0 such that [8,9,11,23] where The nonlinear system (10) can be rewritten as where Based on [11,21], a UIO can be built for the nonlinear discrete-time system as shown below.
Here,x k ∈ R n ,ŷ k ∈ R p , and z k ∈ R n are the estimate of the state vector x k , the estimate of the measurement output vector, and the state vector of the observer, respectively. The observer matrices F ∈ R n×n , Γ ∈ R n×n , H ∈ R n×p , and L ∈ R n×p should be designed according to the state error vector.
The state error can be defined as From Equations (12), and (13), we can write aŝ Equation (12) can be written as: Based on Equations (16) and (17), the estimation error of Equation (16) can be written as where If the following conditions are satisfied: Without loss of generality, the matrix Γ can be defined as where φ 1 ∈ R nxp and φ 2 ∈ R pxp are arbitrary matrices. By solving the system in Equations (20) to (24), we obtain: and The inequality in Equation (11) needs to be satisfied for the nonlinear function ∆φ x k ,u k as: where τ = τI n 0 0 0 p ∈ R n From (26), the matrix Ξ can be written as follows: Lemma 1 ([9]). The necessary and sufficient conditions for the existence of the UIO in (13) for the system in (12) are as follows. (c) A k − zI n D d C k 0 = n + p ∀z with |z| > 1 Lemma 2 ([12]). For the following equation the eigenvalues of the matrix Φ ∈ R n×n belong to the circular region D(α, ρ) with center α + j0 and radius ρ if and only if there exists a symmetric positive definite matrix P ∈ R n×n such that the following condition hold Theorem 1. For system (12), there exists a robust UIO in the form of (13) such that the output error satisfies e yk ≤ µ d k in a prescribed circular region D(α, r) if there exists a positive-definite symmetric matrix P ∈ R n×n , matrix Q ∈ R n×p , and positive scalars µ, and ε such that the following inequalities hold: and where Q = PL. Then, the state observer (13) is asymptotically stable.
Proof of Equation (26). Consider a Lyapunov function V k = e T k Pe k . The difference between two adjacent steps of the Lyapunov function ∆V k = V k+1 − V k is calculated as where Moreover, Equation (27) can be expressed as: Combining Equations (32) and (33) leads to: where To satisfy the measurement error condition e yk ≤ µ d k of the output, the matrix Θ can be represented as Using Equations (32) and (35), the matrix T n can be written as Finally, an inequality of matrix T n can be represented as Applying the Schur Complement Lemma [33] to Equation (38) for Π n < 0, we have: Similarly, Equation (30) is satisfied by applying the Schur Complement Lemma [32] to Equation (39) for Π n < 0.

Proof of Equation (27). Using Lemma 2, inequality (31) is met by applying Equation (18).
In summary, the design procedure for the sensor fault estimator consists of the following steps: Step 1: Build the augmented system (12) for the nonlinear discrete-time system (10). Step 2: Determine the matrices Q, P, F, and L = P −1 Q by solving the LMI matrix inequalities (30) and (31).
Step 3: Obtain the state estimate and the fault estimate asx k = I n o n×p x k , andŝ k = C sxk , respectively, where C s = o r×n I p .

Remark 1.
This UIO reconstruction only applies to observable systems. For systems that are not possible to observe, the reduced-order observer construction shown in [33,34] should be applied.

Remark 2.
This paper uses the sensor fault reconstruction in [8,26] to design a discrete-time nonlinear observer system for the state observer that is asymptotically stable. Here, constructing and proving Theorem 1 to apply the LMI optimization algorithm was made simpler [8] by using Lyapunov's stability under matrix equations. With this sensor fault reconstruction, it is possible to estimate the faults using the UIO model, which has advantages such as directly estimating the sensor fault on the UIO without decoupling the disturbance matrix, as shown in [9].

Remark 3.
Constructing a robust UIO reconstruction method using LMI optimization algorithm was utilized by a function H ∞ as performed in [8]. However, in this paper, by applying Lyapunov stability theory under the formulated matrix equations, a robust UIO reconstruction method for using the LMI optimization algorithm was demonstrated that was simpler than in [8], and in which the state observer is asymptotically stable.

Fault Diagnosis-Based General Residual from the Sensor Fault Signal
The general residual due to a sensor fault is defined as the difference between the response signal and the feedback estimation signal from the UIO. The residual vector is calculated as [11] The FD method consists of two main tasks: fault detection and fault isolation. The fault detection process involves determining whether a fault has occurred based on information from the RS, which means that r k = 0 if s k = 0 without fault, and r k 0 if s k 0 with fault [11]. The fault isolation process is executed to make a binary decision signal based on the fault detection process through executing logic that is constructed via the RS and the threshold value k. The binary decision signal is 0 when |r k | ≤ k, and conversely, this signal is 1 when |r k | > k.

Sensor FTC Compensation
Sensor fault output from the measurements influences the closed-loop behavior and corrupts the state estimation.
The FTC designed to diagnose the position SFs of the MMP system is shown in Figure 2. The PID main controller performs conventional closed-loop position tracking control such that the measurement signal is employed, as in [5] y the state estimation. The FTC designed to diagnose the position SFs of the MMP system is shown in Figure 2. The PID main controller performs conventional closed-loop position tracking control such that the measurement signal is employed, as in [5]  Fault compensation is performed by a logical process wherein fault magnitude is received from the estimated fault signal based on the residual signal to make a control decision based on the binary signal. The fault compensation process will not be executed if the binary decision signal has a value equal to 0; otherwise, if the binary decision signal is equal to 1, the fault compensation process will be executed using the threshold k. [11].

PID Control for MMP System
The control error is one of the most important factors to be chosen for evaluating the PID controller performance. The control error is defined as The maximum position error max k e of the position control error k e shown in Figure 3 can be described as Fault compensation is performed by a logical process wherein fault magnitude is received from the estimated fault signal based on the residual signal to make a control decision based on the binary signal. The fault compensation process will not be executed if the binary decision signal has a value equal to 0; otherwise, if the binary decision signal is equal to 1, the fault compensation process will be executed using the threshold k. [11].

PID Control for MMP System
The control error is one of the most important factors to be chosen for evaluating the PID controller performance. The control error is defined as The maximum position error e kmax of the position control error e k shown in Figure 3 can be described as where y rmax and y ckmax are the maximum values of the reference and response signals in the time period (t 1 , t 2 ).  The maximum control error of the system during the time (t1, t2) is obtained as The control error performance k η of the system is defined as follows based on the maximum control error max ck e during the time period (t1, t2): where max s and min s are the maximum and minimum sensor faults, respectively, during the time To explicitly evaluate the effect of a fault, the sensor fault error sfk e is defined as The minimum position error e kmin of the position control error e k is computed for the time period (t 1 , t 2 ) in Figure 3 as where t 1 and t 2 are the begin time and the end time of the time period used to calculate the position control error. During the time period (t 1 , t 2 ), y rmin and y ckmin are the minimum reference signal and the minimum response signal, respectively, as shown in Figure 3.
The maximum control error of the system during the time (t 1 , t 2 ) is obtained as The control error performance η k of the system is defined as follows based on the maximum control error e ckmax during the time period (t 1 , t 2 ): where s max and s min are the maximum and minimum sensor faults, respectively, during the time period (t 1 , t 2 ). To explicitly evaluate the effect of a fault, the sensor fault error e s f k is defined as where e f k and e 0ck are the position control error in the case of sensor fault and the position control error for the case without sensor fault, respectively. The maximum sensor fault error e s f kmax of the sensor fault error e s f k is computed as Similarly, the control error performance ξ k of the system during the time period (t 1 , t 2 ) is calculated in terms of the sensor fault error e s f kmax as

Parameters of the MMP System
Determining the parameters of the MMP system is complex and difficult when the parameters of the theory system are similar to the parameters of the experimental system. Therefore, we determined the main parameters of the MMP real system to specify the parameters of the real system. These parameters are defined as shown in Table 1 based on the values of a real system, namely, a KYB MMP with a φ40-φ20-250 mm cylinder and a 24V DC motor. After the fundamental parameters were confirmed by normal calculations, the other parameters were determined by using the parameter estimation method in Matlab. Finally, the main parameters of the MMP real system are listed in Table 1. The parameters of the MMP model in Equation (9) were determined to be B k = 0 0 0 0 T ; C k = 1 0 0 0 ; F s = 1;T s = 0.001 where a sampling time T s = 0.001(s) is applied to all of the simulation processes.
The reference signal is given as We assume that the sensor fault s(t) is given as Based on design experience, we chose positive scalars τ = 30, r = 0.05, α = 0.3, ε = 0.2, and µ = 0.1. We can solve for the matrices P, Q, L and F using the LMI defined in (30), and (31) if the solution is feasible. We obtained the following results:

Simulation Results
A model of the sensor FTC for the MMP system was constructed in Simulink to perform numerical simulations using input and output signals where we assumed that the sensor is faulty, as described by Equation (52). Simulation results of the MMP system were compared between a system without UID (d k = 0) and a system with UID d k = 0.0025rand(2, t) (mm), which are shown in Figures 5-8. Here, the PID controller parameters (K P = 1.63193, K I = 2.45106, and K D = 0.01068) are used in all simulations with a set point of the initial state in the system x 0k = [0, 0, 0] T .
The simulation results in Figure 4a show the effects of the fault on the PID position response signal, as well as the fault estimate from 11s to 12.008 s, and from 22.75 s to 30 s for the case without signal, as well as the fault estimate from 11s to 12.008 s, and from 22.75 s to 30 s for the case without FTC. We can also see that the PID position response signal follows the reference signal at locations where there is no fault impact.
In contrast, the PID position response signal does not follow the reference signal at locations where there is a fault change. The PID position response signal shown in Figure 4b is also affected by an unknown disturbance of the actuator. Additionally, the sensor fault and its estimate are similar to one another, as shown in Figure 5a, which means that the fault estimator of the UIO works well. A big difference between the sensor fault and the estimated fault is observed in Figure 5b under the impact of the disturbance k d .   In contrast, the PID position response signal does not follow the reference signal at locations where there is a fault change. The PID position response signal shown in Figure 4b is also affected by an unknown disturbance of the actuator. Additionally, the sensor fault and its estimate are similar to one another, as shown in Figure 5a, which means that the fault estimator of the UIO works well. A big difference between the sensor fault and the estimated fault is observed in Figure 5b under the impact of the disturbance d k .
In contrast, we see in Figure 6a,b that the PID position response signal and the reference signal are approximately the same when the SFTC technique is applied to implement the fault compensation process. The system still operates normally, even when the fault and disturbance are compounded. Figure 6 shows that the response signal nearly replicates the desired signal, and this issue also proves that the PID position response signal with the disturbance and without disturbance is approximately the same, although the system is impacted by the disturbance.  In contrast, we see in Figure 6a,b that the PID position response signal and the reference signal are approximately the same when the SFTC technique is applied to implement the fault compensation process. The system still operates normally, even when the fault and disturbance are compounded. Figure 6 shows that the response signal nearly replicates the desired signal, and this issue also proves that the PID position response signal with the disturbance and without disturbance is approximately the same, although the system is impacted by the disturbance.
In contrast, we see in Figure 6a,b that the PID position response signal and the reference signal are approximately the same when the SFTC technique is applied to implement the fault compensation process. The system still operates normally, even when the fault and disturbance are compounded. Figure 6 shows that the response signal nearly replicates the desired signal, and this issue also proves that the PID position response signal with the disturbance and without disturbance is approximately the same, although the system is impacted by the disturbance.
The simulation results in Figure 7a,b shows that the sensor fault and disturbance of the actuator decreased to approximately zero after the SFTC compensation procedure. This demonstrates that the SFTC technique and the sensor fault estimator work well for the MMP system. Disturbances coming from the actuator can be cancelled after each fault compensation, so we can use only a sensor FTC for fault compensation without needing an additional actuator FTC for fault compensation. The simulation results in Figure 7a,b shows that the sensor fault and disturbance of the actuator decreased to approximately zero after the SFTC compensation procedure. This demonstrates that the SFTC technique and the sensor fault estimator work well for the MMP system. Disturbances coming from the actuator can be cancelled after each fault compensation, so we can use only a sensor FTC for fault compensation without needing an additional actuator FTC for fault compensation.
The results of the PID controller performance are shown in  Table 2. The disturbance showed a pretty big influence in the case without The results of the PID controller performance are shown in Table 2. The results were used to evaluate the effectiveness of the SFTC technique of the PID controller based on data related to the maximum control error e ckmax and the control error performance η k . The evaluation was conducted by numerical simulation with and without a disturbance and corresponding cases with SFTC and without SFTC, respectively. We can clearly see that the maximum control error e ckmax for the case without a disturbance strongly decreased when SFTC was applied, which decreased from 1.424 to 0.175 from 0.1 s to 15 s, 0.431 to 0.041 during 15 s to 25 s, and 0.388 to 0.139 from 25 s to 35 s. The control error performance η k of the PID controller was high when the SFTC was used, as it showed the average error performance e ckmax (η k ) increased from 46.52% to 92.67% for the cases without and with SFTC, respectively. Similar results also show the maximum control error e ckmax for the cases when the disturbance strongly dropped from 1.450 to 0.185, 0.608 to 0.062, and 0.480 to 0.062, corresponding to times of 0.1 s to 15 s, 15 s to 25 s, and 25 s to 35 s, respectively, while the average error performance e ckmax (η k ) significantly increased from 38.77% to 92.58% when SFTC was employed, as shown in Table 2. The disturbance showed a pretty big influence in the case without SFTC, with an average error performance e ckmax from 46.52% to 38.77% which is compared to average error performances e ckmax from 92.67% to 92.58% for the case using SFTC. Overall, most of the impacts of the disturbance and sensor fault were reduced under SFTC technology. Moreover, the results in Figure 8 clearly show the influence of sensor faults on the control error signal e ck in the cases without disturbances and without SFTC. In contrast, when we utilize the SFTC technique to implement a simulation in a case without a disturbance, the effect of the sensor fault is so small that the control error signal e ck in the case with a sensor fault is approximately the control error signal e ck in the case without a sensor fault, which is shown in Figure 8b. fault is so small that the control error signal ck e in the case with a sensor fault is approximately the control error signal ck e in the case without a sensor fault, which is shown in Figure 8b. We can also see the influence of the sensor fault on the control error signal sfk e in the cases without disturbance and without SFTC more clearly in Figure 9a as well as in Figure 9b for the case without disturbance and with SFTC. We can also see the influence of the sensor fault on the control error signal e s f k in the cases without disturbance and without SFTC more clearly in Figure 9a as well as in Figure 9b for the case without disturbance and with SFTC. We can also see the influence of the sensor fault on the control error signal sfk e in the cases without disturbance and without SFTC more clearly in Figure 9a as well as in Figure 9b for the case without disturbance and with SFTC. Similarly, the results of the control error signal ck e show a pretty big influence under disturbance conditions for the case without a SFTC as shown in Figure 10a. Although the system does have a disturbance and sensor fault, the system successfully removed faults when the SFTC was performed as shown in Figure 10b. Similarly, the results of the control error signal e ck show a pretty big influence under disturbance conditions for the case without a SFTC as shown in Figure 10a. Although the system does have a disturbance and sensor fault, the system successfully removed faults when the SFTC was performed as shown in Figure 10b. The influence of sensor fault on the control error signal e s f k in Figure 11a,b showed a result similar to that shown in Figure 9a    Similarly, an evaluation of the SFTC technique based on the maximum sensor fault error e s f kmax and the error performance ξ k is shown in Table 3. The average performance ξ k for the sensor fault error e s f kmax is 44.53% without SFTC and 94.91% with SFTC for the case without disturbance d k . The lowest performance interval (0.61%) is from 0.1 s to 15 s. The average performance ξ k is reduced by increasing the disturbance d k . The system achieves a performance of only 36.74% without SFTC. However, the system achieves a performance of 93.78% when SFTC technology is applied for the disturbance d k case. High average performance ξ k was obtained by applying the SFTC technique to the MMP system in the cases without a disturbance (94.91%) and with a disturbance (93.78%). These results demonstrate that the SFTC technique works well.The results of the control error evaluation shown in Tables 2 and 3 can demonstrate the superiority of the proposed FTC method compared to the traditional PID method. This superiority can be shown in average error performance e ckmax which was increased by 46.15% (from 46.52% to 92.67%) of without disturbance case and by 53.81% (from 38.77% to 92.58%) of with disturbance case that shown in Table 2. In addition, this evaluation can also show in Table 3 by average error performance e s f kmax which was increased by 50.38% (from 44.53% to 94.91%) of without disturbance case and by 57.04% (from 36.74% to 93.78%) of with disturbance case.

Diagram of the Testbed for the MMP System
In this section, the configuration of the EHA testbed setup is presented, where the EHA system includes a hydraulic cylinder, which is adjusted directly by operation of the bidirectional pump, as shown in Figure 12a. This means that changes in the cylinder speed depend on electric power shifts of the motor. Figure 12, the signal will be transferred to amplifier DSI-301B. Then, this signal will be transferred to the computer through a PCI card (NI 6251). Conversely, if the difference between the desired signal and the feedback signal from the LVDT sensor is zero, then the computer will send the analogue signal to the motor driver via PCI card NI 6251. As a result, the system operates under closed-loop control. The controller for the testbed is implemented on a personal computer (core 2 Duo 2.2 GHz) using Matlab Simulink version 2013b (32 bits) and the Real-time Windows Target Toolbox.  Therefore, to control the piston position of the cylinder, the relationship between electrical energy changes of the motor and piston position changes needs to be established. To do this, a linear variable differential transformer (LVDT) sensor is used to obtain data from the piston in the form of a digital signal. When the LVDT sensor TCLA-30A touches the surface of object Mp, as shown in Figure 12, the signal will be transferred to amplifier DSI-301B. Then, this signal will be transferred to the computer through a PCI card (NI 6251). Conversely, if the difference between the desired signal and the feedback signal from the LVDT sensor is zero, then the computer will send the analogue signal to the motor driver via PCI card NI 6251. As a result, the system operates under closed-loop control. The controller for the testbed is implemented on a personal computer (core 2 Duo 2.2 GHz) using Matlab Simulink version 2013b (32 bits) and the Real-time Windows Target Toolbox.

Results
SFTC experimental scheme of the MMP real system is performed using the setup shown in the diagram in Figure 12a,b. This technique is proposed to improve the performance of the PID controller for the MMP real system. The principle of this process is as follows. The feedback signals from the LVDT sensor signal where these sensor signals are filtered by a lowpass filter in combination with the mean value to smooth out the noise, which makes control easier. The experimental results were performed under a predetermined fault condition in (52) and the reference input in (51) for the MMP system. The results are shown in Table 4. The value of the error performance η k for the case without SFTC was −5.69%, which indicates low performance. A negative value for the performance implies that the real system is affected by several faults, disturbances, and noises that are unknown. The experimental results are shown in Figure 13. The PID position response signal is significantly improved when using the SFTC technique, as shown in Figure 14a,b. The effectiveness of this technique was evaluated using error performance η k , as shown in Table 4. The differences between the average  Table 2 and 4 for the cases with and without SFTC, showing the effectiveness of the technique. Specifically, the control error performance η k is 59.82% for the case in which the SFTC technique was applied, which is shown in Figure 14a,b. The experimental results are shown in Figure 13. The PID position response signal is significantly improved when using the SFTC technique, as shown in Figure 14a,b. The effectiveness of this technique was evaluated using error performance k η , as shown in Table 4. The differences between the average error performances are listed in Table 2 and 4 for the cases with and without SFTC, showing the effectiveness of the technique. Specifically, the control error performance k η is 59.82% for the case in which the SFTC technique was applied, which is shown in Figure 14 a,b.

Discussion
In this paper, a mathematical model was developed based on [3]. In addition, a UIO based on the system reconstruction approach is applied to perform state and fault estimation. These techniques were studied and improved upon based on the proofs in previous papers [11,21]. Asymptotic stability of the state observer is guaranteed using the LMI optimization algorithm. Here, the numerical simulation process was conducted successfully on the MMP model. In particular, the SFTC technique was implemented effectively in the simulations and experiments. The experimental results showed that the position response in the case of SFTC was better than without SFTC. This performance of the applied SFTC resulted in an increase of 65.51% (from −5.69% to 59.82%) as shown in Table 4.

Conclusions
The SFTC technique was performed successfully in simulations and experiments for a nonlinear EHA system. The SFTC performance in simulations was approximately 93.78% in the case with a disturbance and 94.91% in the case without a disturbance. The fault and disturbance were nearly cancelled out when the SFTC technique was applied. In experiments using a real MMP system, the average performance was 59.82% when the SFTC technique was used. With the SFTC technique, the average performance increased by 65.51% (from −5.69% to 59.82%) and the control error improved,

Discussion
In this paper, a mathematical model was developed based on [3]. In addition, a UIO based on the system reconstruction approach is applied to perform state and fault estimation. These techniques were studied and improved upon based on the proofs in previous papers [11,21]. Asymptotic stability of the state observer is guaranteed using the LMI optimization algorithm. Here, the numerical simulation process was conducted successfully on the MMP model. In particular, the SFTC technique was implemented effectively in the simulations and experiments. The experimental results showed that the position response in the case of SFTC was better than without SFTC. This performance of the applied SFTC resulted in an increase of 65.51% (from −5.69% to 59.82%) as shown in Table 4.

Conclusions
The SFTC technique was performed successfully in simulations and experiments for a nonlinear EHA system. The SFTC performance in simulations was approximately 93.78% in the case with a disturbance and 94.91% in the case without a disturbance. The fault and disturbance were nearly cancelled out when the SFTC technique was applied. In experiments using a real MMP system, the average performance was 59.82% when the SFTC technique was used. With the SFTC technique, the average performance increased by 65.51% (from −5.69% to 59.82%) and the control error improved, as shown in Table 4 and Figure 14a,b. Thus, the system can work well even when faults occur.
Author Contributions: The mathematical modelling of the MMP system which is constructed to apply to the UIO reconstruction. An inequality under matrix is performed to determine observer gain by LMI optimization algorithm and a procedure for evaluating the tracking performance of the MMP system under disturbances and sensor faults is proposed. Constructing the evaluation process of the error performance during simulations and experiments was performed to determine the level achieved. T.V.N. and C.H.'s major contribution in this paper is that the proposed SFTC technique is successfully applied to reduce minimum impacts of faults and disturbances aimed at stability and safety insurance for the system.