2.1. Variable-structure HFV model

The longitudinal variable-structure HFV model adds variable-structure aerodynamic areas to the classic model. This is described by differential equations with velocity (V), flight path angle (γ), altitude (h), attack angle (α), and pitch rate (q). The nonlinear model is expressed as [30]: (1) where T, D_dr, L, m, ρ, M_yy, and I_yy denote the thrust, drag, lift, aircraft mass, gravitational constant, pitch moment, and moment of inertia, respectively; R represents the earth’s radius; r = R+h. The forces and moments are expressed as (2) where S is the nominal aerodynamic area; S_s is the variable-structure aerodynamic area; δ_e denotes the elevator deflection control signals; C_T, C_D, and C_L are the coefficients of throttle, drag, and lift, respectively; C_M(·) is the pitch moment coefficient with respect to (·). More parameter details are presented in [5]; C_T contains another control signal, δ_T.

In an actual flight, multi-sensor faults may occur when the HFV encounters a harsh environment or its structure changes. However, the HFV longitudinal model (1) has two problems. First, the model is nonlinear and strongly coupled. Multi-sensor fault modeling based on this mode is difficult, and the corresponding high-quality FDI and active FTC are unlikely to be realized. Second, the model parameters are uncertain, as expressed by: (3) where (·₀) are the nominal values, Δ(·) are the parameter uncertainties. Solving these two problems requires type-II fuzzy method, which can linearize the nonlinear model, its upper and lower membership functions can eliminate the parameter uncertainties and ensure the model standardization [31]. The controller designed for the simplified model will not need to consider the above two problems.

2.2. Fuzzy modeling with multisensor faults

As mentioned earlier, the type-II fuzzy technique is utilized in this study to simplify system (1), which is conducive to the design of diagnosis and compensation schemes.

First, the longitudinal model described in (1) can be written in an affine nonlinear form: (4) where x(t) = [x₁(t), x₂(t), x₃(t), x₄(t), x₅(t)]^T = [V, γ, h, α, q]^T∈R⁵ is the state vector and u(t) = [δ_e, δ_T]^T∈R² is the control input vector. Given that the HFV has five sensors (i.e., velocity, flight-path angle, altitude, attack angle, and pitch rate sensors) to measure the real-time flight states, the output vector can be derived as y(t) = [V, γ, h, α, q]^T∈R⁵.

The type-II fuzzy model of variable structure HFV is then expressed as:

Rule i: IF f₁(x₁(t)) is ξ_i1… f₅(x₅(t)) is ξ_i5, THEN (5) where A_i, A_s, C_i∈R^5×5, B_i, B_s∈R^5×2, A_i, B_i, C_i are the linear modal parameter matrices, A_s, B_s are the active variable structure parameters obtained according to S_s in (1) and (2), C_i represents the nature of output measurement and is not affected by the aerodynamic area structure, ξ_ij(i = 1,…, w; j = 1,…, 5) is interval type-II fuzzy set. The triggering strength of each fuzzy rule is expressed as (6) where ω_i,up(x(t))≥ω_i,low(x(t))≥0. ω_i,up(x(t)) is the upper membership, ω_i,low(x(t)) is the lower membership. The upper and lower membership functions are represented by and , respectively. The global type-II fuzzy model is expressed as follows: (7) where (8) (9) and ν_i(x(t)) is the weight coefficient satisfies 0≤ν_i(x(t)) ≤1.

Although the type-II fuzzy technique has simplified the structure and parameters of the nonlinear dynamics of (1), an approximation error between (1) and (5) inevitably exists. Therefore, some nonlinear terms (g_i(x, t)) describing this error are added to improve the model accuracy. In harsh environments, more than one sensor fails at a time. In this study, we only consider two sensors that fail and have bias faults.

In the harsh environment, it is possible that more than one sensor fails at a time. In this study, we only consider two sensors that fail and have bias faults.

Assume that a bias fault in sensor k and another bias fault in sensor s exist (1≤k<s≤5). Moreover, consider the approximation error and FLAD. Then, the variable-structure HFV fuzzy model is described as: (10) where g_i(x, t) is a smooth vector field on R⁵; d(t)∈R¹, f(t)∈R², and ω(t)∈R¹ denote the input FLAD, sensor faults, and output noise, respectively; F∈R^5×2 (F_k1 = 1, F_s2 = 1, and other entries of F are equal to zero) is an unknown matrix; D∈R⁵ and F_ω∈R⁵ are two constant matrices.

Assumption 1. Pairs (A_i, A_s, B_i, B_s) is completely controllable. Pairs (A_i, A_s, C_i) is observable.

Assumption 2. ||d(t)||≤d₀, , d₀, d₁, g₀, and g₁ are positive scalars.

Assumption 3. The nonlinear term g_i(x, t) can satisfy: (11) where μ_i is the Lipschitz constant.

Remark 1. To enable the designed algorithm to affect the system, we assume that the system can be controlled by the input; to be able to use detection errors to design algorithm and observers, we assume that the system can be observed. The disturbance and nonlinear errors need to be assumed to be bounded, and their respective derivatives are also bounded. The boundedness assumption is easy to understand. The amplitude of any disturbance cannot be infinite, and a large enough amplitude will directly destroy the system. It is unrealistic to study such disturbance. It is necessary to assume that the nonlinear function satisfies the Lipschitz condition, that is, the inequality (11), adjust μ_i so that the right side of the inequality is less than zero. Non-linear functions that do not satisfy Lipschitz condition are usually discontinuous, we do not consider the discrete systems.

Lemma 1 [28]. For any positive scalar μ and real matrices X, Y with appropriate dimensions, (12) holds: (12)

2.3. Nominal controller design

A controller is designed to ensure system stability and track the commands of altitude and velocity under variable fuselage conditions.

Under the nominal condition without faults and disturbance, let e_yd = y−y_d, where y_d is the desired output, and a step signal. An global fuzzy variable-structure controller is selected: (13) (14) where K_i∈R^2×5 is the output feedback matrix to be designed and B_i⁺ is the pseudo-inverse of B_i. Then, Theorem 1 is obtained.

Theorem 1. If symmetric positive definite matrices P_c, G, Γ₁ and real matrix K_i exist with appropriate dimensions, such that the following inequality holds for any i = 1, …, w: (15) (16) (17) then, the tracking error e_yd is uniformly ultimately bounded under the controller (13).

Proof: Denote (18)

The Lyapunov function is selected as follows: (19)

The following error dynamics can be deduced from (10), (13), and (18): (20)

The derivative of (19) can be written as follows: (21)

Based on Lemma 1, there exists a matrix G = G^T>0 such that (22)

Let (23) (24)

We have (25)

Let (26)

Then, (25) can be rewritten as follows: (27)

If (15) holds, then denote є_i = min(eig(−Π_i)). Thus, (28)

For i = 1,…, w, when , can be obtained, i.e., the error dynamics converge to an interval: (29)

Therefore, e_yd and e_gi are ultimately uniformly bounded. This completes the proof of Theorem 1.

The subsequent section presents the design of the diagnosis and FTC schemes to ensure the stability and tracking accuracy of the variable-structure HFV with multi-sensor faults.

3.1. Fault detection

To simplify the design of the variable-structure detection observer, the input FLAD and output noise are not considered. The observer is introduced as follows: (30) where H_i∈R^5×5 is the observer gain matrix to be designed.

The actual corresponding functions are subtracted from the observed state/output functions. The errors are denoted as follows: (31)

Theorem 2. Under the fault-free condition, (y^f(t) = y(t)), if there exists a positive scalar κ_d, symmetric positive definite matrices P_d, and Q_d, and a real matrix Y_i with appropriate dimensions, such that the following inequality holds with H_i = P_d⁻¹Y_i, then the estimate error, e_x(t), asymptotically converges to zero.

(32)

Proof: The Lyapunov function is selected as follows: (33) (34)

Referring to (34), the derivative of (33) can be written as follows: (35)

Consider the following inequality: (36)

If the inequality holds, then, can be obtained. The detection system is stable.

According to the Schur complement, if (32) holds, then lim_t→∞e_x(t) = 0.

The detection residual and detection mechanism are defined as follows: (37) (38)

The value of threshold J is important for fault detection because it is associated with missing and false reports. Next, J is designed by considering the observer error (30) and external FLAD.

From Lemma 2, we obtain (39)

Let (40)

Then (41)

By combining (33) and (41), we can derive (42): (42)

Then, (43)

Note that r(0) can be used to substitute ||C_i||||e_x(0)||. Under the fault and disturbance-free circumstances, that is, y^f(t) = y(t), d(t) = 0, and ω(t) = 0, (44)

Denote (45)

When considering d(t) and ω(t), the detection threshold is derived as follows: (46) where β>1 and ε>0 are two constants whose values are determined by the magnitudes of FLAD and noise.

The detection alarm activates when the sensor output residual exceeds the threshold, which is set in advance.

3.2. Fault isolation

Although it can be determined whether sensor faults exist in the variable-structure HFV over time, it cannot be determined which sensors have failed only through detection. Therefore, fault isolation is necessary.

Existing methods are difficult to isolate multiple faults. Accordingly, in this section, an improved isolation process is proposed. In accordance with the concept of combination, more observers are necessary to isolate multi-sensor faults from a single fault. As mentioned, it is necessary to design 10 isolation observers because two sensors are assumed to fail. Next, the improved isolation process of multi-sensor fault isolation is explained in detail.

First, the following is defined: (47) where y^r_ks, C^r_ks,i, and H^r_ks,i are the remaining parts of y, C_i, and H_i, respectively, after the deletion of their kth and sth rows.

For 1≤k<s≤5, the mth (m = 1, …, 10) variable- structure isolation observer is introduced as follows: (48)

Define (49) (50)

Based on Theorem 2, if inequality (44) holds such that H^r_ks,i = P_m⁻¹ C^r_ks,i for any i = 1, …, w, (51) (52) then the estimate error e_xm(t) asymptotically converges to zero.

Similarly, we define (53) (54) where (55)

Thus, the fault isolation mechanism is presented as: (56)

Hence, the mth observer is capable of isolating the faulty sensors, i.e., the kth and sth sensors are known to have failed, and the unknown matrix F is determined.

This study only considers the situation where two sensor faults occur instead of one or more than two sensor failures; hence, the 10 sets of thresholds are independent of each other. When faults occur in sensors k and s, only the observer with the exact matrices, y^r_ksC^r_ks,i and H^r_ks,i, can isolate the faulty sensors because the residual (r_m^r) generated by this observer is always less than J_m^r. The other residuals exceed their corresponding thresholds after the sensor faults occur.

Hence, faulty sensors are identified, and an effective estimation scheme should be designed to obtain their fault values rapidly and accurately.

3.3. Fault estimation

This section presents an adaptive augmented observer developed to accurately estimate the magnitude of these two sensor faults when FLAD exist in preparation for the subsequent fault compensation. To make the magnitude of disturbance unrestricted under the premise that the FLAD is bounded, the type-II fuzzy adaptive estimation method is employed instead of using the performance indicators. Notably, an adaptive law with an improved proportional-differential part estimates a class of FLAD; its performance is compared with the method in [26].

The augmented method is also used for the estimation. Parallel fault estimation considers the combinatorial effects of faulty sensors; this is more reasonable. The variable-structure HFV model with sensor faults, FLAD, and output noise can be rewritten in the following augmented form: (57) where (58)

Let (59) (60)

To estimate multisensor faults f(t) and the FLAD d(t), the following adaptive augmented observer is designed: (61) where ξ is a fictitious variable with respect to the augmented vector ; M_i, N_i and M_s, N_s are the fixed structure gain matrices and variable structure gain matrices to be determined later, respectively. U^* satisfy: (62) where Θ is the fast adaptive compensation function, ϑ₁ and ϑ₂ are the learning rates. Adding the derivative of FLAD estimation in Θ can make the observer more sensitive to the rate of disturbance change, thereby adapting to the speed of FLAD and compensating it, eliminating FLAD’s influence on fault estimation.

The following error vectors are defined: (63)

By considering (60)–(63), we can obtain (64)

Let (65) and (66)

Then (67)

To handle disturbance, robust performance indicators are usually adopted. If there is a high demand for estimation accuracy, the performance indicator must be small when the magnitude of the disturbance is large (For example FLAD). In this case, the LMI and indicator usually have no solution. Furthermore, obtaining the disturbance value aids in the FTC design. Accordingly, an improved adaptive law is proposed to estimate the FLAD as accurately as possible.

The FLAD estimation algorithm is shown as follows: (68)

According to definition (31), e_y can be obtained by subtracting the augmented observer output and the sensor output y^f(t). can be obtained by differentiating e_y with respect to t. Given that e_y and are both available, the estimation of FLAD (68) is possible and practical.

In accordance with (57), (58), and (61), (68) can be rewritten as follows: (69)

Remark 2. FLAD are estimated using (68) with a differential part, whereas only constant disturbances can be estimated by the proportional-integral (PI) observer in [26]. The adaptive augmented observer discussed in this study can simultaneously estimate the sensor and actuator faults if the FLAD is treated as an actuator fault. The selection basis of Γ₂ is (70), because (69) is associated with the estimated value of FLAD and Γ₂, (70) contains this value.

Theorem 3. If there exists a positive scalar κ_e, symmetric positive-definite matrices P₁, P₂, S₁, S₂, Γ₂, and real matrix M_i with appropriate dimensions, such that (70) always holds: (70) (71) (72) (73) then the estimate errors e_au and e_d are uniformly ultimately bounded under the estimation algorithm (68).

Proof. Let (74)

The Lyapunov function is selected as follows: (75) where P_e is a symmetric positive-definite matrix.

Based on (74) and (75), we can obtain (76) where (77) (78)

The derivative of (75) can be written as follows: (79)

Based on Lemma 1, there exists a matrix S = S^T>0 such that (80)

Denote (81)

Then, (82)

Denote (83)

And suppose

Then, the following can be obtained: (84) (85)

Therefore, (82) can be rewritten as (86)

FLAD is bounded, if (70) is true, that is, every linear mode obtained by real-time calculation of the left matrix in (70) is negative, then denote σ_i = min(eig(−Ξ_i)). Thus, (87)

When , we can obtain the following.

(88)

That is, error dynamics can converge to an interval (89)

Therefore, e_au and e_d are ultimately uniformly bounded. This completes the proof of Theorem 3.

Obtain the LMI cluster through real-time calculation (70) and use Schur’s supplementary lemma, feasible solutions (P₁, P₂, M_i, M_s, κ_e) can be obtained. Then, N_i and N_s can be determined using (65). Accordingly, observer (61) is determined.

Through the isolation process, the unknown matrix, F, as well as F_au, is determined; without a loss of generality, F_au is the full column rank. Then, the estimation of the two sensor faults and output noise can be achieved via the following augmented observer.

(90)

Obtaining valuable information about sensor faults, FLAD, and noise can aid in designing an active FTC scheme.

3.4. Fault compensation

The design of the nominal controller allows the variable-structure HFV to fly stably and track the altitude and velocity commands. However, when multi-sensor faults occur, the HFV will be inaccurate. Moreover, because FLAD will also affect flight performance, the HFV must be able to overcome external FLAD. Therefore, a robust variable-structure fault compensation based on the diagnosis results is essential.

When sensors fail, the actual measurement y(t) changes into y^f(t) that provides wrong signals for the basic output feedback controller. Thus, y^f(t) should be substituted by a compensated output y_c(t) with (91)

The FLAD should also be counteracted to make the flight of the variable-structure HFV smooth. Hence, the variable-structure FTC laws are designed as follows: (92) (93) (94)

Remark 3. The FTC laws, (91)–(94), render the HFV robust against both input disturbance and output noise. The control precision of (94) is better than that of because the latter does not consider the output noise.

After introducing (92)–(94), (95) is derived as follows: (95) where (96)

Based on Theorems 1 and 3, e_gi and e_d are known to be uniformly ultimately bounded and can be sufficiently small when suitable parameters are selected. Therefore, the effects of e_gi and e_d are neglected in the process of ensuring the stability of the FTC.

The Lyapunov function is selected as follows: (97) where h is a positive scalar, and P_f1 and P_f2 are two symmetric positive-definite matrices with appropriate dimensions.

Thus, it can be obtained as follows: (98) where κ_f2 is a positive scalar.

If (99) holds, then (99) then there exist some positive scalars б_i such that (100)

Thence, (101)

According to Theorem 1, there exist some positive scalars ς_i, such that (102) holds, (102)

By substituting (102) into (101), the following expression can be derived: (103)

Let (104) then (105)

Thus, under FTC laws (92)–(94), system (10) remains stable, and the HFV can accurately track the given command when sensor faults and disturbances exist. Therefore, efficient variable-structure flight is achieved.

The detection, isolation, and estimation of multi-sensor bias faults and the FTC design for the HFV have been completed thus far. The simulation results explain the validity of the proposed methods in Section 4.

Remark 4. The proposed diagnosis and FTC schemes can deal with more sensor failures. However, the definitions of the matrices (y_au,ks, C_au,ks, and H_au,iks) in the isolation observers and that of matrix F in the estimation algorithm have to be modified according to the number of faulty sensors.

4.1. Fault detection and isolation

Ideally, a threshold exceeding zero indicates a fault. However, because of disturbance, the threshold is no longer zero. Moreover, because the initial state of the observer (30) is set to be consistent with the original system (10), r(0) is zero. Therefore, β can be set as any constant greater than one. The maximum values of the three disturbance/noise channels are 0.05, 0.1, and 0.3; ε is set as 0.5. Hence, 0.5 is selected as the warning threshold to reduce false/missing reports. The FDI results are shown in Figs 3 and 4.

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Fig 3. Detection residual and threshold.

https://doi.org/10.1371/journal.pone.0256200.g003

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Fig 4. Isolation residuals and threshold.

(a) Second isolation residual J₂ (m = 2, k = 1, s = 3), (b) Third isolation residual J₃ (m = 3, k = 1, s = 4).

https://doi.org/10.1371/journal.pone.0256200.g004

Fig 3 shows that the detection residual exceeds the threshold (0.5) after 20 s (i.e., after the sensor faults occur). It should be noted that the residual increases rapidly at 20 and 25 s after the velocity and altitude sensors begin to fail, respectively. Fig 4 only presents the residuals of the second and third observers of the total 10 observers. As shown in Fig 4(A), the residual is less than 0.5; in Fig 4(B), the residual starts to exceed 0.5 at 25 s. Thus, the first and third sensors (velocity and altitude sensors, respectively) are observed to have failed.

4.2. Fault and disturbance estimation

In Section 4.2, we present the estimation results of the two faulty sensors and the disturbance through the adaptive augmented observer to illustrate the effectiveness of the proposed estimation scheme. Among the two sensor faults, one is constant, and the other is time-varying. To demonstrate the superiority of the proposed observer, the disturbance estimation result obtained using the observer is compared with that obtained using the PI observer in [26]. The fault and disturbance estimation results are shown in Figs 5 and 6, respectively.

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Fig 5. Fault estimation with the adaptive augmented observer.

(a) Estimation of f_v, (b) Estimation of f_h.

https://doi.org/10.1371/journal.pone.0256200.g005

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Fig 6. Disturbance estimation results via different observers.

(a) Estimation of d via the PI observer, (b) Estimation of d via the adaptive augmented observer.

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Fig 5 shows that the fault observer can flexibly estimate different types of faults in different control loops. It can stably and accurately track both the velocity constant fault ((a)) and altitude time-varying fault ((b)), create the condition for the active FTC.

Fig 6 shows the superior performance of the observer proposed in this paper when dealing with FLAD. Fig 6(A) shows the estimation result of the PI observer in [26]; evidently, it cannot satisfy the accuracy requirement. The results obtained using the observer presented in this paper are shown in Fig 6(B). The estimation of FLAD is fast and stable, and the tracking error is practically zero. So the proposed observer improves the existing method.

4.3. Fault compensation

To illustrate the importance of fault compensation, this section presents the velocity and altitude curves that are obtained under the nominal control and FTC. The simulation results are shown in Fig 7.

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Fig 7. Outputs with different controllers in sensor fault case.

(a) Velocity response under the nominal control and FTC, (b) Altitude response under the nominal control and FTC.

https://doi.org/10.1371/journal.pone.0256200.g007

Fig 7 shows that under fault conditions, the nominal controller cannot complete the normal velocity and altitude tracking controls. When multi-sensor faults occur, Fig 7(A) shows the velocity deviation under nominal control. The desired speed cannot be tracked at 3850 m/s, and there is a 10 m/s oscillation near the 3820 m/s deviation. Fig 7(B) indicates that the faults will also affect the tracking of the expected altitude (3450 m), and the system will oscillate approximately 16 m after 25 s. The solid line in Fig 7 indicates that the FTC can maintain the stable and accurate tracking of the two expected outputs with multi-sensor faults as well as achieve automatic fault shielding. The curves in Figs 5–7 all have slight random oscillations, the amplitude does not exceed 0.1% of the measured signal values. Due to the robustness of the system and the addition of multiple adaptive vibration suppression functions, these curves look very smooth, fully meet the engineering requirements for suppression of mechanical vibration. Our next step is to study the FTC problem of compound faults at different positions of the fuselage, the HFV flight control problem in different flight stages, and the active-passive hybrid fault repair problem.