1. Introduction

2. Cascade Electro-Hydraulic Control System of Turbofan Engine

3. Fault Diagnosis Scheme

3.1. Optimal Fault Detection Filter Design for the Outer Loop System

Consider the following dynamic control system that incorporates additive disturbances and faults:

$$\{\begin{array}{l}{\dot{x}}_1(t)=A_1{x}_1(t)+B_1{u}_1(t)+E_{d1}{d}_1(t)+E_{f1}{f}_1(t)\\ y_1(t)=C_1{x}_1(t)+D_1{u}_1(t)+F_{d1}{d}_1(t)+F_{f1}{f}_1(t)\end{array}$$

(1)

where, x₁ is the state vector of outer-loop control system, y₁ signifies outer-loop output vector, A₁, B₁, C₁, and D₁ are system matrices, u₁ is the control input, d₁(t) represents the unknown disturbance input vector of the system, f₁(t) denotes the fault vector of the system, E_d1, E_f1, F_d1, and F_f1 are the corresponding disturbance and fault matrices with appropriate dimensions.

To generate fault residuals of the outer-loop control system, the following state observer is designed:

$$\{\begin{array}{l}{\dot{\widehat{x}}}_1(t)=A_1{\widehat{x}}_1(t)+B_1{u}_1(t)+L_{obs}\left[y_1(t)-{\widehat{y}}_1(t)\right]\\ {\widehat{y}}_1^{ }(t)=C_1\widehat{x}(t)+D_1{u}_1(t)\end{array}$$

(2)

where, ${\dot{\widehat{x}}}_1$ and ${\widehat{y}}_1$ are the estimated x₁ and y₁, respectively. L_obs is the observer gain matrix that stabilizes the eigenvalues of the matrix (A₁-L_obsC₁) stable, ensuring that the designed observer can achieve an unbiased estimate of the system state.

Define the observation residuals e_obs1 as follows

$$e_{obs1}(t)=x_1(t)-{\widehat{x}}_1(t)$$

(3)

The dynamic characteristics of the residual generator can be described as

$$\{\begin{array}{l}\dot{\xi }(t)={\tilde{A}}_1\xi (t)+{\tilde{E}}_{d1}{\tilde{d}}_1(t)+{\tilde{E}}_{f1}{f}_1(t)\\ r_1(t)=W\left[{\tilde{C}}_1\xi (t)+{\tilde{F}}_{d1}{\tilde{d}}_1(t)+{\tilde{F}}_{f1}{f}_1(t)\right]\end{array}$$

(4)

where, W is a pre-filter to be designed, r₁ is the generated system residual,

According to Equation (1), the transfer functions from system disturbances ${\tilde{d}}_1$ and system faults f₁ to r₁ are as follows, respectively

$$G_{r_1{\tilde{d}}_1}(s)=W\left[{\tilde{F}}_{d1}+{\tilde{C}}_1\left(sI-{\tilde{A}}_1+L_{obs}{\tilde{C}}_1\right)\left({\tilde{E}}_{d1}-L_{obs}{F}_{d1}\right)\right]$$

(5)

$$G_{r_1{f}_1}(s)=W\left[{\tilde{F}}_{f1}+{\tilde{C}}_1\left(sI-{\tilde{A}}_1+L_{obs}{\tilde{C}}_1\right)\left(E_{f1}-L_{obs}{F}_{f1}\right)\right]$$

(6)

To achieve the optimal balance between the robustness of system disturbances and the sensitivity to system faults, we can solve the following optimal design problem using the theorem presented in [29] to obtain the optimal solution.

$$\underset{L_{obs},W}{\max }J(L_{obs},W)={\displaystyle \frac{{\sigma }_i\left(G_{r_1{f}_1}(\mathrm{j}\omega )\right)}{{\Vert G_{r_1{\tilde{d}}_1}(\mathrm{j}\omega )\Vert }_{\infty }}}, \omega \in [0,\infty )$$

(7)

where, ${\sigma }_i(\ast )$ is the i-th non-zero singular value, j is imaginary unit, ω is the angular frequency.

The matrices L_obs and W that satisfies the solution of the above optimization problem could make the observer robust to disturbances while being as sensitive to faults as possible.

To detect faults, the root mean square (RMS) of the residual signal is utilized to evaluate the energy of the residual signal:

$${\Vert r_1(t)\Vert }_{\mathrm{RMS}}=\sqrt{{\displaystyle \frac{1}{T}}{\displaystyle {\int }_t^{t+T}{r}_1{ }^{\mathrm{T}}(\tau )r_1(\tau )\mathrm{d}\tau }}$$

(8)

where, ${\Vert ·\Vert }_{\mathrm{RMS}}$ represents the average energy of the residual signal within the specified time interval (t, t+T) and it can be determined based on the maximum energy value observed in the system’s historical data under fault-free conditions.

$$J_{th1,RMS}=\underset{\mathrm{fault}-\mathrm{free}}{\sup }{\Vert r_1(t)\Vert }_{\mathrm{RMS}}$$

(9)

By monitoring the RMS value of the residual signal and comparing it against a predetermined threshold J_th1,RMS, the fault detection logic of the outer-loop control system is given by

$$\{\begin{array}{l}{\Vert r_1(t)\Vert }_{\mathrm{RMS}}>J_{th1,RMS}\Rightarrow \mathrm{fault}\\ {\Vert r_1(t)\Vert }_{\mathrm{RMS}}\le J_{th1,RMS}\Rightarrow \mathrm{fault}-\mathrm{free}\end{array}$$

(10)

3.2. Robust Unknown Disturbance Decoupled Residual Generator for the Inner-Loop System

The inner-loop dynamic system can be described as

$$\{\begin{array}{l}{\dot{x}}_2(t)=A_2{x}_2(t)+B_2{u}_2(t)+E_{d2}{d}_{a2}(t)\\ y_2^{ }=C_2{x}_2(t)+D_2{u}_2(t)\end{array}$$

where, x₂ is state vector of the inner-loop control system, y₂ is inner-loop output, A₂, B₂, C₂, and D₂ are system matrices, u₂ is control input, d_a2(t) represents the unknown external disturbance, and E_d1 is disturbance distribution matrix.

For the inner loop fuel control system dominated by multiplicative faults, an unknown disturbance decoupling observer is established as follows:

$$\{\begin{array}{l}\dot{z}(t)=F_{obs}z(t)+T_{obs}{B}_2{u}_2+K_{obs}{y}_2(t)\\ {\widehat{x}}_2(t)=z(t)+H_{obs}{y}_2(t)\end{array}$$

(11)

where, $z(t)$ is the state vector of the unknown disturbance decoupling observer, ${\widehat{x}}_2(t)$ is the estimated state vector of the inner loop, F_obs, T_obs, K_obs, and H_obs are the observer matrices that need to be designed to achieve decoupling of additive perturbation and state estimation.

Fig. 2 shows the structure of the unknown disturbance decoupling observer.

Figure 2. Structure of robust unknown disturbance decoupled residual generator.

Figure 2. Structure of robust unknown disturbance decoupled residual generator.

The estimated state error e_obs2 is

$$e_{obs2}(t)=x_2(t)-{\widehat{x}}_2(t)$$

(12)

The error control equation can be described as

$$\begin{array}{l}{\dot{e}}_{obs2}(t)=(A_2-H_{obs}{C}_2{A}_2-K_{obs1}{C}_2)e_2(t)-[F_{obs}-(A_2-H_{obs}{C}_2{A}_2-K_{obs1}{C}_2)]z(t)\\ -[K_{obs2}-(A_2-H_{obs}{C}_2{A}_2-K_{obs1}{C}_2)H_{obs}]y_2(t)\\ -[T_{obs}-(I-H_{obs}{C}_2)]B_2{x}_{sv}(t)-(H_{obs}{C}_2-I)E_{d2}{d}_{a2}(t)\end{array}$$

(13)

where, K_obs=K_obs1+K_obs2.

When system input and output are decoupled, the state estimation error of the system can be simplified to:

$${\dot{e}}_{obs2}(t)=F_{obs}{e}_{obs2}(t)=\left(A_2-H_{obs}{C}_2{A}_2-K_{obs1}{C}_2\right)e_{obs2}(t)$$

(14)

If all eigenvalues of the matrix $F_{obs}$ are stable, it can be ensured that the estimation error gradually approaches zero and the estimated system states approach the true values.

According to the output of the observer and the actual output of the system, the residuals of the system can be obtained

$$r_2(t)=y_2(t)-C_2{\widehat{x}}_2(t)=(I-C_2{H}_{obs})y_2(t)-C_{2}z(t)$$

(15)

where, I is an identity matrix with appropriate dimensions.

Similarly, RMS index is utilized to evaluate the change of the residual signal, i.e.,

$${\Vert r_2(t)\Vert }_{\mathrm{RMS}}=\sqrt{{\displaystyle \frac{1}{T}}{\displaystyle {\int }_t^{t+T}{r}_2{ }^{\mathrm{T}}(\tau )r_2(\tau )\mathrm{d}\tau }}$$

(16)

The maximum energy value of historical data recorded under fault-free conditions is used as the threshold of the residual signal

$$J_{th2,RMS}=\underset{\mathrm{fault}-\mathrm{free}}{\sup }{\Vert r_2(t)\Vert }_{\mathrm{RMS}}$$

(17)

If the RMS value exceeds this threshold, a fault condition is declared. Therefore, the fault detection logic of the outer loop is

$$\{\begin{array}{l}{\Vert r_2(t)\Vert }_{\mathrm{RMS}}>J_{th2,RMS}\Rightarrow \mathrm{fault}\\ {\Vert r_2(t)\Vert }_{\mathrm{RMS}}\le J_{th2,RMS}\Rightarrow \mathrm{fault}-\mathrm{free}\end{array}$$

(18)

3.3. Steady-State Propagation Characteristics of Faults in Cascade Control Systems

To analyze the propagation characteristics of actuator and sensor faults in the cascade control system, it is first necessary to derive the fault propagation characteristics of the single-loop control system shown in Figure 3.

To simplify the derivation process, multiplicative faults can be equivalently converted into additive faults, and the system dynamics of the inner loop can be described as:

$$\{\begin{array}{l}{\dot{x}}_2=A_2{x}_2+B_2{u}_2+E_{f2}{f}_{a2}\\ y_{2m}=C_2{x}_2+F_{f2}{f}_{s2}\end{array}$$

(19)

where, ${\tilde{E}}_{f2}$ is the actuator fault matrix, ${\tilde{f}}_{a2}$ is the actuator fault vector, ${\tilde{F}}_{f2}$ is the sensor fault matrix, ${\tilde{f}}_{s2}$ is sensor fault vector.

The system matrices of the inner-loop electro-hydraulic FMV are

where, K_L is load stiffness of FMV, mt is mass of the FMV, A_p is the area of the FMV control chamber, B_p is damping coefficient of FMV, C_ip is the internal leakage coefficient, β_e is elastic modulus of the fuel, K_c2 is the pressure gain coefficient of EHSV, K_q2 flow gain coefficient of EHSV, V_t is volume of the FMV control chamber, and K_s2 is the LVDT gain.

The closed-loop transfer matrix from system input v₂ to output y_2m is:

$${\dot{x}}_2=(A_2-B_2{C}_2{K}_2)x_2+B_2{K}_2{v}_2-B_2{\tilde{F}}_{f2}{\tilde{f}}_{s2}{K}_2+{\tilde{E}}_{f2}{\tilde{f}}_{a2}$$

(20)

where, K₂ is inner-loop controller.

Then the transfer function from system input v₂ to output y₂ is

$$G_{v_2,y_2}(s)=C_2{\left[sI-(A_2-B_2{C}_2{K}_2)\right]}^{-1}{B}_2{K}_2 ={\displaystyle \frac{-K_{s2}{K}_2\left[V_t{m}_t{s}^2+\left(4{\beta }_e{K}_{c2}{m}_t+V_t{B}_p\right)s+4{\beta }_e{A}_p^2\right]}{V_t{m}_t{s}^3+\left(4{\beta }_e{K}_{c2}{m}_t+V_t{B}_p\right)s^2+4{\beta }_e{A}_p^{2}s+4{\beta }_e{A}_p{K}_{s2}{K}_2{K}_{q2}}}$$

(21)

From equation (21), it can be observed that feedback sensor gain K_s2 and controller K₂ have direct impacts on the system output under the closed-loop control situation.

Similarly, the transfer functions of the actuator faults to the specific signals in the loop $G_{u_2,f_{a2}}$, $G_{y_2^{ },f_{a2}}$, $G_{y_{2m},f_{a2}}$, $G_{u_2,f_{s2}}$, $G_{y_2^{ },f_{s2}}$, and $G_{y_{2m},f_{s2}}$ can be obtained.

The steady-state outputs of the derived system transfer functions can be calculated respectively using the final value theorem, and results are as follows.

It can be seen from the above calculation results that, for a single-loop closed-loop control system, feedback sensor faults will affect the steady-state control accuracy of the closed-loop control system, and the control error is directly related to the faulty sensor gain. In contrast, actuator faults usually do not affect the actual steady-state output of the system. Furthermore, neither actuator faults nor sensor faults affect the steady-state values of the measured output y_2m and the control input u₂ within the loop.

The fault propagation characteristics of a single closed-loop system are summarized in Table 1.

As evident from Table 1, additive disturbances in the system do not affect the final steady-state control characteristics of the system. This underscores the robustness of the closed-loop control system, which is capable of mitigating the effects of process disturbances in the system, except in cases of feedback sensor faults or other disturbances.

Nonetheless, when considering the presence of the external control loop as depicted in Figure 4, the conclusion changes. In this scenario, the steady-state error of the inner loop system’s actual output is further corrected by the outer controller, thereby altering the steady-state outputs of multiple signals within the loop.

The analytical derivation method can be employed to obtain the transfer functions between different signals in the cascade control system. Additionally, the steady-state propagation characteristics of faults within the cascade fuel control system are presented in Table 2.

According to the previous analysis, when the feedback sensor in the inner loop experiences a gain failure, it results in a deviation in the control input command of the fuel servo mechanism. This, in turn, leads to changes in the actual outputs of the fuel metering valve, fuel flow, and engine rotor speed. However, the outer-loop controller corrects the deviations by adjusting the command of the inner-loop actuator, effectively eliminating the engine speed deviation and maintaining the measured rotation speed at the reference value.

4. Experiments

5. Discussion and Conclusions

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