2.1 Dynamic model of the EMLA system

This paper considers a moving-coil EMLA based on a Halbach permanent magnet array. Although the considered EMLA differs from suspension-oriented EMLAs in structural parameters and operating conditions, it shares the same direct-drive electromagnetic actuation mechanism and similar nonlinear dynamic characteristics. Therefore, it can serve as a representative platform for investigating fault diagnosis and fault-tolerant control problems of EMLA-based actuation systems. The actuator consists of two main components: a mover and a stator. The mover includes the coil framework and the coil winding, while the stator is composed of permanent magnets, an outer yoke, and an inner yoke. The overall structure of the EMLA is illustrated in Figure 1, and detailed operating principles can be found in Reference [29].

Figure 1 Structure of the EMLA

Based on electromagnetic and mechanical principles, the dynamic model of the EMLA can be expressed as:$\{\begin{array}{l}\dot{i}=\frac{u}{L}-\frac{k_{\text{e}}}{L}\dot{x}-\frac{R}{L}i\\ \ddot{x}=\frac{k_{\text{m}}}{m}i-\frac{c}{m}\dot{x}-\frac{F_{\text{s}}+F_{\text{f}}}{m}\end{array}$(1)where m is the mass of the mover, c is the damping coefficient, u and i are the power supply voltage and the current through the coil, respectively. R and L are the resistance and inductance of the entire coil, respectively. x, ẋ, and ẍ represent the position, velocity, and acceleration of the mover, respectively. k_m is the electromagnetic force constant, k_e is the back electromotive force coefficient, with k_m and k_e approximately equal. F_s is the external disturbance, and F_f is the nonlinear friction force.

Define the state variable as $x\left(t\right)={\left[x\dot{x}i\right]}^{\text{T}}$, the input variable as u(t) = [u], the Lipschitz nonlinear term as ϕ(t) = [0 –F_f/m 0]^T, and the external disturbance as d(t) = [F_s]. Then, the state-space representation of the EMLA system can be written as:$\{\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+\varphi \left(x(t)\right)+Dd(t)\\ y(t)=Cx(t)\end{array}$(2)where $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& -\frac{c}{m}& \frac{k_{\text{m}}}{m}\\ 0& -\frac{k_{\text{e}}}{L}& -\frac{R}{L}\end{array}\right]$, $B={\left[\begin{array}{ccc}0& 0& \frac{1}{L}\end{array}\right]}^{\text{T}}$, $C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$, $D={\left[\begin{array}{ccc}0& -\frac{1}{m}& 0\end{array}\right]}^{\text{T}}$.

To facilitate subsequent analysis, the following assumptions are introduced.

Assumption (1): the nonlinear term ϕ(x(t)) is bounded, i.e., ‖ϕ(x(t))‖ ≤ ε, where ε is a positive constant.

Assumption (2): the nonlinear term ϕ(x(t)) satisfies the Lipschitz condition, i.e., $\Vert \varphi \left(x\left(t\right)\right)-\varphi \left(\widehat{x}\left(t\right)\right)\Vert \le \gamma \Vert x\left(t\right)-x\left(t\right)\Vert$, where γ is a positive Lipschitz constant.

Assumption (3): the external disturbance d(t) is bounded such that $d{\left(t\right)}_2\le \overline{d}$, where $\overline{d}$ is a positive constant.

2.2 Fault model of the EMLA system

In practical applications, EMLAs may experience various types of faults [9], including actuator gain faults, bias faults, and sticking faults. Among these, gain faults—typically caused by magnetic leakage or actuator degradation—are one of the most common fault types. Therefore, this paper focuses on actuator gain faults.

When a gain fault occurs, the actual input of the actuator can be modeled as u_f = (1−δ)u, where δ ∈ [0, 1] represents the gain loss coefficient. When δ = 0, the system operates normally; when 0 < δ < 1, a gain fault is present. The faulty input can be rewritten as Bu_f = Bu − Bδu = Bu + Ff(t). Considering external disturbances, actuator gain faults, and measurement noise, the state-space equations of the faulty EMLA system can be described by:$\{\begin{array}{c}\dot{x}(t)=Ax(t)+Bu(t)+\varphi \left(x(t)\right)+Dd(t)+Ff(t)\\ y(t)=Cx(t)+E\eta (t)\end{array}$(3)where F is the fault distribution matrix, E is the measurement noise distribution matrix, f(t) denotes the fault signal,η(t) represents measurement noise.

Assumption (4): the fault signal and measurement noise are bounded. The fault signal f(t) satisfies $\Vert f\left(t\right){\Vert }_2\le \overline{f}$. In the mean square limit sense, the measurement noise η(t) and its derivative N(t) satisfy $\Vert \eta (t){\Vert }_2\le \overline{{\eta }_1}$ and $\Vert N(t){\Vert }_2\le \overline{{\eta }_2}$, where $\overline{f}$, $\overline{{\eta }_1}$, and $\overline{{\eta }_2}$ are all positive constants.

For subsequent stability analysis, the following lemmas are introduced.

Lemma (1): for any matrices X and Z with compatible dimensions, there exists a symmetric positive definite matrix H₁ such that:$X^{\text{T}}Z+Z^{\text{T}}X\le X^{\text{T}}{H}_{1}X+Z^{\text{T}}{H}_1^{-1}Z$(4)

Lemma (2): for a given matrix A₁∈R^n×n, there exists a symmetric positive definite matrix P₁∈R^n×n such that:$\left[\begin{array}{cc}-P_1& P_1(A_1-\phi I_{\text{n}})\\ \ast & -r^2{P}_1\end{array}\right]<0$(5)where I_n denotes the n-dimensional identity matrix. In Equation (5), the eigenvalues of matrix A₁ lie within a specified circular stability region D(φ, r), where the center and radius of the circular stability region are φ and r, respectively.

These assumptions and lemmas provide the theoretical foundation for the observer design and stability analysis presented in the subsequent sections.

3.1 Overall framework of the fault diagnosis and CFTC scheme

For the EMLA system in the presence of external disturbances, gain faults, and measurement noise, the overall framework of the fault diagnosis and CFTC scheme based on the ISMLO is illustrated in Figure 2. First, the ISMLO is designed to perform fault detection, state estimation, and fault estimation simultaneously. Once a fault is detected through the residual evaluation mechanism, the observer can rapidly track the system states and accurately estimate the fault. Based on the obtained fault estimation information, a CFTC strategy is subsequently constructed. According to the fault classification logic, faults are categorized into minor and major faults, for which the primary fault-tolerant control (PFTC) and secondary fault-tolerant control (SFTC) are employed, respectively. Finally, the control input is fed back to both the EMLA system and the ISMLO to compensate for faults, thereby ensuring stable operation of the system. This framework integrates fault diagnosis and control in a unified structure, improving both reliability and robustness.

Figure 2 Overall framework of the fault diagnosis and CFTC for EMLA based on ISMLO

3.2 Design of the ISMLO

To achieve accurate fault estimation and robust state estimation, a PD-type ISMLO is developed. The proposed observer combines the advantages of sliding mode control and iterative learning mechanisms, enabling strong robustness against disturbances and enhanced estimation accuracy. To improve estimation performance, a PD-type learning mechanism with a forgetting factor is introduced for fault estimation.

Define the output estimation residual as e_y(t) = y(t)–ŷ(t). Based on Equation (3), the ISMLO is constructed as:$\{\begin{array}{l}\dot{\widehat{x}}(t)=A\widehat{x}(t)+Bu(t)+\varphi \left(\widehat{x}(t)\right)+F\widehat{f}(t)+L{e}_y(t)+Dv(t)\\ \widehat{y}(t)=C\widehat{x}(t)\\ \widehat{f}(t)=\widehat{f}(t-\tau )+K\left((1-\alpha ){\dot{e}}_y(t-\tau )+e_y(t)\right)\end{array}$(6)where $\widehat{x}(t)$ is the estimated value of x(t), ŷ(t) is the estimated value of y(t), $\Phi \left(\widehat{x}\left(t\right)\right)$ is the estimation of the nonlinear term, $\widehat{f}(t)$ is the fault estimate value, L and K denote the observer gain matrices that require design, v(t) is the sliding mode control term, τ is the iterative learning interval, and α is the forgetting factor. The fault estimation law incorporates a PD-type iterative learning mechanism with a forgetting factor α, which improves convergence speed while preventing error accumulation.

Let e_y(t) be the sliding surface, the sliding mode control law is designed as follows:$v(t)=\{\begin{array}{cc}-\mu \frac{H{e}_y(t)}{H{e}_y(t)+\epsilon }& e_y(t)\ne 0\\ 0& e_y(t)=0\end{array}$(7)where ε denotes a scalar constant with a very small value, H represents the gain matrix that requires design, and μ denotes the gain parameter to be designed.

Define the state estimation error e_x(t) and fault estimation error e_f(t) as:$\{\begin{array}{c}{e}_x(t)=x(t)-\widehat{x}(t)\\ e_f(t)=f(t)-\widehat{f}(t)\end{array}$(8)

Integrating Equation (3) with Equation (6), the error equations can be derived as:$\{\begin{array}{l}{\dot{e}}_x(t)=(A-LC)e_x(t)+\varphi \left(\tilde{x}(t)\right)+D\left(d(t)-v(t)\right)+\\ F\left(f(t)-\widehat{f}(t)\right)-LE\eta (t)\\ e_y(t)=C{e}_x(t)+E\eta (t)\end{array}$(9)where $\varphi \left(\tilde{x}\left(t\right)\right)=\varphi \left(x\left(t\right)\right)-\varphi \left(\widehat{x}\left(t\right)\right)$, $\varphi \left(x\left(t\right)\right)$ satisfies the Lipschitz condition, and $\Vert \begin{array}{c}\varphi \left(\tilde{x}\left(t\right)\right)\end{array}\Vert \le \gamma \Vert \begin{array}{c}x\left(t\right)-\widehat{x}\left(t\right)\end{array}\Vert =$$\gamma \Vert e_x\left(t\right)\Vert$.

Assumption (5): the fault matrix satisfies ${\Vert \tilde{f}\left(t\right)\Vert }_{\infty }\le \overline{f_1}$, where $\tilde{f}\left(t\right)=f\left(t\right)-f\left(t-\tau \right)$, and $\overline{f_1}$ is a positive constant.

3.3 Stability analysis of the ISMLO

The following theorem is presented to establish sufficient conditions for the ultimate boundedness of the observer estimation errors and the corresponding H_∞ performance of the estimation error system.

Theorem (1): consider the faulty EMLA system and ISMLO under Assumptions (1−5), if there exist positive definite matrices P∈R^3×3 and Q∈R^3×3, a matrix Y∈R^3×3, matrices H∈R^1×3 and K∈R^1×3, as well as positive scalars γ and γ₁, then by applying Lyapunov stability theory, Young's inequality, and bounding techniques for time-delay terms, the following linear matrix inequalities (LMIs) and equality constraints are derived:$\left[\begin{array}{cccccc}{\Omega }_{11}& {\Omega }_{12}& {\Omega }_{13}& C^{\text{T}}& P& {\Omega }_{16}\\ \ast & {\Omega }_{22}& 0& 0& 0& 0\\ \ast & \ast & -{\gamma }_1^2{I}_3& 0& 0& 0\\ \ast & \ast & \ast & -I_3& 0& 0\\ \ast & \ast & \ast & \ast & -I_3& 0\\ \ast & \ast & \ast & \ast & \ast & {\Omega }_{66}\end{array}\right]<0$(10)$\left[\begin{array}{cc}-Q+(1-\alpha ){\gamma }^2{I}_3& (A^{\text{T}}{P}^{\text{T}}-C^{\text{T}}{Y}^{\text{T}})F\\ \ast & -I/(1-\alpha )\end{array}\right]<0$(11)$PD=C^{\text{T}}{H}^{\text{T}}$(12)$KC=F^{\text{T}}P$(13)$(1-\alpha )KCF-I=0$(14)where Ω₁₁ = A^TP + PA − YC − C^TY^T + Q + γ²I₃, Ω₁₂ = − PFKE − YE + C^TE, Ω₁₃ = − (1 − α)PFKE, Ω₁₆ = PFKC, Ω₂₂ = − γ₁²I + E^TE, Ω₆₆ = − I₃/2(1 − α), Y = PL. Equations (12–14) are introduced to eliminate bilinear terms. If Equations (10–14) are satisfied, then e_x(t) and e_f(t) ultimately stabilize within a certain interval, and e_y(t) satisfies:$\Vert e_y(t)\Vert \le {\gamma }_1\begin{array}{c}\Vert w_d(t)\Vert \end{array}+k_{\text{q}}$(15)where w_d(t) = [η^T(t) N^T(t)]^T, k_q is a positive constant.

Proof: to analyze the stability of the error system, consider the following Lyapunov function:$V(t)=e_x^{\text{T}}(t)P{e}_x(t)+{\int }_{t-\tau }^t{e}_x^{\text{T}}(s)Q{e}_x(s)\text{d}s$(16)which incorporates both instantaneous and historical information of the estimation error. The integral term captures the effect of the iterative learning interval, which is essential for ISMLO. Taking the derivative of the Lyapunov function yields:$\begin{array}{l}\dot{V}(t)=e_x^{\text{T}}(t)[{(A-LC)}^{\text{T}}P+P(A-LC)]e_x(t)+2e_x^{\text{T}}(t)P\varphi \left(\tilde{x}(t)\right)\\ +2e_x^{\text{T}}(t)PD\left(d(t)-v(t)\right)+2e_x^{\text{T}}(t)PFf(t)-2e_x^{\text{T}}(t)PF\widehat{f}(t)\\ -2e_x^{\text{T}}(t)PLE\eta (t)+e_x^{\text{T}}(t)Q{e}_x(t)-e_x^{\text{T}}(t-\tau )Q{e}_x(t-\tau )\\ =e_x^{\text{T}}(t)[{(A-LC)}^{\text{T}}P+P(A-LC)]e_x(t)+2e_x^{\text{T}}(t)P\varphi \left(\tilde{x}(t)\right)\\ +2e_x^{\text{T}}(t)PD\left(d(t)-v(t)\right)+2e_x^{\text{T}}(t)PFf(t)-2e_x^{\text{T}}(t)PLE\eta (t)\\ +e_x^{\text{T}}(t)Q{e}_x(t)-e_x^{\text{T}}(t-\tau )Q{e}_x(t-\tau )-2e_x^{\text{T}}(t)PF\widehat{f}(t-\tau )\\ -2e_x^{\text{T}}(t)PFKC{e}_x(t)-2e_x^{\text{T}}(t)PFKE\eta (t)-2(1-\alpha )e_x^{\text{T}}(t)PFKEN(t-\tau )\\ -2(1-\alpha )e_x^{\text{T}}(t)PFKC(A-LC)e_x(t-\tau )-2(1-\alpha )e_x^{\text{T}}(t)PFKC\varphi \left(\tilde{x}(t-\tau )\right)\\ -2(1-\alpha )e_x^{\text{T}}(t)PFKCD\left(d(t-\tau )-v(t-\tau )\right)-2(1-\alpha )e_x^{\text{T}}(t)PFKCFf(t-\tau )\\ +2(1-\alpha )e_x^{\text{T}}(t)PFKCF\widehat{f}(t-\tau )+2(1-\alpha )e_x^{\text{T}}(t)PFKCLE\eta (t-\tau )\\ =e_x^{\text{T}}(t)[{(A-LC)}^{\text{T}}P+P(A-LC)-2PFKC+Q]e_x(t)+2e_x^{\text{T}}(t)P\varphi \left(\tilde{x}(t)\right)\\ +2e_x^{\text{T}}(t)PD\left(d(t)-v(t)\right)-2e_x^{\text{T}}(t)PLE\eta (t)-e_x^{\text{T}}(t-\tau )Q{e}_x(t-\tau )\\ -2e_x^{\text{T}}(t)PFKE\eta (t)-2(1-\alpha )e_x^{\text{T}}(t)PFKEN(t-\tau )\\ -2(1-\alpha )e_x^{\text{T}}(t)PFKC(A-LC)e_x(t-\tau )-2(1-\alpha )e_x^{\text{T}}(t)PFKC\varphi \left(\tilde{x}(t-\tau )\right)\\ -2(1-\alpha )e_x^{\text{T}}(t)PFKCD\left(d(t-\tau )-v(t-\tau )\right)+2(1-\alpha )e_x^{\text{T}}(t)PFKCLE\eta (t-\tau )\\ +2e_x^{\text{T}}(t)PF((1-\alpha )KCF-I)\widehat{f}(t-\tau )+2e_x^{\text{T}}(t)PF(I-(1-\alpha )KCF)f(t-\tau )\\ +2e_x^{\text{T}}(t)PF\tilde{f}(t)\end{array}$(17)

Based on Lemma (1) and Assumptions (1−5), the subsequent inequality is satisfied:$2e_x^{\text{T}}(t)PF\tilde{f}(t)\le e_x^{\text{T}}(t)PF{F}^{\text{T}}P{e}_x(t)+{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)$(18)$\begin{array}{l}-2(1-\alpha )e_x^{\text{T}}(t)PFKC(A-LC)e_x(t-\tau )\le \\ (1-\alpha )e_x^{\text{T}}(t)PF{F}^{\text{T}}P{e}_x(t)\\ +(1-\alpha )e_x^{\text{T}}(t-\tau ){[KC(A-LC)]}^{\text{T}}KC(A-LC)e_x(t-\tau )\end{array}$(19)$\begin{array}{l}2e_x^{\text{T}}(t)P\varphi \left(\tilde{x}(t)\right)\le e_x^{\text{T}}(t)PP{e}_x(t)+{\varphi }^{\text{T}}\left(\tilde{x}(t)\right)\varphi \left(\tilde{x}(t)\right)\\ \le e_x^{\text{T}}(t)PP{e}_x(t)+{\gamma }^2{e}_x^{\text{T}}(t)e_x(t)\end{array}$(20)$\begin{array}{l}-2(1-\alpha )e_x^{\text{T}}(t)PFKC\varphi \left(\tilde{x}(t-\tau )\right)\\ \le {\varphi }^{\text{T}}\left(\tilde{x}(t-\tau )\right)\varphi \left(\tilde{x}(t-\tau )\right)+e_x^{\text{T}}(t)PFKC{(PFKC)}^{\text{T}}{e}_x(t)\\ \le (1-\alpha )e_x^{\text{T}}(t)PFKC{(PFKC)}^{\text{T}}{e}_x(t)\\ +(-\alpha ){\gamma }^2{e}_x^{\text{T}}(t-\tau )e_x(t-\tau )\end{array}$(21)$\begin{array}{l}2(1-\alpha )e_x^{\text{T}}(t)PFKCLE\eta (t-\tau )\\ \le e_x^{\text{T}}(t)PFKC{(PFKC)}^{\text{T}}{e}_x(t)+{\eta }^{\text{T}}(t-\tau ){(LE)}^{\text{T}}(LE)\eta (t-\tau )\\ \le (1-\alpha )e_x^{\text{T}}(t)PFKC{(PFKC)}^{\text{T}}{e}_x(t)+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\end{array}$(22)where λ_max is the largest eigenvalue of matrix L. If Equations (12–14) are satisfied, substituting Equations (18–22) into Equation (17) yields:$\begin{array}{l}\dot{V}(t)=e_x^{\text{T}}(t)[{(A-LC)}^{\text{T}}P+P(A-LC)+Q-\alpha PFKC\\ +PP+{\gamma }^2{I}_3+2(1-\alpha )PFKC{(PFKC)}^{\text{T}}]e_x(t)\\ +2\Vert H{e}_y(t)\Vert (d(t)-\Vert \mu \Vert )-2e_x^{\text{T}}(t)PLE\eta (t)\\ -2e_x^{\text{T}}(t)PFKE\eta (t)-2(1-\alpha )e_x^{\text{T}}(t)PFKEN(t-\tau )\\ -2\Vert H{e}_y(t)\Vert (d(t-\tau )-\Vert \mu \Vert )+{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)\\ +e_x^{\text{T}}(t-\tau )\{(1-\alpha ){[KC(A-LC)]}^{\text{T}}KC(A-LC)\\ -Q+(1-\alpha ){\gamma }^2{I}_3\}{e}_x(t-\tau )+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\end{array}$(23)

In the design, let the gain μ > 0, and μ ≥ d(t), μ ≥ d(t−τ), then it follows that:$\begin{array}{l}\dot{V}(t)=e_x^{\text{T}}(t)[{(A-LC)}^{\text{T}}P+P(A-LC)+Q-\alpha PFKC\\ +PP+{\gamma }^2{I}_3+2(1-\alpha )PFKC{(PFKC)}^{\text{T}}]e_x(t)\\ -2e_x^{\text{T}}(t)PLE\eta (t)-2e_x^{\text{T}}(t)PFKE\eta (t)\\ -2(1-\alpha )e_x^{\text{T}}(t)PFKEN(t-\tau )+{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)\\ +e_x^{\text{T}}(t-\tau )\{(1-\alpha ){[KC(A-LC)]}^{\text{T}}KC(A-LC)-Q\\ +(1-\alpha ){\gamma }^2{I}_3\}{e}_x(t-\tau )+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\end{array}$(24)

If Equation (11) holds, then according to the Schur complement lemma, then it follows that:$(1-\alpha ){[KC(A-LC)]}^{\text{T}}KC(A-LC)-Q+(1-\alpha ){\gamma }^2{I}_3<0$(25)

If the forgetting factor α ≥ 0, then –αPFKC ≤ 0, and the following inequalities hold:$\begin{array}{l}\dot{V}(t)=e_x^{\text{T}}(t)[{(A-LC)}^{\text{T}}P+P(A-LC)+Q+PP+{\gamma }^2{I}_3\\ +2(1-\alpha )PFKC{(PFKC)}^{\text{T}}]e_x(t)-2e_x^{\text{T}}(t)PLE\eta (t)\\ -2e_x^{\text{T}}(t)PFKE\eta (t)-2(1-\alpha )e_x^{\text{T}}(t)PFKEN(t-\tau )\\ +{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\\ \le X^{\text{T}}{M}_{1}X+{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\end{array}$(26)where $M_1=\left[\begin{array}{ccc}\Pi & -PFKE-PLE& -(1-\alpha )PFKE\\ \ast & 0& 0\\ \ast & \ast & 0\end{array}\right]$, Π =${\left(A-LC\right)}^{\text{T}}P+P\left(A-LC\right)+Q+PP+{\gamma }^2{I}_3+2\left(1-\alpha \right)PFKC$${\left(PFKC\right)}^{\text{T}}$, and $X={\left[\begin{array}{ccc}{e}_x^T\left(t\right)& {\eta }^{\text{T}}\left(t\right)& N^{\text{T}}\left(t\right)\end{array}\right]}^{\text{T}}$.

Define the H_∞ performance index as:$J_1={\displaystyle {\int }_0^{\infty }\left[e_y^{\text{T}}(t)e_y(t)-{\gamma }_1^2{w}_d^{\text{T}}(t)w_d(t)\right]}\text{d}t$(27)

Under zero initial conditions, it can be obtained that:$\begin{array}{l}{J}_1={\displaystyle {\int }_0^{\infty }\left[e_y^{\text{T}}(t)e_y(t)-{\gamma }_1^2{w}_d^{\text{T}}(t)w_d(t)+\dot{V}(t)\right]}\text{d}t+V(0)-V(\infty )\\ \le {\displaystyle {\int }_0^{\infty }\left[X^{\text{T}}{M}_{2}X+{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\right]}\text{d}t\end{array}$(28)where $M_2=\left[\begin{array}{ccc}\Pi +C^{\text{T}}C& -PFKE-PLE+C^{\text{T}}E& -(1-\alpha )PFKE\\ \ast & -{\gamma }_1^{2}I+E^{\text{T}}E& 0\\ \ast & \ast & -{\gamma }_1^2{I}_3\end{array}\right]$. If M₂ < 0 holds true, then:$\begin{array}{l}{J}_1={\displaystyle {\int }_0^{\infty }\left[e_y^{\text{T}}(t)e_y(t)-{\gamma }_1^2{w}_d^{\text{T}}(t)w_d(t)\right]}\text{d}t\\ \le {\displaystyle {\int }_0^{\infty }\left[{\tilde{f}}^{\text{T}}(t)\tilde{f}(t)+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2{I}_3\right]}\text{d}t\\ \le k_{\text{a}}^2+(1-\alpha ){\lambda }_{\max }^2(LE){\overline{\eta }}^2=k_{\text{q}}^2\end{array}$(29)

where k_a is a positive constant.

Therefore, if Assumptions (3–5) hold, then the H_∞ performance index in Equation (15) is satisfied. If M₂ < 0 holds, let Y = PL. Based on the Schur complement lemma, Equation (10) is satisfied. Proof completed.

It should be emphasized that Theorem (1) establishes the ultimate boundedness of the observer estimation errors, including e_x(t) and e_f(t), under bounded disturbances, measurement noise, and fault variations, rather than their asymptotic convergence to zero. Therefore, under practical operating conditions, the estimation errors e_x(t) and e_f(t) are generally not zero. These estimation errors should be treated as generally nonzero but ultimately bounded signals in the subsequent controller stability analysis rather than as signals that necessarily converge to zero.

Note (1): to guarantee that the designed ISMLO exhibits robust convergence characteristics, the eigenvalues of matrix A−LC are placed within region (φ, r). As established in Lemma (2), the subsequent linear matrix inequality holds:$\left[\begin{array}{cc}-P& PA-YC-\phi P\\ \ast & -r^{2}P\end{array}\right]<0$(30)where r is a positive constant, φ is a negative constant, and |φ| ≥ r.

Note (2): regarding Theorem (1), a sufficient criterion ensuring the existence of an admissible solution for the proposed ISMLO is provided by:$\text{rank}\left[\begin{array}{cc}A-s{I}_{\text{n}}& F\\ C& 0\end{array}\right]=n+m,s\in C$(31)where n = 3, m = 1, s is the complex frequency variable, and Equation (31) is equivalent to the condition that (A, F, C) possesses no invariant zeros located in the right half of the complex plane.

3.4 Fault detection

A residual fundamentally represents the difference between the actual output of a system and the output predicted by a corresponding model. Variations in the residual may indicate abnormal conditions or fault-related features within the system. In the context of fault detection, it is standard practice to develop a residual evaluation function and assess it against a predefined threshold to ascertain the presence of faults in the system.

The residual evaluation function can be expressed as:$r(t)=\Vert e_y(t){\Vert }_2$(32)

The preset threshold for the fault detectability index is denoted as:$J(t)=\underset{f(t)=0}{\sup }r(t)$(33)

The fault detection logic can be expressed as:$\{\begin{array}{c}r(t)\ge J(t)\\ r(t)<J(t)\end{array}\begin{array}{c}\text{fault}\\ \text{no fault}\end{array}$(34)

When r(t) exceeds the preset threshold J(t), it indicates that a fault has occurred in the EMLA. Otherwise, the EMLA is considered to be operating normally.

4.1 Fault classification

The objective of the control design is to ensure accurate trajectory tracking of the EMLA system in the presence of actuator gain faults, external disturbances, and measurement noise. Specifically, the controller should guarantee:

(i) The tracking error converges to a small neighborhood of zero;

(ii) The closed-loop system remains stable under both minor and major faults;

(iii) The control strategy adapts to different fault severities to improve overall performance.

To achieve these goals, a CFTC strategy for faults of varying severity is developed based on the fault estimation results provided by the ISMLO, which includes both PFTC and SFTC. The PFTC is developed utilizing ISMLO and state feedback to manage minor faults, while the SFTC is formulated utilizing state feedback and system output residuals to effectively mitigate major faults.

The SFTC, which incorporates system output residuals, performs better in fault tolerance when handling major faults. Therefore, the SFTC is employed in the occurrence of major faults. However, for systems experiencing minor faults, introducing system output residuals actually degrades fault-tolerant performance. Thus, the PFTC, which has better fault tolerance in such cases, is used when minor faults occur.

When EMLA experiences gain faults, the fault estimate value of ISMLO is $\widehat{f}(t)$, and the fault estimate value during normal system operation is defined as ${\widehat{f}}_0\left(t\right)$ = 0.

The residual evaluation function may be formulated as:$r_1(t)=\widehat{f}(t)-{\widehat{f}}_0(t)$(35)

The preset fault classification threshold is defined as:$J_1(t)=\underset{\delta (t)=0.47}{\sup }{r}_1(t)$(36)

The fault classification logic can be articulated as:$\{\begin{array}{c}{r}_1(t)<J_1(t)\\ r_1(t)\ge J_1(t)\end{array}\begin{array}{c}\text{a minor fault occurred, activating PFTC}\\ \text{a major fault occurred, activating SFTC}\end{array}$(37)

When r₁(t) remains below the preset threshold J₁(t), this signifies a minor fault, prompting the activation of the PFTC. Conversely, if r₁(t) surpasses the preset threshold J₁(t), it indicates a major fault, thereby triggering the engagement of the SFTC.

This switching strategy ensures that:

(i) The PFTC is applied under minor fault conditions to maintain smooth control performance;

(ii) The SFTC is activated under major faults to enhance robustness and compensate for large uncertainties.

As a result, the proposed CFTC strategy achieves a balance between control performance and robustness under varying fault conditions.

4.2 Design of the SFTC

Assumption (6): rang(B, F) = rank(B)

According to Reference [30], the above assumption is equivalent to the existence of a matrix B^*∈ R^m×n, such that:$(I_{\text{n}}-B{B}^{*})F=0$(38)

The SFTC is designed as:$u(t)=-K_1\widehat{x}(t)-K_2{e}_y(t)-B^{*}F\widehat{f}(t)$(39)

Substituting Equation (39) into Equation (3) yields:$\begin{array}{l}\dot{x}(t)=Ax(t)-B{K}_1\widehat{x}(t)-B{K}_2{e}_y(t)-B{B}^{*}F\widehat{f}(t)\\ +\varphi \left(x(t)\right)+Dd(t)+Ff(t)\\ =Ax(t)-B{K}_1\left(x(t)-e_x(t)\right)-B{K}_2{e}_y(t)-B{B}^{*}F\widehat{f}(t)\\ +\varphi \left(x(t)\right)+Dd(t)+F(e_f(t)+\widehat{f}(t))\\ =(A-B{K}_1)x(t)+B{K}_1{e}_x(t)-B{K}_2{e}_y(t)\\ +(I_3-B{B}^{*})F\widehat{f}(t)+\varphi \left(x(t)\right)+Dd(t)+F{e}_f(t)\end{array}$(40)

Substituting Equation (38) into Equation (40) yields:$\begin{array}{ll}\dot{x}(t)\hfill & =(A-B{K}_1)x(t)+B{K}_1{e}_x(t)-B{K}_2{e}_y(t)\hfill \\ \hfill & +\varphi \left(x(t)\right)+Dd(t)+F{e}_f(t)\hfill \end{array}$(41)

The secondary fault-tolerant controller uses the estimated states and estimated faults provided by the ISMLO. Therefore, the observer estimation errors enter the closed-loop dynamics. In the following controller stability analysis, the state estimation error e_x(t), the fault estimation error e_f(t), and the external disturbance d(t) are grouped into the lumped exogenous signal $\omega \left(t\right)={\left[\begin{array}{ccc}{e}_x{}^{\text{T}}\left(t\right)& e_{\text{f}}{}^{\text{T}}\left(t\right)& d^{\text{T}}\left(t\right)\end{array}\right]}^{\text{T}}$. According to Theorem (1), the observer estimation errors are ultimately bounded under bounded disturbances, measurement noise, and fault variations. Therefore, e_x(t) and e_f(t) are treated as exogenous components of ω(t) in the controller stability analysis rather than as signals that necessarily converge to zero.

The case ω(t) = 0 corresponds to an ideal unforced nominal reference case, where the observer estimation errors and external disturbances are absent, i.e., e_x(t) = 0, e_f(t) = 0, and d(t) = 0. In practical Hardware-in-the-Loop (HIL) tests or actual operation, the exact case ω(t) = 0 generally does not occur because disturbances, measurement noise, nonlinear effects, and observer estimation errors are present. Therefore, this case is not regarded as a practical operating condition. It is retained only as an ideal zero-input counterpart of the closed-loop system for examining the internal stability of the controller dynamics. For the practical case with ω(t) ≠ 0, asymptotic stability is not claimed; instead, the prescribed H_∞ disturbance attenuation performance with respect to the lumped exogenous signal ω(t) is analyzed in the standard L₂-gain sense.

Theorem (2): For given scalars η > 0, σ > 0, and ρ > 0, if there exist a positive definite matrix Q ∈ R^3×3 and matrices M ∈ R^3×3 and N ∈ R^3×3 that satisfy the H_∞ performance index ‖x‖₂² < η²‖ω‖₂², and the following conditions are met, then the ideal unforced closed-loop system Equation (41) with ω(t) = 0 is asymptotically stable. When ω(t) ≠ 0, asymptotic stability is not claimed; instead, the closed-loop system satisfies the prescribed H_∞ disturbance attenuation performance with respect to the lumped exogenous signal ω(t) in the L₂-gain sense.$\left[\begin{array}{llllll}{\overline{\Omega }}_{11}\hfill & {\overline{\Omega }}_{12}\hfill & QF\hfill & QD\hfill & Q\hfill & {\overline{\Omega }}_{16}\hfill \\ *\hfill & -\eta I_3\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ *\hfill & *\hfill & -\eta I_3\hfill & 0\hfill & 0\hfill & 0\hfill \\ *\hfill & *\hfill & *\hfill & -\eta I_3\hfill & 0\hfill & 0\hfill \\ *\hfill & *\hfill & *\hfill & *\hfill & -I_3/\sigma \hfill & 0\hfill \\ *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -I_7\hfill \end{array}\right]<0$(42)where ${\overline{\Omega }}_{11}=QA+A^{\text{T}}Q-M-M^{\text{T}}+I_3/\eta$, ${\overline{\Omega }}_{12}=M-NC$, ${\overline{\Omega }}_{16}=\sqrt{{\rho }^2/\sigma \theta {\left(t\right)}_2{}^2}$.

Proof: define the Lyapunov function as:$V_1=x^{\text{T}}(t)Qx(t)$(43)

Taking the derivative of the Lyapunov function, then it follows that:$\begin{array}{l}{\dot{V}}_1=x^{\text{T}}(t)\left(Q(A-B{K}_1)+{(A-B{K}_1)}^{\text{T}}Q\right)x(t)\\ +2x^{\text{T}}(t)QB{K}_1{e}_x(t)-2x^{\text{T}}(t)QB{K}_2{e}_y(t)\\ +2x^{\text{T}}(t)QF{e}_f(t)+2x^{\text{T}}(t)Q\varphi \left(x(t)\right)+2x^{\text{T}}(t)QDd(t)\end{array}$(44)