Algebraic Observer-Based Fault Diagnosis for Proton Exchange Membrane Fuel Cell Air Supply Systems with Sliding-Mode Derivative Estimation

Proton exchange membrane fuel cells (PEMFCs) have gained considerable attention as clean power sources for both automotive and stationary applications, due to their high energy conversion efficiency, fast dynamic response, and negligible local pollutant emissions [1, 2]. The air supply subsystem constitutes a critical component of the PEMFC power plant, since it governs oxygen delivery to the cathode and directly influences the oxygen excess ratio, stack voltage, and overall system efficiency. Insufficient air supply can lead to cathode oxygen starvation, which accelerates membrane degradation and shortens the fuel cell lifetime [3].

Reliable fault detection and isolation in the air path is therefore of practical importance for safe and efficient PEMFC operation. In the recent literature, three main families of fault diagnosis approaches have been developed and evaluated for PEMFC systems. Data-driven methods include neural networks, support vector machines, and deep learning architectures. They show promising classification performance using electrochemical impedance spectroscopy (EIS) data or voltage/current measurements without a physical model [4-6]. However, these methods typically require large labelled training datasets, suffer from limited generalization over different stacks and operating conditions, and offer only limited physical insight into the diagnosed faults.

Model-based fault diagnosis methods exploit the physics‑based dynamics to generate fault‑sensitive information, making them effective for PEMFC air supply problems [7, 8]. Among the model-based strategies, observer-based approaches are particularly interesting since they can reconstruct unmeasured internal states and estimate fault signals in real time [9]. In the PEMFC context, various observer architectures have been studied, including adaptive Kalman filters [10] and linear parameter-varying (LPV) observers [11]. While these methods provide valuable diagnostic information, they typically rely on asymptotic convergence, meaning that the fault estimate improves gradually over time and may respond slowly during rapid transients, and often require residual evaluation with manually tuned detection thresholds. More recently, Yang et al. [12] proposed an augmented LPV observer that simultaneously reconstructs component faults and performs active fault-tolerant control of the oxygen excess ratio in the PEMFC air management system. In the domain of finite-time and sliding-mode approaches, Guo et al. [13] introduced an adaptive prescribed-performance controller that handles unknown air compressor faults, while Wang et al. [14, 15] developed fixed-time and finite-time sliding-mode fault-tolerant control frameworks for the supply manifold and compressor subsystems.

Another possible approach is to use algebraic observer techniques, based on the differential-algebraic framework of Diop and Fliess [16] and Fliess and Sira-Ramirez [17]. In this approach, system states and fault signals are expressed as functions of the measured outputs and their time derivatives. This eliminates the need for asymptotic convergence and yields fault estimates that are available as soon as accurate derivative information is obtained. Algebraic observers have been applied to PEMFC systems by Baroud et al. [18, 19], who demonstrated finite-time state estimation and also combined algebraic observers with output-feedback control. Jing et al. [20] proposed multi-objective sliding-mode control with adaptive algebraic observers, while Liu et al. [21] developed a robust model-based fault diagnosis scheme for the PEMFC air-feed system. Building on such algebraic frameworks, Flores-Mendez et al. [22] combined algebraic identification with extended state observers to improve the robustness of fault diagnosis in nonlinear systems, demonstrating non-asymptotic fault estimation under realistic perturbation levels.

A common practical challenge when using algebraic methods is the sensitivity of numerical differentiation to measurement noise. Direct finite-difference approximations lead to the amplification of high-frequency disturbances and compromise the quality of the reconstructed signals. To address this issue, Levant introduced higher order sliding mode (HOSM) differentiators [23, 24], which provide exact derivative estimation in finite-time for signals with bounded higher-order derivatives. The theoretical properties of these differentiators have been rigorously established through strict Lyapunov functions by Moreno and Osorio [25].

The present work combines the direct algebraic fault reconstruction framework with a robust exact differentiator based on the second-order sliding-mode algorithm. The differentiator is parameterized so that only a single design parameter an upper bound on the second derivative of the measured signal requires tuning. The diagnosis focuses on actuator faults (compressor motor degradation) and process faults (supply manifold leakage), which represent the most common failure modes in the air supply subsystem. Unlike observer-based approaches that achieve fault estimation only asymptotically, the proposed algebraic framework yields fault reconstruction formulas that are instantaneous functions of measured signals and their derivatives. Combined with the finite-time convergence of the robust exact differentiator, this results in a fault diagnosis scheme that achieves finite-time reconstruction with a guaranteed finite settling time.

The remainder of this paper is organized as follows. Section 2 describes the PEMFC air supply system model and the fault parameterization. Section 3 develops the algebraic observability analysis and the direct fault reconstruction formulas with explicit robustness considerations. Section 4 presents the robust exact differentiator and its application to the PEMFC diagnosis problem. Section 5 presents simulation results using the Pukrushpan PEMFC model. And Section 6 concludes the paper.

3.1 Algebraic observability framework

A nonlinear system is algebraically observable [30, 31]

$\dot{x}=f(x, u), y=h(x)$          (17)

If there exist finite positive integers $\mu$ and $v$ such that $x(t)=\Phi(y, \dot{y}, \ldots, y(\mu), u, \dot{u}, \ldots, u(v))$, where $\Phi(\cdot)$ is a (possibly nonlinear) vector-valued function. In other words, each state component can be written as a function of measurable quantities and their derivatives.

3.2 Direct signal recovery from measured outputs

The algebraic reconstructibility of the signals required for fault diagnosis in the PEMFC air supply system is established using only the measured output vector Eq. (11) and known inputs $V_{c m}, I_{s t}$.

Step 1: Compressor speed and manifold pressure. The outputs $y_3$ and $y_4$ provide two signals directly: $\omega_{c p}=y_3$, $P_{s m}=y_4$.

Step 2: Sum of cathode partial pressures. From the definition of $P_{c a}$ and the output $y_2=P_{c a}, P_{O_2}+P_{N_2}=y_2-\gamma_2$. Thus, the sum of partial pressures is directly measurable at every instant.

It is worth noting that recovering the individual partial pressures $P_{O_2}$ and $p_{N_2}$ separately is not required for fault reconstruction. The actuator fault formula depends only on $y_3, y_4, V_{c m}$, while the process fault formula depends only on $y_1, y_2$ and $y_4$. The full output vector therefore provides all information necessary for fault diagnosis without any additional simplifying assumptions on the cathode gas composition.

3.3 Actuator fault reconstruction

The fault reconstruction Eq. (18) uses a part of the model that is independent of the rest. Consider the compressor dynamics with the actuator fault. Since $y_3=\omega_{c p}$ is directly measured, we can differentiate $y_3$ with respect to time to obtain:

$\dot{y}_3=\Theta_1\left(y_3, y_4, u\right)+\Omega_1\left(y_3\right) \theta_a(t)$            (18)

where,

$\begin{aligned} \Theta_1\left(y_3, y_4, u\right)= & -\gamma_9 y_3-\gamma_{10} \gamma_{17} \Psi+\gamma_{13} V_{c m}-\gamma_{13} k_v y_3\end{aligned}$            (19)

$\Omega_1\left(y_3\right)=-\gamma_{13} y_3$           (20)

Both $\Theta_1$ and $\Omega_1$ depend on the measurable signals $y_3, y_4$, and the known input $V_{c m}$. Solving for the actuator fault:

$\hat{\theta}_a(t)=\frac{\left(\dot{y}_3-\Theta_1\left(y_3, y_4, u\right)\right)}{\Omega_1\left(y_3\right)}$          (21)

3.4 Process fault reconstruction

Similarly, for the supply manifold dynamics with leakage, since $y_4=P_{S m}$ is measured:

$\dot{y}_4=\Theta_2\left(y_1, y_2, y_4\right)+\Omega_1\left(y_2, y_4\right) \theta_p$           (22)

where,

$\Theta_2\left(y_1, y_2, y_4\right)=\gamma_{14}\left[1+\gamma_{15} \Psi\right]\left(y_1-\gamma_{16}\left(y_4-y_2\right)\right)$           (23)

$\Omega_1\left(y_2, y_4\right)=-\gamma_{14}\left[1+\gamma_{15} \Psi\right]$           (24)

with $\Psi=\left(y_4 / \gamma_{11}\right)^{\wedge} \gamma_{12}-1$. All quantities are expressed through the measured outputs $y_1, y_2, y_4$ only. The process fault reconstruction is:

$\hat{\theta}_p(t)=\frac{\left(\dot{y}_4-\Theta_2\left(y_1, y_2, y_4\right)\right)}{\Omega_1\left(y_2, y_4\right)}$            (25)

The actuator fault Eq. (21) depends only on $y_3, y_4, V_{c m}$, and $\dot{y}_3$, while the process fault Eq. (25) depends only on $y_1, y_2, y_4$ and $\dot{y}_4$. The two reconstruction expressions are therefore structurally decoupled in the sense that each fault is recovered from a distinct subset of measurements and a distinct output derivative. However, the underlying PEMFC dynamics are physically coupled: the compressor drives the manifold pressure, and the manifold pressure influences the cathode, which in turn affects the compressor load. Consequently, while the reconstruction formulas are algebraically independent, dynamic coupling interactions between the fault estimates may still produce transient errors during rapid operating-point transitions or simultaneous fault occurrences.

The fault reconstruction formulas involve a division by the sensitivity terms $\Omega_1\left(y_3\right)$ and $\Omega_2\left(y_2, y_4\right)$, respectively. For Eq. (21), $\Omega_1$ becomes zero only if the compressor speed $y_3=0$, which is outside any realistic operating regime. For Eq. (25), $\Omega_2$ reaches zero when $y_2=y_4$ (zero pressure gradient across the manifold), which cannot occur under normal or faulty operating conditions since the compressor ensures a positive pressure differential.

The proposed fault diagnosis framework is validated on the nonlinear Pukrushpan PEMFC model [27], implemented in MATLAB/Simulink with the parameters of Tables 1 and 2. The simulation environment and software specifications are summarized in Table 3. A proportional‑integral (PI) controller regulates the oxygen excess ratio $\lambda_{O_2}$ to its reference value of 2 by adjusting the compressor motor voltage $V_{c m}$. The stack current $I_{s t}$ follows a step-wise profile from 150 A to 300 A over a 30-second window, as shown in Figure 4.

Table 3. Simulation environment and software specifications, central processing unit (CPU), random access memory (RAM), operating system (OS)

Two fault scenarios are considered: an actuator fault $\theta_a$ (compressor motor degradation) modelled as a $30 \%$ reduction in the back-EMF constant from $t=8 \mathrm{~s}$ to $t=14 \mathrm{~s}$, followed by a $60 \%$ reduction from $t=14 \mathrm{~s}$ to $t=22 \mathrm{~s}$; and a process fault $\theta_p$ (supply manifold leakage) modelled as a $20 \%$ increase in the cathode inlet orifice coefficient from $t=6 \mathrm{~s}$ to $t=12$ s , followed by a $50 \%$ increase from $t=12 \mathrm{~s}$ to $t=20 \mathrm{~s}$. Introduced simultaneously to assess the decoupled reconstruction capability of the scheme.

Figure 4. System inputs, control signal, and derivative estimation performance

Figure 5. Physical system behavior under fault conditions

Figure 4 also shows the derivative estimation performance of the two differentiators. Brief transient peaks appear at each load step, which is consistent with the abrupt changes imposed on the system. These errors decay rapidly, and accurate tracking of both $\dot{\omega}_{c p} \quad$ and $\quad \dot{P}_{s m}$ is achieved within approximately 2 seconds, in agreement with the finite-time convergence.

The physical effect of the faults on system behavior is illustrated in Figure 5. The manifold pressure and compressor speed deviate from their nominal trajectories once the faults are active. As a consequence, the oxygen excess ratio drops noticeably during load transients, indicating that undetected faults directly compromise cathode oxygen supply and fuel cell efficiency.

The air leakage fault $\theta_p$ is reconstructed as shown in Figure 6. The estimated signal $\hat{\theta}_p$ faithfully tracks the ramp reference after a short initial transient of approximately 1 to 2 seconds. The brief spikes at current step instants are due to the differentiator reconvergence interval and do not affect the steady-state reconstruction accuracy.

Figure 6. Air leakage fault $\hat{\theta}_p$ reconstruction

The motor fault $\theta_a$ is reconstructed as shown in Figure 7. The estimated signal $\hat{\theta}_a$ follows the fault scenario with high fidelity. The proposed scheme is capable of reconstructing both actuator and process fault signals simultaneously and accurately in the presence of load variation, without requiring threshold tuning or asymptotic observer convergence.

Figure 7. The motor fault reconstruction $\hat{\theta}_a$

5.1 Sensitivity and robustness analysis

To test the robustness of the proposed algebraic observer, simulations were carried out in the presence of measurement noise on $y_3=\omega_{c p}$ and $y_4=P_{s m}$. Let $y_{(f a)}=y_3+f_a(t)$ be the noisy measurement of $y_3$ and $y_{\left(f_p\right)}=y_4+f_p(t)$ the noisy measurement of $y_4$, where $f_a$ and $f_p$ are noisy signals shown in Figure 8. It is possible to see that the algebraic observer estimates both faults well enough in spite of the noise in the measurements.

To quantify robustness, 30 independent runs were performed at a signal to noise ratio (SNR) of 40 dB. The resulting fault estimates were identical across all runs (standard deviation $<10^{-20}$), because the robust exact differentiator [24] yields exact derivatives in finite-time, making the algebraic reconstruction deterministic. Table 4 summarises the mean settling times, steady state errors, 95% confidence intervals, and root mean square error (RMSE). The zero standard deviation and degenerate confidence intervals reflect the perfect repeatability of the method. For a more severe scenario (SNR = 20 dB), result is shown in Figure 9. Even with this higher noise level, the estimated faults accurately track the true faults, and the absolute errors remain below $1 \times 10^{(-7)}$ bar for the process fault and $4 \times 10^{(-5)}$ V·s/rad for the actuator fault.

Figure 8. Noise affecting the system outputs (signal to noise ratio, SNR = 20 dB)

Table 4. Statistical performance of fault reconstruction over 30 independent runs (standard deviation (SD), signal to noise ratio (SNR) of 40 dB and root mean square error (RMSE))

The zoomed-in insets (5.9 – 6.2  s) confirm that transients are handled without oscillations. The proposed algebraic observer is highly robust to measurement noise, providing deterministic, finite-time fault reconstruction with negligible steady-state error. To visualise the statistical spread, Figure 10 presents the results under elevated noise (SNR = 20 dB). The shaded area (mean ± 1σ) remains very narrow, demonstrating the method’s robustness even under severe measurement noise.

Figure 9. Fault estimation under noise (SNR = 20 dB)

Figure 10. Estimated faults (mean ± 1σ) vs. true faults under measurement noise (SNR = 20 dB)

5.2 Comparison with the super-twisting sliding-mode observer

To further evaluate the proposed scheme, Table 5 compares its performance with a super-twisting sliding-mode observer (ST-SMO) applied to the same PEMFC air-feed system in terms of convergence, simultaneous fault capability, and chattering. The ST-SMO is able to reconstruct the supply manifold leakage fault $\theta_p$ when considered alone, with a settling time of approximately 0.4 s (Figure 11). However, it does not allow simultaneous estimation of the actuator fault $\theta_a$, and therefore requires a second observer specifically designed for the compressor dynamics.

Table 5. Comparison of the super-twisting sliding-mode observer (ST-SMO) and proposed algebraic fault reconstruction method

In contrast, the proposed algebraic scheme reconstructs both faults simultaneously in approximately 2 s using two decoupled expressions that do not rely on shared state estimation. This significantly simplifies the design and reduces the tuning effort to only two scalar bounds, $L_3$ and $L_4$.

Figure 11. Reconstruction of the supply manifold leakage fault $\hat{\theta}_p$ using the super-twisting sliding-mode observer (ST-SMO)