Fault-tolerant control of nonlinear systems: An inductive synthesis approach

TL;DR

This work tackles fault-tolerant control for nonlinear systems subject to actuator faults and input saturation. It advances a counterexample guided inductive synthesis (CEGIS) framework to design fixed-gain, saturation-aware passive fault-tolerant controllers for nonlinear dynamics by reformulating the problem as an uncertain system and solving Lipschitz-stable LMIs within a learning-verification loop. The approach delivers finite-time convergence guarantees, outperforms conventional $\mathcal{H}_{\infty}$ and nonlinear MPC baselines in robustness and domain of attraction, and is computationally efficient enough for embedded deployment. The method is demonstrated on hover-capable AUVs in both simplified 5D models and higher-fidelity OpenMAUVe simulations, including fault injections on multiple thrusters, with substantial reductions in memory usage and computation time compared to MPC. This yields practical, energy-conscious fault-tolerant control suitable for cyber-physical systems operating under strict resource constraints.

Abstract

Actuator faults heavily affect the performance and stability of control systems, an issue that is even more critical for systems required to operate autonomously under adverse environmental conditions, such as unmanned vehicles. To this end, passive fault-tolerant control (PFTC) systems can be employed, namely fixed-gain control laws that guarantee stability both in the nominal case and in the event of faults. In this paper, we propose a counterexample guided inductive synthesis (CEGIS)-based approach to design reliable PFTC policies for nonlinear control systems affected by partial, or total, actuator faults. Our approach enjoys finite-time convergence guarantees and extends available techniques by considering nonlinear dynamics with possible fault conditions. Extensive numerical simulations illustrate how the proposed method can be applied to realistic operational scenarios involving the velocity and heading control of autonomous underwater vehicles (AUVs). Our PFTC technique exhibits comparatively low synthesis time (i.e. minutes) and minimal computational requirements, which render it is suitable for embedded applications with limited availability of energy and onboard power resources.

Figures (14)

• Figure 1: At every iteration $k \in \mathbb{N}_+$, the learner proposes a candidate function $h_k(z)$, while the verifier checks its validity through $r(h(z)) \le 0$.
• Figure 2: The region identified by the union of the green "balls" centred on each vertex pair of matrices $\{(\hat{A}_i, \hat{B}_i)\}_{i=1}^{V_k}$ (red dots, among all the other samples) and the associated set of matrices with eigenvalues $(\epsilon/\lambda_{\textrm{max}}^2(P))$-distant from those belonging to $\textrm{cl}(\textrm{conv}(\mathcal{S}_k))$ (green line and its interior, with $\textrm{conv}(\mathcal{S}_k)$ represented by the blue dashed line), denotes the actual portion of volume where it is guaranteed that no counterexample can be found. As the iterations of Algorithm \ref{['algo:OverallCegis']} progress, the resulting area will cover the whole of $\Omega$, regardless of its shape.
• Figure 3: Hover-capable AUV with three (fixed) thrusters moving in the horizontal plane.
• Figure 4: Comparison of our proposed controller $K_\mathrm{IS-sat}$ with the two $\mathcal{H}_\infty$ controllers for tracking a constant reference $\bar{x} = [0.5, 0]$. The top plot shows the norm of the control input over time. The middle plot shows the control action applied to each input channel, with gray lines indicating saturation limits. The bottom plot shows the norm of the tracking error (on both components). The three vertical coloured regions indicate the three fault modes.
• Figure 5: Comparison of the our controller $K_\mathrm{IS-sat}$ with the aggressive $\mathcal{H}^a_\infty$ controller from grande2024passive for tracking a sinusoidal reference signal. In the third plot, the solid line represents the reference trajectory, with $x_1$ in blue and $x_2$ in orange.

Theorems & Definitions (6)

• Lemma 2.1
• Lemma 3.1: horn1994topics
• Theorem 3.2
• Remark 3.3
• Theorem 3.4
• Remark 3.5