Closed-loop stability of control systems can be undermined by actuator faults. Redundant actuator sets and Fault-Tolerant Control (FTC) strategies can be exploited to enhance system resiliency to loss of actuator efficiency, complete failures or jamming. Passive FTC methods entail designing a fixed-gain control law that can preserve the stability of the closed-loop system when faults occur, by compromising on the performance of the faultless system. The use of Passive FTC methods is of particular interest in the case of underwater autonomous platforms, where the use of extensive sensoring to monitor the status of the actuator is limited by strict space and energy constraints. In this work, a machine learning-based method is formulated to systematically synthesise control laws for systems affected by actuator faults, encompassing partial and total loss of actuator efficiency and control surfaces jamming. Differently from other methods in this category, the closed-loop stability is formally certified. The learning architecture encompasses two Artificial Neural Networks, one representing the control law, and the other resembling a Control Lyapunov Function (CLF). Periodically, a Satisfiability Modulo Theory solver is employed to verify that the synthesised CLF formally satisfies the theoretical Lyapunov conditions associated to both the nominal and faulty dynamics. The method is applied to three marine test cases: first, an Autonomous Underwater Vehicle performing planar motion and subjected to full loss of actuator efficiency is investigated. Next, a study is conducted on a hybrid Underwater Glider with a pair of independent twin stern planes jamming at a fixed position. Finally, partial loss of effectiveness is considered. In all three scenarios, the system is able to synthesise stabilising control laws with performance degradation prescribed by the user. Unlike other machine-learning based techniques, this method offers formal stability certificates and relies on limited computational resources rendering it possible to be run on unassuming office laptops. An open-source software tool is developed and released at: https://github.com/grande-dev/pFT-ANLC.

Machine learning-based methodologies have recently been adapted to solve control problems. The Neural Lyapunov Control (NLC) method is one such example. This approach combines Artificial Neural Networks (ANNs) with Satisfiability Modulo Theories (SMT) solvers to synthesise stabilising control laws and to prove their formal correctness. The ANNs are trained over a dataset of state-space samples to generate candidate control and Lyapunov functions, while the SMT solvers are tasked with certifying the correctness of the Lyapunov function over a continuous domain or by returning a counterexample. Despite the approach's attractiveness, issues can occur due to subsequent calls of the SMT module at times returning similar counterexamples, which can turn out to be uninformative and may lead to dataset overfitting. Additionally, the control network weights are usually initialised with pre-computed gains from state-feedback controllers, e.g. Linear-Quadratic Regulators. To properly perform the initialisation requires user time and control expertise. In this work, we present an Augmented NLC method that mitigates these drawbacks, removes the need for the control initialisation and further improves counterexample generation. As a result, the proposed method allows the synthesis of nonlinear (as well as linear) control laws with the sole requirement being the knowledge of the system dynamics. The ANLC is tested over challenging benchmarks such as the Lorenz attractor and outperformed existing methods in terms of successful synthesis rate. The developed framework is released open-source at: https://github.com/grande-dev/Augmented-Neural-Lyapunov-Control.

Performance and closed-loop stability of control systems can be jeopardised by actuator faults. Actuator redundancy in combination with appropriate control laws can increase the resiliency of a system to both loss of efficiency or jamming. Passive Fault-Tolerant Control (FTC) systems aim at designing a unique control law with guaranteed stability in both nominal and faulty scenarios. In this work, a novel machine learning-based method is devised to systematically synthesise control laws for systems affected by actuator faults, whilst formally certifying the closed-loop stability. The learning architecture trains two Artificial Neural Networks, one representing the control law, and the other resembling a Control Lyapunov Function (CLF). In parallel, a Satisfiability Modulo Theory solver is employed to certify that the obtained CLF formally guarantees the Lyapunov conditions. The method is showcased for two scenarios, one encompassing the stabilisation of an inverted pendulum with redundant actuators, whilst the other covers the control of an Autonomous Underwater Vehicle. The framework is shown capable of synthesising both linear and nonlinear control laws with minimal hyperparameter tuning and within limited computational resources.