Fault-Tolerant Control of Two-Time-Scale Systems

Abstract

This work presents fault-tolerant control of two-time-scale systems in both linear and nonlinear cases. An adaptive approach for fault-tolerant control of singularly perturbed systems is used in linear case, where both actuator and sensor faults are examined in the presence of external disturbances. For sensor faults, an adaptive controller is designed based on an output-feedback control scheme. The feedback controller gain is determined in order to stabilize the closed-loop system in the fault-free case and vanishing disturbance, while the additive gain is updated using an adaptive law to compensate for the sensor faults and the external disturbances. To correct the actuator faults, a state-feedback control method based on adaptive mechanism is considered. The both proposed controllers depend on the singular perturbation parameter \(\varepsilon \) leading to ill-conditioned problems. A well-posed problem is obtained by simplifying the Lyapunov equations and subsequently the controllers using the singular perturbation method and the reduced subsystems yielding to an \(\varepsilon \)-independent controller. In the nonlinear case, an additive fault-tolerant control for nonlinear time-invariant singularly perturbed system against actuator faults based on Lyapunov redesign principle is presented. The full-order two-time-scale system is decomposed into reduced slow and fast subsystems by time-scale decomposition using singular perturbation method. The time-scale reduction is carried out by setting the singular perturbation parameter to zero, which permits to avoid the numerical stiffness due to the interaction of two different dynamics. The fault-tolerant controller is computed in two steps. First, a nominal composite controller is designed using the reduced subsystems. Second, an additive part is appended to the nominal controller to compensate for the effect of an actuator fault. In both cases, the Lyapunov stability theory is used to prove the stability provided the singular perturbation parameter is sufficiently small. The designed control schemes guarantee asymptotic stability in the presence of additive faults. Finally, the effectiveness of the theoretical results is illustrated using numerical examples.