Adaptive Actuator Failure Compensation Control of Uncertain Nonlinear PDE–ODE Cascaded Systems

In this article, the stabilization problem is addressed for a class of reaction–diffusion equation described by a cascaded partial differential equation (PDE)–ordinary differential equation (ODE) system. Because of the inherent unknown nonlinearity and actuator failure existing in the considered system, it is fundamentally different from the systems studied in most related studies. To simplify the stability analysis, the original PDE–ODE cascaded system is redescribed as a new target system by leveraging finite- and infinite-dimensional backstepping techniques. In the newly transformed system, the compensation term in the form of the smoothing function is used to eliminate the influence caused by actuator faults. Then, a novel fault-tolerant control strategy is further developed, which can not only eliminate the influence of uncertainty, but also overcome the difficulties of actuator faults. The system stability and the asymptotic convergence of all states in the original system are proved by applying the Lyapunov function theory. The simulation example shows that the presented control method ensures the realization of the control objectives.