Optimal sinusoidal disturbances damping for singularly perturbed systems with time-delay

This paper considers an optimal damping vibration control problem for linear time-delay singularly perturbed systems affected by external sinusoidal disturbances. By using the slow–fast decomposition theory of singularly perturbed systems, the system is first decomposed into a fast subsystem without time-delay and a slow subsystem with time-delay affected by external sinusoidal disturbances. Then, the successive approximation approach (SAA) and the feedforward compensation techniques are proposed to solve the slow-time scale time-delay optimal control problem and damp the external sinusoidal disturbances, respectively. The conditions of existence and uniqueness of the finite- and infinite-horizon feedforward and feedback composite control (FFCC) laws are presented, and the design approaches are given. The FFCC laws consist of nondelay feedback terms, disturbances compensation terms and a time-delay compensation term which is the limit of the solution sequence of the adjoint vector equations. Simulation examples indicate that the SAA is valid, and the FFCC laws are easy to implement, and more effective with respect to damping the external sinusoidal disturbances than that of the classical feedback composite control (FCC) laws.