Practical finite time vibration suppression of mechatronic systems using proportional–integral–derivative variable structure controls with dead-band nonlinearity

Vibration is an intrinsic phenomenon in many mechanical and mechatronic applied devices and undesirable vibration can either degrade the performance of the system or lead to unpredictable outputs. The main purpose of this article is to introduce a novel second-order proportional–integral–derivative sliding mode control methodology to suppress the undesirable vibrations of a class of applied dynamical systems with applications to mechatronic and mechanical devices. After designing a nonlinear proportional–integral–derivative terminal sliding manifold, rigorous mathematics are provided to guarantee that the origin is a practical finite time stable equilibrium point. Consequently, two efficient control laws are proposed to ensure the occurrence of the sliding motion with and/or without system unknown parameters. Motivated by situations encountered in practice, unknown lumped uncertainties are also added to the system and their impacts are tackled using adaptive control techniques. Furthermore, a hard nonlinear dead-band function is utilized in the control input and its effects such as lags and delays appeared on the control signals as well as on the system outputs are dealt with by the proposed proportional–integral–derivative variable structure controller. The proposed second-order variable structure controller not only utilizes the simple effective design approach of the proportional–integral–derivative controllers to ensure a reasonable transient performance, but also displays fast convergence properties demonstrated in non-singular terminal sliding modes. Finally, through simulation studies, it is confirmed that the proposed control strategy is effective in vibration attenuation of microelectromechanical resonators.