Proportional-integral-derivative-acceleration robust controllers for vibrating systems

This paper presents a design framework to obtain a robust multivariable Proportional-Integral-Derivative (PID) controller for second-order linear vibrating systems. A Proportional-Integral-Derivative plus acceleration (PIDA) controller is also proposed to deal with the regularization problem. Relevant control challenges, such as modeling error, regulatory performance optimization, regional pole placement, saturation avoidance, and constant reference tracking are handled within the proposed Linear Matrix Inequality (LMI) design approach. The design strategy is obtained from a linear transformation that can be applied to achieve constant reference tracking for an actuated subspace of underactuated systems. Moreover, the integral action has two additional objectives: (1) to improve regulatory performance in the presence of constant disturbance and (2) to increase the design degree of freedom in order to robustly achieve closed-loop specifications. Three simulation case studies are used to highlight the benefits of the PID and PIDA controllers.