PID Control Design for Second-Order Systems with Time-Delay via Frequency Response
Second Order Systems, Time-Delay Systems, Receptance Matrix, Genetic Algorithm.
Phenomena such as mechanical vibrations, resonance and oscillations can be mathematically described by systems of second-order differential equations, these systems being commonly referred to as second-order systems. Working with this type of model, rather than first-order state models, has numerical benefits, but there are inherent difficulties in determining its physical parameters. The challenges are even more significant when considering the existence of delays between state measurements and actuation signals, leading some approaches to the need for further analysis to determine the stability of the calculated solutions.
An alternative to overcome the difficulties of measuring parameters is the frequency response approach that uses models based on receptance. It is recognized in the literature that more accurate receptance models can be obtained experimentally when compared to models by differential equations. In addition, the receptance models allow to treat the delay explicitly, without resorting to approximations.
In recent works, a regulatory control method by state feedback was developed for this type of system, using the receptance approach, which guarantees robust stability of the closed-loop system in the face of model uncertainties, in addition to partial pole allocation. Optimization via Genetic Algorithm is used to determine controller gains.
The objective of this work is to develop a method based on receptance for the robust tracking of references of second-order systems with delay, using PID - Proportional-Integral-Derivative controllers. The integral action of this controller guarantees constant reference tracking, as long as the closed loop system is stable. The robustness conditions and the search via optimizationsdescribed above must, therefore, be extended to this type of controller. Both the single-variable case and the case of systems with multiple inputs and outputs must be analyzed. The effectiveness of the technique must be evaluated through numerical simulations, using models available in the literature.