Projeto de controladores PID multivariáveis para sistemas de segunda ordem com atraso

Abstract

Second-order systems, which originate from second-order differential equations, play a crucial role in control system studies due to their broad capability to model practical situations, such as mechanical vibrations, resonance, and even specific processes in areas like fermentation in certain biological applications. Working with this type of model, instead of first-order state models, offers advantages such as enabling the design of more sophisticated controllers and providing a clearer frequency-domain analysis. In practice, these systems are affected by inherent delays in various components, such as actuators and sensor measurements, which can lead to performance degradation, instability, and oscillations. Since even small delays can cause significant disturbances, it is beneficial to adopt an approach based on the system’s receptances, which accurately represent delays through a frequency-domain analysis. This dissertation aims to develop multivariable PID controllers for linear dynamic systems with multiple inputs and outputs (MIMO) that exhibit actuation delays and are described by second-order differential equation systems. The proposed technique employs a frequency-domain model known as the receptance matrix, which can be precisely obtained experimentally and allows for the exact treatment of delay effects without approximations. Closed-loop stability is ensured using the socalled generalized Nyquist criterion, verified through eigenloci diagrams, with a given stability margin imposed by enforcing a minimum distance between the eigenloci and the critical stability point. An optimization problem is formulated to compute the PID controller gains that ensure this margin while minimizing a time-domain performance index, which is solved using a Genetic Algorithm. Different delays for each actuator and various PID controller configurations are considered, with numerical examples illustrating the proposed approach.