Proportional integral derivative (PID) controllers are widely used to solve different control engineering problems. To know the dynamic behaviour of a working plant by mathematical modelling is quite challenging. Finite element method (FEM) is a well-known technique and broadly used for the modelling of engineering systems. This article presents the FEM-based heuristic approach to design and optimize the PID controllers. The ‘allowed area method’ has been used for the formulation of the objective function followed by the tuning of the PID controller. First, the proposed approach is tested on 2-degree of freedom (DOF) mass-spring-damper (MSD) system. FEM modelling of 2-DOF MSD system with PID controller has been carried out in COMSOL Multiphysics and coupling of particle swarm optimization (PSO) has been carried out with the FEM model of the MSD system, for the optimization of PID controller gain. The FEM results are in good agreement with the analytical one. Next, the established method is applied to design and optimize the PID controller gain to control the vibration of a cantilever beam using piezoelectric actuator. Similar to the MSD system, FEM modelling of PID controller for the smart beam has been carried out in COMSOL Multiphysics, and the coupling of PSO is carried out with the FEM model of the smart beam for the optimization of PID controller gain. Simulation of the uncontrolled and controlled responses of the smart beam is carried out at the optimum controller gain for free vibration and step excitation. The piezoelectric actuator of smart beam has successfully damped the vibration within approximately 2.5 s.
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