Integrating of LMIs with Buckingham Π theorem for control of uncertain oscillating systems

This paper introduces a novel framework for robust controller synthesis in oscillating systems by synergistically combining generalized linear matrix inequalities (GLMIs) with the Buckingham Π theorem. The primary goal is to enhance the design of robust controllers for systems with uncertain parameters through the exploitation of dimensionless groups derived from the Buckingham Π theorem. By establishing a dimensionless state-space representation, the proposed approach demonstrates adaptability to a broad spectrum of similar systems. The LMI-based formulation facilitates the systematic design of controllers capable of effectively mitigating the impact of parametric uncertainties. Simulation results for an uncertain inverted pendulum and a vehicle active suspension system validate the superior performance of the proposed method compared to conventional control techniques, such as MPC, LQG, SMC, PID, and LQR. For the uncertain inverted pendulum system, the proposed robust control scheme achieves a cart position stabilization in 5.35 s and pendulum stabilization in 6.3 s, which is faster than standard control algorithms. In the context of the vehicle active suspension system, the performance indices for sprung mass acceleration, suspension deflection, and tire deflection are 0.19, 0.47, and 0.07, respectively. These values indicate significant reductions when compared to conventional passive suspension systems and standard robust control methodologies.