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Optimal Tuning of Linear Quadratic Regulators Using Quantum Particle Swarm Optimization

Linear Quadratic Regulator (LQR) is an optimal multivariable feedback control approach that minimizes the excursion in state trajectories of a system while requiring minimum controller effort. The behaviour of a LQR controller is determined by two parameters: state and control weighting matrices. These two matrices are main design parameters to be selected by designer and greatly influence the success of the LQR controller synthesis. However, it is not a trivial task to decide these two matrices. The classic approaches such as trial-and-error, Bryson’s method, and pole placement are labour-intensive, time consuming and do not guarantee the expected performance. Furthermore, these techniques only aim to minimize the quadratic performance index and do not consider other control objectives such as minimizing the overshoot, rise time, settling time, and steady state error. In this paper, for the first time, we apply quantum particle swarm optimization (QPSO) algorithm to automatically and optimally adjust weighting matrices. QPSO is an extension of conventional PSO algorithm, in which particles obey the quantum mechanics rather than Newtonian mechanics. We applied the proposed approach to stabilize an inverted pendulum system. The results suggest that QPSO-based LQR outperforms LQR tuned by trial-and-error, genetic algorithm and conventional PSO methods in terms of rising time, settling time and quadratic performance index. Also, it is competitive with mentioned approaches in terms of maximum overshoot percentage and steady-