text: A multi-objective design approach for tuning LQR controllers is proposed.A new reinforced quantum-behaved PSO algorithm is introduced.Comparative studies among nine various techniques are conducted. A novel and generic multi-objective design paradigm is proposed which utilizes quantum-behaved PSO (QPSO) for deciding the optimal configuration of the LQR controller for a given problem considering a set of competing objectives. There are three main contributions introduced in this paper as follows. (1) The standard QPSO algorithm is reinforced with an informed initialization scheme based on the simulated annealing algorithm and Gaussian neighborhood selection mechanism. (2) It is also augmented with a local search strategy which integrates the advantages of memetic algorithm into conventional QPSO. (3) An aggregated dynamic weighting criterion is introduced that dynamically combines the soft and hard constraints with control objectives to provide the designer with a set of Pareto optimal solutions and lets her to decide the target solution based on practical preferences. The proposed method is compared against a gradient-based method, seven meta-heuristics, and the trial-and-error method on two control benchmarks using sensitivity analysis and full factorial parameter selection and the results are validated using one-tailed T-test. The experimental results suggest that the proposed method outperforms opponent methods in terms of controller effort, measures associated with transient response and criteria related to steady-state.
  @cite_4: Aiming at the difficulty of designing weighting matrices for linear quadratic regulator (LQR), a multi-objective evolution algorithm (MOEA) based approach is proposed. The LQR weighting matrices, state feedback control rate and optimal controller are obtained by means of establishing the multi-objective optimization model of LQR weighting matrices and applying MOEA to it, which makes control system meet multiple performance indexes simultaneously. Controller of double inverted pendulum system is designed using the proposed approach. Simulation results show that it has shorter adjusting time, smaller amplitude value deviating from steady-state than the pole assignment LQR weighting matrices design approach. So the validity of the proposed approach is confirmed.
  @cite_22: This paper introduces an application of Multi-Objective Evolution Algorithm (MOEA) to design Q and R weighting matrices in Linear Quadratic regulators (LQR). Considering the difficulty of designing weighting matrices for a linear quadratic regulator, a multi-objective evolutionary algorithm based approach is proposed. The LQR weighting matrices, state feedback control rate and consequently the optimal controller are obtained by means of establishing the multi-objective optimization model of LQR weighting matrices and applying MOEA to it, which makes control system meet multiple performance indexes simultaneously. Controller of double inverted pendulum system is designed using the proposed approach. Simulation results show that it has shorter adjusting time and smaller amplitude value deviating from steady-state than a Non-dominated Sorting Genetic Algorithm LQR ( NSGA- LQR )weighting matrices design approach.
summary: They are a few studies that exploit multi-objective design for tuning LQR controllers. In @cite_4 a multi-objective evolutionary algorithm is presented to control a double inverted pendulum system. They concluded that multi-objective approach results in shorter adjusting time and smaller amplitude value deviating from steady-state in comparison with the pole assignment design approach. @cite_22 also applies a multi-objective evolutionary algorithm to stabilize a double inverted pendulum system. Their results suggest that multi-objective approach results in shorter adjusting time and smaller amplitude value deviating from steady-state in comparison to non-dominated sorting GA. As far as the authors' knowledge is concerned, it is the first time that a multi-objective derivative of the PSO algorithm is applied for optimal design of LQR controllers.