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_61: With the development of construction techniques, it is possible to build large-span bridges, pipelines, dams, and other essential structures in seismically active regions or in active faults. A new method to find the optimal control forces for active tuned mass damper is presented in this paper. Three algorithms: discrete wavelet transform (DWT), particle swarm optimization (PSO), and linear quadratic regulator (LQR). DWT are used to obtain the local energy distribution of the motivation over the frequency bands. PSO is used to determine the gain matrices through the online update of the weighting matrices used in the LQR controller while eliminating the trial and error. The method is tested on a 10-story structure subject to several historical pulse-like near-fault ground motions. The results presented in this paper indicate that the proposed method is more effective at reducing the displacement response of the structure in real time than conventional LQR controllers.
  @cite_38: This paper presents the design of an active suspension control of a two-axle railway vehicle using an optimized linear quadratic regulator. The control objective is to minimize the lateral displacement and yaw angle of the wheelsets when the vehicle travels on straight and curved tracks with lateral irregularities. In choosing the optimum weighting matrices for the LQR, the Particle Swarm Optimization (PSO) method has been applied and the results of the controller performance with weighting matrices chosen using this method is compared with the commonly used, trial and error method. The performance of the passive and active suspension has also been compared. The results show that the active suspension system performs better than the passive suspension system. For the active suspension, the LQR employing the PSO method in choosing the weighting matrices provides a better control performance and a more systematic approach compared to the trial and error method.
  @cite_47: The application of LQR controller in inverted pendulum system is common, but the choice of weight matrix Q and R of LQR controller has always been a difficult. This paper presents a new method to achieve the double inverted pendulum system LQR controller parameters optimization through the particle swarm optimization algorithm. The simulation result shows that the method used in the choice of matrix Q and R achieves very good results.
  @cite_64: One of the main observed problems in the control of automatic generation control systems is the limitation to access and measurement of state variables in the real world. In order to solve this problem, an optimal output feedback method, the linear-quadratic regulator controller, is used. In the output feedback method, only measurable state variables within each control area are required to use for feedback. But in order to improve dynamic performance and provide a better design for this controller, the concept of an intelligent regulator is added to the linear-quadratic regulators; as a result, the particle swarm optimization based linear-quadratic output feedback regulator and the imperialist competitive algorithm based linear-quadratic output feedback are proposed to calculate the global optimal gain matrix of con- troller intelligently. The optimal control law of this controller must be determined by minimizing a performance index under the output feedback conditions leading to coupled matrix equations. In conventionalmethods, the control law is handled by pole placement, iterative, or trial-and-error methods for choosing controller gains; thus, intelligent optimization techniques are applied to solve this problem. The proposed controllers are tested on a two-area automatic generation control power system, and a complete comparison between the proposed output feedback controllers with adaptive weighted particle swarm optimization, particle swarm optimization, the imperialist competitive algorithm, and a conventionaloutput feedbackcontroller is presented.The results show that the proposed intelligent controller improved the dynamic response of the system faster than the conventional controller and provided a control system that satisfied the load frequency control requirements.
  @cite_7: The main focus of this paper is to develop an optimization method for the automatic fighter tracking (AFT) problem. The AFT problem is similar to a general evader-pursuer maneuvering automation problem between the dynamic systems of two highly interactive objects. This paper proposes a particle swarm optimizer-based variable feedback gain controller (PSO-based VFGC) for dealing with AFT problems. The PSO-based VFGC is designed to obtain the control value of a pursuer through an error-feedback gain controller. Once conditions of system closed-loop stability have been satisfied, the optimal feedback gains can be obtained through PSO, and the actual control values can be derived from the obtained values. Simulation results confirm the capabilities of the proposed method by comparing the results against two other methods in the field: the weight matrix value defined Ricatti equation, and the linear matrix inequality (LMI) based linear quadratic regulator (LQR). The performance of the proposed method is superior to that of its alternatives.
  @cite_9: Abstract This paper presents an optimized reconfigurable control design methodology by separating control commands distribution task from flight controller for different types of fault handling. The proposed strategy improves the flight control performance in normal and fault situations. The particle swarm optimization (PSO) based multi-input multi-output (MIMO) linear quadratic regulator (LQR) is used to produce virtual command signals. A modified weighted pseudo-inverse (WPI) based cascaded re- allocation technique is employed for effective implementation of commands to redundant control surfaces in a realistic nonlinear aircraft benchmark model. Control surface fault modelling is performed for the evaluation of optimized reconfiguration based modular flight control strategy. Simulation results show that acceptable fault tolerant control (FTC) performance can be achieved by using swarm intelligence based optimization technique for modular control design.
  @cite_21: 
  @cite_57: The unified power-quality conditioner (UPQC) is used to provide balanced and sinusoidal source currents and load voltages under unbalanced and distorted three-phase supply voltages and load currents in a power distribution network. In this paper, a particle swarm optimization-based feedback controller is designed for UPQC. This feedback controller has optimal performance in various operating conditions and is robust to parametric uncertainties when compared to the conventional feedback controllers, such as the linear quadratic regulator. The proposed method is implemented on a distribution system and detailed simulation and experimental results are presented.
  @cite_59: In this paper, state feedback control design for tracking control of a flexible link manipulator is considered. The computation of state feedback control gains is conventionally handled by pole placement method or LQR method via Riccati equation. Unfortunately, they still possess trial-and-error approach in choosing their parameters. Particularly, choosing elements of Q and R matrices in the state feedback control design using LQR method has to be done by trial. Therefore, an intelligent method to resolve this problem is proposed by adopting PSO-based optimization. Experimental work is carried out to evaluate effectiveness of the proposed method.
  @cite_2: This paper established a mathematical model of the inverted pendulum system and put forward a design method of LQR based on PSO. A LQR is designed by this method according to the computer simulation experiments. The simulation results show that the inverted pendulum system with the LQR has good performance. Then the real-time control of the pendulum has been accomplished successfully based on the real-time control model built in Simulink. The LQR designed in the computer simulation experiments is applied to real-time controls. The real-time control experiments also validated the correctness of the theoretical analysis and the rightness of the computer simulation.
  @cite_15: In this paper, the state feedback control design problem for tracking control of a flexible link manipulator is considered. The calculation of state feedback control gains is conventionally handled by pole placement method or LQR method via Riccati equation. Unfortunately, they still possess trial and error approach of choosing some parameters. Particularly, choosing elements of Q and R matrices in the state feedback control using LQR method has to be done by trial. Therefore, an intelligent method to resolve this problem is proposed by adopting PSO algorithm. The experimental work is carried out to evaluate the effectiveness of the proposed method.
summary: In @cite_15 a method is proposed to determine the weighting matrices using PSO with pole region constraint for controlling a flexible-link manipulator. In @cite_38 it is suggested that PSO-based LQR produces better result compared to trial-and-error approach for the active suspension system. In @cite_64 authors applied ordinary LQR, PSO-based LQR, AWPSO-based LQR and ICA-based LQR for optimal load frequency control and concluded that AWPSO-based LQR outperforms other approaches in terms of settling-time and maximum overshoot. A PSO-based optimal real-time LQR controller for stabilizing an inverted pendulum system is proposed in @cite_2 . In @cite_61 wavelet-PSO is proposed for tuning LQR controllers and is applied to an optimal structural control. In @cite_57 it is concluded that contrary to trial-and-error approach, a PSO-based state feedback controller does not have sub-optimal performance in case of partial state feedback. @cite_21 proposed a quantum-behaved PSO algorithm for tuning a given LQR controller. They exploited a conventional weighted aggregation of control objectives which can only provide the designer with one solution. More studies on PSO-based LQR design can be found in @cite_9 @cite_7 @cite_59 @cite_47 .