Comparison of the Performance of Metaheuristic Algorithms in Optimal Selection of LQR Weighting Matrices

Active control acts as an effective strategy to improve the seismic behavior of structures by calculating and applying the external forces and leads to the adaptive change in dynamic properties of the structure during the earthquake. In the LQR method as the most common control algorithm, the second-order performance index has been optimized to establish a balance between response and control force reduction. In defining the performance index, the relative importance of response reduction and control force reduction is regulated by the selection of Q and R weight matrices. Q and R are the weight matrices for the response and control forces, respectively, and their values indicate the relative importance of response or control force decrement. There is no systematic way to determine these matrices so far these weights are chosen based on trial and error and designer experience. This study has concentrated on metaheuristic algorithms in optimal active control of structures. In this paper, to facilitate the process of controller design, R and Q weighting matrices are considered as design variables and wavelet-based performance index as objective function of metaheuristic optimization. Imperialist competitive algorithm (ICA) as a socio-politically motivated algorithm, differential evolution (DE) as an evolutionary method, bat (BA) and firefly (FA) as natural-inspired algorithms, colliding bodies (CB) as physics-inspired algorithm and harmony search (HS) as music-inspired metaheuristic are used in this study to compare different aspects of metaheuristics in dealing with the optimal active control problem. Some of these metaheuristics have not been explored in control problem yet. In this study, in order to enhance the controller performance, DWT has been applied to decompose the excitation into different frequency bands. For decomposition process, signal is decomposed using Daubechies wavelet of order 10 (db10) mother wavelet in five levels. Defined wavelet-based performance index has been selected as objective function and weighting matrix elements considered as design variables of optimization process. Then, metaheuristic algorithm has been employed to search the optimum weights for each frequency band to calculate the control force in each domain. In addition to wavelet-based LQR performance index used as objective function of optimization, nine benchmark indices obtained from the results are calculated to evaluate the controller-performance by reduction of different design parameters such as interstory drift, story displacement, acceleration, base shear and needed control force compared to the uncontrolled cases. The best solution, convergence rate and computational effort of aforementioned optimization methods for a three-story structure under different artificial earthquakes have been compared. According to the results of simulations, the effectiveness of each algorithm in vibration reduction through various seismic excitations has been evaluated by numerical simulations. This formula is able to be easily performed with any metaheuristic algorithm providing the great flexibility in the active controller design. The results for active control of structures have shown the potential of wavelet and metaheuristic algorithms in vibration control for building structures depending on the user to select the appropriate algorithm or on the problem types and its requirements such as the quality of solution, convergence rate, computational effort, and consuming time. The results indicate the effective role of metaheuristic methods in reducing responses and control forces simultaneously over the LQR method. Considering total cost value, CB, ICA and DE have produced better results compared to all the mentioned algorithms. FA and BA have good convergence tendency compared to other algorithms. Each metaheuristic algorithm has advantages and disadvantages in solving active control problem, whiles the superiority of CB, ICA and DE over other methods in finding the optimal responses for active control problem has been shown as well.