To improve the controllability of an electro-hydraulic position servo control system while simultaneously enhancing the anti-jamming ability of a PID controller, a compound PID controller that combines the beetle antennae search algorithm with PID strategy was proposed, and used to drive the position servo control system of the electro-hydraulic servo system. A BAS-PID controller was designed, and the beetle antennae search algorithm was used to tune PID parameters so that the disturbance signal of the system was effectively restrained. Initially, the basic mathematical model of the electro-hydraulic position servo control system was established through theoretical analysis. The transfer function model was obtained by identifying system parameters. Then, the PID parameter-tuning problem was converted into a class of three-dimensional parameter optimization problem, and gains of PID controllers were adjusted using the beetle antennae search algorithm. Finally, by comparing the effectiveness of different algorithms, simulation and experimental results revealed that the BAS-PID controller can greatly enhance the performance of the electro-hydraulic position servo control system and inhibit external disturbances when different interference signals are used to test the system’s robustness.

For shortcomings of poor exploaration and parameter complexities of the butterfly optimization algorithm, an improved butterfly optimization algorithm based the self-adaption method (SABOA) was proposed to extremely enhance the searching accuracy and the iteration capability. SABOA has advantages of having fewer parameters, the simple algorithm structure, and the strong precision. First, a new fragrance coefficient was added to the basic butterfly optimization algorithm. Then, new iteration and updating strategies were introduced in global searching and local searching phases. Finally, this paper tested different optimization problems including low-high functions and constrained problems, and the obtained results were compared with other well-known algorithms, this paper also drew various mathematical statistics figures to comprehensively analyze searching performances of the proposed algorithms. The experimental results show that SABOA can get less number of function evaluations compared to other considered algorithms, which illustrates that SABOA has great searching balance, large exploration, and high iterative speed. INDEX TERMS Butterfly optimization algorithm, global optimization, constrained problem. I. INTRODUCTION In applied mathematics and engineering fields, there are numerous optimization problems whose calculated solutions are in a large and complex searching space. Traditional optimization methods, including the steepest descent method, the conjugate gradient method, the variable scale method, and Newton method, can only deal with objective functions that are simple, continuously differentiable, and high order differentiable [1]-[3]. With the increasing of problem diversities and problem complexities, traditional optimization methods can not meet different requirements of higher calculation speed and lower average percentage error, so it is crucial to find for new optimization methods that have fast calculation speed and perfect convergence abilities [4], [5]. With the development of artificial intelligence, digitization, and computer technologies, numerous meta-heuristic optimization algorithms have been increasingly proposed and applied in science and engineering fields [6]-[8]. Meta-heuristic algorithms own characteristics of selforganizing, mutual compatibility, simplicity, parallelism, wholeness and harmony. Meta-heuristic algorithms work The associate editor coordinating the review of this manuscript and approving it for publication was Jagdish Chand Bansal.

To improve the control ability of proportional–integral–derivative (PID) controllers and increase the stability of force actuator systems, this paper introduces a PID controller based on the self-growing lévy-flight salp swarm algorithm (SG-LSSA) in the force actuator system. First, the force actuator system model was built, and the transfer function model was obtained by the identification of system parameters identifying. Second, the SG-LSSA was proposed and used to test ten benchmark functions. Then, SG-LSSA-PID, whose parameters were tuned by SG-LSSA, was applied to the electro-hydraulic force actuator system to suppress interference signals. Finally, the temporal response characteristic and the frequency response characteristic were studied and compared with different algorithms. Ten benchmark function experiments indicate that SG-LSSA has a superior convergence speed and perfect optimization capability. The system performance results demonstrate that the electro-hydraulic force actuator system utilized the SG-LSSA-PID controller has a remarkable capability to maintain the stability and robustness under unknown interference signals.

To enhance the anti-interference capability of an electrohydraulic force servo control system and increase the efficiency of the PID controller, this paper proposes a LBAS-PID controller. In LBAS, the random step created by the Lévy flight trajectory was used in the original algorithm to enhance the diversity of the population and convergence speed. In the force servo control system, LBAS-PID can enhance the performance of the system. First, the basic mathematical model of an electrohydraulic force servo control system was built based on theoretical analysis. The transfer function model was obtained by identifying the system parameters. Second, the introduced Lévy flight beetle antennae search algorithm was introduced and applied to ten benchmark functions, and the results were compared with those of other algorithms. Then, the proposed algorithm was applied in the PID controller to tune PID parameters in the force servo control system. To comprehensively evaluate performances of an electrohydraulic force servo control system that is controlled by the LBAS-PID controller, the frequency response analysis and temporal response analysis were obtained. The numerical analysis results indicate that an electrohydraulic force servo control system with an LBAS-PID controller could substantially increase the control characteristics of the system and restrain the external disturbances when different interference signals are examined.

Metaheuristic algorithms are often applied to global function optimization problems. To overcome the poor real-time performance and low precision of the basic salp swarm algorithm, this paper introduces a novel hybrid algorithm inspired by the perturbation weight mechanism. The proposed perturbation weight salp swarm algorithm has the advantages of a broad search scope and a strong balance between exploration and exploitation and retains a relatively low computational complexity when dealing with numerous large-scale problems. A new coefficient factor is introduced to the basic salp swarm algorithm, and new update strategies for the leader position and the followers are introduced in the search phase. The new leader position updating strategy has a specific bounded scope and strong search performance, thus accelerating the iteration process. The new follower updating strategy maintains the diversity of feasible solutions while reducing the computational load. This paper describes the application of the proposed algorithm to low-dimension and variable-dimension functions. This paper also presents iteration curves, box-plot charts, and search-path graphics to verify the accuracy of the proposed algorithm. The experimental results demonstrate that the perturbation weight salp swarm algorithm offers a better search speed and search balance than the basic salp swarm algorithm in different environments.