Piezoelectric-actuated precision positioning stages are widely used in high-precision instruments and high-end equipment due to their advantages of high resolution, fast response, and compact size. However, due to the strong nonlinearity of hysteresis, the presence of hysteresis in piezoelectric actuators seriously affects the positioning accuracy of the system. In addition, it is challenging to identify the model parameters for hysteresis. In this paper, an adaptive backstepping time delay control method is proposed for piezoelectric devices system with unknown hysteresis. Firstly, the Bouc–Wen model is used to describe the hysteresis characteristics, and the model is interpreted as a linear term and a bounded uncertain hysteresis term. Then, the time delay estimation technique is used to estimate the hysteresis term of the Bouc–Wen model online, and the unknown parameters of the system and hysteresis model are obtained through adaptive updating laws. Furthermore, the stability of the control scheme is proved based on Lyapunov stability theory. Finally, the effectiveness and superiority of the proposed control scheme are demonstrated by comparing it with two typical hysteresis compensation control algorithms through three different sets of input signals.

An optimization technique seeks to identify the best solution for a given problem. When the problem involves a single objective function, the optimal solution yields the best value for that objective. However, in cases with multiple objectives, selecting the best solution becomes more complex, as these objective functions often conflict with one another. For such multi-objective optimization problems, relying on analytical or exact methods is often not feasible. This paper proposes a modified MFO called multi-objective moth–flame optimization based on self-organizing mapping (MMFOSOM) to solve multi-objective optimization problems. For attaining high-quality initial moths, dual opposition-based learning mechanism is employed to generate two distinct populations, referred to as the exploration moth (EM) and the auxiliary moth (AM). In order to enhance the search capabilities of the algorithm, each population is assigned a SOM network, the adaptive neighborhood structure of which contributes to forming modal clusters in the mapping, organizing similar solutions together. In order to select the excellent individual as the training data for SOM, a novel environmental selection mechanism is introduced. To validate the effectiveness of the algorithm, experiments will be conducted on a set of multi-objective optimization test functions. The experimental results indicate that the proposed algorithm can effectively address multi-objective optimization problems.

Due to the complexity of multi-objective engineering problems, solutions obtained by many algorithms often exhibit poor distribution, and the algorithms tend to fall into local optima. To effectively alleviate these issues, an improved multi-objective moth–flame optimization algorithm (IETMFO) is proposed in this paper, with three core novelties: A hybrid mutation mechanism (integrating two mutation techniques) is first used to generate a new population, and then an indicator-based selection mechanism is adopted to produce a high-quality initial population, enhancing solution distribution. Enhanced Brownian motion is introduced as a local search strategy to reduce the risk of falling into local optima. An improved flame update mechanism is incorporated to maintain flame diversity, boosting the algorithm’s adaptability. The IETMFO is tested on 49 benchmark functions and 6 constrained engineering problems, and then compared with seven well-known algorithms (including NSGA-II, MOEA/D, and traditional MFO). The experimental results show the following: in benchmark function tests, IETMFO reduces the IGD value by an average of 32.7% and increases the HV value by an average of 28.5% compared with NSGA-II; on ZDT series functions, it outperforms the seven contrast algorithms in solution distribution uniformity; in the six engineering problems, its optimal solution proportion reaches 66.7%, and the risk of falling into local optima is reduced by 41.2%. These results demonstrate that the IETMFO achieves competitive performance in addressing multi-objective engineering problems.