HSEs, as critical vertical transport, face escalating HV amplitudes due to continuously increasing operating speeds (Li et al., 2024; Fan et al., 2024). This not only degrades passenger ride quality but also threatens the system’s long-term stability. Traditional passive damping systems, such as spring systems, have clear shortcomings. They are ineffective against uneven guideways, dynamic loads, and complex excitations, which leads to imprecise vibration control. Crucially, the optimization of quantization factors in variable-domain fuzzy PID control has relied on empirical trial and error. This has resulted in poor parameter demand matching, vague adjustment ranges, and inadequate condition adaptability. To resolve these limitations, this study proposes an intelligent control method integrating a BPNN with an optimized NSGA-II-A. This approach ensures perfect harmony between the optimization algorithm and control strategy to address the limited adaptability of traditional methods in nonlinear vibration management. The study firstly improves the performance of the algorithm through an adaptive mechanism, and designs dynamically adjusted formulas for calculating the crossover probability P c and the variance probability P m , as shown in Equation 1. The fairness of comparative algorithm tuning is ensured through a rigorous parameter optimization protocol: All comparison algorithms (standard NSGA-II, MOEA/D, and SPEA2) undergo identical Bayesian optimization procedures with the same computational budget of 1,000 function evaluations per algorithm. Each algorithm’s key parameters are independently optimized using the same hyperparameter tuning framework, with convergence criteria standardizes across all methods to eliminate tuning bias. The parameter selection process employs a systematic approach where key algorithm parameters are determined through extensive sensitivity analysis. The crossover probability P c is set to 0.9 based on Pareto efficiency tests that balances exploration and exploitation capabilities, while the mutation probability P m is optimized to 0.1 through convergence stability analysis to maintain population diversity without compromising search efficiency.

Unlike traditional adaptive NSGA-II variants, such as ε-NSGA-II and adaptive crowding NSGA-II, which use fixed or semi-fixed thresholds to control diversity, the proposed mechanisms use adaptive thresholds and dynamic probabilities. These are better suited to handling the variable nature of objective functions and real-time control requirements in elevator vibration control. This ensures better adaptability and efficiency in parameter optimization. Through grid search optimization, the adjustment coefficients a and b in Equation 1 are calibrated to 0.5 and 0.3, respectively, ensuring a smooth transition between the global and local search phases across generations. P c = P c m a x − P c m a x − P c m i n × g / G a P m = P m m i n + P m m a x − P m m i n × g / G b (1)

In Equation 1, g displays the current generation. G displays the maximum quantity of generations. a and b are the adjustment coefficients. This formula enables the algorithm to maintain a strong global search capability at the initial stage and enhance the local development capability at the later stage. Parameter sensitivity analysis reveals that increasing a beyond 0.7 accelerates convergence but risks premature termination at local optima, while decreasing a below 0.3 enhances global search but significantly prolongs convergence time. Similarly, b values above 0.5 strengthen local exploitation but reduce population diversity, whereas values below 0.1 maintain diversity but slow down convergence rate. Second, the traditional congestion calculation adopts a fixed threshold, which is difficult to adapt to the dynamic change of population diversity. Therefore, the study proposes a dynamic threshold adjustment strategy, as shown in Equation 2 (Chen et al., 2023). The real-time computing feasibility is quantitatively assessed through comprehensive performance profiling: the average computation time per control cycle is measured at 2.3 ms with a standard deviation of 0.4 ms, well within the 10 ms real-time constraint for 100 Hz control frequency. The worst-case execution time is bounded at 5.1 ms through code optimization. The computational load is quantified at 15.7 MIPS (million instructions per second), utilizing only 35% of the available processing capacity on the target embedded platform (ARM Cortex-A53@1.2 GHz). Memory requirements are optimized to 2.3 MB RAM and 1.7 MB flash, fitting within the resource constraints of industrial elevator controllers. Anti-aliasing filters and a fixed sampling rate of 1 kHz address sampling limitations, providing 10 samples per control cycle. The 12-bit ADC resolution ensures sufficient signal fidelity for vibration control applications. The dynamic crowding distance threshold parameters are carefully selected through MOO: σ i n i t i a l is set to 0.2 to ensure adequate diversity preservation in early generations, while σ f i n a l is optimized to 0.05 to provide precise convergence in final stages. The decay rate is calibrated to ensure smooth transition between these extremes. σ d y n a m i c = σ i n i t i a l × 1 − g / G + σ f i n a l × g / G (2)

Equation 2 defines the dynamic crowding distance threshold σ d y n a m i c that decreases adaptively from the initial value σ i n i t i a l to the final target σ f i n a l as generations progress. The σ range from 0.2 to 0.05 is optimized through extensive parameter sensitivity tests, where values below 0.05 lead to premature convergence while values above 0.2 results in insufficient selection pressure. This range ensures 85% population diversity maintenance in early stages and 92% convergence precision in final generations. Experimental analysis indicates that when the σ i n i t i a l value falls below 0.15, population diversity declines rapidly while exploration efficiency on the Pareto front deteriorates. Conversely, when the σ i n i t i a l value exceeds 0.25, computational overhead increases without yielding significant performance gains. Similarly, σ f i n a l values below 0.03 lead to over-crowding and solution clustering, while values above 0.08 reduce the precision of final solution selection. This dynamic adjustment promotes population diversity during early evolutionary stages while enhancing convergence precision in later stages. Crowding is only accounted for in the congestion calculation when the inter-individual distance is less than σ d y n a m i c . This improvement improves the uniformity of the distribution of the Pareto front. After improving the NSGA-II-A for adaptive cross-variance and dynamic congestion calculation, the problems of low efficiency of objective function calculation and high parameter sensitivity still exist.

This study introduces adaptive crossover-mutation probabilities and dynamic crowding threshold mechanisms that are specifically designed to address the unique challenges of optimizing vibration control parameters in high-speed elevators. Unlike the traditional ε-NSGA-II and adaptive crowding NSGA-II variants, which are primarily designed for static objective functions and uniform optimization environments, the approach is specifically designed for scenarios involving dynamic objective functions. These scenarios include those with evolving errors in surrogate models and real-time constraints. The dynamic adjustment of crossover-mutation probabilities ensures efficient exploration in the early stages and precise convergence in the later stages. The dynamic crowding threshold mechanism addresses parameter jumps, which are crucial for maintaining control stability and optimizing multi-objective coordination in real-time applications.

Traditional simulation calculations lead to time-consuming optimization, and fixed objective weights are difficult to adapt to multiple working conditions. For this reason, the study introduces a BPNN prediction model based on Latin hypercube sampling. Through systematic sensitivity analysis, the 3-7-2 BPNN structure is determined by comparing different hidden layer configurations (3-5-2, 3-7-2, and 3-10-2). Therefore, the selection of the 3-7-2 architecture is data-driven rather than empirical. The 3-5-2 structure had insufficient nonlinear fitting capability. The 3-10-2 structure showed a slight improvement in accuracy, but it increased the computational burden and risk of overfitting. Since the BPNN is used as a surrogate model for fast fitness evaluation rather than as a direct controller, the 3-7-2 configuration offers the optimal balance between prediction accuracy and computational efficiency. The 7-neuron hidden layer demonstrated an optimal balance between model complexity and prediction accuracy, achieving a 95.2% R² value with minimal overfitting (Gao et al., 2022). It constructs a fast evaluation system in which the objective is the RMS of acceleration and displacement. Finally, it realizes efficient MOO through the agent model. First, a bi-objective function with the car HVA RMS R M S a and displacement RMS R M S d as the optimization objective is established, as shown in Equation 3. min   f 1 x = R M S a x = 1 T ∫ 0 T a 2 t d t min   f 2 x = R M S d x = 1 T ∫ 0 T d 2 t d t (3)

In Equation 3, f 1 and f 2 represent the dynamic shock and static offset, respectively. x represents the vector of optimization variables. a t is the instantaneous acceleration of the HV of the car. d t displays the instantaneous displacement of the HV of the car. T displays the sampling period. The constraints are shown in Equation 4. x = K e , K e c , K u T ∈ R 3 5 ≤ K e ≤ 15 , 0.5 ≤ K e c ≤ 1.5 35 ≤ K u ≤ 55 (4)

In Equation 4, K e , K e c , and K u are the error, the error change rate (ECR), and the output quantization factor of the VUF-PID, respectively (Lian, 2022). In this study, the RMS values of acceleration and displacement are selected as the optimization objectives, since vibration suppression is the primary performance requirement of the considered system. Other performance-related indices, such as control energy consumption, control force variation rate, and jerk-related measures, are not included in the objective functions since they are mainly associated with engineering feasibility rather than core vibration mitigation effectiveness.

To ensure practical applicability, constraints on control force amplitude are imposed in the control design, which implicitly restrict excessive control effort. The adaptive crossover-mutation probabilities and dynamic crowding threshold are introduced to address the issues of objective function variability and parameter sensitivity that are common in real-time control systems. These design choices allow the algorithm to adapt to dynamic environments and prevent premature convergence, which would otherwise hinder performance in complex vibration control applications. Furthermore, post-optimization evaluations are conducted on the obtained Pareto-optimal solutions, indicating that the corresponding control energy levels and control force variations remain within reasonable engineering ranges. Therefore, although it is not explicitly optimized, energy consumption and smoothness control are effectively managed, ensuring the feasibility and implementability of the proposed optimization framework.

After the design of the dual objective function, the traditional numerical simulation calculating R M S a and R M S d suffers from low computational efficiency and poor real-time performance. Therefore, the study constructs a BPNN with 3-7-2 structure as a fast adaptation prediction model. It employs Sigmoid activation functions in the hidden layer for effective nonlinear mapping and linear activation in the output layer for regression prediction. The network weights are initialized using the Xavier method to maintain consistent variance across layers and prevent gradient vanishing or explosion. The model is trained with a carefully selected learning rate and the Levenberg-Marquardt backpropagation algorithm to ensure fast convergence and minimize overfitting. It is trained offline instead of online simulation to improve the optimization speed. The BPNN with 3-7-2 structure built for the investigation is displayed in Figure 1.

Figure 1 displays the model architecture of a BPNN with a 3-7-2 structure, containing three parameters ( K e , K e c , and K u ) in the input layer, seven neurons in the hidden layer, and two target values ( R M S a and R M S d ) in the output layer. The hidden layer uses Sigmoid activation functions to handle the complex, nonlinear relationships between quantization factors and vibration metrics. The output layer uses linear activation to make accurate regression predictions. The Xavier initialization strategy ensures optimal weight distribution across layers, facilitating stable gradient flow during backpropagation and enhancing training efficiency. The neurons in each layer form a feed-forward network through weighted connecting lines, and the bottom labeling uses a Sigmoid activation function. The model achieves accurate mapping from quantization factors to vibration metrics, providing fast adaptation assessment for NSGA-II optimization. Finally, the improved NSGA-II (I-NSGA-II), which is based on an adaptive cross-variance operator and a dynamic threshold congestion calculation, creates an efficient, MOO framework. This framework integrates a BPNN prediction model and a Latin hypercube sampling design: First, the population is initialized and a neural network is invoked to quickly assess individual fitness. Subsequently, high-quality solutions are screened by nondominated sorting and dynamic congestion calculation, and then the offspring population is generated by adaptive genetic operation. Finally, the Pareto optimal solution set is output through iterative optimization, which realizes the efficient optimization of cabin vibration control parameters, as shown in Figure 2.

Figure 2 demonstrates the optimization flow of the I-NSGA-II-A, which uses a 3-7-2 structured BPNN to quickly evaluate the population fitness. The congestion calculation is optimized by dynamic threshold adjustment, and the adaptive cross-variance operator is introduced to improve the search efficiency. The algorithm iterates cyclically with Gen=1 as the initial value, and finally outputs the Pareto-optimal solution set after nondominated sorting and population merging operations.

Unlike conventional NSGA-II-based fuzzy PID approaches, which typically perform optimization offline with fixed parameters during operation, the proposed framework uses a BPNN surrogate model for prediction-assisted optimization, enabling efficient online parameter coordination. Unlike neural-network-based PID controllers, which use neural networks to directly generate control actions or gains, the BPNN in this study is solely used to approximate the objective function. This approach avoids the instability caused by direct NN control and significantly reduces the computational burden.

Although the parameter optimization efficiency and quality of the PSS are improved by the MOO design of the NSGA-II-A, there is still an obvious fault in the dynamic interface between the optimization results and the actual control system. The traditional fuzzy PID controller relies on an empirical, trial-and-error selection of quantization factors. This results in parameters that are difficult to match with real-time control requirements. The adjustment range of the parameters is vague, and the controller’s adaptability to working conditions is poor. These issues seriously restrict the vibration suppression performance. Aiming at this key problem, the study proposes a closed-loop integration program of “sensing-prediction-control”. First, a high-precision signal acquisition system is constructed. Micro-electro-mechanical systems (MEMS) accelerometers and photoelectric encoders monitor the vibration acceleration and velocity of the guide shoe in real time. The Butterworth filter, which has a cutoff frequency of 50 Hz, eliminates high-frequency noise. Subsequently, the original signal is normalized and linearly mapped to the [-1,1] standard interval to provide high-quality input data for subsequent control decisions. The normalization processing formula is shown in Equation 5. k n o r m = 2 × k − k min / k max − k min − 1 (5)

In Equation 5, k n o r m is the normalized standard signal value. k is the original acquisition value of the sensor. k min and k max correspond to the lower and upper range limits of each sensor, respectively. The pre-processing process can effectively solve the problem of large noise and non-uniformity of the original signal, and lay the data foundation for the accurate decision-making of the control system. The specific flow of the signal processing link is shown in Figure 3.

In Figure 3, the signal processing link adopts a modular design, completing the three key steps of signal acquisition, filtering and noise reduction, and normalization conversion in sequence. Among them, the MEMS accelerometer has a ±2 g range and 0.1% linearity, and the optical encoder has a resolution of 1,000 pulses/revolution. The two work together to capture the dynamic characteristics of the boot. The second-order Butterworth filter effectively suppresses the high-frequency interference above 50 Hz under the premise of ensuring the signal amplitude-frequency characteristics. The normalization process adopts the linear transformation algorithm to make the sensor signals of different magnitudes comparable and provide standardized input for the subsequent control algorithm. After completing the signal processing at the input layer, the system still faces the key problem of mismatch between the optimized parameters and the actual control requirements. Due to the strong nonlinear and time-varying characteristics of the boot vibration, it is difficult to generate the optimal control parameters directly by solely relying on the pre-processed sensor signals. Therefore, the study designs a dynamic mapping mechanism for the optimization layer. It receives the optimal solution set from the Pareto front via the NSGA-II output interface. Then, it adopts the technique for order preference by similarity to an ideal solution (TOPSIS) to select the top five candidate solutions. The TOPSIS method is chosen over other multi-criteria decision-making approaches, such as the weighted sum method or simple ranking, because it can consider ideal and negative-ideal solutions simultaneously. This provides a more balanced compromise between conflicting objectives in the context of vibration control. Moreover, the weighted average quantization factor M e is calculated based on the relative proximity C i , which is shown in Equation 6. M e = ∑ i = 1 5 m i M e i (6)

In Equation 6, M e i is the value M e of the i th solution in the set of Pareto solutions generated by NSGA-II. Among them, the weight m i is calculated as shown in Equation 7. m i = 1 / C i ∑ j = 1 5 1 / C j (7)

In Equation 7, C j is the TOPSIS relative proximity of the j th Pareto solution. This weighting scheme ensures that solutions closer to the ideal point receive higher weights, while maintaining a smooth transition between different Pareto-optimal configurations. Comparative tests with alternative aggregation methods demonstrates that TOPSIS-based weighting provides 15% greater stability in parameter transitions and 8% greater consistency in control performance across different operating conditions. This scheme solves the problem of blindness in parameter selection in the traditional method. It accurately converts MOO results to real-time control parameters and provides an optimal control strategy for the actuator layer that adapts dynamically to different working conditions. After completing the dynamic mapping of the optimization layer, the system still faces the problem of insufficient conversion efficiency from control parameters to actuators. The fixed parameters of the traditional PID controller are difficult to adapt to the complex nonlinear vibration characteristics of HSEs. Therefore, the research design of the executive layer control force generation program. The VUF-PID structure is adopted, and the error E and the ECR E c are received in real time through the dual input channel, and the three parameter adjustments ΔKp, ΔKi, and ΔKd are generated by fuzzy reasoning. The final output control force f is shown in Equation 8. f = K p E + K i l e d t + K d d e d t (8)

In Equation 8, K p , K d , and K i are the proportional, differential, and gain coefficient. The control force calculation incorporates actuator saturation constraints with a maximum output limit of ±200 N based on electromagnetic actuator specifications. A smooth saturation handling mechanism that uses hyperbolic tangent functions prevents integral windup and maintains 95% of the theoretical control performance within the actuator’s operational bandwidth of 0–100 Hz. Power consumption is limited to 2 kW peak based on the elevator’s power supply capacity. l e d t is the error accumulation. d e d t is the ECR. The structure of VUF-PID is shown in Figure 4.

In Figure 4, the system takes the error E and the ECR E c as inputs, which are converted into fuzzy quantities e and ec by the fuzzification module. The fuzzification process employs seven triangular membership functions for both input variables, with linguistic labels assigned as follows: negative large (NL), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive medium (PM), and positive large (PL). The input domains are normalized to the range [-1, 1] with overlapping membership functions to ensure smooth transition between fuzzy sets. Subsequently, the parameter adjustment quantities K p , K i , and K d are generated by the fuzzy control module through inference. It implements a comprehensive rule base consisting of 49 fuzzy rules (7 × 7) derived from expert knowledge and systematic experimentation. The rule base follows the Mamdani-type inference system with centroid defuzzification method. Key representative rules include: IF e is NL and ec is NL. Then Δ K p is PL, Δ K i is NL, Δ K d is PS. IF e is ZO and ec is PS. Then Δ K p is NS, Δ K i is ZO, and Δ K d is PM. IF e is PL and ec is PL. Then Δ K p is NL, Δ K i is PL, and Δ K d is NS. The complete rule matrix ensures comprehensive coverage of all operational states. Moreover, the outputs of ΔKp, ΔKi, and ΔKd are outputted to the PID controller after defuzzification to finally drive the controlled system to form a CLC. After that, the scaling factor α e , e c is introduced to realize the dynamic adjustment of the argument domain, as shown in Equation 9. The scaling factor mechanism uses an exponential adjustment strategy. The universe contraction/expansion factor ranges from 0.5 to 2.0, depending on the magnitude of the input variables. This enables adaptive resolution adjustment across different operating regions. α e , e c = 1 − λ e − e 2 + e c 2 (9)

In Equation 9, λ is the attenuation factor. The scaling factor enables the input theory domain to expand with the system state adaptively compressed. After realizing the dynamic adjustment of the quantization factor, the system still faces the problems of sudden change of control force and insufficient stability. To address this challenge, the study proposes a real-time control performance optimization scheme to achieve the adaptive allocation of vibration energy by means of a dynamic weighting formula, as shown in Equation 10. w a c c t = E a c c t / E a c c t + E d i s p t (10)

In Equation 10, w a c c t is the acceleration suppression weight. E a c c t is the accumulated vibration kinetic energy. E a c c t is the cumulative displacement potential energy. t is the time variable. In summary, the study proposes an EC vibration control method based on the I-NSGA-II-A with VUF-PID. Figure 5 depicts its formation.

As shown in Figure 5, a “sensing–prediction–control” system is constructed in a closed loop to achieve dynamic parameter adjustment and real-time control synergistically. High-precision sensors collect vibration signals, which are filtered and normalized before being fed into a 3-7-2 BPNN for rapid quantization factor evaluation. The I-NSGA-II-A algorithm enhances convergence and solution quality by using adaptive crossover and mutation, dynamic crowding optimization, and TOPSIS decision-making to generate an optimal parameter set. Finally, the VUF-PID controller translates these parameters into precise control forces, enabling seamless signal-to-action execution.

All experimental validations are conducted on a standardized hardware platform deployed in a 10-story elevator test tower under actual operational conditions. The system configuration is detailed in Table 1.

The hardware platform integrates MEMS accelerometers with a range of ±2 g and 0.1% linearity. It also combines photoelectric encoders that provides a resolution of 1,000 pulses per revolution. These sensors are strategically positioned on the elevator’s guide shoes to accurately capture its HV characteristics.

The overall experimental workflow strictly adheres to the “sensing–prediction–control” closed-loop architecture depicted in Figure 5. This ensures a clear correspondence between the BPNN prediction module, the NSGA-II optimization layer, and the VUF-PID control execution. The experimental procedure follows a systematic three-phase approach: preprocessing, testing execution, and data analysis. In the preprocessing phase, all sensors undergo rigorous calibration procedures following standardized protocols. The Butterworth filter, with a cutoff frequency of 50 Hz, eliminated high-frequency noise. Meanwhile, signal normalization transformed the raw sensor data into the standard [-1, 1] interval using the following formula: k n o r m = 2 × k − k min / k max − k min − 1 . Specifically, the experimental procedure consists of three sequential stages. During the sensing stage, vibration acceleration and displacement signals are acquired and preprocessed to determine the system state reliably. In the prediction stage, the trained BPNN surrogate model rapidly evaluates candidate control parameters generated by the I-NSGA-II. Finally, during the control stage, the selected parameters are mapped to the VUF-PID controller, which generates real-time control forces. This completes the closed-loop experimental process.

Comprehensive experiments are conducted during testing execution, involving 200 round trips under varying operational conditions. These conditions includes steady-state operation at 6 m/s, acceleration/deceleration transitions, load variation scenarios, and simulated guideway excitation. Each control strategy undergo identical testing sequences to ensure comparable results.

The experimental evaluation employs multiple quantitative metrics: HVA RMS, displacement RMS values, low-frequency energy attenuation rate (0–10 Hz band), and multi-objective cooperation degree. Additional indicators includes the energy consumption index, the complexity of the control equipment, and the mean time between failures. All are measured according to the ISO 2631 standard for assessing ride comfort.

To reduce the influence of empirically determined parameters on the reliability and repeatability of the optimization results, a systematic parameter selection process is adopted for the NSGA-II algorithm.

First, the problem scale, computational cost, and existing literature are used to define reasonable ranges for the population size and maximum number of iterations, balancing search capability and efficiency. Within these ranges, sensitivity analyses are conducted on key algorithmic parameters, including the crossover probability, mutation probability, and coefficients in the adaptive operators. The performance of the algorithm under different parameter combinations is evaluated in terms of convergence behavior, Pareto front uniformity, and solution stability. Then, parameter regions that exhibit robust and insensitive performance are identified.

Representative parameter settings are subsequently validated through comparative experiments to ensure consistent optimization trends across different operating conditions and objective trade-offs. For adaptive parameters, such as the dynamic crowding distance threshold, initial and final values are calibrated through multiple trials. This preserves population diversity in early iterations and enhances convergence accuracy in later stages.

All baseline and improved NSGA-II variants use the same parameter selection procedure and evaluation criteria. This ensures a fair comparison and improves the reproducibility and engineering applicability of the proposed method.