Adaptive LQR active control of pantograph based on MDO algorithm

Abstract

With the continuous increase in the operating speed of high-speed railways, the fluctuation of the contact force in pantograph–catenary systems becomes increasingly significant, posing higher requirements for current collection quality and operational reliability. Conventional passive control methods mainly rely on structural parameter optimization and lack online adaptability, which limits their effectiveness under complex operating conditions. Although the Linear Quadratic Regulator (LQR) provides good response performance and robustness, its weighting matrices are typically selected through empirical tuning, and the fixed-gain structure cannot adequately address the nonlinear and time-varying characteristics of pantograph–catenary systems. To overcome these limitations, this paper proposes an Adaptive Linear Quadratic Regulator based on the Multi-strategy Dandelion Optimization Algorithm (ALQR-MDO). First, the original Dandelion Optimization (DO) algorithm is enhanced by integrating several improvement strategies, including boundary reflection, sub-elite guidance, Gaussian–Cauchy hybrid mutation, and dynamic population adjustment, thereby improving global search capability and convergence efficiency. Second, an adaptive LQR controller based on pantograph head displacement-segmented gain scheduling is developed, in which smooth transitions of feedback gains under different operating conditions are achieved using linear interpolation. Subsequently, the weighting matrices of the ALQR controller are treated as optimization variables, and a fitness function considering both pantograph head displacement and contact force fluctuations is constructed. The optimal control parameters are obtained using the MDO algorithm. Finally, a pantograph–catenary coupled dynamic model is established and validated through simulations under different operating speeds. Additionally, control energy consumption analysis and closed-loop stability verification are conducted. Simulation results show that under 200 km/h operating conditions, the proposed ALQR-MDO method reduces the standard deviation and range of contact force as well as the standard deviation and range of pantograph head displacement by 49.72%, 41.92%, 59.69%, and 50.96%, respectively, compared with passive control. Under 300 km/h, the corresponding improvements reach 50.48%, 48.14%, 54.95%, and 48.60%. These results demonstrate that the proposed method significantly improves current collection performance while exhibiting strong adaptability to high-speed operating conditions and robust disturbance rejection capability.

Introduction

With the continuous increase in train operating speeds, the reliability of pantograph–catenary contact and the stability of current collection have become increasingly critical issues in high-speed railway systems¹. The fluctuation amplitude of the contact force between the pantograph and the catenary increases significantly with train speed, which severely affects the current collection quality of the system². Both insufficient and excessive contact force can adversely influence the stability of the pantograph–catenary system and reduce the service life of the pantograph. Therefore, maintaining the contact force within a reasonable range is a key requirement for ensuring stable and reliable current collection in high-speed railways³.

In passive pantograph control strategies, vibration suppression is typically achieved by optimizing structural parameters such as springs or pneumatic cylinders. Wilk et al.⁴ proposed a novel method for estimating inertial and dissipative parameters of a railway pantograph model by combining multibody dynamic simulation with experimental measurements, thereby enabling accurate parameter identification and resolving friction issues in pantograph joints. Gil et al.⁵ developed a nonlinear pantograph model by introducing two Z-shaped springs between the pantograph head and the frame, forming a multi-degree-of-freedom system. The model parameters were optimized using a genetic algorithm, effectively suppressing system oscillations. Although passive control methods are relatively simple to implement, they lack the ability to adapt to changing operating conditions.

Against this background, researchers have continuously explored and investigated active control strategies. In recent years, a wide range of advanced control architectures has been explored, many of which enhance structural flexibility by augmenting or extending the classical PID framework. The current state-of-the-art control strategies include the Gudermannian PID controller, softsign fractional order controller, sigmoid PID controller, and BELBIC PID controller, among others.

Reference⁶ proposed a Gudermannian function-based proportional-integral-derivative (G-PID) controller to enhance the transient performance of automatic voltage regulator (AVR) systems operating under highly dynamic conditions. By embedding the smooth, bounded nonlinear mapping of the Gudermannian function into the classical PID structure, the proposed controller improves adaptability to large signal variations while effectively suppressing overshoot. Reference⁷ proposed a novel fractional-order proportional-integral-derivative (softsign-FOPID) controller incorporating the softsign function, optimized using a fungal growth optimizer (FGO), for the stabilization and precise position control of an unstable magnetic ball suspension system. The proposed controller introduces the smooth nonlinear softsign function into the conventional FOPID structure to constrain abrupt control actions and enhance transient smoothness, while retaining the flexibility of fractional-order dynamics. Reference⁸ proposed a novel control method, the sigmoid-based fractional-order PID (SFOPID), aimed at improving the dynamic response and robustness of AVR systems. The parameters of the proposed SFOPID control strategy are optimized using the dandelion optimizer (DO). Simulation results demonstrate that the proposed DO-SFOPID control technique is highly successful in terms of stability and robustness. Reference⁹ proposed an optimal fractional-order BELBIC to improve the small-signal stability of an interconnected hybrid power system. The FOBELBIC is introduced to enhance the damping capability of automatic generation control (AGC). Simulation results clearly confirm the damping performance of the MOALO-based FOBELBIC, enhancing the small-signal stability of the hybrid power system. However, although many specialized PID variants exhibit excellent performance in single-loop stability, they typically require complex decoupling or structural modifications to match the high-dimensional state feedback employed for the pantograph-catenary system in this study.

Active control strategies, in contrast, improve the current collection quality of pantograph–catenary systems by applying external control forces through dedicated controllers. Various control approaches have been investigated, including sliding mode control, PID control, H∞ control, model predictive control (MPC), and linear quadratic regulator (LQR) control. These controllers are fundamentally based on state-space representations, which naturally accommodate the multi-input multi-output (MIMO) characteristics of the three-mass model. Walters¹⁰ applied a fuzzy control strategy to reduce fluctuations in pantograph–catenary contact force; however, the control accuracy remained limited and the controller parameters were manually tuned. Wen et al.¹¹ proposed a pantograph active control algorithm based on Temporal Difference–Model Predictive Control (TD-MPC), in which MPC performs short-term trajectory optimization using a learned environmental dynamic model, while temporal difference reinforcement learning estimates long-term rewards to determine optimal control strategies. Chang¹² established a pantograph–catenary dynamic model based on a three-mass pantograph representation and derived the equivalent excitation force generated by fluctuating wind acting on the contact wire, followed by the design of a sliding-mode active controller. Chen et al.¹³ designed an optimal output tracking controller based on the LQR method, demonstrating through simulations that the stability of pantograph contact force was significantly improved.

Overall, active control methods generally provide higher control accuracy but exhibit strong dependence on system models. Different control approaches have their own advantages and limitations in terms of robustness, computational efficiency, and constraint handling. Among them, LQR control combined with intelligent optimization algorithms provides an effective approach for improving control performance while maintaining relatively low online computational complexity.

Although the Riccati equation can provide the optimal state feedback gain for a given set of weighting matrices, traditional pantograph LQR active control strategies typically rely on manual experience to determine the weighting matrices, obtaining the optimal state feedback control law through repeated trial and error. This approach consumes significant time and effort, and the resulting optimal solution can only improve pantograph-catenary performance within a specific speed range, making it difficult to achieve global optimization. To overcome this limitation, intelligent optimization algorithms have been widely employed to determine optimal LQR weighting matrices. For instance, Yang¹⁴ optimized the weighting coefficients of an active suspension LQR controller using the Whale Optimization Algorithm (WOA), significantly improving ride comfort and vehicle stability. Panjwani et al.¹⁵ designed an LQR controller for an inverted pendulum system and further optimized the weighting matrices using WOA and a multi-objective genetic algorithm, demonstrating superior reference tracking and robustness to parameter uncertainties. Feng et al.¹⁶ addressed the limitation of empirically selected weighting parameters in pantograph LQR control by optimizing controller parameters using an improved artificial bee colony algorithm, which significantly improved current collection quality. Mao and Luo¹⁷ combined a genetic algorithm with an LQR controller based on a three-mass pantograph model and verified through simulations that the optimized controller effectively reduced fluctuations in contact force and pantograph head displacement. Although these studies demonstrate the effectiveness of combining optimization algorithms with LQR control, most existing approaches focus on fixed-gain controllers, which cannot adequately adapt to the nonlinear and time-varying characteristics of pantograph–catenary systems under different operating speeds. Moreover, the convergence speed and susceptibility to local optima remain challenging issues for many optimization algorithms.

In recent years, several low-complexity optimization methods—such as the improved smooth function algorithm¹⁸, the norm-constrained SPSA (simultaneous perturbation stochastic approximation) algorithm¹⁹, and the safe experimental dynamics algorithm²⁰—have attracted attention in the field of controller parameter tuning. The improved smooth function algorithm focuses on "denoising and smoothing" at the mathematical level, enhancing search stability by addressing the issue of non-differentiable objective functions. The norm-constrained SPSA algorithm, while maintaining the efficiency of high-dimensional stochastic search, ensures that controller parameters always remain within physically realizable ranges through hard constraints. The safe experimental dynamics algorithm shifts the perspective from “optimal results” to “process safety”, safeguarding system security during online tuning by employing probabilistic models or stability criteria. These three approaches target the three critical dimensions of function differentiability, parameter compliance, and experimental safety, respectively, providing a complete solution for controller tuning in complex systems that bridges mathematical computation and engineering practice. However, the optimization problem considered in this study involves high-dimensional weighting parameters. Furthermore, the evaluation of candidate solutions relies on time-domain closed-loop simulations, resulting in an objective function that is highly non-convex, multimodal, and irregular. Compared with traditional gradient-based optimization methods, the dandelion optimizer (DO)²¹ offers advantages such as a simple structure, few parameters, and strong global search capability. These characteristics make it suitable as the fundamental optimization framework for this study.

Despite the progress achieved in previous studies, several limitations remain:

1. Existing intelligent optimization studies for pantograph active control mainly focus on algorithms such as genetic algorithms, artificial bee colony algorithms, and whale optimization algorithms, while research on recently developed swarm intelligence algorithms with superior performance remains limited.

2. Most optimized LQR methods are designed for fixed weighting matrices, making them difficult to adapt to the nonlinear and time-varying characteristics of pantograph–catenary systems under varying speeds and vibration amplitudes.

3. Current studies primarily focus on time-domain fluctuation indices, while comprehensive multidimensional evaluations of closed-loop control performance remain insufficient.

To address these issues, this study focuses on the Dandelion Optimizer (DO), a recently proposed swarm intelligence algorithm characterized by a simple structure, a small number of parameters, and strong global search capability. Based on this algorithm, several improvement strategies are introduced to enhance convergence performance and optimization efficiency. Furthermore, an Adaptive Linear Quadratic Regulator (ALQR) based on pantograph head displacement-segmented gain scheduling is proposed, and the improved optimization algorithm is employed to determine optimal controller parameters.

The main contributions of this work are summarized as follows:

1. An adaptive LQR active control strategy based on pantograph head displacement-segmented gain scheduling is proposed to address the nonlinear and time-varying characteristics of pantograph–catenary systems during high-speed operation, enabling smooth switching of control gains under different vibration conditions.

2. A controller parameter optimization model considering both contact force fluctuations and pantograph head displacement responses is established to overcome the dependence of traditional LQR weighting matrices on empirical tuning.

3. A Multi-strategy Dandelion Optimization Algorithm (MDO) is proposed to overcome the limitations of the original DO algorithm, including boundary aggregation, insufficient population diversity in later iterations, and susceptibility to local optima. The proposed algorithm incorporates boundary reflection, dual-elite guidance, Gaussian–Cauchy hybrid mutation, and dynamic population adjustment, and its effectiveness is verified through convergence analysis and benchmark function tests.

4. A pantograph–catenary coupled dynamic model based on the DSA380 pantograph and an elastic chain catenary system is established. Simulation experiments under 200 km/h and 300 km/h operating conditions demonstrate that the proposed ALQR-MDO control strategy significantly outperforms existing methods in suppressing contact force fluctuations and controlling pantograph head displacement.

Pantograph–catenary coupled model

Pantograph model

Based on the pantograph–catenary electrical contact experimental platform developed by the research group (Fig. 1), systematic dynamic characteristic tests were conducted on the DSA380 pantograph.

Fig. 1.

Pantograph–catenary electrical contact experimental platform.

In this experimental platform, a pure copper wire with a nominal cross-sectional area of 120 mm² is embedded in the groove along the edge of a rotating disk to simulate the contact wire in the catenary system. A metal-impregnated carbon strip is mounted on a transverse sliding platform to simulate the pantograph contact strip on the locomotive. The relative sliding motion between the wire and the strip reproduces the zigzag contact trajectory observed in real pantograph–catenary interactions.

By simulating pantograph–catenary contact processes under different operating speeds, multidimensional dynamic data—including pantograph head displacement, contact force, and frame vibration—are collected for subsequent analysis.

The pantograph, which serves as the current collection device of an electric train, consists of several key components, including the pantograph head, frame, base, and transmission mechanism²². In pantograph–catenary dynamic analysis, the contact between the pantograph and the catenary is commonly modeled as the interaction between the pantograph and a variable-stiffness spring.

In the study of pantograph dynamic characteristics, lumped-mass equivalent models are commonly employed for modeling and analysis²³. Using frequency response methods, the pantograph system can be simplified into two or three equivalent concentrated masses to describe its dynamic behavior. Based on the experimental data obtained from the platform, this study identifies the equivalent mass, stiffness, and damping parameters of the pantograph system using a combination of the frequency response method and least-squares parameter identification.

Previous studies have shown that when the train speed exceeds 250 km/h, the three-mass model provides higher simulation accuracy and more accurately reflects the dynamic interactions among different pantograph components²⁴. Therefore, the pantograph is modeled using a three-mass lumped parameter model, as shown in Fig. 2. The key parameters of the pantograph obtained from experimental data are summarized in Table 5. These parameters have been validated using multiple experimental datasets and can accurately represent the dynamic characteristics of the DSA380 pantograph.

Fig. 2.

Three-mass pantograph dynamic model.

Table 5.

Test Results of Different Algorithms.

Serial Number	MDO		DO		PSO		GA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
	1.44e-38	2.53e-38	3.71e-07	1.90e-07	2.71e-04	1.89e-04	1.25e + 04	3.03e + 03
	6.42e-29	8.15e-29	3.58e-04	2.98e-04	3.43e-03	2.08e-03	5.30e + 01	8.17e + 00
	1.07e-01	4.79e-02	4.16e-01	3.03e-01	4.95e + 00	1.35e + 00	6.40e + 01	2.65e + 00
	2.81e-03	5.29e-02	1.27e-02	8.57e-01	4.84e-02	5.77e + 00	6.67e-01	3.07e + 01
	5.41e-15	1.94e-15	1.86e-04	7.26e-05	5.43e-01	7.28e-01	1.85e + 01	1.15e + 00
	0.00e + 00	0.00e + 00	1.49e-02	1.41e-02	2.96e-02	6.28e-03	3.45e + 01	6.54e + 01
	9.98e-01	0.00e + 00	9.98e-01	0.00e + 00	9.98e-01	0.00e + 00	9.63e-01	5.11e-10
	−1.02e + 01	0.00e + 00	−1.02e + 01	0.00e + 00	−1.02e + 01	0.00e + 00	−5.61e-01	3.61e-01

Using the above three-mass pantograph model and applying Newton’s second law, the coupled dynamic equations of the pantograph system can be expressed as:

1

where denotes the catenary stiffness; represent the masses of the pantograph head, upper frame, and lower frame, respectively; denote the equivalent stiffness between the pantograph head and the frames; represent the damping coefficients between the pantograph components and the train body; denotes the locomotive excitation signal; represents the static uplift force; and denotes the active control force applied to the pantograph head.

Catenary model

Catenary systems can be classified into several types, including simple suspension, simple catenary suspension, compound catenary suspension, and elastic catenary suspension. Among these, the elastic catenary suspension optimizes the distribution of structural elasticity by incorporating elastic droppers, making the elasticity at support points and mid-span more uniform. At speeds of 160 km/h and above, it significantly reduces contact force fluctuations and improves the dynamic current collection performance of the pantograph-catenary system. Additionally, it offers unique advantages in vibration suppression and wave propagation control in scenarios involving multiple pantographs or complex current collection. Its parametric characteristics align well with the high reliability and stability requirements of modern high-speed railways and urban rail transit power supply systems.

Furthermore, this type of catenary has been validated in engineering applications on multiple high-speed lines both domestically and internationally. Modeling based on its parameters effectively enhances the engineering applicability and accuracy of simulation results, providing a scientifically reliable analytical basis for the matching design, performance optimization, and fault prediction of pantograph-catenary systems. Therefore, this paper selects the elastic catenary suspension as the modeling type for the contact wire. The contact wire stiffness k(t) is derived through finite element analysis, and the corresponding formula is fitted using the least squares method²⁵:

2

where , , , . ν is the train speed (m/s); L is the span length of the catenary (m); is the spacing between adjacent droppers (m); is the average stiffness of the catenary (N/m); to are the stiffness variation coefficients of the catenary.

The parameters for the elastic catenary suspension are as follows: =1925; =0.0755; = −0.0735; = −0.1459; =-0.0575; = −0.0699.

Pantograph-catenary coupling model

Based on the catenary model and dynamic equations described above, the state vector is defined as: , the state-space equation can be expressed as:

3

where

Adaptive LQR controller design

Adaptive gain scheduling strategy

The conventional LQR controller generally exhibits satisfactory convergence and stability under ideal operating conditions. However, its control performance is highly dependent on the selection of the weighting matrices. In pantograph–catenary systems, the contact stiffness varies nonlinearly with contact force, and significant geometric nonlinearities may occur during large-amplitude vibrations. Consequently, a fixed weighting matrix cannot effectively accommodate the full-speed operating conditions of the system, which may lead to degraded control performance.

To address this issue, an adaptive gain scheduling strategy based on pantograph head displacement is proposed in this study. The pantograph head displacement is divided into several operating regions, and a local linear controller is designed for each region. By dynamically adjusting the weighting matrices according to the operating region, smooth switching between controllers can be achieved, thereby improving the stability and reliability of the pantograph control system under varying operating conditions.

The vertical displacement of the pantograph head directly reflects the vibration level of the system and is closely related to the fluctuation of the contact force during train operation. Therefore, the pantograph head displacement amplitude is selected as the scheduling variable in this study. Based on the vibration characteristics of the pantograph and the service standards of the catenary system, three gain scheduling points are defined.

Adaptive LQR controller design

For each scheduling point, a corresponding LQR controller is designed. In the pantograph–catenary system model, the catenary stiffness is time-varying, which results in time-varying system matrices. For the i-th scheduling point, the optimal control performance index is defined as

4

where Q and R are the weighting matrices of the LQR controller.

By introducing the Algebraic Riccati Equation (ARE) and assuming that the control time horizon approaches infinity, the matrix solution converges to a constant steady-state matrix. The steady-state solution can then be obtained, from which the optimal state feedback vector is derived as

5

The optimal feedback gain matrix can be calculated as

6

Finally, three gain matrices are obtained: , , 。Direct switching of control gains may cause sudden changes in the control force, which may induce system shocks. Therefore, linear interpolation is adopted to achieve smooth transitions between different control gains .

7

where the interpolation coefficient is calculated as

8

Based on the above design, the gain-scheduled LQR control law can be expressed as

9

Expanding the control law yields

10

where u represents the control signal, i.e., the active control force applied to the pantograph system. The physical meanings of each feedback component are as follows:

• The displacement feedback term corresponds to proportional control, similar to a restoring spring force.

• The velocity feedback term provides damping effects, suppressing oscillations.

• The multi-point displacement feedback represents coordinated motion of different pantograph components.

• The multi-point velocity feedback reflects coordinated damping effects within the pantograph structure.

MDO algorithm and optimization of ALQR parameters

Dandelion optimization algorithm

The Dandelion Optimization (DO) algorithm is inspired by the natural dispersal process of dandelion seeds. The population evolution process consists of three main stages:

1. Rising stage

2. Descending stage

3. Landing stage

Rising stage

The rising stage considers two environmental scenarios: sunny weather and rainy weather. During sunny weather, the wind speed is assumed to follow a log-normal distribution. Under the influence of wind, vortices above the dandelion seeds continuously adjust, causing the seeds to rise in a spiral motion. The corresponding position update equation is

11

where t denotes the current iteration number and represents the position of the i-th individual at iteration t. represents the new position of the i-th individual after rising. The parameter represents a randomly selected position within the search space, which is initialized using the same method as the population initialization. The lifting coefficients generated by vortex separation are defined as

12

where θ is a random number within the range [-π, π].

follows a log-normal distribution with μ = 0 and . The mathematical expression is:

13

In Eq. (13), y represents the standard normal distribution . Additionally, α in Eq. (11) is an adaptive parameter used to adjust the search step size, calculated as follows:

14

where rand() denotes a random number between [0, 1], and T represents the maximum number of iterations.

During rainy weather, due to factors such as air resistance and humidity, the dandelion seeds cannot rise with the wind. Instead, local exploitation is performed, and the position update rule becomes

15

In the equation, k is used to adjust the local search region of the dandelion, and its calculation method is as follows:

16

In summary, the position update formula for the rising phase of dandelion seeds is:

17

where randn() is a random number following a standard normal distribution (Table 1).

Table 1.

Selection of Scheduling Points.

Scheduling Point	Pantograph Head Displacement Range/m	Operating Condition Characteristics	Control Objective
Scheduling Point 1		Small Disturbance/Steady State	Maintain stability and conserve energy
Scheduling Point 2		Medium Disturbance	Balance response speed and energy consumption
Scheduling Point 3		Large Disturbance/Resonance	Rapid suppression to prevent detachment

Descending stage

In the descending stage, the DO algorithm emphasizes exploration and uses Brownian motion to simulate the motion trajectories of dandelion seeds. The position update rule is

18

where β denotes a Brownian motion random variable. The average population position after the rising stage is defined as

19

where N denotes the population size.

Landing stage

In the landing stage, the algorithm focuses on local exploitation. The search process combines the Levy flight mechanism to enhance exploration ability. The position update equation is

20

where denotes the best solution obtained at the current iteration and represents the Levy flight distribution.

Multi-strategy Dandelion optimization algorithm (MDO)

During the search process of the original DO algorithm, issues such as slow convergence and susceptibility to local optima may occur. To overcome these limitations, a Multi-strategy Dandelion Optimization Algorithm (MDO) is proposed.

The corresponding relationships between the deficiencies of the original DO algorithm and the proposed improvement strategies are summarized in Table 2.

Table 2.

Improvements of the MDO Algorithm.

Problem in Original DO	Proposed Improvement	Effect
Boundary aggregation	Boundary reflection	Maintain population diversity
Single elite guidance	Dual-elite cooperative guidance	Improve robustness of search direction
Premature convergence	Gaussian–Cauchy hybrid mutation	Enhance ability to escape local optima
Fixed population size	Dynamic population adjustment	Balance exploration and exploitation

From the perspective of optimization mechanisms, the proposed improvements correspond to four aspects:

1. Constraint boundary handling

2. Guidance mechanism during exploitation

3. Balance between global exploration and local exploitation

4. Dynamic allocation of computational resources

These improvements enhance the search stability, population diversity, and convergence efficiency of the algorithm when solving high-dimensional control parameter optimization problems.

Improved boundary handling strategy

In the original DO algorithm, boundary handling is performed after each position update in the three evolutionary stages (rising, descending, and landing). The original boundary handling rule is defined as

21

where represents the position of the individual before boundary checking, denotes the updated position after boundary correction, lb and ub represent the lower and upper bounds of the search space, respectively.

Based on the processing method described in the above equation, seeds that fall below the lower bound of the search space are updated to the lower boundary position, while those exceeding the upper bound are moved to the upper boundary. This approach tends to cause out-of-bounds individuals to cluster near the boundaries, which is detrimental to maintaining population diversity.

Inspired by the lens opposition-based learning strategy²⁶, a boundary reflection mechanism is introduced in this study. When an individual exceeds the search boundary, it is reflected back into the feasible region while preserving a random displacement proportional to the distance beyond the boundary. This strategy maintains population diversity while preventing excessive concentration of individuals at the boundaries. The improved boundary handling rule is expressed as

22

where denotes the new reflected position after boundary handling, is the original position of the individual exceeding the boundary in the current iteration and r is a random number in (0,1].

Figure 3 illustrates the comparison of the boundary occupation ratio between the original algorithm and the improved boundary strategy when the population size is set to 100 and the maximum iteration number is 200. The boundary occupation ratio represents the proportion of individuals located on the boundary during each iteration.

Fig. 3.

Comparison of boundary occupation ratios during iterations.

The results show that in the original algorithm, especially during the early and middle stages of iteration, a considerable number of individuals accumulate at the boundary, which may lead to premature convergence. In contrast, the proposed boundary reflection strategy effectively prevents such aggregation and maintains population diversity throughout the optimization process, thereby enhancing the global search capability of the algorithm.

Search strategy optimization

In the landing stage of the original DO algorithm, the search process relies solely on the current global best individual to guide the population. Although this strategy can accelerate convergence, it may also cause individuals to cluster around a single elite solution, increasing the risk of premature convergence and trapping the algorithm in local optima.

To overcome this limitation, a dual-elite cooperative guidance mechanism is introduced in this study. In the proposed strategy, both the best individual and the second-best individual in the current population are used to guide the search direction.

Compared with the original single-elite strategy, the dual-elite mechanism provides a more robust search direction and reduces the risk of misleading guidance caused by an inaccurate elite solution. Consequently, individuals are more likely to move toward promising regions in the search space, thereby improving the efficiency and stability of the optimization process.

The improved position update rule for the landing stage is defined as

23

where m is a fixed coefficient set to 0.55, represents the global elite individual position in the t-th iteration, and denotes the global sub-elite individual position in the t-th iteration.

This strategy effectively balances exploration and exploitation, allowing the algorithm to maintain diversity while accelerating convergence.

Gaussian–Cauchy hybrid mutation

During the later stages of optimization, individuals in the population tend to cluster around the current best solution. If this solution is not the global optimum, the population may stagnate in a local optimum, preventing further improvement of the solution quality.

To alleviate this issue, a Gaussian–Cauchy hybrid mutation strategy is introduced to perturb elite individuals and increase population diversity. Gaussian mutation mainly enhances local search capability, as the generated perturbations are usually small and concentrated around the current solution. In contrast, Cauchy mutation produces occasional large perturbations, which helps the algorithm escape from local optima and improves global exploration ability²⁷.

Therefore, a hybrid mutation strategy combining Gaussian mutation and Cauchy mutation with equal probability (0.5) is adopted in this study. The probability density functions of the Gaussian, Cauchy, and hybrid distributions are shown in Fig. 4.

Fig. 4.

Probability density functions of Gaussian–Cauchy hybrid mutation.

From the comparison, it can be observed that Gaussian distribution generates small perturbations with high probability, which is beneficial for fine local search, while Cauchy distribution has heavier tails and is more likely to generate large perturbations, facilitating global exploration.

The Gaussian–Cauchy hybrid mutation strategy can therefore enhance the overall search capability of the algorithm by combining the advantages of both distributions. The mutation equation is expressed as

24

where represents the current mutated individual, represents the elite individual at the current iteration, and are perturbation coefficients, with and , represents the Gaussian mutation operator, and represents the Cauchy mutation operator.

Dynamic population adjustment strategy

During the natural dispersal process of dandelion seeds, environmental conditions such as climate changes may influence the success rate of seed propagation. Adverse weather conditions may reduce the number of seeds that can successfully spread and survive.

Inspired by this natural phenomenon, a dynamic population adjustment strategy is introduced in the landing stage of the MDO algorithm. In the early stage of the optimization process, a larger population is beneficial for global exploration, enabling the algorithm to search a wider region of the solution space. However, in the later stage of optimization, excessive population size may reduce convergence speed and increase computational cost.

Therefore, the number of individuals in the landing stage is gradually reduced as the iteration progresses, allowing the algorithm to focus more on local exploitation in the later stage. The dynamic population adjustment rule is defined as

25

where is the population size during the landing phase at the current iteration, is the population size after the previous iteration, and a and b are fixed parameters set to 0.02 and 0.3, respectively.

Convergence analysis

To provide a rigorous theoretical foundation for the proposed MDO algorithm, we analyze the population’s evolutionary behavior as a discrete-time linear time-varying (LTV) dynamical system^28–30. Specifically, focusing on the landing stage where the final convergence occurs, the position update rule for a seed across each dimension can be simplified into a one-dimensional form by treating the elite positions and as local constants. The update equation is expressed as:

26

This can be generalized into the standard state-space representation:

27

where represents the state variable, is the time-varying coefficient matrix, and B(t) denotes the external input term influenced by the dual-elite guidance. According to Lyapunov stability theory for discrete systems, a sufficient condition for the asymptotic stability and convergence of the individual state sequence x(t) is that the spectral radius of the transition matrix, represented here by the eigenvalue λ (t), satisfies throughout the iterative process.

As illustrated in Fig. 5, the eigenvalue analysis reveals a distinct contrast between the original and improved algorithms. In the original DO, the magnitude of λ (t) frequently exceeds the stability boundary (> 1) during early iterations, leading to erratic search trajectories and potential divergence. In contrast, the MDO eigenvalues remain strictly within the unit circle |λ (t)|<1 and monotonically approach zero as . This mathematical framework confirms that the dual-elite cooperative guidance and adaptive step-size scaling ensure stable convergence toward the global optimum while eliminating the unstable search regions inherent in the baseline algorithm.

Fig. 5.

Comparison of Eigenvalue Changes.

Test function simulation analysis

Algorithm parameter settings

To validate the search capability of the proposed MDO algorithm, a comparative performance analysis is conducted using the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), the Dandelion Optimization Algorithm (DO), and MDO. The basic parameters are set as follows: population size N=30, dimension , and maximum iteration count . The specific parameter settings for each algorithm are detailed in Table 3 below.

Table 3.

Algorithm Parameter Settings.

Algorithm Name	Basic Parameters
GA	,
PSO	, ,
DO	–
MDO

Test functions introduction

Eight benchmark functions were selected from the CEC 2017 test suite for evaluation. Among them, to are unimodal functions, while to are multimodal functions. The names and specific parameters of these test functions are listed in Table 4 below:

Table 4.

Test Function Parameters.

Sequence Number	Function Name	Domain	Optimum Value
	Sphere Function	[−100,100]	0
	Schwefel 2.22	[−10,10]	0
	Schwefel 2.21	[−100,100]	0
	Quartic Function with Noise	[−1.28,1.28]	0
	Ackley Function	[−32,32]	0
	Griewank Function	[−600,600]	0
	Shekel’s Foxholes Function	[−65.536,65.536]	0.998
	Shekel Function	[0,10]	−10.1532

Test function analysis

To more intuitively illustrate the convergence characteristics of the MDO algorithm, 20 independent experiments were conducted for each test function. The iterative convergence curves of the fitness values for the eight benchmark functions under the four algorithms are plotted in Fig. 6. Additionally, the mean (Mean) and standard deviation (Std) of each test function are calculated and presented in Table 5.

Fig. 6.

Convergence Curves of the Test Functions.

The results show that:

• For some functions, none of the algorithms reach the theoretical optimum, but MDO consistently achieves values closer to the optimal solution.

• For certain functions, only MDO successfully finds the global optimum with fewer iterations and faster convergence.

• For other functions, MDO, DO, and PSO exhibit comparable optimization performance, but MDO shows better stability.

Overall, the experimental results demonstrate that the proposed MDO algorithm provides superior optimization performance in terms of search accuracy, convergence speed, and robustness, validating its effectiveness for high-dimensional control parameter optimization.

Optimization procedure of the MDO-ALQR controller

To obtain the optimal control force for the pantograph system, the Multi-strategy Dandelion Optimization Algorithm (MDO) is employed to optimize the weighting matrices of the Adaptive Linear Quadratic Regulator (ALQR) controller. The overall control system block diagram is shown in Fig. 7.

Fig. 7.

Block diagram of the MDO-ALQR system.

The proposed optimization process integrates the MDO algorithm with the pantograph–catenary dynamic simulation model, enabling intelligent tuning of the ALQR weighting parameters. The detailed implementation steps are described as follows.

Step 1 Initialization

An initial population is generated randomly within the predefined search boundaries:

28

where.

• represents the i-th candidate solution in the population

• and denote the weighting matrices of the LQR controller corresponding to the j-th scheduling region

• lb and ub represent the lower and upper bounds of the search space.

Step 2 Gain calculation

For each candidate solution , the corresponding weighting matrices and are constructed. The optimal feedback gain matrices are then calculated using the LQR formulation:

29

where A and B are the system state-space matrices of the pantograph–catenary dynamic model.

Step 3 Closed-loop simulation

The obtained feedback gains are applied to the pantograph–catenary coupled dynamic model, and time-domain simulations are performed.

During the simulation process, the following system responses are obtained:

• Pantograph head displacement x(t)

• Pantograph–catenary contact force

These signals are used to evaluate the performance of the controller.

Step 4 Fitness evaluation

A fitness function is constructed to evaluate the quality of each candidate solution. The objective function considers both the contact force fluctuation and the pantograph head displacement response:

30

where.

• is the standard deviation of the pantograph head displacement

• is the standard deviation of the contact force

• and are reference values obtained from the passive control case

• and are weighting coefficients.

To balance the mechanical dynamic characteristics and the current collection stability during the controlled process, a weighted objective function f is defined. This formulation incorporates the initial standard deviations and for normalization, effectively eliminating the influence of inconsistent physical dimensions. The weight coefficients are set to ., reflecting the engineering priority given to contact force stability. Since fluctuations in contact force are directly linked to the contact loss rate, arcing, and component wear—which are more critical to current collection quality than the pantograph head displacement —a higher weight is assigned to to ensure the optimization process aligns with the fundamental requirements of operational safety and reliability.

To ensure numerical robustness, a penalty mechanism is introduced. If abnormal simulation values occur during the evaluation process, a large penalty fitness value is assigned to the candidate solution to avoid invalid solutions affecting the optimization process.

Step 5 Population update

The candidate solutions are updated using the exploration and exploitation mechanisms of the MDO algorithm, including.

• Boundary reflection strategy

• Dual-elite guidance

• Gaussian–Cauchy hybrid mutation

• Dynamic population adjustment.

These strategies collectively improve the global search capability and convergence stability of the algorithm.

Step 6 Iterative optimization

Steps 2–5 are repeated until the termination criterion is satisfied: t=T.

During the iterative process, the algorithm continuously updates the population and searches for the optimal solution.

Step 7 Output of optimal solution

After the optimization process converges, the optimal weighting matrices are obtained: . And the corresponding optimal feedback gain matrices are calculated as .

These optimal parameters are then used to construct the ALQR-MDO controller, which is applied to the pantograph–catenary system for subsequent simulation and performance evaluation.

Computational complexity analysis

A formal analysis of the computational complexity is conducted to evaluate the efficiency of the MDO algorithm relative to the original DO and other metaheuristics. The total complexity is primarily determined by three factors: initialization, fitness evaluation, and position updating.

• Original DO Complexity For a population size N, iteration count T, and problem dimension D, the basic complexity is .

• MDO Complexity The proposed enhancements—namely boundary reflection, dual-elite cooperative guidance, and Gaussian–Cauchy hybrid mutation —are all designed as element-wise vector operations. Consequently, each strategy maintains a linear complexity of O(D) per individual. The aggregate complexity of MDO thus remains , consistent with the original algorithm’s growth rate, ensuring that the performance gains do not come at the cost of excessive computational overhead.

• Impact of Dynamic Population Adjustment Unlike traditional algorithms with a fixed N, the MDO incorporates a Dynamic Population Adjustment strategy that gradually reduces the number of active seeds as iterations progress. Mathematically, the population size N(t) follows a decaying exponential function:

  31

This results in a total number of Function Evaluations (FEs) given by , which is strictly less than the evaluation count of the standard DO. Therefore, MDO not only provides higher search accuracy but also reduces the overall computational burden, particularly in the later stages of optimization where the search focuses on local exploitation with a leaner population.

Simulation results and analysis

To verify the effectiveness and superiority of the proposed Adaptive LQR control strategy based on the Multi-strategy Dandelion Optimization Algorithm (ALQR-MDO), a pantograph–catenary coupled dynamic simulation model is established on the MATLAB/Simulink platform. Multiple control strategies are compared through systematic simulations. All results are based on the average of 30 independent optimization runs.

The performance of the controllers is comprehensively evaluated from several perspectives, including:

• Time-domain response

• Control energy consumption

• Frequency-domain characteristics

• Closed-loop stability.

Simulation parameter settings

In the simulation study, the DSA380 pantograph, which is widely used in Chinese high-speed railways, is selected together with the Jingshen high-speed railway catenary parameters as the simulation case.

The span of the catenary is set to 50 m, and the spacing between adjacent droppers is 10.5 m. The key pantograph parameters are listed in Table 6.

Table 6.

Pantograph Parameters.

Parameter Name	Unit	Numerical Value
Pantograph head mass	kg	7.12
Upper frame mass	kg	8
Lower frame mass	kg	10
Equivalent stiffness of the pantograph head	N/m	3540
Equivalent stiffness between upper and lower frames	N/m	13,800
Equivalent damping coefficient of the pantograph head	N·s/m	10
Equivalent damping coefficient between upper and lower frames	N·s/m	20
Equivalent damping coefficient between the lower frame and the train car body	N·s/m	70

Time-domain response analysis

Simulation results at 200 km/h

Under the operating speed of 200 km/h, the weighting matrices of the ALQR controller were optimized using the Multi-strategy Dandelion Optimization (MDO) algorithm. The convergence of the MDO algorithm compared with the standard DO algorithm is shown in Fig. 8. The MDO algorithm reaches the optimal solution at the 5th iteration, with a best fitness value of 1.72163, indicating faster convergence and superior optimization performance.

Fig. 8.

Comparison of Algorithm Iteration Curves at 200 km/h.

The optimal weighting matrices obtained through MDO optimization are as follows: The optimal weighting coefficients obtained through the MDO algorithm are as follows: Small displacement interval: =diag (483.63, 5.68, 251.40, 9.96, 545.68, 7.13); Medium displacement interval: =diag (586.32, 4.87, 812.89, 10.00, 3624.80, 6.21); Large displacement interval: =diag (510.84, 5.47, 1452.43, 1.46, 1563.88, 10.00).

These optimized matrices were applied to the pantograph–catenary simulation model. Figure 9 presents the contact force fluctuations under different control strategies, and Fig. 10 shows the corresponding pantograph head displacement responses. Quantitative simulation results are summarized in Table 7.

Fig. 9.

Comparison of Contact Force Fluctuations Under Different Control Methods at 200 km/h.

Fig. 10.

Comparison of Pantograph Head Displacement Fluctuations Under Different Control Methods at 200 km/h.

Table 7.

Simulation Results of Different Control Methods at 200 km/h.

Control Method	Standard deviation of contact force	Range of contact force	Standard deviation of pantograph head displacement	Range of pantograph head displacement
Passive Control	15.203	106.80	0.008842	0.065661
LQR-DO	17.425	103.21	0.010064	0.064020
LQR-MDO	15.529	89.61	0.009009	0.055744
ALQR-DO	11.698	106.36	0.0065104	0.065345
ALQR-MDO	7.6439	62.028	0.0038291	0.032197

Compared with passive control, the ALQR-MDO controller reduces:

• Contact force standard deviation by 49.72%

• Contact force range by 41.92%

• Pantograph displacement standard deviation by 59.69%

• Displacement range by 50.96%

This demonstrates a significant improvement in current collection performance.

Simulation results at 300 km/h

To further assess the controller performance at higher speeds, simulations were conducted at 300 km/h. The convergence curves for MDO and DO algorithms are shown in Fig. 11. The MDO algorithm reaches the optimal fitness value of 1.6666 at iteration 12, while the DO algorithm converges to 1.9834 at iteration 18, highlighting the superior optimization capability of MDO.

Fig. 11.

Comparison of Algorithm Iteration Curves at 300 km/h.

The optimal weighting coefficients obtained through the MDO algorithm are as follows: Small displacement interval: =diag (48.84, 30.37, 49.51, 19.92, 327.53, 35.37); Medium displacement interval: =diag (126.00, 10.00, 14.58, 10.00, 199.29, 12.87); Large displacement interval: =diag (984.58, 10.00, 842.05, 18.24, 125.63, 24.51.

Figures 12 and 13 illustrate the contact force and pantograph head displacement under different control strategies. Quantitative simulation results are summarized in Table 8.

Fig. 12.

Comparison of Contact Force Fluctuations Under Different Control Methods at 300 km/h.

Fig. 13.

Comparison of Pantograph Head Displacement Fluctuations Under Different Control Methods at 300 km/h.

Table 8.

Simulation Results of Different Control Methods at 300 km/h.

Control Method	Standard deviation of contact force	Range of contact force	Standard deviation of pantograph head displacement	Range of pantograph head displacement
Passive Control	15.625	124.35	0.008369	0.062946
LQR-DO	17.733	118.79	0.009665	0.060887
LQR-MDO	15.924	107.95	0.008661	0.055692
ALQR-DO	10.180	106.37	0.005223	0.053644
ALQR-MDO	7.738	64.50	0.003771	0.032354

The ALQR-MDO controller shows significant improvement:

• Contact force standard deviation reduced by 50.48%

• Contact force range reduced by 48.14%

• Displacement standard deviation reduced by 54.95%

• Displacement range reduced by 48.60%

This demonstrates that ALQR-MDO can more effectively reduce contact force fluctuations and significantly enhance current collection quality.

In summary, the simulation results not only validate the effectiveness of the MDO algorithm in parameter optimization but also highlight the superiority of the adaptive gain scheduling strategy over fixed-gain LQR control in adapting to the system’s nonlinear characteristics. This provides a comprehensive theoretical foundation and numerical support for the engineering implementation of active pantograph control systems.

Frequency-domain analysis

To further investigate the dynamic characteristics of the pantograph–catenary system, frequency-domain analysis was performed on the contact force signals under the 200 km/h operating condition. The power spectral density (PSD) was estimated using the Welch method. The results for different control strategies are shown in Fig. 14.

Fig. 14.

Comparison of contact force power spectral density under different control methods.

From the figure, it can be observed that the dominant vibration energy of the system is concentrated around 4 Hz, which corresponds to the primary vibration mode of the pantograph–catenary system. Table 9 summarizes the PSD characteristics of each control strategy.

Table 9.

Power Spectral Density Analysis.

Control Strategy	Dominant Frequency (Hz)	PSD Peak	Energy (0–20 Hz)
Passive control	0	916.8	3218
LQR-DO	4.00	580	3687
LQR-MDO	4.00	504	3215
ALQR-DO	0	774	2710
ALQR-MDO	0	607	2120

The results indicate:

1. The ALQR-MDO controller effectively suppresses low-frequency vibration energy.

2. Compared to passive control, the PSD peak is reduced by approximately 45%, indicating a significant improvement in dynamic performance.

3. Adaptive gain scheduling further improves the low-frequency vibration suppression compared to fixed-gain LQR controllers.

Control energy analysis

To assess the workload on the pantograph actuator, control energy consumption was evaluated. The control energy is defined as:

32

where u(t) is the actuator control force over time.

The simulation results are summarized in Table 10.

Table 10.

Control Energy Comparison.

Control Strategy	Control Energy	Control Force RMS
Passive control	0	0
LQR-DO	23,585	68.63
LQR-MDO	20,840	64.52
ALQR-DO	40,805	63.88
ALQR-MDO	31,244	55.89

Key observations:

1. The MDO-optimized LQR controller reduces control energy by approximately 11.6% compared with DO-optimized controllers while achieving superior vibration suppression.

2. The adaptive ALQR strategy slightly increases total control energy compared to fixed-gain LQR, but it significantly reduces pantograph displacement and contact force fluctuations, indicating a worthwhile trade-off between performance and actuator effort.

Closed-loop stability analysis

The stability of the closed-loop system under different control strategies was evaluated by analyzing the eigenvalues of the system matrix. For the fixed-gain LQR controller, the closed-loop matrix is written as . For the ALQR controller, the local closed-loop matrices corresponding to the three scheduling points are analyzed separately. Table 11 presents the maximum real parts of the eigenvalues for each control strategy.

Table 11.

Closed-loop Stability Analysis.

Control Strategy	Maximum Real Part of Eigenvalues
LQR-DO	− 1.50
LQR-MDO	− 2.04
ALQR-DO	− 1.36 ~  − 1.60
ALQR-MDO	− 1.36 ~  − 4.14

Observations:

1. All eigenvalues have negative real parts, indicating that the closed-loop system is asymptotically stable under all active control strategies.

2. The ALQR-MDO strategy achieves the largest negative eigenvalue, suggesting enhanced robustness and faster convergence of system responses.

Furthermore, considering that linear interpolation is employed to achieve smooth gain transitions between scheduling intervals, a stability scan was conducted across the transition regions, as shown in Fig. 15. The results indicate that the maximum real part of the closed-loop eigenvalues remains consistently below zero throughout the entire interpolation interval, demonstrating that the proposed smooth switching strategy does not compromise system stability.

Fig. 15.

Closed-loop stability scanning across the gain scheduling transition intervals.

Controller robustness verification

The external disturbance is set as a multi-frequency disturbance , and the operating condition is 350 km/h. Simulations are conducted using Simulink software for the MDO-ALQR active control model, with a simulation time of 10 s. The comparison of pantograph-catenary contact pressure fluctuations is shown in the figure (Fig. 16).

Fig. 16.

Comparison of Pantograph-Catenary Contact Pressure Fluctuations Before and After Introducing Disturbance.

In the MDO-ALQR model, the standard deviation of pantograph head displacement changes by 0.14%, and the standard deviation of contact force changes by 0.18%. In summary, the external disturbance has minimal impact on the control performance of MDO-ALQR, indicating that MDO-ALQR exhibits strong disturbance rejection capability.

Sensitivity analysis

In practical operation, due to mechanical wear of the carbon strip and tension fluctuations of the catenary under varying ambient temperatures, the system’s physical parameters—such as the equivalent mass of the pantograph head and the equivalent stiffness of the catenary —often deviate from their nominal design values. To comprehensively evaluate the robustness of the proposed ALQR-MDO controller, a series of sensitivity experiments were conducted. In these experiments, and were perturbed within a range of 20% of their nominal values at a step size of 10%. By comparing the standard deviation of contact force under passive control and ALQR-MDO under parameter drift, the controller’s ability to handle system uncertainties was verified. The sensitivity analysis results are shown in Fig. 17, from which the following conclusions can be drawn:

• Robustness to mass variation When the pantograph head mass increases by 20%, the contact force fluctuation of the passive control intensifies significantly. In contrast, leveraging the adaptive gain scheduling strategy, the ALQR-MDO controller can perceive changes in vibration amplitude in real time and adjust the control gain accordingly, resulting in far smaller fluctuations in compared to passive control.

• Adaptability to stiffness variation Since the catenary stiffness directly affects the system’s damping characteristics and frequency response, its fluctuation has a substantial impact on the passive system. Experimental results show that even with a 20% deviation in ALQR-MDO maintains a stable suppression effect, with all performance indicators deviating by less than 5%.

• Summary The above analysis strongly demonstrates that the ALQR-MDO control strategy not only performs excellently under nominal operating conditions but also exhibits strong robustness and engineering practicality in the face of model parameter uncertainties and random disturbances, fully satisfying the stringent requirements of high-speed railways for current collection stability.

Fig. 17.

Sensitivity analysis results.

Conclusion

An Adaptive Linear Quadratic Regulator (ALQR) controller optimized via the Multi-strategy Dandelion Optimization (MDO) algorithm is proposed for pantograph active control in high-speed railway systems.

The main conclusions are as follows:

1. A multi-strategy improved Dandelion Optimization (MDO) algorithm was developed, integrating boundary reflection, dual-elite guidance, Gaussian–Cauchy hybrid mutation, and dynamic population adjustment. Benchmark function tests indicate that MDO outperforms DO, PSO, and GA in terms of convergence speed and optimization accuracy.

2. An adaptive LQR controller with displacement-based gain scheduling was designed to dynamically adjust control gains under different vibration conditions. This method effectively accommodates the nonlinear and time-varying characteristics of the pantograph–catenary system, outperforming conventional fixed-gain LQR controllers.

3. Simulation results under 200 km/h and 300 km/h show that the ALQR-MDO controller significantly reduces pantograph vibration and contact force fluctuations. Quantitative improvements include: Compared with passive control, the standard deviation of contact force and pantograph head displacement are reduced by 49.72% and 59.69%, respectively, under the 200 km/h operating condition; under the 300 km/h condition, the reductions are 50.48% and 54.95%, respectively.

4. Frequency-domain analysis confirms effective suppression of dominant vibration modes (~ 4 Hz). Control energy analysis shows that the MDO-optimized controller reduces actuator energy by 11.6% compared with DO-optimized LQR.

5. Closed-loop eigenvalue analysis and robustness verification confirm the stability and robustness of the ALQR-MDO controller, validating its suitability for high-speed railway pantograph systems.

Future work will consider experimental validation, track irregularities, and time-delay effects to further verify practical performance.

List of symbols

Equivalent masses of pantograph head, upper frame, and lower frame

Equivalent stiffness of the catenary

Equivalent stiffness between pantograph head and frames

Equivalent damping coefficients of the pantograph system

Pantograph–catenary contact force

Active control force applied to the pantograph head

Active control force applied to the pantograph head

Vertical displacement of pantograph head, upper frame, and lower frame

Q,R

Weighting matrices of the LQR controller

K

State feedback gain matrix

A,B

System state-space matrices

ρ

Gain scheduling variable (pantograph head displacement)

N

Population size in the optimization algorithm

T

Maximum iteration number

Author contributions

**[Hui Lichuan]:** Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources; **[Zhang Hanlin]:** Software, Supervision, Validation, Visualization, Writing–original draft, Writing–review and editing.

Funding

This work was supported by the Education Department of Liaoning Province basic scientific research project of colleges and universities of Liaoning Province [grant number LJ2017QL009].

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Code availability

The custom code used to generate the results in this study have been deposited in Zenodo and can be accessed via the following link: [https://doi.org/10.5281/zenodo.20120651].

Declarations

Competing interests

The authors declare no competing interests.