Research on Parameter Tuning of Electro-Hydrostatic Actuator Position Sliding Mode Controller Based on Enhanced Dynamic Sand Cat Search Optimization Algorithm

Abstract

This paper proposes an Enhanced Dynamic Sand Cat Search Optimization algorithm (EDSCSO) designed to address the high-order nonlinearities and strong coupling issues in the parameter tuning of the position sliding mode controller for electro-hydrostatic actuators (EHAs). Traditional swarm intelligence optimization algorithms often struggle with the transition from global to local search, which leads to being trapped in local optima and results in lower computational efficiency. To overcome these challenges, the EDSCSO algorithm introduces an escape mechanism, a stochastic elite cooperative bootstrap strategy, and a multi-path differential perturbation strategy. These enhancements significantly increase the diversity of the population, facilitate a smooth transition from global to local search, avoid local optimum traps, and better balance the exploration and exploitation capabilities of the algorithm. Based on this algorithm, the sliding mode surface and convergence rate parameters within the sliding mode controller are optimized. Simulation validations conducted on the combined platform of MATLAB/Simulink and AMESim demonstrate that the sliding mode PID controller optimized by the EDSCSO algorithm achieves smaller steady-state and tracking errors, exhibits greater robustness, and offers enhanced computational efficiency compared to other swarm intelligence optimization algorithms. This study provides an effective optimization strategy to improve the control performance of the EHA position sliding mode controller.

1. Introduction

An electro-hydrostatic actuator (EHA) is an integrated closed-loop pump-controlled electro-hydraulic servo system that consists of a permanent magnet synchronous motor (PMSM), a plunger pump, and a hydraulic cylinder [1]. Its working principle involves adjusting both the speed and direction of the motor to control the plunger pump, which, in turn, regulates the flow of hydraulic oil, thereby controlling the extension and retraction of the hydraulic cylinder’s piston and achieving displacement output. By combining the advantages of both electric drive and hydraulic transmission, the EHA offers faster response speeds and greater power output. In contrast to traditional electro-mechanical actuators (EMAs), an EHA eliminates the risk of mechanical jamming and does not require components such as servo valves or oil reservoirs [2]. This results in a highly integrated system characterized by high power density, enhanced reliability, and ease of maintenance. Consequently, EHAs have found widespread applications in fields such as aerospace, marine valve control, and robotics [3,4,5].

Despite their advantages, EHAs are complex systems that integrate mechanical, electrical, and hydraulic components, making them high-order nonlinear systems characterized by significant uncertainties and intricate electro-mechanical–hydraulic coupling behavior. Notably, as closed-loop hydraulic systems, temperature rises in EHAs are more pronounced than in traditional hydraulic systems. During operation, structural parameters such as the bulk modulus of hydraulic oil and the internal leakage coefficient of the hydraulic cylinder undergo significant variations in response to temperature changes. These variations subsequently increase the challenges associated with designing high-precision and high-performance EHA controllers.

Current control strategies for EHAs typically include PID control, fuzzy control, sliding mode control (SMC), adaptive control, and model predictive control (MPC) [6,7,8,9,10]. [While traditional PID control does not necessitate an accurate system model, its dynamic performance in the face of unknown disturbances is relatively suboptimal, and its applicability is limited. Conversely, SMC exhibits exceptional performance in systems with uncertain parameters or substantial external disturbances due to its robust characteristics and rapid response.] To enhance the speed and robustness of EHA control, numerous researchers have proposed hybrid control strategies. For instance, Ren et al. [11] combined fuzzy control with PID control, employing fuzzy logic to optimize PID parameters and improve system response speed, although this approach increases the complexity of parameter tuning. Similarly, Zhang et al. [12] proposed a composite control strategy that integrates PID and sliding mode control, demonstrating significant advantages in speed and robustness; however, it still relies heavily on empirical knowledge, complicating parameter tuning.

In recent years, swarm intelligence algorithms (SIAs), inspired by natural phenomena, have gained popularity across various fields such as path planning, image segmentation, feature selection, scheduling optimization, and parameter optimization [13,14,15,16,17,18]. These algorithms leverage the cooperative interactions among individuals in a population, promoting decentralized control and self-organization. To address the limitations of traditional EHA control strategies and enhance the computational efficiency and performance of EHA position sliding mode controllers, this paper proposes a novel Sand Cat Swarm Optimization (SCSO) algorithm for tuning the sliding surface and reaching rate parameters. The algorithm, based on the hunting behavior of sand cats, incorporates an escape mechanism (EM) and a random elite collaboration guiding strategy (RE-CGS) to improve global search capabilities. Furthermore, a multi-path differential perturbation strategy (MDPS) is implemented to enhance population diversity, which [not only prevents premature convergence but also achieves] a balance between global exploration and local exploitation, thereby improving search accuracy and speed. Subsequent experiments, conducted using Matlab/Simulink 2024a and AMESim 2023 software for co-simulation, demonstrate that the optimized sliding mode PID controller exhibits significant improvements in response speed, steady-state error, dynamic tracking performance, and robustness.

This article is organized as follows: Section 2 presents the mathematical modeling of the electro-hydrostatic actuator system. Section 3 details the design of the electro-hydrostatic actuator system. Section 4 elaborates on the improved sand cat optimization search algorithm. Section 5 provides experimental results of the improved sand cat optimization search algorithm for parameter tuning of the EHA position sliding film controller, alongside an analysis of the proposed method. Finally, Section 6 summarizes the primary contributions of this article.

2. The Mathematical Model of the Electro-Hydrostatic Actuator System

The basic components and working principle of the (EHA) system are illustrated in Figure 1. The system consists of three main subsystems: the permanent magnet synchronous motor, the speed control subsystem (referred to as the motor subsystem), the plunger pump and hydraulic cylinder subsystem (referred to as the mechanical–hydraulic subsystem), and the coupling mechanism. Specifically, (1) the motor subsystem is composed of the EHA controller, DC power supply, inverter, and PMSM (shown as “1” in the figure); (2) the coupling mechanism (shown as “2”) connects the motor subsystem and the mechanical–hydraulic subsystem; and (3) the mechanical–hydraulic subsystem consists of the plunger pump (shown as “3”), check valves (shown as “4-1” to “4-4”), accumulator (shown as “5”), relief valves (shown as “6-1” and “6-2”), pressure sensors (shown as “7-1” and “7-2”), hydraulic cylinder (shown as “8”), and displacement sensor (shown as “9”). The pressure sensors provide pressures $p_1$ and $p_2$ on both sides of the hydraulic cylinder, while the displacement sensor provides the displacement x of the hydraulic cylinder’s piston. The EHA system outputs a control signal based on the given position reference $x_d$ and the position feedback $x$ from the displacement sensor. This control signal adjusts the motor speed and rotational direction, thereby controlling the flow rate and direction of the pressurized oil output by the plunger pump. The pressurized oil enters the hydraulic cylinder to drive the piston, which in turn drives the load.

2.1. Mathematical Modeling of Motor Link

The motor is a surface-mounted permanent magnet synchronous motor, and the stator winding voltage equation expression is shown in Equation (1).

$$\left\{\begin{array}{l}{U}_{\mathrm{d}}=R\cdot i_{\mathrm{d}}+{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{d}}}{\mathrm{d}t}}-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{q}}\\ U_{\mathrm{q}}=R\cdot i_{\mathrm{q}}+{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{q}}}{\mathrm{d}t}}-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{d}}\end{array}\right.$$

(1)

where $U_{\mathrm{d}}$, $U_{\mathrm{q}}$ are the voltages of the motor d and q axes, respectively, in V; $R$ is the equivalent resistance of the motor in $\Omega$; $i_d$, $i_q$ are the currents of the motor d and q axes, respectively, in A; ${\psi }_{\mathrm{d}}$, ${\psi }_{\mathrm{q}}$ are the fluxes of the motor d and q axes, respectively, in Wb; and ${\omega }_e$ is the electromechanical electrode angle in $\mathrm{rad}\cdot {\mathrm{s}}^{-1}$.

The stator magnetic chain equation is shown in Equation (2).

$$\left[\begin{array}{c}{\psi }_{\mathrm{d}}\\ {\psi }_{\mathrm{q}}\end{array}\right]=\left[\begin{array}{cc}{L}_{\mathrm{d}}& 0\\ 0& L_{\mathrm{q}}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+\left[\begin{array}{c}{\psi }_{\mathrm{f}}\\ 0\end{array}\right]$$

(2)

where $L_{\mathrm{d}}$ and $L_{\mathrm{q}}$ are the d and q axes inductances in H; ${\psi }_{\mathrm{f}}$ is the rotor permanent magnet chain in Wb.

By combining Equations (1) and (2), the dynamic equations of the motor can be obtained as shown in Equation (3).

$$\left\{\begin{array}{l}{U}_{\mathrm{d}}=R\cdot i_{\mathrm{d}}-{\omega }_{\mathrm{e}}\cdot L_{\mathrm{d}}\cdot i_{\mathrm{q}}+L_{\mathrm{d}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{d}}}{\mathrm{d}t}}\\ U_{\mathrm{q}}=R\cdot i_{\mathrm{q}}-{\omega }_{\mathrm{q}}\cdot L_{\mathrm{d}}\cdot i_{\mathrm{d}}+L_{\mathrm{q}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{q}}}{\mathrm{d}t}}+{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\end{array}\right.$$

(3)

Since the inductance values of the d-axis and q-axis of the surface-mounted motor are equal, the expression for the electromagnetic torque $T_{\mathrm{e}}$ of the motor is shown in Equation (4).

$$T_{\mathrm{e}}={\displaystyle \frac{3}{2}}{P}_{\mathrm{n}}{\psi }_{\mathrm{f}}{i}_{\mathrm{q}}$$

(4)

where is the number of motor pole pairs. The equation of motion of the motor is shown in Equation (5).

$$T_{\mathrm{e}}-T_{\mathrm{L}}=J{\dot{\omega }}_{\mathrm{m}}+B_{\mathrm{m}}{\omega }_{\mathrm{m}}$$

(5)

where $T_L$ is the load torque in $N\cdot m$; $J$ is the motor moment of inertia in $\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^3$; ${\omega }_{\mathrm{m}}$ is the mechanical angular velocity of the motor rotor in $\mathrm{rad}\cdot {\mathrm{s}}^{-1}$; and $B_{\mathrm{m}}$ is the motor viscous friction coefficient in $N\cdot \mathrm{m}\cdot \mathrm{s}\cdot {\mathrm{rad}}^{-1}$.

The expression for the motor link can be found using Equations (3)–(5), as shown in Equation (6).

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{q}}}{\mathrm{d}t}}={\displaystyle \frac{1}{L_{\mathrm{q}}}}\left(-R{i}_{\mathrm{q}}-{\psi }_{\mathrm{f}}{P}_{\mathrm{n}}{\omega }_{\mathrm{m}}+U_{\mathrm{q}}\right)\\ {\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}={\displaystyle \frac{1}{J}}\left({\displaystyle \frac{3}{2}}{P}_{\mathrm{n}}{\psi }_{\mathrm{f}}{i}_{\mathrm{q}}-B_{\mathrm{m}}{\omega }_{\mathrm{m}}-T_{\mathrm{L}}\right)\end{array}\right.$$

(6)

2.2. Mathematical Modeling of Machine-Fluid Chain

The piston pump must be selected to meet the requirements of response speed and its load force in the EHA system, ignoring the change in the volume of the hydraulic fluid, the expression for the output flow rate of the piston pump, $Q_{\mathrm{a}}$, and the input flow rate, $Q_{\mathrm{b}}$ is shown in Equation (7).

$$\left\{\begin{array}{l}{Q}_{\mathrm{a}}=D_{\mathrm{p}}{\omega }_{\mathrm{m}}-\xi \left(p_{\mathrm{a}}-p_{\mathrm{b}}\right)-L_{\mathrm{c}}\left(p_{\mathrm{a}}-p_0\right)\\ Q_{\mathrm{b}}=D_{\mathrm{p}}{\omega }_{\mathrm{m}}-\xi \left(p_{\mathrm{a}}-p_{\mathrm{b}}\right)+L_{\mathrm{c}}\left(p_{\mathrm{b}}-p_0\right)\end{array}\right.$$

(7)

where $D_{\mathrm{p}}$ is the displacement of the piston pump in ${\mathrm{m}}^3\cdot {\mathrm{rad}}^{-1}$; $p_{\mathrm{a}}$, $p_{\mathrm{b}}$, $p_0$ are the piston pump outlet pressure, inlet pressure, and drain port pressure in Pa; and $\xi$, $L_c$ are the internal leakage coefficient and external exposure coefficient of the piston pump in ${\mathrm{m}}^3\cdot {\mathrm{s}}^{-1}\cdot {\mathrm{Pa}}^{-1}$.

When the piston pump and hydraulic cylinder seal is good, ignoring the accumulator and drain port oil loss, at this time the piston pump and actuator cylinder input flow and output flow to meet the following:

(1)

Piston pump output flow $Q_{\mathrm{a}}$ = hydraulic cylinder input flow $Q_1$;

(2)

Plunger pump input flow $Q_{\mathrm{b}}$ = hydraulic cylinder output flow $Q_2$.

According to the mechanical structure of the hydraulic cylinder, the expressions for the input and output flow rates $Q_1$ and $Q_2$ of the hydraulic cylinder are shown in Equation (8).

$$\left\{\begin{array}{l}{Q}_1=A{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}+{\displaystyle \frac{V_{\mathrm{a}}+Ax}{E_{\mathrm{y}}}}\cdot {\displaystyle \frac{\mathrm{d}{p}_1}{\mathrm{d}t}}+L_{\mathrm{ep}}{p}_1=Q_{\mathrm{a}}\\ Q_2=A{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}-{\displaystyle \frac{V_{\mathrm{a}}-Ax}{E_y}}\cdot {\displaystyle \frac{\mathrm{d}{p}_2}{\mathrm{d}t}}-L_{\mathrm{ep}}{p}_2=Q_{\mathrm{b}}\end{array}\right.$$

(8)

where $A$ is the effective area of the hydraulic cylinder piston in ${\mathrm{m}}^2$; $x$ is the displacement of the hydraulic cylinder piston in $\mathrm{m}$; $V_{\mathrm{a}}$ is the effective volume of the hydraulic cylinder in ${\mathrm{m}}^3$; $E_{\mathrm{y}}$ is the modulus of elasticity of the fluid in $N\cdot {\mathrm{m}}^{-2}$; $T_{\mathrm{ep}}$ is the leakage coefficient of the hydraulic cylinder piston in ${\mathrm{m}}^3\cdot {\mathrm{s}}^{-1}\cdot {\mathrm{Pa}}^{-1}$; and $p_1$, $p_2$ are the inlet pressure and outlet pressure of the hydraulic cylinder, respectively, in Pa.

In general, the piston pump and hydraulic cylinder leakage coefficient is equal, that is, $L_{\mathrm{c}}=L_{\mathrm{ep}}$. The joint Equations (7) and (8) can be obtained from the expression of the equation of motion of the hydraulic cylinder, as shown in Equation (9).

$$\begin{array}{ll}{D}_{\mathrm{p}}{\omega }_{\mathrm{m}}=& A{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}+{\displaystyle \frac{V_{\mathrm{a}}}{2E_{\mathrm{y}}}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(p_1-p_2\right)+{\displaystyle \frac{Ax}{2E_{\mathrm{y}}}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(p_1+p_2\right)\\ & +\xi \left(p_{\mathrm{a}}-p_{\mathrm{b}}\right)+{\displaystyle \frac{L_c}{2}}\left(p_1-p_2\right)+{\displaystyle \frac{L_c}{2}}\left(p_{\mathrm{a}}-p_{\mathrm{b}}\right)\end{array}$$

(9)

The relationship among the internal pressures, $p_1$, $p_2$, $p_{\mathrm{a}}$, and $p_{\mathrm{b}}$ among the piston pump and the hydraulic cylinder is shown in Equation (10).

$$\left\{\begin{array}{c}{p}_1=p_a-p_{pipe}\\ p_2=p_b-p_{pipe}\end{array}\right.$$

(10)

where $P_{pipe}$ is the flow loss between the piston pump and the hydraulic cylinder in ${\mathrm{m}}^3\cdot {\mathrm{s}}^{-1}\cdot {\mathrm{Pa}}^{-1}$.

The expression for the pressure difference $\Delta p$ between the inlet and outlet of the hydraulic cylinder can be derived from Equation (10), as shown in Equation (11).

$$\Delta p=p_1-p_2=p_a-p_b$$

(11)

Since the hydraulic cylinder is a symmetric structure, then $d{p}_1/dt\approx -d{p}_2/dt$ and then associated with Equations (9) and (11), the simplified expression of the equation of motion of the hydraulic cylinder can be obtained as shown in Equation (12).

$$D_{\mathrm{p}}{\omega }_{\mathrm{m}}=A{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}+{\displaystyle \frac{V_{\mathrm{a}}}{2E_{\mathrm{y}}}}{\displaystyle \frac{\mathrm{d}\left(\Delta p\right)}{\mathrm{d}t}}+L_{\mathrm{a}}\Delta p$$

(12)

where $L_{\mathrm{a}}=\xi +L_c$ is the overall leakage coefficient of the system in ${\mathrm{m}}^3\cdot {\mathrm{s}}^{-1}\cdot {\mathrm{Pa}}^{-1}$.

The piston of a hydraulic cylinder is loaded during operation and its mechanical equation is expressed as shown in Equation (13).

$$A\Delta p=M{x}^{″}+B_{\mathrm{c}}{x}^{\prime }+K_{\mathrm{t}}x+F_{\mathrm{L}}$$

(13)

where $M$ is the load mass in $\mathrm{k}\mathrm{g}$; $B_c$ is the coefficient of viscous friction of the hydraulic cylinder in $N\cdot \mathrm{s}\cdot {\mathrm{m}}^{-1}$; $K_t$ is the coefficient of elastic load in $N\cdot {\mathrm{m}}^{-1}$; and $F_L$ is the load force in N.

Combine Equations (12) and (13) to obtain the expression of the mathematical model of the hydraulic link, as shown in Equation (14).

$$\left\{\begin{array}{l}{D}_{\mathrm{p}}{\omega }_{\mathrm{m}}=L_{\mathrm{a}}\Delta p+A{x}^{\prime }+{\displaystyle \frac{V_{\mathrm{a}}}{2E_{\mathrm{y}}}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\Delta p\\ A\Delta p=M{x}^{″}+B_{\mathrm{c}}{x}^{\prime }+K_{\mathrm{t}}x+F_{\mathrm{L}}\end{array}\right.$$

(14)

2.3. Equation for the State of the EHA System

The state variable $X$ of the system is selected and its expression is shown in Equation (15).

$$X={\left[x_1 x_2 x_3 x_4 x_5\right]}^T={\left[i_{\mathrm{q}} {\omega }_{\mathrm{m}} x \dot{x} \mathsf{\Delta }p\right]}^T$$

(15)

Therefore, the state equation of the EHA system is shown in Equation (16).

$$\left\{\begin{array}{l}{\dot{x}}_1={\displaystyle \frac{1}{L_{\mathrm{q}}}}\left(-R{x}_1-{\psi }_{\mathrm{f}}{p}_{\mathrm{n}}{x}_2+U_{\mathrm{q}}\right)\\ {\dot{x}}_2={\displaystyle \frac{1}{J}}\left({\displaystyle \frac{3}{2}}{p}_{\mathrm{n}}{\psi }_{\mathrm{f}}{x}_1-B_{\mathrm{m}}{x}_2-D_{\mathrm{p}}{x}_5-T_{\mathrm{f}}\right)\\ {\dot{x}}_3=x_4\\ {\dot{x}}_4=-{\displaystyle \frac{K_{\mathrm{t}}}{M}}{x}_3-{\displaystyle \frac{B_{\mathrm{c}}}{M}}{x}_4+{\displaystyle \frac{A}{M}}{x}_5-{\displaystyle \frac{F_{\mathrm{L}}}{M}}\\ {\dot{x}}_5={\displaystyle \frac{2E_{\mathrm{y}}}{V_{\mathrm{a}}}}\left(D_{\mathrm{p}}{x}_2-A{x}_4-L_{\mathrm{a}}{x}_5\right)\end{array}\right.$$

(16)

3. The EHA Controller Design

For the EHA, which is a high-order, nonlinear, variable structure system whose own structural parameters change with the operation time, the sliding mode controller has more advantages than the traditional PID controller, so the sliding mode control is used in cascade with the PID controller.

3.1. Design of Machine–Hydraulic Link Controllers

By analyzing Equation (16), the EHA system is essentially composed of a second-order motor link and a third-order machine–hydraulic link. For the hydraulic cylinder in the machine–hydraulic link, its position is closed-loop controlled using a sliding mode controller, and since there are non-matching perturbations in the machine–hydraulic link and the sliding mode control method is not robust to non-matching-type perturbations, it is necessary to perform a coordinate transformation of the machine–hydraulic link to redefine the expressions of the state variables of this subsystem, as shown in Equation (17).

$${\left[\begin{array}{ccc}{z}_1& z_2& z_3\end{array}\right]}^T={\left[\begin{array}{ccc}{x}_3& {\dot{x}}_3& {\ddot{x}}_3\end{array}\right]}^T$$

(17)

The expression corresponding to the equation of state is shown in Equation (18).

$$\left\{\begin{array}{l}{\dot{z}}_1=z_2\\ {\dot{z}}_2=z_3\\ {\dot{z}}_3=g\left(z_3,z_2,z_1,u,F_{\mathrm{L}}\right)\end{array}\right.$$

(18)

According to the redefined state variables, as shown in Equation (19), the following occurs:

$${\dot{z}}_3=-{\displaystyle \frac{K_{\mathrm{t}}}{M}}{z}_2-{\displaystyle \frac{B_{\mathrm{c}}}{M}}{z}_3+{\displaystyle \frac{A}{M}}{\dot{x}}_5-{\displaystyle \frac{{\dot{F}}_{\mathrm{L}}}{M}}$$

(19)

Calculate Equation (19) and then combine Equation (16) to obtain the expression of the equation of state of the hydraulic link after the transformation of the coordinates, as shown in Equation (20).

$$\left\{\begin{array}{l}{\dot{z}}_1=z_2\\ {\dot{z}}_2=z_3\\ {\dot{z}}_3=g_{3}u-A_3{z}_3-A_2{z}_2-A_1{z}_1-f_{\mathrm{d}}\left(t\right)\end{array}\right.$$

(20)

where $u={\omega }_{\mathrm{m}}$, $g_3=2A{E}_y{D}_p{\left(M{V}_{\mathrm{a}}\right)}^{-1}$, $A_3=\left(B_{\mathrm{c}}{V}_{\mathrm{a}}+2E_{\mathrm{y}}{L}_{\mathrm{a}}M\right){\left(M{V}_{\mathrm{a}}\right)}^{-1}$, $A_2=\left(K_{\mathrm{t}}{V}_{\mathrm{a}}+2A_2{E}_{\mathrm{y}}+2E_{\mathrm{y}}{L}_a{B}_c\right){\left(M{V}_{\mathrm{a}}\right)}^{-1}$, $A_1=\left(2E_{\mathrm{y}}{L}_{\mathrm{a}}{K}_{\mathrm{t}}\right){\left(M{V}_{\mathrm{a}}\right)}^{-1}$ and $fd\left(t\right)=\left(2E_y{L}_a{F}_L+V_a\times d{F}_L/\mathrm{dt}\right){\left(M{V}_{\mathrm{a}}\right)}^{-1}$. Here $F_L$ is a bounded value.

Let the system track a given reference position signal as xd. Then, the expressions for the position error, the rate of change in the position error, and the acceleration of the position error are shown in Equation (21).

$$\left\{\begin{array}{l}{e}_1=e=z_1-x_{\mathrm{d}}\\ e_2=\dot{e}=z_2-{\dot{x}}_{\mathrm{d}}\\ e_3=\ddot{e}=z_3-{\ddot{x}}_{\mathrm{d}}\end{array}\right.$$

(21)

The design slip mold surface is shown in Equation (22).

$$s=c_{1}e+c_2\dot{e}+\ddot{e}$$

(22)

where $c_1$ and $c_2$ are greater than zero and satisfy the Hurwtiz condition.

Using the exponential convergence rate, an expression for the control rate can be obtained, as shown in Equation (23).

$$u={\displaystyle \frac{\left(A_3-c_2\right)z_3+\left(A_2-c_1\right)z_2+A_1{z}_1}{g_3}}+{\displaystyle \frac{c_1{\dot{x}}_{\mathrm{d}}+c_2{\ddot{x}}_{\mathrm{d}}+{\stackrel{⃛}{x}}_{\mathrm{d}}}{g_3}}+{\displaystyle \frac{f_{\mathrm{d}}\left(t\right)}{g_3}}-{\displaystyle \frac{\xi \mathrm{sgn}\left(s\right)+ks}{g_3}}$$

(23)

where the $\mathrm{sgn}$ function is a sign function; $\xi$ and $k$ denote the rate at which the moving point of the system converges to the switching surface $s=0$, and both $\xi$ and $k$ are greater than zero.

Construct the Lyapunov function as shown in Equation (24).

$$V={\displaystyle \frac{1}{2}}{s}^2$$

(24)

Associative Equations (23) and (24) can be obtained to satisfy the condition as shown in Equation (25).

$$\dot{V}=s\dot{s}=-\xi \left|s\right|<0$$

(25)

Equation (25) shows the existence and accessibility of the slip mode surface. In order to avoid the occurrence of sliding mode jitter vibration, the continuous obtained saturation function sat is used to replace the sign function, and the expression of the saturation function is shown in Equation (26).

$$sat\left(*\right)=\left\{\begin{array}{c}\begin{array}{ccc}1& & *\ge 1\end{array}\\ \begin{array}{ccc}*& & -1<*<1\end{array}\\ \begin{array}{ccc}-1& & *\le 1\end{array}\end{array}\right.$$

(26)

3.2. Design of Motor Link Controller

For the motor link, its structural parameters in the process of system operation change range are relatively small and can be approximated as a linear link. Additionally, the use of PID on the motor current and speed control is easier to realize, and can realize the motor output without static difference, as the system’s dynamic response is fast.

The expression for the PID control of the speed–current double loop is shown in Equation (27).

$$\left\{\begin{array}{l}{u}_{\mathrm{speed}}=K_{\mathrm{ps}}(v^{*}-v)+K_{\mathrm{is}}\int (v^{*}-v)\mathrm{d}t\\ u_{\mathrm{current}}=K_{\mathrm{pc}}({i_{\mathrm{q}}}^{*}-i_{\mathrm{q}})+K_{\mathrm{ic}}\int ({i_{\mathrm{q}}}^{*}-i_{\mathrm{q}})\mathrm{d}t\end{array}\right.$$

(27)

where $u_{\mathrm{speed}}$ and $u_{\mathrm{current}}$ are the controller outputs of the speed and current loops, respectively; $K_{\mathrm{ps}}$ and $K_{\mathrm{is}}$ are the proportionality and integration coefficients of the speed controller; $K_{\mathrm{pc}}$ and $K_{\mathrm{ic}}$ are the proportionality and integration coefficients of the current controller; $v^{*}$ is the target value of the speed, and $v$ is the value of the actual speed; $i^{*}$ is the target value of the speed, and $i$ is the value of the actual speed.

According to the state Equation (16) and the sliding mode control law Equation (23) of the EHA and the double loop PID control law Equation (27), the block diagram of the EHA servo system is obtained as shown in Figure 2. Figure 2 demonstrates the comprehensive framework of the EHA servo system, embodying the design of a three-closed-loop servo control system, in which the outermost loop is the position sliding mode controller of the hydraulic cylinder, and the inner loop consists of the motor’s current and velocity dual-loop PID controller. The system monitors the phase currents of the motor through Hall sensors and utilizes a resolver to obtain the position and angular velocity of the motor, forming a current loop and a velocity loop to ensure that the motor can accurately respond to control commands in actual operation. In addition, the position sensor feeds back the position of the hydraulic cylinder piston in real time and works with the sliding mode controller to form a position loop, so as to realize the precise control of the hydraulic cylinder. The system supports a variety of permanent magnet synchronous motor drive modes, including simple square wave control, sinusoidal wave control (e.g., SPWM or SVPWM) to reduce torque fluctuations, and the most advanced vector control (FOC), which decouples the three-phase currents into the d-axis and q-axis components through a coordinate transformation to realize the independent control of magnetic flux and torque. By applying the Parker transform and the inverse Parker transform, the system simplifies the process of current control, while the SVPWM technique converts the control signals into pulse-width modulated signals suitable for driving three-phase motors. Real time speed and position feedback of the generator is provided by an encoder, which strengthens the accuracy of the closed-loop control and the stability of the system, thus ensuring that the overall EHA servo system operates efficiently and reliably under various operating conditions.

4. EDSCSO-Based EHA Position Sliding Mode Controller

4.1. Original SCSO

The Sand Cat Search Optimization (SCSO) algorithm mimics the hunting behavior of sand cats in a low-frequency noise environment at 2 kHz. The algorithm consists of two primary phases: exploration and hunting. In this context, each sand cat represents a problem variable, and these variables are modeled as vectors within the algorithm’s structure. For a D-dimensional optimization problem, each sand cat is represented as a $1\times D$ array. The value of each variable is constrained to a floating-point number within a specified lower and upper bound. To initialize the SCSO algorithm, an initial candidate matrix of size ($N\times D$) is created, where N denotes the number of sand cats. The fitness of each sand cat is then evaluated by optimizing the corresponding fitness function. The sand cat with the best fitness is selected as the optimal solution, while the other sand cats continuously update their positions in each iteration, aiming to find a more optimal position and potentially replace the current best sand cat. This iterative process continues until the desired level of optimization is achieved. The search mechanism of the SCSO algorithm relies on the low-frequency noise detected by the sand cats. Each sand cat is modeled as a vector, and due to their ability to perceive low-frequency noise below 2 kHz, their hearing quality is set to 2. Moreover, the sensitivity range of each sand cat is determined by its ability to detect this low-frequency noise. To prevent the sand cat from losing its target during pursuit, the sensitivity variable decreases linearly from 2 to 0 with each iteration, as shown in Equation (28).

$$r_g=S_m-\left({\displaystyle \frac{S_m\cdot t}{T_{\max }}}\right)$$

(28)

where $t$ represents the current iteration count, and $T_{\max }$ represents the maximum number of iterations. To avoid being stuck in a local optimum, each sand cat is assigned a unique sensitivity range at random, as demonstrated in Equation (29).

$$r_i=r_g\cdot \mathrm{rand}$$

(29)

where $\mathrm{rand}$ represents a random number ranging from 0 to 1, and $r_i$ is the sensitivity of the i-th individual. The vector $R$ is an important parameter for the control algorithm to switch between different search phases, and its value is influenced by the random value $r_g$, as shown in Equation (30).

$$R=2\cdot r_g\cdot \mathrm{rand}-r_g$$

(30)

To control the balance between local and global search, SCSO uses the parameters $r_g$ and $R$. The parameter $R$ is generated randomly within the interval [−2, 2]. If $R$ is greater than 1, the sand cat enters the search phase and looks for prey using Equation (31). If $R$ is less than or equal to 1, the sand cat enters the predation phase and attacks prey found during the exploration phase using Equation (32).

$$X_{ij}\left(t+1\right)=r_i\cdot \left(X_{bj}\left(t\right)-\mathrm{rand}\cdot X_{ij}\left(t\right)\right)$$

(31)

In Equation (31), $X_{ij}\left(t\right)$ denotes the j-th dimension of the i-th individual in the current iteration, $X_{bj}\left(t\right)$ is the j-th dimension of the current optimal individual, and $X_{ij}\left(t+1\right)$ denotes the j-th dimension of the i-th individual in the next iteration.

$$X_{ij}\left(t+1\right)=X_{bj}\left(t\right)-r_i\cdot \left|\mathrm{rand}\cdot X_{bj}\left(t\right)-X_{ij}\left(t\right)\right|\cdot \cos \left(\theta \right)$$

(32)

In Equation (32), the sand cat’s sensitivity range can be visualized as a circle, with the moving direction determined by randomly selecting an angle $\theta$ using roulette. The value of the random angle falls within the range of $0°$ to $360°$, allowing each sand cat to move in different directions within the search space.

4.2. Proposed EDSCSO

4.2.1. Escape Mechanism (EM)

In the original SCSO algorithm, the process is divided into two distinct phases: exploration and hunting. For simplicity, the hunting phase is referred to as the exploitation phase in the following sections. However, this division leads to an abrupt transition between exploration and exploitation, which may result in premature convergence and limit the algorithm’s global search capability. To address this issue, the EDSCSO algorithm introduces an escape mechanism that simulates the natural behavior of sand cats, where they alter their direction when prey escapes during a pursuit. This mechanism extends the exploration and exploitation phases of the original SCSO into three distinct phases and refines the calculation formulas for each phase. Additionally, the hearing quality of the sand cat is increased to 2.8, thereby enhancing its range of motion and allowing for a more dynamic search process. Furthermore, the structure of the algorithm was modified. Instead of perturbing each dimension of an individual variable, as conducted in the original SCSO, the new algorithm perturbs the entire individual with a vector that contains identical random values for all dimensions. This adjustment reduces the running time of the algorithm to some extent. As a result, the parameter value range has been updated to [−2.8, 2.8]. The procedure for the escape mechanism is outlined as follows:

$$r_i=r_g\cdot \mathrm{rand}$$

(33)

$$R=2\cdot r_g\cdot \mathrm{rand}-r_g\cdot a$$

(34)

$$P_r=\mathrm{rand}\left(0,1\right)$$

(35)

In Equations (33)–(35), $\mathrm{rand}$ is a vector of random vectors with the same values from 0 to 1 for all dimensions, $a$ is a unit vector, and $r_i$ is a vector of sensitivities for the i-th individual. $P_r$ is a random number from 0 to 1 that is used to control the way in which individuals are updated.

If $R$ is greater than or equal to 1.5, it indicates that the algorithm is in the exploration phase. According to Equations (36) and (37), the population individuals will be explored in two ways, which are mathematically modeled as follows:

$$X_i^{\prime }\left(t\right)=r_i\cdot \left(X_a\left(t\right)-\mathrm{rand}\cdot X_i\left(t\right)\right)$$

(36)

$$X_i^{\prime }\left(t\right)=X_i\left(t\right)-r_i\cdot \left|r_c\cdot X_b\left(t\right)-r_1\cdot X_i\left(t\right)\right|\cdot \cos \left(\theta \right)$$

(37)

where, when $P_r$ is greater than 0.5, the position obtained by the sand cat through Equation (36) is located between the current position $X_i\left(t\right)$ and the random position $X_a\left(t\right)$. It is independent of the current optimal individual and highly randomized. $X_i^{\prime }\left(t\right)$ denotes the current individual after the update. On the contrary, if $P_r$ is less than or equal to 0.5, the position obtained by the sand cat through Equation (37) will be close to $X_i\left(t\right)$, which is influenced by the random angle $\theta$ and the optimal individual $X_{\mathrm{b}}\left(t\right)$. Furthermore, to increase the activity range of the sand cat, $r_c$ and $r_1$ are used as learning parameters for $X_{\mathrm{b}}\left(t\right)$ and $X_i\left(t\right)$, where $r_c$ and $r_1$ are random values generated according to the Cauchy and Levy distributions, respectively. By cross-searching the solution space, these two update equations enhance the global search capability of the algorithm.

If $R$ is greater than 0.75 or less than 1.5, the algorithm enters the selection phase, where the sand cats selectively engage in exploration or exploitation. As shown in Equation (38), when $P_r$ is greater than 0.5, the sand cat continues to explore the solution space with reference to the current optimal individual position $X_{\mathrm{b}}\left(t\right)$ and the current individual $X_i\left(t\right)$ position. As shown in Equation (39), when $P_r$ is less than or equal to 0.5, the sand cat feeds on prey with reference to the optimal individual position $X_{\mathrm{b}}\left(t\right)$ and the current individual $X_{\mathrm{b}}\left(t\right)$ position, as well as the random angle $\theta$. The algorithm smoothly transitions from exploration to exploitation during the selection phase. The mathematical model is as follows:

$$X_i^{\prime }\left(t\right)=r_i\cdot \left(X_b\left(t\right)-\mathrm{rand}\cdot X_i\left(t\right)\right)$$

(38)

$$X_i^{\prime }\left(t\right)=X_{\mathrm{b}}\left(t\right)-r_i\cdot \left|\mathrm{rand}\cdot X_b\left(t\right)-X_i\left(t\right)\right|\cdot \cos \left(\theta \right)$$

(39)

If $R$ is less than or equal to 0.75, the algorithm enters the predation phase. The sand cat will prey in two ways, as shown in Equations (40) and (41). When $P_r$ is greater than 0.5, the sand cat exploits the local region between the current optimal individual position $X_{\mathrm{b}}\left(t\right)$ and the current individual position $X_i\left(t\right)$, as shown in Equation (40). Otherwise, the sand cat will develop the target area with reference to the current optimal individual position $X_{\mathrm{b}}\left(t\right)$ and the population center position $X_{mean}\left(t\right)$, as shown in Equation (41). The random angle $\theta$ ensures that the sand cat exploits the target area at different angles. The two exploitation methods cause a misalignment in the explored areas, aiding the algorithm in escaping local optima. The updated equations are as follows:

$$X_i^{\prime }\left(t\right)=X_b\left(t\right)-r_i\left|\mathrm{rand}\cdot X_b\left(t\right)-X_i\left(t\right)\right|\cdot \cos \left(\theta \right)$$

(40)

$$X_i^{\prime }\left(t\right)=X_{\mathrm{b}}\left(t\right)-r_i\cdot \left|\mathrm{rand}\cdot X_b\left(t\right)-X_{mean}\left(t\right)\right|\cdot \cos \left(\theta \right)$$

(41)

Using Equation (42), the updated individual ($X_i^{\prime }\left(t\right)$) is compared to the pre-update individual ($X_i\left(t\right)$), and the better individual is retained.

$$X_i\left(t+1\right)=\left\{\begin{array}{c}{X}_i\left(t\right)\\ X_i^{\prime }\left(t\right)\end{array}\right.\begin{array}{cc}& \end{array}\begin{array}{c}fitness\left(X_i\left(t\right)\right)\le fitness\left(X_i^{\prime }\left(t\right)\right)\\ fitness\left(X_i\left(t\right)\right)>fitness\left(X_i^{\prime }\left(t\right)\right)\end{array}$$

(42)

4.2.2. Random Elite Cooperative Guidance Strategy (RE-CGS)

The current optimal individual affects the position-updating process of SCSO during both the exploration and exploitation stages. The optimal individual attracts and clusters other individuals, restricting the algorithm’s search space during exploration and possibly making it miss the global optimum. This limitation also restricts SCSO’s potential to exploitation. EDSCSO utilizes an RE-CGS, which entails the following actions, to handle this problem:

As shown in Figure 3, first, the top $\mathsf{\alpha }$ individuals with the highest fitness values are classified as the elite population $Xe$, with the value of $\mathsf{\alpha }$ being one-sixth of the total number of individuals while the remaining individuals belong to the ordinary population $Xo$. Then, random elite individuals are selected to guide the search of k (Equation (43)) randomly selected individuals from the ordinary population as in Equation (44).

$$k={\displaystyle \frac{\left(N-\alpha \right)}{\alpha }}$$

(43)

$$X{o}_{rk\left(s\right)}^{\prime }\left(t\right)=X{e}_r\left(t\right)\pm C\cdot r_g\cdot \left(X{e}_r\left(t\right)-X{o}_{rk\left(s\right)}\left(t\right)\right)$$

(44)

where $X{e}_r\left(t\right)$ represents a randomly selected elite individual from $Xe$, and $X{o}_{rk\left(s\right)}\left(t\right)$ denotes the s-th individual out of $k$ randomly selected common individuals. $X{o}_{rk\left(s\right)}^{\prime }\left(t\right)$ denotes the individual after $X{o}_{rk\left(s\right)}\left(t\right)$ is updated. Step size factor $C$ has a value of 0.5. Each random elite individual guides $k$ ordinary individuals. After being guided by the elite individuals, the ordinary individuals undergo a greedy selection to retain the best individual, as shown in Equation (45).

$$X{o}_{rk\left(s\right)}\left(t+1\right)=\left\{\begin{array}{c}X{o}_{rk\left(s\right)}\left(t\right)\\ X{o}_{rk\left(s\right)}^{\prime }\left(t\right)\end{array}\right.\begin{array}{cc}& \end{array}\begin{array}{c}fitness\left(X{o}_{rk\left(s\right)}\left(t\right)\right)\le fitness\left(X{o}_{rk\left(s\right)}^{\prime }\left(t\right)\right)\\ fitness\left(X{o}_{rk\left(s\right)}\left(t\right)\right)>fitness\left(X{o}_{rk\left(s\right)}^{\prime }\left(t\right)\right)\end{array}$$

(45)

The pseudo-code for this process is shown in Algorithm 1.

Algorithm 1 Pseudo-Code of the Each Random Elite Individual Guides k Ordinary Individuals

Inputs: Random elite individuals $X{e}_r\left(t\right)$, $k$ random ordinary individuals.

Output: $X{o}_{rk\left(s\right)}\left(t+1\right)$.

For$\mathrm{s}=1:k$

$\mathrm{The} \mathrm{position} \mathrm{of} X{o}_{rk\left(s\right)}\left(t\right)$$\mathrm{is} \mathrm{updated} \mathrm{according} \mathrm{to} \mathrm{Equation} \left(44\right) \mathrm{and} \mathrm{the} \mathrm{updated} \mathrm{position} \mathrm{is} X{o}_{rk\left(s\right)}^{\prime }\left(t\right)$

$\mathrm{According} \mathrm{to} \mathrm{Equation} \left(45\right), \mathrm{comparing} \mathrm{the} \mathrm{fitness} \mathrm{of} X{o}_{rk\left(s\right)}\left(t\right)$$\mathrm{with} X{o}_{rk\left(s\right)}^{\prime }\left(t\right)$ $\mathrm{and} \mathrm{retaining} \mathrm{the} \mathrm{better} \mathrm{individual} \mathrm{yields} X{o}_{rk\left(s\right)}\left(t+1\right)$.

End for

Finally, in the mid to later stages of the algorithm, cooperative computation is employed to improve the elite individuals. The idea of cooperative computation is inspired by the concept of context vector introduced by Gao et al. [19]. The basic idea is to use the current global optimum as the context vector and replace the corresponding dimension of the context vector with the dimension of the individuals. This approach allows each individual to contribute to the global optimum in each dimension. The pseudo-code for the cooperative computation is shown in Algorithm 2.

Algorithm 2 Pseudo-Code of the Cooperative Computation

Inputs: Elite population ($\alpha$ individuals), $X_b\left(t\right)$

Output:  $X_b\left(t+1\right)$

For e = 1:  $\alpha$

$CC$ = $X_{re.e}\left(t\right)$

For d = 1:  $D$

$DD$ = $X_b\left(t\right)$; $D{D}_d$ = $C{C}_d$;

If $fitness\left(DD\right)<fitness\left(X_b\left(t\right)\right)$

$X_b\left(t+1\right)$ = $DD$

End if

End for

End for

However, cooperative computation requires accessing information from each dimension, which significantly increases the algorithm’s time complexity. To better leverage the cooperative computation strategy while minimizing time costs, a novel segmented cooperative computation method is employed in RE-CGS. Specifically, during the algorithm’s iteration from T/5 to 4T/5, to prevent premature convergence, cooperative computation is performed only once on the elite population after updating the positions of all individuals in the population. In the later stages of the algorithm (from 4T/5 to T), cooperative computation is carried out on the elite population every time one-third of the individuals’ positions are updated, thereby improving the efficiency of exploitation. This situation-based cooperative computation strategy effectively balances the algorithm’s performance and time complexity. As an example, consider 500 iterations with a population size of 30. If the traditional cooperative computation method is used, where elite cooperative computation is performed for each updated individual’s position, it would require 15,000 cooperative computations. In contrast, the situation-based strategy proposed in this study requires only 600 cooperative computations.

4.2.3. Multi-Path Differential Perturbation Strategy (MDPS)

During the computational process of the algorithm, the individuals in the population gradually concentrate, reducing the diversity of the population and increasing the chance of the algorithm to become trapped in local optima. To overcome this limitation, a MDPS is proposed. This strategy consists of multiple perturbation stages, where each stage performs a different mutation operator to perturb the positions of individuals. The basic calculation formulas for the two mutation operators are as follows:

$$V_i=X_{r1}+F\left(X_{r2}-X_{r3}\right)+F\left(X_{r4}-X_{r5}\right)$$

(46)

$$V_i=X_{r1}+F\left(X_b-X_m\right)+F\left(X_{r2}-X_{r3}\right)$$

(47)

where $X_{r1}$, $X_{r2}$, $X_{r3}$, $X_{r4}$, and $X_{r5}$ represent random individuals, $X_b$ denotes the current best individual, $X_m$ represents the average individual in the population, and $F$ is the scaling factor. The vector $V_i$ represents the new position vector generated by the mutation operator. In Equation (46), $V_i$ is generated by adding the weighted difference vector between four random individuals in the population and another random individual. This mutation operator can maintain the exploitative capability of the population while increasing its diversity. In Equation (47), $V_i$ is generated by adding the weighted difference vector among the current best individual, the average individual $X_m$, and two random individuals in the population, with another random individual. This mutation operator has the advantage of maintaining a balance between exploration and exploitation in the population.

MDPS introduces the perturbation for the first stage based on Equation (48).

$$X_{one}\left(t\right)=X_{r1}\left(t\right)+R\cdot \left(X_{r2}\left(t\right)-X_{r3}\left(t\right)\right)+R\cdot \left(X_{r4}\left(t\right)-X_{r5}\left(t\right)\right)$$

(48)

$$X_i\left(t+1\right)=\left\{\begin{array}{c}{X}_i\left(t\right)\\ X_{one}\left(t\right)\end{array}\right.\begin{array}{cc}& \end{array}\begin{array}{c}fitness\left(X_i\left(t\right)\right)\le fitness\left(X_{one}\left(t\right)\right)\\ fitness\left(X_i\left(t\right)\right)>fitness\left(X_{one}\left(t\right)\right)\end{array}$$

(49)

where $X_{one}\left(t\right)$ represents the variant individual, and others are random individuals ($X_{r1}$, $X_{r2}$, $X_{r3}$, $X_{r4}$, $X_{r5}$). The control factor $R$ in Equation (34) is used as the scaling factor to scale the different vector and control the step length of searching, which decreases with the iteration number. In the initial stage of the algorithm, a larger value of $R$ can encourage the sand cat to explore unknown areas globally. As $R$ decreases, the sand cat’s search behavior becomes more focused on local areas, allowing for fine-grained exploration of the solution space and the search for better local solutions. Finally, a greedy selection strategy is employed to eliminate individuals with lower fitness.

A perturbation strategy for the second stage is proposed using Equation (50).

$$X_{two}\left(t\right)=X_i\left(t\right)+\mathrm{rand}\cdot \left(X_b\left(t\right)-X_m\left(t\right)\right)+\mathrm{rand}\cdot \left(X_{r1}\left(t\right)-X_{r2}\left(t\right)\right)$$

(50)

$$X_i\left(t+1\right)=\left\{\begin{array}{c}{X}_i\left(t\right)\\ X_{two}\left(t\right)\end{array}\right.\begin{array}{cc}& \end{array}\begin{array}{c}fitness\left(X_i\left(t\right)\right)\le fitness\left(X_{two}\left(t\right)\right)\\ fitness\left(X_i\left(t\right)\right)>fitness\left(X_{two}\left(t\right)\right)\end{array}$$

(51)

The symbols in the equations denote the mutant individual ($X_{two}\left(t\right)$), the current population $X_i\left(t\right)$, the mean individual ($X_m\left(t\right)$), the best individual in the population $X_b\left(t\right)$, and random individuals ($X_{r1}\left(t\right)$ and $X_{r2}\left(t\right)$).

The first stage of perturbation will be executed for each individual when its position is updated, and the second stage will be conducted after position updating of all individuals. In order to push the population towards better solutions, better individuals are selectively retained according to Equations (49)–(51). The pseudo-code for EDSCSO is given by Algorithm 3.

Algorithm 3 Pseudo-Code of the EDSCSO

Inputs:$\mathrm{The} \mathrm{population} \mathrm{size}$$N$$\mathrm{and} \mathrm{the} \max \mathrm{iterations} T$ $\mathrm{and} \mathrm{the} \mathrm{dimension} \mathrm{of} \mathrm{the} \mathrm{problem} D$.

Output:$\mathrm{The} \mathrm{best} \mathrm{solution} X_b$. Initialize the population. Calculating the fitness function based on the objective function. $\mathrm{Initialize} \mathrm{the}$$S_m$ $r_g$ $\mathrm{and} R$. For $t=1:T$     $\mathrm{For} \mathrm{each} \mathrm{search} \mathrm{agent}, \mathrm{obtain} \mathrm{a} \mathrm{random} \mathrm{angle} \mathrm{based} \mathrm{on} \mathrm{the} \mathrm{Roulette} \mathrm{Wheel} \mathrm{Selection} [0° \le \theta$ < 360°).     $X_b\left(t\right)=best\left(X\left(t\right)\right)$.       For $i=1:N$          If $(R$ > 1.5)              Update the population according to Equation (36) or Equation (37).              Elseif$(R$ > 0.75)              Update the population according to Equation (40) or Equation (41).              Else              Update the population according to Equation (42) or Equation (43).          End if              Retention of the more numerous individuals according to Equation (44).              Update the position of the best individual so far.              Calculating, comparing and updating the fitness.              Each random elite individual guides k ordinary individuals according to Algorithm 1.          If (((t < 4/5T)&&(t > T$/5\left)\right)\&\&i$ == N)||((t > 4/5T)&&(t == N/3$\left|\right|i$ == 2N/3|| $i$ == N)).              Cooperative Computation according to Algorithm 2.          End if          The first differential perturbation stage is performed according to Equations (50) and (51).          End for       For $i=1:N$          The second differential perturbation stage is performed according to Equations (50) and (51).       End for End for Return $\mathrm{the} \mathrm{best} \mathrm{solution} X_b$.

4.3. Effectiveness Analysis of Improvement Strategies

In this section, the effectiveness of the three proposed improvement strategies is analyzed. The algorithm that incorporates only the escape mechanism (EM) is referred to as A_1SCSO, the algorithm that includes the RE-CGS is named A_2SCSO, and the algorithm featuring MDPS is designated as A_3SCSO. Subsequently, the performance of A_1SCSO, A_2SCSO, and A_3SCSO is analyzed and evaluated in terms of exploration–exploitation balance, exploitation capability, and population diversity, respectively.

4.3.1. Effectiveness of RE-CGS

To analyze the impact of the RE-CGS on the exploitation capability of the SCSO algorithm, both SCSO and A_2SCSO are used to solve the Schwefel’s Problem 2.21 function. Equation (52) represents Schwefel’s Problem 2.21 function, which is a unimodal function commonly used to evaluate the exploitation capability of algorithms. The optimal value of this test function is 0 and the global minimum coordinate in the 3D search space is ${\mathrm{x}}^{\mathrm{*}}=(0,0,0)$. Figure 4a–d, respectively, depicts the three-dimensional plots of Schwefel’s Problem 2.21 and the spatial distribution of the populations for the two algorithms at different stages (early, mid, and late). The black pentagram in the figures represents the location of the global optimum, while the yellow and green circles denote the populations corresponding to SCSO and A_2SCSO, respectively.

$$F\left(x\right)=MAX\left\{\begin{array}{cc}\left|x_i\right|& 1\le x_i\le 30\end{array}\right\}$$

(52)

The population distributions of A_2SCSO (green) and SCSO (yellow) in the solution space are similar in the early iterations. This is due to the fact that RE-CGS has little effect on the population diversity in the early stages. In the middle of the iteration, the population distribution of A_2SCSO becomes more concentrated in the solution space compared to SCSO. This is due to the fact that the elite individuals guide the common individuals towards the neighborhood of the optimal solution, which speeds up the convergence of the algorithm. In later iterations, the population of A_2SCSO is closer to the global optimum in the solution space. Furthermore, after 30 repeated optimization experiments on Schwefel’s Problem 2.21, the results show that the coevolutionary strategy using stochastic elite guidance successfully identifies better individuals with an average success rate of 65.37%. In other words, A_2SCSO outperforms SCSO in more than half of the iterations throughout the optimization process. In summary, RE-CGS steers the search direction of the SCSO population and overcomes the limitation of SCSO to be limited to a specific search.

It is clear in Figure 4 that the improved strategy of the algorithm gradually approaches the optimal value as the number of iterations increases. The convergence accuracy reaches 10⁻⁵⁷ in the middle of the iteration and further improves to 10⁻⁷⁷ in the late iteration. This result indicates that the search agent is able to find better solutions as the number of iterations increases, which effectively validates the effectiveness of the improved strategy.

4.3.2. Effectiveness of MDPS

To investigate the influence of the dual-layer differential perturbation strategy on population diversity, in this section, A_3SCSO and SCSO are employed to solve the Schwefel 2.26 function. Equation (53) represents the mathematical expression of the Schwefel 2.26 function, which has multiple local minima and a global minimum at ${\mathrm{x}}^{\mathrm{*}}=(420.97,420.97,420.97)$ in a three-dimensional space. Figure 5a–d depicts the three-dimensional plots of the Schwefel 2.26 function and the spatial distribution of the populations for both algorithms in the early, mid, and later iterations. The black pentagram in the figures indicates the location of the global optimum, while the yellow and blue circles represent the populations optimized using SCSO and A_3SCSO, respectively.

$$F\left(x\right)=-{\sum }_{i=1}^n\left(x_i\sin \sqrt{\left|x_i\right|}\right)$$

(53)

From Figure 5, it can be observed that as the iterations progress, the SCSO population (yellow) quickly converges toward the current optimum, leading to a rapid reduction in population diversity. In contrast, due to the dual-layer differential perturbation strategy, A_3SCSO (blue) maintains a larger search range than SCSO during the early and mid iterations, resulting in better population diversity. In the later iterations, most individuals in both A_3SCSO and SCSO are concentrated near the current optimum.

However, A_3SCSO outperforms SCSO in terms of both the breadth of the search and the accuracy of the optimal solution. Specifically, the blue circles exhibit a more scattered spatial distribution and are closer to the global optimum compared to the yellow circles. This can be attributed to the effective expansion of the population’s search range through the dual-layer differential perturbation strategy, which facilitates the discovery of better solutions. Furthermore, after conducting 30 repeated optimization experiments on the Schwefel 2.26 function, the results indicate that the average probability of obtaining better individuals through the first-path differential perturbation is 38.29%, while the average probability of obtaining improved populations through the second-level differential perturbation is 100%. (The second-path differential perturbation is applied to the entire population, and the improvement in population quality is determined by comparing the overall fitness of the population before and after the perturbation). In conclusion, the MDPS enhances population diversity.

4.3.3. Complementary Analysis of Improvement Strategies

In this article, three improvement strategies are proposed: (1) escape mechanism (EM) to balance the searching ability of the algorithm at different stages; (2) randomized elite-guided coevolutionary strategy (RE-CGS) to enhance the exploitation ability of the algorithm; and (3) multi-dimensional perturbation strategy (MDPS) to improve the diversity of the populations. Compared to the original SCSO, the inclusion of EM provides a more balanced trade-off between exploration and exploitation. RE-CGS addresses the lack of exploitation capability in SCSO by bootstrapping ordinary individuals with elite individuals and performing cooperative computation with elite populations at different stages. However, when dealing with specific challenges such as multimodal functions, it is still possible for the population to fall into a local optimum. To mitigate this, MDPS is used to continuously perturb the population, thus maintaining the diversity of the population.

To conduct a comprehensive complementary analysis of the improvement strategies, the algorithm incorporating the EM and random elite-guided strategy is denoted as A_12SCSO, the algorithm incorporating the EM and MDPS is denoted as A_13SCSO, and the algorithm incorporating the random elite-guided strategy and MDPS is denoted as A_23SCSO. Comparative experiments are conducted on four test functions (F3, F5, F7–F8) from CEC2017 for SCSO, six combinatorial algorithms, and EDSCSO that incorporates all three strategies. The F3 test function is named Shifted and Rotated Zakharov Function, F5 test function is named Shifted and Rotated Rastrigin’s Function, F7 test function is named Shifted and Rotated Lunacek Bi_Rastrigin Function, and F8 test function is named Shifted and Rotated Non-Continuous Rastrigin’s Function. The parameter settings are as follows: dimension D = 30, population size N = 30, maximum number of iterations T = 500. The results are presented in

Table 1. Comparative results of different strategies for improving SCSO.

		SCSO	A_1SCSO	A_2SCSO	A_3SCSO	A_12SCSO	A_13SCSO	A_23SCSO	EDSCSO
F3	AVG	5.73 × 104	5.59 × 104	2.09 × 104	5.73 × 104	4.41 × 102	1.11 × 104	1.21 × 104	3.00 × 102
	STD	9.58 × 103	6.34 × 103	7.92 × 103	9.58 × 103	3.21 × 102	3.58 × 103	4.30 × 103	1.76 × 10−3
F5	AVG	7.69 × 102	7.69 × 102	6.89 × 102	7.13 × 102	6.93 × 102	5.92 × 102	6.36 × 102	5.80 × 102
	STD	4.09 × 101	3.55 × 101	4.28 × 101	4.57 × 101	4.37 × 101	2.00 × 101	3.16 × 101	2.47 × 101
F7	AVG	1.16 × 103	1.20 × 103	1.00 × 103	1.09 × 103	1.09 × 103	8.70 × 102	9.29 × 102	8.48 × 102
	STD	1.01 × 102	5.83 × 101	8.38 × 101	7.82 × 101	6.86 × 101	3.81 × 101	5.15 × 101	3.70 × 101
F8	AVG	1.01 × 103	1.01 × 103	9.60 × 102	9.91 × 102	9.52 × 102	8.77 × 102	9.11 × 102	8.69 × 102
	STD	3.53 × 101	3.66 × 101	3.28 × 101	3.09 × 101	3.08 × 101	2.14 × 101	2.91 × 101	1.96 × 101

.

Table 1 shows that in all test functions, A_12SCSO and A_13SCSO perform better than A_1SCSO, which just employs the selection method. A_12SCSO and A_23SCSO show better performance in all test functions than A_2SCSO, which only uses the random elite-guided method. In all test functions, A_13SCSO and A_23SCSO perform better than A_3SCSO, which solely employs the dual-path perturbation approach. Lastly, EDSCSO with all three strategies outperforms the other algorithms in most of the test functions.

In conclusion, the EDSCSO algorithm incorporates an EM that balances exploration and exploitation, enabling the algorithm to unleash its full potential. The RE-CGS enhances the algorithm’s exploitation capability by guiding the population toward promising search directions. The MDPS helps the algorithm escape from local optima. The synergistic effect of these three strategies greatly improves the overall performance of the algorithm.

5. Intermodulation Simulation Verification of EHA System

5.1. EHA Position Slide Mold Controller Parameter Setting

The parameters to be optimized for the sliding mode controller are set to be $c_1$, $c_2$, $\xi$, $k$. The initial number of populations npop = 20, the lower boundary of the initial solution is lb = [0, 0, 0, 0], the upper boundary of the initial solution is ub = [15,000, 570,576, 600,000, 82,000], the number of iterations Max_iter = 100, and the expression for calculating the degree of adaptation is given as follows:

$$Z=a_1\cdot t_r+a_2\cdot t_s+a_3\cdot e_{ave}+a_4\cdot o_s$$

(54)

where $Z$ is the objective function fitness; $a_1$, $a_2$, $a_3$, $a_4$ are all weighting factors; $t_r$ is the rise time; $t_s$ is the regulation time; $e_{ave}$ is the steady-state mean error; and $o_s$ is the amount of overshoot.

Take the weighting factors $a_1$, $a_2$, $a_3$, $a_4$, respectively, the following occurs:

$$\left[\begin{array}{cccc}{a}_1& a_2& a_3& a_4\end{array}\right]=\left[\begin{array}{cccc}100& 50& 200000& 10\end{array}\right]$$

(55)

Matlab/Simulink and AMEsim software are utilized to carry out the joint simulation verification work. The simulation models of the controller and motor are built in Simulink, and the simulation model of the hydraulic link is built in AMEsim. Six sets of comparison experiments will be carried out in the following: (1) traditional three-closed-loop PID; (2) sliding mode PID before optimization; (3) sliding mode PID after optimization of SSA algorithm; (4) sliding mode PID after optimization of PSO algorithm; (5) sliding mode PID after optimization of GWO algorithm; and (6) sliding mode PID after optimization of EDSCSO algorithm.

The key simulation parameters of the EHA system are summarized in

Table 2. EHA hydraulic part model parameters.

Paramete	Value
Effective area of hydraulic cylinder piston A/m2	5.6352 × 10−3
Effective stroke of hydraulic cylinder piston x/m	0.8
Hydraulic cylinder internal leakage coefficient La/(${\mathrm{m}}^3\cdot {\mathrm{s}}^{-1}\cdot {\mathrm{Pa}}^{-1}$)	2.25 × 10−12
Modulus of elasticity of the fluid Ey/($\mathrm{N}\cdot {\mathrm{m}}^{-2}$)	6.86 × 108
Hydraulic cylinder effective volume Va/m3	4.5 × 10−3
Hydraulic cylinder viscous friction coefficient Bc/($\mathrm{N}\cdot \mathrm{s}\cdot {\mathrm{m}}^{-1}$)	1000
Hydraulic cylinder and its load mass M/kg	200
Displacement of piston pump Dp/(${\mathrm{m}}^3\cdot {\mathrm{rad}}^{-1}$)	2.387 × 10−6
Motor viscous friction coefficient Bm/($\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{s}\cdot {\mathrm{rad}}^{-1}$)	6 × 10−4
Phase resistance R/Ω	0.0485
Phase inductance L/mH	0.395
Motor moment of inertia J/($\mathrm{kg}\cdot {\mathrm{m}}^2$)	0.0027
Magnetic chain Ψf/Wb	0.1194
Elastic load factor Kt/($\mathrm{N}\cdot {\mathrm{m}}^{-1}$)	108
Bus voltage VDC/V	750

.

5.2. Comparative Analysis of Simulation Results

The controller parameters used in the simulation are as follows:

(1)

PID-related parameters: $k_p$ = 32.4976, $k_i$ = 90.2094.

(2)

Pre-optimization sliding mode PID-related parameters: $c_1$ = 69,300; $c_2$ = 260, $x_i$ = 666,520; $k$ = 79,235.

(3)

SSA post-optimization sliding mode PID-related parameters: $c_1$ = 570,576; $c_2$ = 5.4, $x_i$ = 600,000; $k$ = 78,217.

(4)

PSO optimized sliding mode PID-related parameters: $c_1$ = 570,576; $c_2$ = 11,395, $x_i$ = 6556, $k$ = 65,325.

(5)

GWO optimized sliding mode PID-related parameters: $c_1$ = 418,783; $c_2$ = 9395, $x_i$ = 329,354, $k$ = 64,040.

(6)

Parameters related to sliding mode PID after EDSCSO optimization: $c_1$ = 328,309; $c_2$ = 11,649, $x_i$ = 479,610, $k$ = 43,754.

Three kinds of input commands $x_d$ are used in the simulation, i.e., (1) 0.06 m step command; (2) 0.6 m step command; and (3) sinusoidal command $x_d=0.03\sin \left(2\prod \times 0.1t\right)+0.03$.

Figure 6 represents the six comparative step response curves at 0.06 m and 0.6 m step commands, and

Table 3. Dynamic performance indexes of six groups of comparison experiments.

Comparison	Time Rising (tr/s)	Time Setting (ts/s)	Error Steady-State (ess/mm)	Overshoot (os/%)
SSA	0.55	0.58	0.18	1.18
PSO	0.55	0.58	0.15	0.82
GWO	0.55	0.58	0.15	0.42
EDSCSO	0.55	0.58	0.15	0.32
Pre-Optimization	0.63	0.75	0.82	0.72
PID	0.96	1.19	1.3	1.6

represents the comparison results of the six dynamic performance indexes at 0.06 m step commands.

An analysis of Figure 6 and Table 3 shows that the sliding mode PID improves the response speed by 34.4% compared with the traditional three-closed-loop PID, the regulation time is shortened by 0.44 s, and the steady-state error during operation is reduced by 36.9%. The optimized sliding mode PID has a significant improvement in response speed compared with the pre-optimization sliding mode PID, and the regulation time is also shorter, and the amount of overshooting and the steady-state error are smaller. Among the optimization algorithms, the EDSCSO algorithm has the best optimization effect.

An analysis of Figure 7 and

Table 4. Six groups of comparative experimental data under external disturbance.

Comparison	Maximum Deviation (xe/mm)	Recovery Time (s)
SSA	2.09	1.58
PSO	2.11	1.62
GWO	2.09	2.19
EDSCSO	2.09	1.5
Pre-Optimization	2.18	4.11
PID	2.2	6.38

shows that the sliding mode PID is more robust than the traditional PID and the optimized sliding mode PID has a shorter recovery time. The EDSCSO algorithm is optimized to be more robust and has a shorter recovery time of only 1.5 s.

By analyzing the tracking curves of the six sets of comparison experiments given a sinusoidal signal of 0.1 Hz, as shown in Figure 8, the following can be determined: The maximum tracking error of the traditional three-closed-loop PID is 8.5 mm, and the maximum tracking error of the sliding mode PID before optimization is 4.2 mm. The maximum tracking errors of the optimized sliding mode PIDs of SSA, PSO, GWO, and EDSCSO are 0.83 mm, 1.11 mm, 0.7 mm, and 0.5 mm, respectively. The maximum tracking error of sliding mode PID after optimization of SSA, PSO, GWO, and EDSCSO is 0.83 mm, 1.11 mm, 0.7 mm, and 0.5 mm, respectively.

Table 5. Running time and minimum fitness of SSA, PSO, GWO, and EDSCSO algorithms.

Comparison	Running Time (h)	Best Fit Value
SSA	2.02	131.7444
PSO	3.05	121.7082
GWO	1.59	118.3554
EDSCSO	1.41	117.7145

shows the running time and best fitness values of the SSA, PSO, GWO, and EDSCSO algorithms. An analysis of the data in Table 5 shows that EDSCSO has the least running time, consumes the least computational resources, and achieves better fitness values for the problem of rectifying and optimizing the EHA position sliding mode controller. This experimental data fully proves that the EDSCSO-optimized EHA position sliding mode controller has better overall performance compared to several other swarm intelligence optimization algorithms.

6. Conclusions

This study analyzes the numerous challenges encountered by EHA as higher-order nonlinear systems, with particular emphasis on their strong coupling and inherent uncertainties. Given these characteristics of the EHA system, traditional optimization algorithms, such as Particle Swarm Optimization, are predisposed to becoming trapped in local optima during the parameter tuning of the EHA position sliding mode controller, while their computational efficiencies are generally inadequate. Such deficiencies hinder the ability to meet the high efficiency demands of practical applications. Consequently, to effectively address this complex issue, EDSCSO is proposed, designed specifically to optimize and accurately tune the parameters of the EHA position sliding mode controller.

This research is underpinned by rigorous theoretical analysis and employs a joint simulation platform comprising Matlab/Simulink and AMEsim to thoroughly evaluate the effectiveness of the proposed EDSCSO algorithm. Experimental results indicate that the EDSCSO-optimized position sliding mode controller significantly reduces steady-state errors and demonstrates improved robustness when compared to traditional swarm intelligence optimization algorithms. These findings provide substantial empirical support for enhancing the control accuracy and reliability of the EHA system, contributing positively not only to the EHA itself but also offering valuable insights for research in related fields.

Although EDSCSO has been comprehensively validated for controller parameter optimization, its potential for practical applications still needs to be deeply explored, and the future will focus on investigating the broad applicability of the improved EDSCSO in several fields. These include the following: in the field of machine learning for hyper-parameter optimization and feature selection, thus improving model performance. In robotics research, to help solve path planning and multi-robot collaboration problems in dynamic environments. In energy management, to optimize renewable energy allocation and demand response strategies for sustainable development. In conclusion, the improved EDSCSO has shown great application value in specific research directions in several fields. However, it must be acknowledged that despite the superior performance of the EDSCSO optimization scheme, certain limitations and constraints still exist. For example, the algorithm may require more computational resources and time when facing extreme uncertainty or dynamically changing environments, which undoubtedly poses a great challenge for practical applications. In addition, ensuring the stability of the optimization process, especially in resource-constrained situations, remains a focus of future research.