Intelligent sliding mode tracking control for electro-hydraulic position servo systems based on variable exponential power reaching law with CGWO-PSO

Abstract

Aiming at the various nonlinear factors, external disturbance in electro-hydraulic servo position systems, a novel sliding mode tracking control method based on the variable exponential power is proposed, which not only improves the speed of reaching the sliding mode surface, but also ensures the smooth approach performance as much as possible: the chattering phenomenon of the sliding surface is reduced effectively as a result. Furthermore, in order to optimize the parameters of the sliding mode controller, a new optimization algorithm is proposed by combining Grey Wolf optimizer with particle swarm optimization (CGWO-PSO), skillfully. The primary advantages of the new CGWO-PSO algorithm include faster convergence speed and higher performance. Finally, the co-simulation platform for the electro-hydraulic servo position system is built. The results of co-simulation show that the sliding mode tracking control method proposed in this paper can realize that the electro-hydraulic position servo system has excellent tracking accuracy, strong robustness, and smooth approach performance.

Keywords

Electro-hydraulic position servo systems nonlinear control sliding mode control variable exponential power reaching law Grey Wolf optimizer mixed particle swarm optimization

Introduction

Electro-hydraulic position servo systems (EHPSSs) offer advantages such as high-power density ratio, lightweight construction, compact size, and high transmission efficiency. These systems have demonstrated promising applications in a wide range of fields, spanning from defense to civil industries (Chen et al., 2005). Although hydraulic systems employ liquids for power transmission and control, despite their higher costs and certain inconveniences relative to electrical systems, they bring forth a plethora of distinct benefits. Among these are rapid response speed, a notable power-to-mass ratio, and impressive load-bearing stiffness. As a result, electro-hydraulic servo systems occupy a distinct and advantageous position within control domains that demand exceptional precision and substantial output power. Amid the array of hydraulic servo systems, the EHPSS emerges as the quintessential and widely adopted variant. Its extensive applications encompass critical areas such as aircraft attitude control, aircraft engine speed regulation, radar antenna azimuth control, robotic joint manipulation, radar and artillery control systems, as well as vibration test benches.

However, EHPSSs exhibit strong uncertainty and nonlinear characteristics, including variable gain, time-varying parameters, and modeling uncertainties. They are also susceptible to external disturbances. Therefore, achieving high-performance tracking control for EHPSSs remains a challenging task in both academic and engineering domains (Shen et al., 2022; Zhang et al., 2022). Various advanced control strategies have been employed to address the position tracking problem of EHPSSs, such as sliding mode control (SMC; Feng et al., 2021), adaptive control (Xu et al., 2017), robust control (Nguyen et al., 2022), and intelligent control (Jiang et al., 2023), among others.

So far, SMC has become a research hotspot because of its low requirement for model accuracy and strong robustness to external interference. Wide applications of SMC can be found in areas concerned with robustness or nonlinear systems (Zhang 2021), including permanent magnet synchronous motors (Qu et al., 2020), microgrid system (Bada et al., 2022), piezoelectric micro-nano platform (Xu et al., 2020), electric fuel pumps (Huang et al., 2022), robotic manipulators (Sai et al., 2021), underwater vehicles (Su et al., 2021), and so on. However, in the realm of research related to SMC, there still remain several areas that require further improvement. These include enhancing the speed at which the sliding surface is attained, mitigating chattering phenomena, and maximizing controller performance to the greatest extent possible. Various scholars have dedicated significant efforts to addressing these challenges and have made substantial contributions in these areas.

A continuous faster terminal SMC is proposed for the dynamic performance of a permanent magnet synchronous motor, which can make the system converge to the sliding mode surface quickly, but the sliding mode jitter problem is not solved (Junejo et al., 2018). Liu et al. (2022) illustrated a new type of slid approach rate that solves the problem of weakening the sliding mode jitter, and the shortcoming is that the speed of reaching the sliding mode surface needs to be improved. Banazadeh et al. (2020) proposed an optimization method based on linear quadratic regulator (LQR) and evolutionary computation techniques for sliding mode controller parameters; although the method can make the results close to optimal, the process is complex and the convergence speed is slow. Sun et al. (2022) developed a position control method of the electro-hydraulic system based on adaptive reaching law, which can suppress the chattering problem and reduce the disturbances effectively, however, which does not significantly improve compared with the traditional approach rate. Yu and Yuan (2022) used an adaptive fuzzy proportional–integral–derivative (PID) control algorithm to adapt to the control of EHPSS. This method has good performance for step response and sinusoidal tracking, but does not involve the effect of external load disturbances. A robust linear approximation method for accurate position control of electro-hydraulic servo systems is proposed (Liu et al., 2020). The method designed different operational modes for the EHPSS and implements rapid switching control between them, achieving satisfactory tracking performance. A comparative table is shown in Table 1 to compare the advantages and disadvantages of the various methods.

Table 1.

Comparative table of some methods.

Literatures	Objects	Control methods	Advantages	Parameters tuning	Pending issues	Difficulty level
Junejo et al. (2018 )	PMSM	Terminal slidingmode control	Rapid convergence, ESO feedforward compensation	No	Chattering, singular point, parameters optimization et.al	Moderate
Liu et al. (2022 )	PMSM	Exponentialreaching lawsliding mode control	Weakening chattering	No	Parameters optimization et.al	Moderate
Banazadeh et al.(2020 )	satellite	Sliding mode control	LQR and evolutionaryalgorithmsto optimize parameters	LQR and evolutionaryalgorithmsto optimize parameters	Chattering, approach speed,optimization speed.	Complex
Yu and Yuan (2022 )	EHPSS	Fuzzy PID	Online tuning of PID gains	Fuzzy Control	Tracking speed and tracking error	Relatively simple
Liu et al. (2020 )	EHPSS	Switch H_∞	Switching control of differentoperational modes	No	Parameters optimization	Complex

Therefore, the purpose of this paper is to present a novel variable exponential power reaching law tracking SMC method with combining Grey Wolf optimizer with particle swarm optimization (CGWO-PSO) algorithm. The primary contributions are summarized as follows:

A novel sliding mode tracking controller with variable exponential power reaching law is proposed to overcome the nonlinear factors and disturbances in EHPSSs and to improve their.

In order to better tune the parameters of the sliding mode tracking controller, a new CGWO-PSO algorithm, which is combined GWO with particle swarm organization (PSO), is presented. The CGWO-PSO algorithm can realize the improvement in finding the optimal controller parameters and the speed of optimization.

The mathematical model of EHPSSs is developed and the co-simulation platform of AMESim/MATLAB for the EHPSSs is built. Co-simulation results show that the proposed sliding mode tracking control method can realize that the EHPSSs has excellent tracking accuracy, strong robustness, and smooth approach performance.

The remainder of this paper is organized as follows: Section “Mathematical model of EHPSS” shows the mathematical model of EHPSSs. Section “Sliding mode tracking controller based on variable exponential power reaching law” presents the design of sliding mode tracking controller based on variable exponential power reaching law, and robust stability analysis is given in this section as well. Intelligent parameters optimization for sliding mode tracking controller by CGWO-PSO is illustrated in Section “CGWO-PSO intelligent optimization algorithm.” In Section “AMESim/MATLAB co-simulation analysis”, the co-simulation platform of AMESim/MATLAB for the EHPSSs is built and co-simulation results are shown. Section “Conclusion, future outlook, and challenges” draws the conclusions of this paper.

Mathematical model of EHPSS

As depicted in Figure 1(a), a physical representation of an EHPSS is presented. In addition, Figure 1(b) illustrates the simplified structural composition of the EHPSS, encompassing the hydraulic oil source, electro-hydraulic servo valve, hydraulic cylinder, displacement sensor, and load.

According to the system kinematic equation, the hydraulic cylinder output force and load force balance equation are obtained as

A_{p} P_{L} = m \frac{d^{2} x_{p}}{d t^{2}} + B \frac{d x_{p}}{dt} + K x_{p} + F_{L}

(1)

where m is the total mass of the equivalent piston rod, F_L is the external load force acting on the piston, A_p is the area of the effective piston of the hydraulic cylinder, x_p is the piston displacement, B is the equivalent viscous damping coefficient, and K is equivalent elastic stiffness. P_L = P₁−P₂, P₁, and P₂ are the pressure in the left and right cavities of the cylinder, respectively. Assuming that the dynamics of the pipeline and the pressure loss in the pipe can be ignored, the pressure in each working chamber of the hydraulic cylinder is equal, the oil temperature and the modulus of volume elasticity are constants, and there is no leakage, then the continuous equation of the hydraulic cylinder flow is shown in equation (2) as follows

Q_{L} = A_{p} \frac{d x_{p}}{dt} + C_{tp} P_{L} + \frac{V_{t}}{4 β_{e}} \frac{d P_{L}}{dt}

(2)

where C_tp is the total leakage coefficient of the hydraulic cylinder, β_e is the effective bulk elastic modulus of oil, Q_L = (Q₁ + Q₂)/2 is the load flow of the hydraulic cylinder, Q₁ and Q₂ are the flow of the left and right chambers of the hydraulic cylinder, V_t = V₁ + V₂ is the total volume of the system control chamber, V₁ and V₂ are the volume of the left and right chambers of the hydraulic cylinder, and x_p is the piston displacement. The flow equation for servo valves is as follows

Q_{L} = C_{d} w x_{v} \sqrt{\frac{P_{s} - sign (x_{v}) P_{L}}{ρ}}

(3)

By using Taylor expansion, the linearization of equation (3) is shown as equation (4)

Q_{L} = K_{q} x_{v} - K_{c} P_{L}

(4)

where $K_{q} = \frac{\partial Q_{L}}{\partial x_{v}}$ and $K_{c} = \frac{\partial Q_{L}}{\partial p_{L}}$ are the flow gain of servo valve and flow pressure amplification factor, respectively, which are expressed as

K_{q} = \frac{\partial Q_{L}}{\partial x_{v}} = C_{d} w \sqrt{\frac{P_{s} - sign (x_{v}) P_{L}}{ρ}}

(5)

K_{c} = - \frac{\partial Q_{L}}{\partial p_{L}} = C_{d} w | x_{v} | \sqrt{\frac{1}{4 ρ (p_{s} - sign (x_{v}) p_{L})}}

(6)

The input and output characteristics of servo amplifier and electro-hydraulic servo valve can be equivalent to a proportional unit as

x_{v} = K_{sv} i

(7)

i = K_{e} u

(8)

where K_sv is the flow gain of the system, i is the input current of the system, K_e is the gain of servo amplifier, and u is the control voltage for servo valve.

Define ${[\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}]}^{T} = {[\begin{matrix} x_{p} & {\overset{\cdot}{x}}_{p} & {\overset{\cdot\cdot}{x}}_{p} \end{matrix}]}^{T}$ , combining equations (1)–(8) to obtain the state space equation of the EHPSS as equation (9)

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ {\overset{\cdot}{x}}_{3} = f (x) + bu + d \end{matrix}

(9)

where $f (x) = a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3}$ , $a_{1} = - \frac{4 β_{e} C_{d} K}{m V_{t}}$ , $a_{2} = - \frac{K}{m} - \frac{4 β_{e}}{m V_{t}} ({A_{p}}^{2} + C_{t} B)$ , $a_{3} = - \frac{B}{m} - \frac{4 β_{e} C_{t}}{V_{t}}$ , $b = \frac{4 β_{e} A_{p}}{M_{t} V_{t}} C_{d} w K_{v} \sqrt{\frac{p_{s} - sign (x_{v}) p_{L}}{ρ}}$ , $d = - \frac{4 β_{e} C_{tp}}{M_{t} V_{t}} F_{L} - \frac{1}{M_{t}} \frac{d F_{L}}{dt}$ is the external disturbance and it is bounded.

Remark 1. In practice $\sqrt{p_{s} - sign (x_{v}) p_{L}}$ is seldom an imaginary number or zero. In the rare case that $p_{s} - sign (x_{v}) p_{L} \leq 0$ , $\sqrt{\frac{p_{s} - sign (x_{v}) p_{L}}{ρ}}$ is set to a small positive number in order to avoid the problem of dividing by an imaginary number or zero (Andrew and Liu, 2000).

Sliding mode tracking controller based on variable exponential power reaching law

Variable exponential power reaching law

The tracking error of the system is defined as

E = {[\begin{matrix} e_{1} & e_{2} & e_{3} \end{matrix}]}^{T} = {[\begin{matrix} x_{p} - x_{d} & {\overset{\cdot}{x}}_{p} - {\overset{\cdot}{x}}_{d} & {\overset{\cdot\cdot}{x}}_{p} - {\overset{\cdot\cdot}{x}}_{d} \end{matrix}]}^{T}

(10)

where x_d is the desired position.

The control objective is

lim e = lim (x_{d} - x_{1}) = 0

(11)

For the EHPSSs (equation (9)), the sliding surface is designed to be of the following form

s = c_{1} e_{1} + c_{2} e_{2} + e_{3}

(12)

Based on the fundamental theory of SMC (Drakunov and Utkin, 1992), the sliding mode controller ensures that the system’s state trajectory reaches the predetermined sliding surface initially, and then drives the tracking error states to converge toward the origin along the sliding surface. However, the stability condition outlined in the basic theory of SMC does not address how the system reaches the sliding surface. To address this reachability issue, one approach is to employ a reaching law, with the exponential reaching law (ERL) being a popular choice. For the scalar form of the sliding mode variable s, the distribution of the ERL is denoted as

\frac{ds}{dt} = - ε sgn (s) - ks

(13)

Although ERL solves the reachability problem, once ε and k are chosen, the arrival velocity of the tracking error to the sliding surface will be slow when s is close to zero. Besides, chattering is notorious.

Therefore, this paper designs a new variable exponential power reaching law (VEPRL) as

\frac{ds}{dt} = - ε {| x |}^{α (s)} \tan h (\frac{s}{g}) - k \cdot s \cdot {| s |}^{β \cdot sgn (| s | - 1)}

(14)

where ε > 0, k > 0, 0 < β < 1, g > 0, x is e in equation (11) and $α (s) = {\begin{matrix} p_{1} / q_{1}, | s | > 1 \\ q_{2} / p_{2}, | s | \leq 1 \end{matrix}$ . $p_{1}$ , $q_{1}$ , $p_{2}$ , $q_{2}$ are positive scalars, and $p_{1}$ > $q_{1}$ > 0, $p_{2}$ > $q_{2}$ > 0.

When |s|> 1, it is viewed as the tracking error is far away from the sliding surface, sgn(|s|−1) = 1 and α(s) > 1. At this time, the tracking error approaches the sliding surface under the combined action of $ε {| x |}^{α (s)} \tan h (\frac{s}{g})$ and ks|s|^β.

When |s|≤ 1, the reaching velocity in ERL decreases greatly due to the decrease of s. On the contrary, sgn(|s|−1) = −1, then the exponential term in equation (14) becomes ks|s|^−β. Obviously, ks|s|^−β > ks|s|^β, thus, the speed approaching to the sliding surface can be increased.

Hence, under the VEPRL (14), when the tracking error is far away from the sliding surface, it can approach the sliding surface with a higher velocity, while the tracking error near the sliding surface, α(s), can be small in order to reduce chattering.

Control law and robust stability

Derivation of sliding surface (12)

\begin{matrix} \overset{\cdot}{s} = c_{1} {\overset{\cdot}{e}}_{1} + c_{2} {\overset{\cdot}{e}}_{2} + {\overset{\cdot}{e}}_{3} \\ = c_{1} e_{2} + c_{2} e_{3} + {\overset{\dots}{y}}_{p} - {\overset{\dots}{y}}_{d} \end{matrix}

(15)

According to the model of EHPSSs (equation (9))

\begin{matrix} c_{1} e_{2} + c_{2} e_{3} + a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + d + bu - {\overset{\dots}{y}}_{d} \\ = - ε {| e |}^{α (s)} \tan h (\frac{s}{g}) - k \cdot s \cdot {| s |}^{β \cdot sgn (| s | - 1)} \end{matrix}

(16)

Then, the control law for EHPSSs is

\begin{matrix} u = \frac{1}{b} (- ε {| e |}^{α (s)} \tan h (\frac{s}{g}) - ks {| s |}^{β \cdot sgn (| s | - 1)} - c_{1} e_{2} - c_{2} e_{3} \\ - a_{1} x_{1} - a_{2} x_{2} - a_{3} x_{3} + d + {\overset{\dots}{y}}_{d}) \end{matrix}

(17)

The Lyapunov function is chosen as

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

(18)

The derivation of equation (18) is

\begin{matrix} \overset{\cdot}{V} = s \overset{\cdot}{s} \\ = s (c_{1} e_{2} + c_{2} e_{3} + a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} \\ + bu - \overset{\dots}{y} + d) \\ = s (- ε {| e |}^{α (s)} \tanh (\frac{s}{g}) - ks {| s |}^{b sgn (| s | - 1)}) \\ = - ε {| e |}^{α (s)} s \cdot \tanh (\frac{s}{g}) - k s^{2} {| s |}^{b sgn (| s | - 1)} \end{matrix}

(19)

where $ε$ > 0, $s \cdot \tanh (\frac{s}{g})$ > 0, then $ε {| e |}^{α (s)} s \cdot \tanh (\frac{s}{g})$ < 0; k> 0, thus, $- k s^{2} {| s |}^{b sgn (| s | - 1)} < 0$ , $\overset{\cdot}{V} < 0$ .

Therefore, the tracking dynamic of EHPSSs is stable robustly.

Remark 2. Under digital computer, signals in the system are all in discrete-time, so from the discrete-time point of view, VEPRL is compared with ERL in the following.

The discrete-time expression of ERL is

s_{1} (n + 1) - s_{1} (n) = - ε_{1} T sgn (s (n))

(20)

where T is the sampling period, n is time interval, $s (n)$ is the corresponding variable in discrete-time domain of $s_{1} (t)$ .

It is assumed that the trajectory of the system can reach the sliding surface in finite steps; it can be described as $\lim_{n \to \infty} s_{1} (n) = 0^{+}$ or $\lim_{n \to \infty} s_{1} (n) = 0^{-}$ .

Assuming $s_{1} (n) = 0^{-}$ , then $s_{1} (n + 1) - s_{1} (n) = - T ε_{1}$ .

Similarly, assuming $s_{1} (n) = 0^{+}$ , $s_{1} (n + 1) - s_{1} (n) = T ε_{1}$ .

Thus, the bandwidth of tracking error around sliding surface is $Δ_{1} = 2 ε_{1} T$ .

Differently, in the VEPRL, when the sliding variable s approaches zero, equation (14) can be simplified to $\frac{ds}{dt} \approx - ε {| x |}^{q_{1} / q_{2}} \tanh (\frac{s}{g})$ , then $s_{1} (n + 1) - s_{1} (n) = - T ε {| x |}^{q_{2} / p_{2}} \tan h (\frac{s_{1} (n)}{g})$ .

It is assumed that the trajectory of the system can reach the sliding surface in finite steps, it can be described as $\lim_{n \to \infty} s_{1} (n) = 0^{+}$ or $\lim_{n \to \infty} s_{1} (n) = 0^{-}$ .

Similarly, assuming $s_{1} (n) = 0^{-}$ , then $s_{1} (n + 1) = - T {| x |}^{q_{2} / p_{2}} \tanh (\frac{s_{1} (n)}{g}) < T {| x |}^{q_{2} / p_{2}}$ . If $s_{1} (n) = 0^{+}$ , then $s_{1} (n + 1) = - T {| x |}^{q_{2} / p_{2}} \tan h (\frac{s_{1} (n)}{g})$ .

The bandwidth of tracking error around sliding surface is $Δ < 2 T ε {| x |}^{\frac{q_{2}}{p_{2}}}$ .

In this case, assuming $Δ = 2 T ε {| x |}^{\frac{q_{2}}{p_{2}}}$ . The sliding mode trajectory based on VEPRL and ERL are shown in Figure 2. Obviously, in the ERL, because the bandwidth is a fixed value, the tracking error cannot reach the origin, but will cross sliding surface repeatedly in the fixed. And, it can be seen from Figure 2 that the tracking trajectory stably approaches to the origin in a sector area, which is conducive to reducing the position tracking error and chattering. In fact $Δ < 2 T ε {| x |}^{\frac{q_{2}}{p_{2}}}$ therefore, SMC-based VEPRL offers a more effective means of mitigating sliding mode oscillations.

Here giving an example, consider the control object model represented as

\overset{\cdot\cdot}{θ} (t) = - f (θ, t) + bu (t) + d (t)

(21)

where $f (θ, t) = 25 \overset{\cdot}{θ}$ , b = 133, d(t) is external disturbance.

Let position command $θ_{d} = \sin t$ , the initial state of the controlled object is [-0.2, -0.2], and the simulation results of VERPL and ERL are as shown in Figure 3. Clearly, by comparing with the SMC based on ERL shown in Figure 3(a) and (b), it can be seen that the SMC based on VEPRL can approach the sliding surface faster and the error converges faster in Figure 3(c) and (d).

CGWO-PSO intelligent optimization algorithm

Basic principle of Grey Wolf optimization

The Grey Wolf optimization (GWO) algorithm is inspired by the process of grey wolf hunting prey (Mirjalili et al., 2014). The grey wolves in the wolf group are divided into four categories, which are expressed as α_wolf, β_wolf, δ_wolf, and ω_wolf. In the grey wolf algorithm, the fitness value indicates the proximity of a grey wolf’s location to the location of the prey, and the self-adaptation value for each grey wolf is calculated sequentially. The top three wolves in the population with the highest fitness (i.e. the three optimal solutions in each population) are denoted as α_wolf, β_wolf, and δ_wolf in descending order of their fitness values. From a different perspective, where α_wolf denotes the position of the current optimal solution, the second and third optimal solutions are β_wolf and δ_wolf, respectively. The remaining solutions are ω_wolves, ω_wolves will follow the evolution of the three best wolves, that is, α_wolf, β_wolf, and δ_wolf.

The process of wolf hunting prey can be expressed as

\vec{X} (t_{GWO} + 1) = {\vec{X}}_{p} (t_{GWO}) - \vec{A} \cdot | \vec{C} \cdot {\vec{X}}_{p} (t_{GWO}) - \vec{X} (t_{GWO}) |

(22)

where $t_{GWO}$ denotes the number of iterations, ${\vec{X}}_{p}$ and $\vec{X} (t_{GWO})$ denote the position vectors of the current prey and the grey wolf, respectively, $\vec{A}$ and $\vec{C}$ are expressed as

\vec{A} = 2 \vec{a} \cdot {\vec{r}}_{1} - \vec{a}

(23)

\vec{C} = 2 {\vec{r}}_{2}

(24)

where $\vec{a}$ is a linear decreasing parameter that guarantees the search and approach ability, ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ are random vectors, ${\vec{r}}_{1} \in [0, 1]$ , ${\vec{r}}_{2} \in [0, 1]$ .

The hunting process is led by the wolf α_wolf, and the position of the prey is determined according to the distance between αwolf, β_wolf and δ_wolf, with the following expressions

{\vec{X}}_{1} = {\vec{X}}_{α} - {\vec{A}}_{1} \cdot | {\vec{C}}_{1} \cdot {\vec{X}}_{α} - \vec{X} |

(25)

{\vec{X}}_{2} = {\vec{X}}_{β} - {\vec{A}}_{2} \cdot | {\vec{C}}_{2} \cdot {\vec{X}}_{β} - \vec{X} |

(26)

{\vec{X}}_{3} = {\vec{X}}_{δ} - {\vec{A}}_{3} \cdot | {\vec{C}}_{3} \cdot {\vec{X}}_{δ} - \vec{X} |

(27)

The process of updating the location is

X_{ω} (t_{GWO} + 1) = \frac{X_{1} + X_{2} + X_{3}}{3}

(28)

PSO primary idea

The PSO is initialized as a population of random particles (random solutions). The optimal solution is found by iterations. In each iteration, the particles update themselves by tracking two extremes (Kennedy and Eberhart, 1995). This speed directly affects the position of the next update of the particles, and the evaluation position is evaluated by the distance of the optimal solution, which is the only criterion to evaluate the particles.

In a D_PSO-dimensional space, there is a swarm of N particles, in which the position of the particles can be recorded as X _i = (x_i₁, x_i₂, … $x_{i D_{PSO}}$ ), the velocity of particles can be expressed as V _i = (v_i₁, v_i₂, … $v_{i D_{PSO}}$ ), the optimal value is found by particle $p_{bes t_{i}}$ = (p_i₁, p_i₂, … $p_{i D_{PSO}}$ ), and the global optimal value searched by particle swarm is g _best = (g₁, g₂, … $g_{D_{PSO}}$ ). After finding the two optimal values, the particle updates its speed and position through equations (29) and (30)

\begin{matrix} v_{i} (k_{PSO} + 1) = ω_{PSO} \cdot v_{i} (k_{PSO}) + c_{PSO 1} rand () \\ \times (p_{bes t_{id}} (k_{PSO}) - x_{id} (k_{PSO})) \\ + c_{PSO 2} \times rand () \times (g_{bes t_{d}} (k_{PSO}) - x_{id} (k_{PSO})) \end{matrix}

(29)

x_{id} (k_{PSO} + 1) = x_{id} (k_{PSO}) + x_{id} (k_{PSO} + 1)

(30)

where k_PSO is current number of iterations, $ω_{PSO}$ is initial weight coefficient, subscript d = 1, 2, 3, …, D_PSO, subscript i = 1, 2, 3, …, M, $c_{PSO 1}$ and $c_{PSO 2}$ are acceleration constant parameters, and rand() is random number in [0, 1].

Combined GWO with PSO

For the traditional GWO, it does not incorporate any mechanism to enhance the best position of the grey wolf, which can result in a suboptimal solution quality. In addition, it may also lead to decreased performance when dealing with multidimensional problems. To overcome these limitations and enhance the search capability, a new optimization algorithm called CGWO-PSO is developed for solving multidimensional and complex optimization problems. In CGWO-PSO, GWO serves as the search tool, initiating the optimization process, while the velocity vector of PSO is introduced into GWO to improve the position of the αwolf grey wolf. The improved formulas for velocity and position updates are given by equations (31) and (32)

\begin{matrix} V_{i} (t_{GWO} + 1) = ω [V_{i} (t_{GWO}) \\ + ς_{1} ζ_{1} ({\vec{X}}_{1} - X_{i} (t_{GWO})) \\ + ς_{2} ζ_{2} ({\vec{X}}_{2} - X_{i} (t_{GWO})) \\ + ς_{3} ζ_{3} ({\vec{X}}_{3} - X_{i} (t_{GWO}))] \end{matrix}

(31)

X_{i} (t_{GWO} + 1) = X_{i} (t_{GWO}) + V_{i} (t_{GWO} + 1)

(32)

where $ω$ is initial weight coefficient, $ς_{1}$ , $ς_{2}$ , $ς_{3}$ are acceleration constant parameters, and $ζ_{1}$ , $ζ_{2}$ , $ζ_{3}$ are random numbers in [0, 1].

The flow chart of the presented intelligent optimization algorithm of CGWO-PSO is shown in Figure 4(a). And to validate the performance of the proposed CGWO-PSO algorithm, tests were conducted on the five-dimensional Rastrigin test function using PSO, GWO, and ACGWO algorithms. After running the algorithms 100 times, the average objective function values obtained were 2.511, 0.091, and 0.002, respectively. Figure 4(b) illustrates the change in objective function values for each algorithm. It can be observed that the CGWO-PSO algorithm outperforms the other two optimization algorithms in terms of both convergence speed and accuracy. It exhibits better stability, indicating superior optimization performance of the designed CGWO-PSO algorithm.

CGWO-PSO for the optimization of controller

In this paper, the absolute error integral index is selected as optimization index, that is

F = \int_{0}^{t} | e (t) | d t

(33)

where e(t) is the output displacement tracking error of EHPSSs.

The block diagram of the new variable exponential power reaching law sliding mode tracking control strategy based on CGWO-PSO optimization for EHPSSs is shown in Figure 5.

AMESim/MATLAB co-simulation analysis

The co-simulation between AMESim and MATLAB combines the simulation capabilities of the hydraulic system in AMESim with the numerical processing advantages of MATLAB. The advantages of AMESim/MATLAB are as follows:

Both of them can establish a more accurate mechanical model and control system model for electro-hydraulic servo hydraulic system, respectively.

Through the controlled object model established by AMESim and the control system model established by MATLAB, coupling relationship between them can be analyzed intuitively.

The S-function module in Simulink provides a point-to-point joint simulation interface for AMESim and MATLAB, effectively avoiding the reconstruction of complex models between different platforms.

In order to verify the effectiveness and practicality of the control strategy proposed in this paper, the AMESim software is utilized to construct the component-level model of the EHPSS (electro-hydraulic power steering system). Subsequently, the co-simulation S-function module is developed in MATLAB/Simulink for designing the control system. Finally, the co-simulation is conducted to analyze the tracking performance of the proposed EHPSS controller. Figure 6 illustrates the AMESim model of the EHPSS and the co-simulation control experiment platform. The AMESim model of EHPSS mainly consists of the following parts: displacement sensors, co-simulation interface (this section transfers the data compilation program S-function from AMESim to MATLAB), hydraulic cylinder (driving the load), servo valve (the opening of the valve controls the displacement of the piston rod of the hydraulic cylinder), and so on. Before conducting co-simulation experiments, specific settings are needed to meet the requirements of co-simulation, including setting the environment variables required for co-simulation and ensuring that the AMESim model and the control algorithm model are located in the same path.

The main technical parameters of the EHPSS are as follows: M_t = 80 kg, C_tp=2×10⁻¹³ m³/(s×Pa), β_e = 8×10⁸ Pa, V_t=3.13×10⁻³ m³, A_p=6.26×10⁻³, K_q = 1.02×10⁻⁷, P_s = 2.1×10⁷ Pa, K_c = 1.4×10⁻¹² m³/(s×Pa).

The CGWO-PSO is used for parameter optimization, after undergoing optimization via CGWO-PSO, the parameters of controller have been refined to a more optimal state, with the resultant values for $ε$ = 52, $q_{1}$ = 11, $q_{2}$ = 1, $p_{1}$ = 31, $p_{2}$ = 3, $c_{1}$ = 610, $c_{2}$ = 55. The criterion for adjusting the β is to ensure that the performance of the controller is satisfactory within the desired range of β values. In other words, the β value should be adjusted in a way that guarantees the controller’s performance meets the required standards and objectives. This criterion is crucial in achieving effective control and attaining the desired outcomes.

Static load tracking control experiment

Here, the static load is simulated and three displacement instruction signals (Case 1) are considered separately as follows:

Multi-frequency sinusoidal displacement command signal (MSDCS)

\begin{matrix} y_{d 1} = 0.01 \sin t + 0.005 \sin 2 t + 0.007 \sin 0.5 t \\ y_{d 2} = {\begin{matrix} 0.003 t t \leq 5 \\ 0.015 t > 5 \end{matrix} \end{matrix}

Slope displacement command signal (SLCS)

Sinusoidal displacement command signal (SCS)

y_{d 3} = 0.016 \sin t

The simulation of the VEPRL, ERL sliding mode position control effects based on CGWO-PSO optimization and high order sliding mode (HSMC) are shown in Figure 7. Clearly, by comparing Figure 7(a), (d) and (g), it is obvious that the VEPRL provides a better tracking effect for MSDCS with less tracking error and less jitter than the ERL and HSMC; while comparing Figure 7(b), (e) and (h), one can see that the VEPRL has smaller steady-state error and faster response time for SLCS tracking; and from Figure 7(c), (f) and (i), similarly, when the tracking target is SCS, VERPL exhibits stronger tracking performance.

Therefore, the tracking effect of EHPSSs by VERPL is better than that based on ERL, with smaller tracking error, faster tracking of the upper displacement command signal in the initial stage, and less chattering.

External interference forces tracking control experiment

Now, considering the effect of sinusoidal disturbance force, its tracking effect simulation results are shown in Figure 8(a), and the tracking error is shown in Figure 8(b), where the value of sinusoidal disturbance force is F₁ = 200 sin t N and the command signal is y_d = 0.01 sin t + 0.005 sin 2t + 0.007 sin 0.5t N (Case 2). Obviously, Figure 8(a)–(c) gives the tracking comparison among PI, SMC with VEPRL, HSMC, and SMC with ERL. From Figure 8(c), it can be observed that the control signal of VEPRL exhibits some initial fluctuations within a very short time at the beginning of the simulation, but quickly stabilizes thereafter. On the contrary, HSMC shows periodic fluctuations in its control signal. While considering the influence of sinusoidal disturbance, the sliding mode tracking control error based on VEPRL is the smallest and the control effect is the best, and the control method with the largest tracking error is the PI.

Two performance indexes are introduced to quantitatively analyze the position tracking effect of the above controllers, and the performance evaluation of the controller is realized by using the displacement tracking error data. The absolute error mean index is as follows

E_{MAE} = \frac{1}{N} \sum_{i = 1}^{N} | e (i) |

The standard deviation of absolute error is as follows

E_{RMSE} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(| e (i) | - E_{MAE})}^{2}}

The performance indexes of each controller are shown in Table 1. By comparing the EMAE and ERMSE values of different controllers, the performance in terms of control accuracy can be assessed. Lower EMAE and ERMSE values indicate higher control accuracy. The HSMC controller falls between the other controllers in terms of EMAE and ERMSE. Although it is not as accurate as the VEPRL controller, it still outperforms the PI and ERL controllers.

Finally, the hydraulic cylinder is given a disturbance force at the 10th second, where the step disturbance force is F_step = 400 N and the displacement command signal is 0.016sint (Case 3). The simulation results are shown in Figure 8(d)–(f). The performance indicators of each controller are shown in Table 2.

Table 2.

Performance comparisons among controllers.

	E _MAE	E _RMSE
PI (Case 2)	5.65 × 10⁻⁴	3.98 × 10⁻⁴
ERL (Case 2)	1.96 × 10⁻⁴	1.38 × 10⁻⁴
HSMC (Case 2)	1.13 × 10⁻⁴	5.96 × 10⁻⁵
VEPRL (Case 2)	4.68 × 10⁻⁵	3.64 × 10⁻⁵
PI (Case 3)	8.86 × 10⁻⁴	4.53 × 10⁻⁴
ERL (Case 3)	2.85 × 10⁻⁴	1.96 × 10⁻⁴
HSMC (Case 3)	1.08 × 10⁻⁴	9.59 × 10⁻⁵
VEPRL (Case 3)	5.74 × 10⁻⁵	3.22 × 10⁻⁵

The VEPRL-based sliding mode controller exhibits the smallest error and achieves the best position tracking among the controllers analyzed. It is followed by HSMC. On the contrary, the PI controller demonstrates the largest error. Notably, the tracking error of the traditional exponential convergence law-based sliding mode controller exhibits significant fluctuations when the actuator changes direction. The VEPRL-based sliding mode controller shows an average control error level that is 93.5% lower than that of the PI controller, 79.9% lower than that of the ERL-based sliding mode controller, and 46.8% lower than HSMC. The standard deviation of the absolute error reflects the dispersion of the control error, with the VEPRL-based sliding mode controller exhibiting the lowest dispersion, followed by the HSMC controller. Conversely, the PI controller shows the highest dispersion of control error. Figure 8(f) is a behavior plot of the control signals. It can be observed that all four control strategies exhibit fluctuations in the initial stages, while the control signal of HSMC shows periodic fluctuations followed by a return to stability. These results highlight the effectiveness of the sliding mode controller based on the new variable exponential power convergence law in controlling EHPSSs and suppressing external disturbances.

Conclusion—future outlook and challenges

Conclusion

A novel intelligent sliding mode tracking control method is proposed for EHPSSs in this paper based on new VEPRL and CGWO-PSO optimization algorithm.

The main characteristic of the intelligent sliding mode tracking control method includes the following:

The new VEPRL can reduce jitter and increase the speed of reaching the slid model surface.

CGWO-PSO has the advantages as follows: better time efficiency, stronger adaptability, and higher performance.

Besides, the AMESim model and the state-space model of the EHPSS are established, and the presented intelligent tracking control method for EHPSSs is varified on the AMESim/MATLAB co-simulation platform. By comparing with the ERL-based SMC and the PI control method, it is shown that the proposed intelligent tracking control method which is on the basis of VEPRL and CGWO-PSO, can guarantee that EHPSSs has excellent tracking accuracy, strong robustness for both sinusoidal disturbance force, and step disturbance force.

Future outlook

Enhanced control performance: Future research can further improve the intelligent sliding mode tracking control method to enhance control performance and response speed. Through in-depth investigations and optimization of the variable exponential power reaching law, combined with more powerful optimization algorithms such as improved CGWO-PSO, more precise position tracking and faster response can be achieved.

Expansion of applications: In addition to EHPSSs, intelligent sliding mode tracking control methods can be applied to other fields and systems such as robot control, industrial automation, and transportation systems. Future research can extend this approach to a wider range of applications and conduct practical engineering validations.

Experimental validation of control algorithms that this paper proposed: The article did not perform experimental validation on the control algorithm. However, due to the unique characteristics of EHPSS, experimental validation is also challenging. Therefore, future research should prioritize conducting hardware-in-the-loop experiments.

Challenges

There are some challenges and limitations for implementing the control algorithm that this paper proposed in real-world.

Hardware constraints: Hardware limitations can include processing power, memory capacity, communication bandwidth, or even power consumption. Adapting the control algorithm to work within these constraints while maintaining acceptable performance is crucial.

Practical considerations

Practical considerations: Implementing a control algorithm in real-world scenarios involves practical considerations that go beyond theoretical models. Real-world environments are prone to uncertainties, disturbances, and noise, which can affect the performance of the control algorithm. Robustness and adaptability become crucial to ensure the algorithm can handle variations, unexpected events, and changing conditions.

Sensor and actuator limitations: The accuracy, reliability, and availability of sensors and actuators can impose limitations on the control algorithm. The algorithm’s effectiveness heavily relies on the quality and precision of the sensor data it receives and the responsiveness and capabilities of the actuators it controls. Dealing with sensor noise, calibration issues, and actuator limitations requires careful consideration during implementation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This manuscript was supported by the National Natural Science Foundation of China (No. 51876089).

ORCID iDs

Lingfei Xiao

Wei Liu

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.

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