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Speed tracking control for electrohydraulic system considering variable load disturbance
Journal of Engineering and Applied Science volume 70, Article number: 15 (2023)
Abstract
Forming speed is one of the key factors affecting metal forming performance. Aluminum in different forming stages need to control different extrusion speed, and withstand different extrusion load. The accuracy and stability of forming speed directly affect the quality of aluminum. Aiming to improve the stability of the forming speed under variable load, the paper established segmented model according to the forming speed and load force conditions. And the unified switching control strategy is proposed based on robust feedback linearized control. In order to improve the robustness of the system, the control law adding compensation items to the variable load disturbance is designed based on the sliding mode control. The boundary layer function is introduced instead of the symbolic function to reduce the system flutter. The proportional derivative control (PD) control, robust control and unified switching control are compared with that of simulation and experiment. The results show that proposed method not only maintains good tracking performance but also has strong robustness under external load disturbance.
Introduction
The servo valvecontrolled asymmetric cylinder system is widely used in the industry, such as in the military, ocean, aerospace and other fields. It has many advantages, such as the compact structure, the small workspace, the large driving capacity, the fast response speed, and the good positioning function [1,2,3]. However, the highaccuracy control of electrohydraulic systems is still challenging owing to the highly nonlinear characteristics, the parametric uncertainties, the modeled uncertainties, and the external disturbances [4, 5]. A synchronous control structure with error feedback to solve the problem of insufficient control accuracy of the double cylinder synchronous system for forging hydraulic press [6]. The hybrid fuzzyPID controller with coupled rules was proposed for position control of an asymmetric hydraulic system [7]. Aiming at the problem of insufficient tracking accuracy caused by nonlinear unknown parameters and unknown external disturbance, an adaptive sliding control method [8], a backstepping design strategy [9], and an adaptive robust controller [10] were presented for an electrohydraulic system. A cascade controller which was composed of active disturbance rejection controller and dead zone inverse compensation was designed to improve dynamic performances and position tracking accuracy [11]. Aiming for guaranteeing a better tracking performance in the presence of timevarying uncertainties and external disturbances, a novel backstepping controller based on Levant’s differentiatorbased disturbance observers had been proposed for a hydraulic system [12]. Aiming for the issue of position tracking control with loadsensing for a valvecontrolled cylinder system using speedcontrolled fixed displacement pump, a slidingfunctionbased feedforward controller with a feedback control system was presented to accomplish significant reduction in the input energy [13]. A combination of load sensing technology to regulate the valve control system was used to overcome the problem of low energy efficiency in the operation of the hydraulic press valve control system [14]. Focusing on the position control of lowvelocity servo controlled electrohydraulic actuator system in order to increase the system productivity, an efficient sliding mode control was designed based on the selection of appropriate sliding surface and control law to overcome the nonlinearities and uncertainties caused by friction and internal leakages [15]. In order to avoid wrinkling, an adaptive simulation approach integrated with a fuzzy control algorithm was used to optimize the loading path of hydroforming a Tshaped tube [16]. Aiming at the problems of low power recovery and large heat release of the conventional pumpcontrolled system free forging hydraulic press, a fast hydraulic forging press with open variable pumpcontrolled system was proposed to achieve a great energysaving advantage [17]. An adaptive sliding mode faulttolerant controller was designed to improve faulttolerant control ability and control accuracy of heavyduty hydraulic press under the condition of actuator fault [18]. For a class of multicylinder hydraulic presses with overdrive characteristics，a new sliding mode faulttolerant dynamic allocation method based on disturbance observer was proposed, as showed strong robustness regardless of whether a specific fault occurs in a single hydraulic cylinder or multiple hydraulic cylinders [19]. A nonlinear parallel control algorithm is developed for an electrohydraulic actuator to improve the velocity tracking performance for a valvepump parallel controlled electrohydraulic actuator under uncertainties and disturbance [20]. A novel robust backstepping control strategy was introduced to achieve highaccuracy tracking performance for electrohydraulic servo systems without velocity information in the presence of uncertainties and disturbances [21]. Forming speed is one of the key factors affecting the forming performance. In the process of metal deformation, the quality of the product is closely related to the speed of forming, especially under the variable load. In order to obtain highquality extrusion products, forming speed should be as far as possible to be maintained continuously stable, and to avoid fluttering. In this paper, a unified switching control strategy is proposed to improve the speed tracking accuracy by considering the variable load disturbance of the electrohydraulic system, remaining the forming speed as stable as possible, so as to obtain highquality products.
Main text
This paper is organized as follows: “System modeling” section presents the studied electrohydraulic system. The unified switching control strategy based on robust feedback linearized control, and the system stability analysis are introduced in “Control design” section. “Simulation and experiment” section contains the simulation and experiment results. The conclusions is provided in “Results and discussion” section.
System modeling
The valvecontrolled electrohydraulic system is illustrated in the Fig. 1. A_{A}, A_{B} are the piston areas, V_{A}, V_{B} are the volumes of the cylinder ports, p_{A}, p_{B} are the pressures at the cylinder ports, q_{A}, q_{B} are the oil inlet flow of the cylinder ports, P_{s} is the output pressure of the pump, P_{o} is the return pressure of oil, m is the load mass (including piston mass), F_{L} is the forming load force, and x is the piston displacement.
The kinetics model
The driving force of the hydraulic system is composed of the system pressure of the two chambers, the viscous resistance of the medium and the inertia force of the moving parts. The equation of the motion for the piston is written as
where b_{c} is the load viscous damping coefficient.
During the motion of the electrohydraulic system at a given reference speed, the forming load force is composed of the friction force and the deformation resistance, which is balanced with the driving force.
where F_{f} is the friction force, F_{σ} is the deformation resistance.
According to Eqs. (12),the system kinetics model is expressed as follows:
The flow model
The equations of the flow through the proportional servo valve can be written as follows:
The equations of the flow through the hydraulic cylinder can be written as follows:
where x_{v} is the servo valve spool displacement; C_{d} is the flow coefficient of proportional valve; w is the opening degree of proportional valve; ρ is the density of hydraulic oil; C_{i} is the internal leakage coefficient in hydraulic cylinder; C_{e} is the external leakage coefficient in hydraulic cylinder; V_{A0} and V_{B0} are the volumes of the cylinder ports as the hydraulic cylinder in the middle position; β_{e} is the effective bulk modulus of the hydraulic oil.
The multistate model of the system
The segmentation of the load
The metal forming process is the result of the combination of deformation resistance and forming speed. The essence is the electrohydraulic speed servo control under timevarying load force, which maintains the desired speed. Based on relationship between forming speed and load force, the forming process can be segmented, as shown in Fig. 2. The numbers of segment can be determined by the variation law of load force.
where S_{1} ∼ S_{n} are the segments, v_{1} ∼ v_{n} the forming speeds; F_{L1} ∼ F_{Ln} the load force.
The multistate model based on the segmentation of the load
The state variables is defined as \(x={\left[{x}_1\;{x}_2\;{x}_3\;{x}_4\right]}^T={\left[x\;\dot{x}\;{p}_A{p}_B\right]}^T\). Based on that, the statespace equation of the system can be expressed as follows:
where: \({f}_i\left(x,t\right)=\left[\begin{array}{c}{x}_2\\ {}1/m\cdot \left({A}_B\left(\varepsilon {x}_3{x}_4\right){F}_{fi}\left({x}_2\right){F}_{\delta i}\right)\\ {}{\beta}_e/{V}_A\left({x}_1\right)\cdot \left({A}_A{x}_2\right)\\ {}{\beta}_e/{V}_B\left({x}_1\right)\cdot \left({A}_B{x}_2\right)\end{array}\right],{g}_i\left(x,t\right)=\left[\begin{array}{c}0\\ {}0\\ {}{\beta}_e/{V}_A\left({x}_1\right)\cdot {B}_v\sqrt{\Delta \left({x}_3\right)}\\ {}{\beta}_e/{V}_B\left({x}_1\right)\cdot {B}_v\sqrt{\Delta \left({x}_4\right)}\end{array}\right],i=1,2\cdots, {F}_{\delta i}\) is the deformation resistance in the corresponding state, F_{fi}(x_{2}) is the friction force, V_{A}(x_{1}),V_{B}(x_{1}) are the working volumes of the cylinder, \(\varepsilon =\frac{A_A}{A_B}\), \(\Delta \left({x}_3\right)=\left\{\begin{array}{l}{p}_s{x}_3\kern0.5em {x}_v\ge 0\\ {}{x}_3{p}_0\begin{array}{cc}& {x}_v<0\end{array}\end{array}\right.\), \(\Delta \left({x}_4\right)=\left\{\begin{array}{l}{\textrm{x}}_4{p}_0\begin{array}{cc}& {x}_v\ge 0\end{array}\\ {}{p}_s{x}_4\begin{array}{cc}& {x}_v<0\end{array}\end{array}\right.\), \({B}_v={C}_dw\sqrt{\frac{2}{\rho }}\).
From the expression of f_{i}(x, t), It can be seen that the deformation resistance in different stages of the forming is inconsistent, equivalent to one forming condition of the electrohydraulic system. The multimodels of the system can be constructed by establishing the submodel of each condition.
Control design
The structure of the switching control
The statespace equation of the system can be expressed as the Eq. (9).
where: A_{σ} is n × nmatrix,B_{σ} is n × 1 matrix,C_{σ} is 1 × n matrix.
Assumed the system meets the following conditions:

(1)
For all \(i\in \Lambda, 0\le k\le n2,{C}_i{A}_i^k{B}_i=0\);

(2)
For all \(j\in \Lambda, {C}_i{A}_i^{n1}{B}_i={C}_j{A}_j^{n1}{B}_j=\gamma \ne 0\);

(3)
For all \(\tau \in \Lambda, \left({C}_1{A}_1^n,\cdots {C}_{\tau }{A}_{\tau}^n\right)\in I\tau \left({G}_1,\cdots {G}_{\tau}\right)\).
where: Conditions (1) and (2) indicate that the output is only a statedependent function for any subsystem. Condition (3) indicates that \({C}_i{A}_i^n\) can be reconstructed by \(\sum \limits_{j=0}^{n1}{C}_i{A_i}^j\), that is \(C_iA_i^n\;x=\sum_{j=0}^{n1}l_jC_iA_i^j\;x\).
If the control law shown in Eq.(10) is selected, the system has a unified control structure, and the system is stable [22].
where γ is a positive constant.
Assumed \({\dot{e}}_{\sigma (t),i}={e}_{\sigma (t),i+1}={y}^{i+1},{e}_{\sigma (t),1}=y={C}_{\sigma (t)}x\), then the follow equation can be obtained from Eq. (9)–Eq. (10).
where \({e}_{\sigma (t)}={\left[\begin{array}{cccc}{e}_{\sigma (t),1}& {e}_{\sigma (t),2}& \cdots & {e}_{\sigma (t),n}\end{array}\right]}^T\).
For the Eq. (11), the control law is expressed as
where k_{j} ∈ R and its value is set to ensure that the polynomial k_{1} + k_{2}s + ⋯ + k_{n}s^{n − 1} = 0 be Hurwitz. According to Eq. (11)–Eq. (12), The following expression can be obtained as the Eq. (13).
where \(A=\left(\begin{array}{c}0\\ {}\vdots \\ {}0\\ {}{k}_1\end{array}\begin{array}{c}1\\ {}\vdots \\ {} 0\\ {}{k}_2\end{array}\begin{array}{c}\cdots \\ {}\ddots \\ {}\cdots \\ {}\cdots \end{array}\begin{array}{c}0\\ {}\vdots \\ {} 1\\ {}{k}_n\end{array}\right)\).
Defined that \({T}_i={\left[{C}_i\;{C}_i{A}_i{C}_i\kern0.24em {A}_i^{n1}\right]}^T\), then the Eq. (13) can be converted as follows:
For the electrohydraulic system shown in Eq. (8), the output of the system is only related to the state of the system and there is an independent inputoutput relationship for any subsystem. Therefore, a unified switching control structure can be applied to the system. Through differential homeomorphism transformation, the Eq. (8) can be converted as the Eq. (15).
where \(\alpha (x)=\left({\varepsilon}^2\frac{F}{V_A\left({x}_1\right)}+\frac{E}{V_B\left({x}_1\right)}\right)\cdot \frac{A_B^2}{m}{x}_2{\dot{F}}_{Li}/m,\beta (x)=\frac{A_B{B}_V}{m}\left(\varepsilon \frac{E}{V_A\left({x}_1\right)}\cdot \sqrt{\Delta {x}_3}+\frac{E}{V_B\left({x}_1\right)}\cdot \sqrt{\Delta {x}_4}\right)\)
The control law is expressed as
where K is the gain of the control.
From Eq. (15), the following result can be obtained.
For each submodel, the control law is expressed as
Then, the system structure can be expressed as
The switching controller
For any submodel, it can be written as
Relative order calculating
Given∇h(x) = [0 1 0 0], ∇h(x) is the gradient of h(x), \({g}_1\left(\textbf{x}\right)=E/{V}_A\left({x}_1\right)\cdot {B}_V\sqrt{\Delta \left({x}_3\right)}\), and \({g}_2\left(\textbf{x}\right)=E/{V}_B\left({x}_1\right)\cdot {B}_V\sqrt{\Delta \left({x}_4\right)}\). By the definition of the relative order, the Eq. (8) has the relative order r.
If the relative order r = 1, then \({L}_gh(x)=\left[0\ 1\ 0\ 0\right]\left[\begin{array}{c}0\\ {}0\\ {}{g}_1(x)\\ {}{g}_2(x)\end{array}\right]=0\).
If the relative order r = 2, then
The switching control law
In order to simplify the analysis process, the relative order of the system is selected as 2. Through differential homeomorphism transformation to the Eq. (8), the Eq. (21) can be obtained as follows:
The Eq. (21) is converted into Brunovsky standard type as
where ς is the internal dynamics of the system and stable [23].
The feedback linearized control law is selected as
Defined \(v={\upsilon}_d^{(2)}+{K}_{\upsilon }e+{K}_a\dot{e}\), the controller can be designed according to the design method of the linear system.
where e = υ_{d} − υ, υ_{d} is the desired velocity, K_{υ} is the feedback coefficients of the velocity error, K_{a} is the feedback coefficients of the acceleration error, strictly positive, its value configures the poles in the left halfplane.
In order to improve the robustness of the system, the compensation is added on base of on sliding mode control, and the compensate part is expressed as the following.
where s is the sliding mode variable, sgn(s) is the symbolic function, K_{i} is the compensation for the uncertain parameters and load disturbance, and speed convergence control.
In order to reduce control energy and the fluctuation of the control law, K_{i} sgn(s) ≜ η_{i} + K sgn(s), where η_{i} is the equivalent compensation of variable load, K sgn(s) is the switch component.
Then, the Eq. (24) can be converted as
From the Eq. (25), a uniform control structure exists for any subsystem, the difference is that η_{i} needs to be adjusted according to the subsystem. The boundary layer function \(\left[\operatorname{sgn}\textrm{s}\right]=\left\{\begin{array}{l}\operatorname{sgn}\left(\frac{s}{\Phi}\right),\kern0.5em \left\frac{s}{\Phi}\right\ge 1\\ {}\frac{s}{\Phi},\kern2.25em \left\frac{s}{\Phi}\right<1\end{array}\right.\) (Φis the boundary layer thickness) is introduced to replace [sgn(s)] to reduce the system flutter.
The analysis of the stability
The common Lyapunov function of the system is constructed as \(V=\frac{1}{2}{s}^2\), \(\dot{s}={K}_i\left[\mathit{\operatorname{sgn}}\;s\right]\), then \(\dot{V}=s\cdot \dot{s}\le {s}^2{K}_is\cdot \left[\mathit{\operatorname{sgn}}\;s\right]\).
In the boundary layer outside, the system is stable as long as \({K}_i>0,\dot{V}<0\). In the boundary layer inside, the system is also stable for \(\dot{V}=s\cdot \dot{s}\le {s}^2{K}_i\cdot s\left[\mathit{\operatorname{sgn}}\;s\right]={s}^2{K}_i\cdot {s}^2/\Phi <0\). According to the Eq. (25), the system is always stable when K > 0.
As a conclusion, the common Lyapunov function exists for K_{i} > 0, so the system is stable.
Simulation and experiment
Simulation results
A joint simulation platform of the AMESIM and Simulink is built for the valvecontrolled asymmetric cylinder system as the Fig. 3.
The simulation load is shown in Fig. 4. ① indicates no load, ② and ③ indicate two different load states. Piston itinerary is 100 mm, and the specific parameters are listed in Table 1. The unified switching control parameters are as follows: K_{v} = 12675, K_{a} = 195, Φ = 200,η_{1} = 100, η_{2} = 150.
The speed response and errors of electrohydraulic system through the PD control, the robust control and the unified switching control respectively are shown in Fig. 5. Under condition ①, the steadystate error of the PD control is about 0.5 mm/s, while that of the robust control and the switching control is very small. Under conditions ②② and ③, the steadystate error of the PD control and the robust control increases, while that of the unified switching control remains unchanged. It shows that the unified switching control achieves better speed response performance of electrohydraulic system under variable load.
Experiment results
Experiment structure of electrohydraulic system is shown in Fig. 6.
The system parameters of the hydraulic system are listed in the Table 2. The load in the experimental process is shown in Fig. 7. The unified switching control parameters are as follows: K_{v} = 32000, K_{a} = 225, Φ = 3200, K^{′} = 50, and η_{1} = 100, η_{2} = 200. The desired speed of the system is 0.01 m/s.
The speed responses of the PD control, the robust control and the unified switching control are compared in Fig. 8. Under the PD control, as the load increases, the error increases, and the steadystate reaches 0.001 m/s. Under the robust control, the steadystate error increases with the increase of load, as reaches 0.0005 m/s when under the condition ③. While the steadystate error with the unified switching control remains within 0.0002m/s. It demonstrates that unified switching control can achieve higher speed tracking performance under variable load and is robust to load change.
The system responses of the switching control to different desired speeds are shown respectively in Figs. 9 and 10. The steadystate errors are both about 0.0002 m/s in 0.01 m/s and 0.02 m/s. It demonstrates that the unified switching control has better robustness to the speed change under a certain range of external load.
Conclusions
The control of the electrohydraulic system due to variable load disturbance is studied to improve the speed control accuracy and robustness of the system. In view of the variable load of the electrohydraulic system, it is proposed to treat different working conditions as one load state and establish the multistate model of the system. Aiming to improve the stability of the speed control under variable load, a unified switching control of the electrohydraulic system is proposed. Simulation and experiment are carried out to compare the response of the PD control and the robust control. The results demonstrate that the unified switching control can effectively suppress the speed tracking problem under variable load disturbance.
Methods/experimental
Velocity was one of the key factors affecting the forming quality. The authors focused on the stability and robustness of the electrohydraulic system running speed under variable load disturbance. By analyzing the relationship between the operating condition and the speed of the system, the relevant state variables were introduced, the mathematical model of the system was established, and the unified switching control was applied to control the system running state. In the simulation process, the operating speed of the system under different operating conditions was set by simulation and the load forces under different operating conditions were adjusted to obtain the actual output speed of the system, and the speed values of the system under different operating conditions were obtained. The corresponding results were obtained by AEMSIM software. At the same time, the experimental verification was completed through the asymmetric top cylinder experimental platform, which was built with 125MN extruder as the prototype. The relevant experimental data were collected, and the corresponding conclusions were acquired.
Results and discussion
The control of system stability and robustness was explored in the study, which can provide some reference for engineering application. By adopting the unified switching strategy, the stability of the system running speed under different working conditions could be improved with less vibration. However, as the dynamic characteristics of the system were affected by various factors such as the nonlinearity of the system itself, variable load, and variable parameters, the study focused on the variable load, so there was still a certain gap compared with the actual application. The next research will optimize the system model on this basis, and select more realistic parameters through repeated simulation and experiment to further improve the stability and robustness of the system.
Availability of data and materials
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
First and foremost, I appreciate my college who give me a comfortable working and researching atmosphere. Second, I would like to show my deepest gratitude to my supervisor, Prof. Tan Jianping, who has walked me through all the stages of the writing of this thesis. Without his illuminating instruction and patience, this thesis could not have reached its present form. I am also greatly indebted to all my teams who have helped me to develop the fundamental and essential academic competence. My sincere appreciation also goes to all my colleagues, who are my proud of my life. Last but not least, I want to thank all my friends, for their encouragement and support.
Funding
The research was funded by the National Key Basic Research Development Program (2014 CB049400) and The Natural Science Foundation of Hunan Province (2020 JJ7084).
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Xu, W., Zeng, L. Speed tracking control for electrohydraulic system considering variable load disturbance. J. Eng. Appl. Sci. 70, 15 (2023). https://doi.org/10.1186/s4414702300185w
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DOI: https://doi.org/10.1186/s4414702300185w