Quasi-synchronous control of uncertain multiple electrohydraulic systems with prescribed performance constraint and input saturation

Shui LI, Qing GUO,*, Yn SHI, Yo YAN, Dn JIANG

a School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, China

b School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

c School of Mechanical and Electrical Engineering,University of Electronic Science and Technology of China,Chengdu 611731,China

KEYWORDS

Abstract This article focuses on the high accuracy quasi-synchronous control issue of multiple electrohydraulic systems (MEHS).In order to overcome the negative effects of parameter uncertainty and external load interference of MEHS,a kind of finite-time disturbance observer(FTDO)via terminal sliding mode method is constructed based on the MEHS model to achieve fast and accuracy estimation and compensation ability.To avoid the differential explosion in backstepping iteration, the dynamic surface control is used in this paper to guarantee the follower electrohydraulic nodes synchronize to the leader motion with a better performance.Furthermore, a timevarying barrier Lyapunov function(tvBLF)is adopted during the controller design process to constraint the output tracking error of MEHS in a prescribed performance with time-varying exponential function.As the initial state condition is relax by tvBLF, the input saturation law is also adopted during the controller design process in this paper to restrain the surges of input signals,which can avoid the circuit and mechanical structure damage caused by the volatile input signal.An MEHS experimental bench is constructed to verify the effectiveness of the theoretical conclusions proposed in this paper and the advantages of the proposed conclusions in this paper are illustrated by a series of contradistinctive experimental results.

1.Introduction

Electro-hydraulic servo system(EHS) is a typical mechatronic motion system,which has been widely used in mechanical engineering.The EHS integrates the advantages of both hydraulic transmission and electric driving, which has high energy density, high control accuracy and low delay.Thus, EHS has been widely used in engineering practice such as: aeroengine afterburner system1and high-speed actuator.2Meanwhile,multiple electrohydraulic systems (MEHS) are often used for cooperative transmission task, such as shaking table,3spatial electrohydraulic robot4and crane.5Hence, the synchronous control performance is an important evaluation index of MEHS in cooperative transmission task.In previous work,lots of nonlinear control methods have been proposed to overcome the nonlinearity of EHS, such as robust controller,6backstepping controller,7sliding mode controller,8fault diagnosis and fault-tolerant control9and feedback linearization controller.10Actually, backstepping method is the general technique in different controller design for EHS.However, a repeated iteration is conducted and several virtual control variables emerge to increases the complicated calculation of controller, which leads to output violent oscillation.This phenomenon is called differential explosion in backstepping iteration.To avoid this phenomenon to maintain the control performance,Swaroop et al.proposed a dynamic surface control method,11which essentially smoothes the virtual control variables.Then this design has been applied in engineering12–13and also in the EHS by Duraiswamy and Chiu,14which improve the control performance of EHS.

In the past decade, the cooperative control of nonlinear multi-agent systems has attracted much attention.Firstly,multiple agent system(MAS)with nonlinear dynamics under directional communication and delay has been investigated to obtain the consensus condition.15Then the distributed consensus problems of MAS under Lipschitz nonlinear dynamics have been addressed by adaptive relative state consensus protocols.16To realize the fast synchronization control of nonlinear MAS,Du et al.17proposed a nonlinear multi-agent output feedback synchronous controller based on finite-time convergence law.Subsequently, Zuo et al.18presented an adaptive fault-tolerant tracking control of nonlinear MAS with Lipschitz dynamics.Hence, lots of cooperative control results on nonlinear MAS with adaptive control techniques have been proposed19–23as reference researches in different application field.

Due to some uncertainties such as parametric uncertainty and external load disturbance existed in mechatronic system,many disturbance observers are presented to estimate them and be compensated in control design, which include adaptive observer,24extended state observer,25and center-based transfer feature learning.26These disturbance observers can effectively suppress uncertainty negative effects and improve the control robustness of EHS.However,these disturbance observers just focus on the estimation accuracy rather convergence speed.Hence, recently a disturbance novel observer called terminal sliding mode observer27has been proposed to improve both the estimation accuracy and convergence speed for many uncertain systems.

To further improve the dynamic and steady performance of EHS, the barrier Lyapunov function28is adopted to restrict the output tracking error.However, the barrier Lyapunov function often requires the initial position condition of EHS,which is also limited in a specified range.Fortunately, the tvBLF29–30is used to relax the initial EHS condition.In fact,if the initial condition of EHS can be moderately easy,the control input signal might be sharply oscillated during the transient response.To address this problem, an input saturation strategy31–32has been presented to guarantee the stability margin as the input saturation emerges in dynamic response.

Inspired by the references aforementioned,this paper plans to design a control algorithm combining tvBLF and input saturation law and hopes that the tracking performance of the system can be guaranteed under this control algorithm while the control input signal can be better limited.Different from the synchronization controller designed in,10this paper considers the MEHS under a stationary communication topology and the controller of third-order MEHS nonlinear system is designed directly based on backstepping iteration process instead of feedback linearization method.The main contributions are listed as follows:

(i) To address the lumped uncertainty in MEHS, a FTDO via terminal sliding mode is designed to guarantee the uncertainty estimation error with fast convergence speed.Then a quasi-synchronous controller is designed by dynamic surface technique to avoid the differential explosion in backstepping iteration.

(ii) A tvBLF is constructed in the controller design process to constrain the output synchronous error of MEHS in a prescribed performance.Meanwhile, since the initial state condition of MEHS is relax by tvBLF, an input saturation law is designed to guarantee the output stability of EHS under the limited control voltage of servo valve and initial large state bias from the demand.

The remainder of this paper is organized as follows.The MEHS model with lumped uncertainty is constructed and the basic graph theory is introduced in Section 2.Then FTDO is designed for MEHS in section 3.Subsequently, a quasisynchronous controller is proposed in Section 4 with timevarying outputs constraint and input saturation.The comparative experimental results are given in Section 5.Finally, the conclusion is drawn in Section 6.

2.Preliminaries

2.1.Multiple electrohydraulic system model

The MEHS is composed by N(N ≥2) isomorphic EHSs,which has the composition and control mechanism as shown in Fig.1.Since the cylinder motion frequency is far less than the cutoff frequency of servo valve, the dynamics of servo valve is neglected in this study.Hence, a state vector for i-th node is defined asand then a three-orders dynamic model of i-th MEHS node is given by

where i ? ｛1 ,2,???,N｝, K denotes the spring stiffness coefficient of the cylinder,b denotes the viscous damping coefficient,FLidenotes the external load of i-th node,βedenotes the effective bulk modulus,Apdenotes the annulus area of the cylinder chamber, Vtdenotes the total volume of the hydraulic power mechanism, Ctldenotes the coefficient of the total leakage of the cylinder, Cddenotes the discharge coefficient, w denotes the area gradient of the servo valve,Ksvdenotes the gain voltage of the servo valve,ρ denotes the density of hydraulic oil,psdenotes the supply pressure from bench, sgn(?) is the signum function, i.e., sgn(?)=1 for ui>0, sgn(?)=0 for ui=0 sgn(?)<0 for ui<0.

Fig.1 Composition of each isomorphic EHS node and its control mechanism.

Remark 2.1.The hydraulic parameters Cd, ρ, K, b, βeand Ctlare all unknown positive constants.

Assumption 2.1.The external load FLiis bounded as｜FLi(t )｜≤FLmaxwhere FLmaxis an uncertain constant.

To avoid the violent control during the transient process,an input saturation law for the control variable uiis designed as

According to Remark 2.1 and Assumption 2.1, together with Eq.(2), the MEHS model Eq.(1) is rewritten as follow

where the nominal model functions of Eq.(3) are

and the two lumped uncertainties are

Assumption 2.2.33The uncertain items Δi2and Δi3are bounded such that.

2.2.Basic graph theory

3.FTDO design

Inspired by the disturbance observer designed in,27a rapidly disturbance observers are designed in this paper to estimate two lumped uncertainties Δi2, Δi3of MEHS.To guarantee the estimation error with fast convergence speed in a finite time, two terminal sliding mode surfaces are defined such that

The two auxiliary variables vi2and vi3in Eq.(4) yield that

Lemma 3.1.36If there exists a continuous positive definite function V(t ) such that.

where a>0, b>0 and 0

Theorem 3.1.The estimation errors of the lumped uncertainties in Eq.(3) converge to zero in a finite-time, by using the disturbances observers Eq.(6).

Proof.See Appendix A.

4.Dynamic surface quasi-synchronization controller design

Here a dynamic surface quasi-synchronous controller with input saturation for the MEHS to guarantee the follower electrohydraulic nodes synchronize to the leader motion.Firstly,the synchronous position error of i-th node is defined as

where ydis the desired position trajectory of virtual leader.

Then the system state errors for i-th EHS node are given by

where αi1and αi2for i ? ｛1 ,2,???,N｝ are virtual control variables in the backstepping iteration.

Together Eq.(9) with Eq.(10), the synchronous position error of i-th node is rewritten as

To guarantee the output stability of EHS under the limited control voltage of serv valve and initial large state bias from the demand, an auxiliary variable ν is design as follow

where kνand νcare positive constants, and Δu=? (u )-u.

To realize the prescribed performance constraint purpose of the synthesized synchronous error, a time-varying exponential function is given by

where a is the convergence rate constant of the synthesized synchronous error,φ0and φ∞are positive initial and terminal constraints.

Then, the prescribed performance constraint of the synthesized synchronous error is described as:

Theorem 4.1.If the initial condition of the synthesized synchronous error Eq.(12) satisfies that eT(0 )e(0 )≤φ0+φ∞,and the control parameters meet the following inequalities:

Then the output synchronous error of MEHS is constrained in a prescribed performance with time-varying exponential function such that eT(t )e(t )

Proof.See Appendix B.

The proposed quasi-synchronization control diagram Eq.(18) for MEHS is shown in Fig.2, which includes dynamics surface Eq.(17)and the FTDO Eq.(6)with the input saturation law Eq.(2).The prescribed performance constraint of the synthesized synchronous error e is restricted in the time-varying exponential function Eq.(16).

5.Experimental results

Fig.2 The proposed quasi-synchronization control diagram for MEHS.

Fig.3 Experimental platform of MEHS.

In this section, a MEHS experimental bench has been constructed to verify the effectiveness of the proposed controller.The main components of the MEHS bench are shown in Fig.3, which includes a pump station (HY-36CC-01/11kw),a servo valve with nozzle flappers (D633-R04K01M0NSM2),three cylinders (UG1511R25/16–100) and a computer (Intel Core i7-12700 K).Furthermore, the positions of three cylinders are measured by three displacement sensors (JHQ-GA-50),the cylinder load pressures are measured by three pressure sensors (BD-sensors-DMP-331).The nominal hydraulic parameters of the MEHS in experiment are shown in Table 1.Considering the range of hydraulic cylinders is ±50 mm, the demand trajectory is selected asmm.The cylinders initial conditions are selected as the maximum negative value.Furthermore, the communication relationship between the virtual leader and following EHS nodes are given by Fig.4, and the corresponding matrix H is

Fig.5 shows the cylinder position responses of 3 follower EHS nodes by using the proposed controller, which illustratesthat all the follower electrohydraulic nodes synchronize to the virtual leader trajectory.Fig.6 shows the corresponding tracking errors of each nodes yi-ydfor i ? ｛1,2,3｝.Furthermore,the experimental results by using the dynamic surface controller (DSC) without prescribed performance constraint are also provided.Comparing with these two methods, we can see that the proposed controller has the higher tracking accuracy of the MEHS.Then Fig.7 shows the comparative result between the synthesized synchronous error of MEHS and the corresponding time-varying constraint boundary Fφwhere a=10, φ0=60 and φ∞=10.Furthermore, the uncertainty estimations by the FTDO and the corresponding estimation errors are shown in Figs.8 and 9, which demonstrate the estimation errors converge to 0 with fast speed.Fig.10 shows the estimation errors during the transient response.According to the estimation parameters of the FTDO, it can be calculated that the estimation instants=0?012 and=0?015 for i ? ｛1 ,2,3｝.Hence, the estimation errors converge to 0 before these instants.Fig.11 shows the dynamic surface Sij(i ? ｛1,2,3｝ and j ? ｛1,2 ｝) of the proposed controller.Finally, the control input uiwith the input saturation law are given in Fig.12.Since the control voltage saturation is umax=5 V, the control input uiare all constrained in [-5,5].Hence, the output stability of the MEHS is guaranteed under the limited control voltage of servo valve and initial large state bias from the demand.

Fig.4 Communication relationship between a virtual leader and following EHS nodes.

Fig.5 Cylinder position responses of following EHS nodes and expected trajectory.

Fig.6 Output tracking errors of following EHS nodes with correlation data under dynamic surface controller (DSC).

Fig.7 Compositive Output Error (COE) of MEHS and corresponding time-varying constraint.

Fig.8 Estimation values and corresponding estimated errors of FTDO.

Fig.9 Estimation values and corresponding estimated errors of FTDO.

Fig.10 Estimated errors during transient process.

Fig.11 Dynamic surface values of proposed control strategy.

Fig.12 Control input signals and corresponding saturated boundaries.

6.Conclusion

This study proposed a quasi-synchronous control algorithm of MEHS with prescribed performance constraint and input saturation.The FTDO is designed for MEHS model to rapidly estimate the lumped uncertainties of the MEHS, which are compensated in the proposed controller.The dynamic surface controller is designed to avoid the differential explosion in backstepping iteration and guarantee the follower electrohydraulic nodes synchronize to the leader position with a better performance.By using the tvBLF method, the output synchronous error of MEHS is constrained in a prescribed performance and thus realize a better tracking performance.Furthermore, the input saturation law is designed to address the limited control voltage of servo valve and initial large state bias from the demand, especially the output stability of the MEHS is guaranteed.Finally,the effectiveness of the proposed controller is verified on the MEHS experimental bench.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (Nos.52175046, 51975024, and 12072068), Sichuan Science and Technology Program (Nos.2022JDRC0018 and 2022YFG0341).

Appendix A.Proof of Theorem 3.1:

For the lumped uncertainties Δi2,a group of candidate Lyapunov functions are selected as follows

and then the time derivative of VSi2is given by

Based on Assumption 2.1, Eq.(A2) yields that

By using Lemma 3.1, the terminal sliding mode surface si2converges to zero in a finite time t

i2such that

Meanwhile, the disturbance estimation error is given by

Similarly, the estimation errors→0 before the finite time, i.e.,→Δi3.

Appendix B.Proof of Theorem 4.1.

The candidate tvBLF for the MEHS Eq.(3) is constructed as follow

which indicates that the MEHS (3) reaches the uniformly ultimate boundedness (UUB).Hence, the synchronous error e is bounded as t →∞, and can be restricted by the control gainsto a zero neighborhood with arbitrarily small size σ/.