Barrier Lyapunov functions-based dynamic surface control with tracking error constraints for ammunition manipulator electrohydraulic system

Shou-Cheng Nie,Lin-Fang Qian,Long-Miao Chen,Ling-Fei Tian,Quan Zou

School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing,China

Keywords:Ammunition manipulator Electro-hydraulic system Error constraints Tracking control

ABSTRACT This paper focuses on the dynamic tracking control of ammunition manipulator system.A standard state space model for the ammunition manipulator electro-hydraulic system(AMEHS)with inherent nonlinearities and uncertainties considered was established.To simultaneously suppress the violation of tracking error constraints and the complexity of differential explosion,a barrier Lyapunov functionsbased dynamic surface control(BLF-DSC)method was proposed for the position tracking control of the ammunition manipulator.Theoretical analysis prove the stability of the closed-loop overall system and the tracking error converges to a prescribed neighborhood asymptotically.The effectiveness and dynamic tracking performance of the proposed control strategy is validated via simulation and experimental results.

1.Introduction

Ammunition loading is a strenuous job for soldiers to carry a projectile weighting up to 45 kg into the gun bore and accomplish the operation of bayonet-chamber by manpower,especially in poor environments.In order to automate the loading process and provide high rate of fire,the ammunition manipulator is widely used in autoloader system of large caliber howitzer[1,2].

Motor device and hydraulic cylinder are the two main forms of driving mechanisms for ammunition coordinator.The high power motor drive system is unable to satisfy the requirement of the ammunition manipulator(e.g.the large load inertia and compact installation space[3])with a relatively large size in general[4].On the contrary,the electro-hydraulic servo system is suitable to the ammunition manipulator due to its small size-to-power ratio and large driving force[5].Besides,structural stability of ammunition manipulator is enhanced as a spatial triangle is formed by top carriage,coordinator body and hydraulic cylinder.Compared with motor driving system,the AMEHS has an additional driving arm which decreases the required driving forces with the help of the mounting of hydraulic cylinder.

However,there are quantities of nonlinearities and uncertainties problems in the ammunition manipulator system.Nonlinearities mainly contain the changes of load,friction and the position relationship between coordinator and cylinder[6].Meanwhile uncertainties consist of leakage,external disturbances,and bulk modulus of the oil etc.[7].To address above-mentioned nonlinearities and uncertainties which occur in industrial robots and manipulators[8],hydraulic load simulator[9],automotive active suspensions[10],and all kinds of engineering machineries[11],several advanced control theories were presented by scholars who applied themselves to the hard work.

Universally,backstepping is one of the most widely used methods dealing with nonlinear control of electro-hydraulic servo systems[12].Nevertheless,the phenomenon of“explosion of terms”associated with backstepping method is unavoidable because of the numerical differentiation.With parametric uncertainties considered,a nonsmooth dynamic surface was designed by Duraiswamy et al.[13]to overcome the repeatedly calculated derivative of virtual control by means of a stabilizing low-pass filter.With both parametric uncertainties and uncertain nonlinearities considered,Yao et al.[5,14,15]developed a critical adaptive robust control(ARC)theory which combined adaptive and robust control.The theory and experiments showed prescribed transient performance and desired final tracking accuracy were guaranteed by the ARC theory.In order to suppress unmatched uncertainties in electro-hydraulic servo systems,all kinds of observers such as disturbance observer[16],high-gain disturbance observer[17],extended disturbance observer[18]and extended state observer[19]were proposed to improve the control accuracy.

Recently,the barrier Lyapunov function(BLF)was widely used to prevent the state and output constraint violation due to physical or performance limitations in nonlinear control systems.A BLF is a continuous and positive definite scalar function which has continuous first-order partial derivatives.And the boundedness of the BLF is derived by choosing an appropriate control input to make its time derivative negative semidefinite.Ngo et al.[20]firstly introduced a barrier-function to suppress the propagation of the errors at each stage of the backstepping procedure.Subsequently,Tee et al.[21,22]and Ren et al.[23]applied the BLF to output constraint and achieved asymptotic tracking without violation of the constraint.Aiming at nonlinear systems in strict feedback form,Li et al.[23]and Li et al.[24]were devoted to full state constrains with the help of the BLF.An innovative work which introduced the BLF into nonlinear stochastic control systems was carried out by Liu et al.[25],and all the states of the stochastic systems were not to transgress their constraints.In time-varying nonlinear systems,Liu et al.[26]and Wang et al.[27]utilized the BLF to ensure the boundedness of unknown states and achieved asymptotic convergence.Wu et al.[28]presented a Control Lyapunov-Barrier Function-based model predictive control method for the stabilization of nonlinear systems with input constraint considered.In order to eliminate the differentiation in backstepping iteration and guarantee the full state constraints,Wang et al.[29]and Zhang et al.[30]combined the dynamic surface control(DSC)and the BLF.Further,the BLF-based dynamic surface control method with full-state error constraints was adopted in electro-hydraulic system by Guo et al.[31].

Researches did numerious efforts to improve control performance of electro-hydraulic servo systems with the same motion law for hydraulic actuator and load such as translational loads driven by hydro-cylinders[5-7]or rotating loads driven by hydraulic motors[9].Here,the ammunition manipulator driven by a translational cylinder rotates around the trunnion.During the rotating process,both the load and driving arm vary nonlinearly.Considering inherent nonlinearities and uncertainties of electrohydraulic servo systems,it is an ambitious project to ensure the AMEHS tracks the desired trajectory well.In this paper,the mathematical model of electro-hydraulic servo system for ammunition manipulator is constructed.The DSC is applied to replace the derivatives of virtual control variables with the help of stabilizing filter functions during backstepping iteration.The BLF is adopted to prevent the violation of tracking error constraints and guarantee the boundedness of all closed loop signals.Then,a BLF-DSC method is presented for the position tracking control of ammunition coordinator with tracking error constraints.The effectiveness and practicability of the proposed BLF-DSC strategy is verified by simulation and experimental results.

We discuss the mathematical model and problem formulation and the design procedure and stability proof of the proposed BLFDSC controller in Section 2 and Section 3 respectively.The simulation results are demonstrated in Section 4 followed by the experimental results in Section 5.Finally,the sixth section draws the conclusions.

2.Mathematical model and problem formulation

The position map of ammunition manipulator is depicted in Fig.1.The trunnion is fixed at point O.Both ends of hydrauliccylinder are hinged spectacularly with top carriage at point M and coordinator body at point N.The mathematical model was established through theoretical formulation and derivation in this section.

The load moment balance equation of ammunition manipulator can be presented as

whereJis the equivalent rotational inertia of the load.αis the angular displacement of ammunition manipulator.The equivalent forceFof hydraulic cylinder is presented asF=P1A1-P2A2.P1andP2represent the pressure inside two chambers of the cylinder.A1andA2represent the piston areas inside two chambers of the cylinder.bis the combined coefficient of modeled damping and viscous friction forces on the cylinder rod.xpis the displacement of the cylinder rod.Gis the equivalent gravity of ammunition manipulator.Tdrepresents the lumped uncertainties due to load variation,unmodeled friction forces,external disturbances and other unmodeled terms.h(α)andhg(α)represent the arm ofFandGrespectively.hg(α)can be presented ashg(α)=rgsin(α+φ-β)whererg=｜OA｜.φis the angular between OA and ON.βis the angular between OM and the perpendicular OB.

The distance between two hinge joints of hydraulic cylinder canbe presented as according to the cosine theorem wherer1=｜OM｜andr2=｜ON｜.l0represents the initial distance between two hinge joints of hydraulic cylinder.We can deductthrough a derivation of the above formula.In addition,the area formula for triangle is given asThen we haveTherefore,the relationship betweenxpandαcan be presented as

According to the liquid flow continuity equation,the actuator of dynamic is given by Ref.[32].

whereQ1andQ2are the supply flow rate of the forward chamber and the return flow of the return chamber,respectively.CipandCepare the coefficients of internal leakage and external leakage respectively.βeis the effective bulk modulus.

The volumes of the two chambers of the cylinder can be presented as whereV01andV02are the initial volumes of the two chambers of the cylinder including the hose volume from the valve to the cylinder.

The forward and return flow through the servo-proportional directional valve can be presented as

wherekqvis the flow gain coefficient of the valve,uis the control voltage.R1andR2are given as

wherePsis the supply pressure.Pris the return pressure.And

Then,equation(3)can be abbreviated as

Define the state variables x= [x1，x2，x3]T=[α，˙α，P1A1-P2A2]T.On the basis of equations(1)-(7),the entire state space can be written as

Remark 1.The practical system parametersJ,b,βe,V01,V02,Cip,Cepare always treated as uncertain positive constants.

The parameter uncertainties are summarized as ΔhJ(x1),Δhg(x1),ΔB,Δg(x)andΔf(x).The state space form can be rewritten as

whered2(t)=ΔhJ(x1)x3-ΔBh2(x1)x2-GΔhg(x1)+d1(t),d3(t)=Δg(x)u-Δf(x).

Assumption 1.The uncertaintiesd2(t)andd3(t)are bounded by｜d2(t)｜≤D2and｜d3(t)｜≤D3whereD2andD3are positive constants.

In this paper,parameter uncertainties and unmodeled lumped uncertainties are integrated asd2(t)andd3(t).The hydraulic uncertain parameters and unmodeled lumped uncertainties in practical AMEHS are all bounded[31].Besides,actual system states such asx1,x2,P1andP2are also physically bounded[17].Consequently,uncertaintiesd2(t)andd3(t)are bounded,which means that Assumption 1 is physically reasonable.

Assumption 2.The functiong(x)satisfies 0＜gmin≤g(x)≤gmaxwheregminandgmaxare positive constants.

3.Controller design

The state errorsei(i=1，2，3)of the AMEHS are defined as

wherex1drepresents desired trajectory,x2dandx3drepresent the virtual control variables to be designed later.

The symmetric BLF is adopted to restrain the tracking errore1-which is given as

wherekc1represents the allowable accuracy range of the tracking error andkc1＞0.

The derivative ofe1with respect to time is given by

Design the virtual control variablex2d

wherek1is a positive control gain.

Then,e2can be presented as

The transfer functionGe(s)[5]frome1toe2is given by

From equation(11),we havek2c1-e21＞0.Thus the transfer functionGe(s)is stable,which means that to makee1small or converging to zero is equivalent to makinge2small or converging to zero.

Similar to equation(11),e2is restrained with the BLF by

wherekc2represents the allowable range of the tracking speed error andkc2＞0.

Assumption 3.[29,31]:For?kc1＞0,there exist positive constantsXl,Xh,X1,X2,δx1d,such that the reference signalx1dand its time derivatives˙x1d,¨x1dsatisfyXl≤x1d≤Xh,｜˙x1d｜≤X1,｜¨x1d｜≤X2,?t≥0,implying that these variables are continuous and available in a compact setΩx1d:=

Lemma 1.[21]:For any positive constantskc1,kc2,letΩe:=｛e??:｜ei｜＜kci｝(i=1，2)be open sets.Consider the system

whereη:=[ω，ei]T∈Ωeandh:?+×Ωe→?3is piecewise continuous intand locally Lipschitz inΩe,uniformly int,on?+×Ωe.Suppose that there exist continuous differentiable and positive definite functionsU:?3→?+andV(e):Ωe→?+in their respective domains,such that

whereγ1andγ2are classK∞functions.Let V(η):=V(e)+U(ω)andei(0)belong to the setΩe.If the flowing inequality holds

wherecandδare positive constants,thene(t)remain in the open setΩe,?t≥0.

Lemma 2.[33]:For all｜ei｜＜kci,kci＞0(i=1，2),the following inequality holds

Theorem 1.Considering the system(9),(10),(11)and(16),suppose the BLF-DSC controller for the AMEHS is given by

then the tracking errore1enters its convergence domain in the finite timet＞0 and stays within the bounded ballBr

whereεandλare positive constants,V(0)is the initial value of the hypersphereH

Proof.

Step 1:Select the candidate BLF of the first subsystem[21].

The derivative ofV1with respect to time is given by

Substituting(12),(14),(20)into(24),˙V1yields

According to Lemma 2,the Eq.(25)can be written as

It is obvious that the differentiation of the virtual control variablex2dis essential in the backstepping iteration.Thus,researches usually adopted the DSC to prevent the differential computation ofx2din this step[29-31].In fact,the calculation of˙x2dis easy.In addition,a boundary layer error is introduced because of the filtering calculation.

Step 2:The derivative ofe2with respect to time is given by

To prevent the explosion of the complex differential calculation forin Eq.(20),a stable first-order filter is given as follows

whereτ3is the positive time constant of the filter.

Define the boundary layer error as

Then,the derivative ofx3dcan be written as

The derivative ofz3with respect to time is given by

Select the candidate BLF of the second subsystem

The derivative ofV2with respect to time is given by

whereσ2＞0.

Applying Young’s inequality,the following inequalities are given as

whereσi＞0(i=1，2，3).

By Eq.(34),Eq.(33)can be rewritten as

According to Lemma 2

Step 3:The derivative ofe3with respect to time is given by

Fig.2.Co-simulation system of the AMEHS.

Table 1Parameters of the system.

The control input is designed as

Select the candidate BLF of the system

Fig.3.Schematic diagram of the BLF-DSC controller.

Fig.4.Desired tracking position.

Fig.5.Tracking errors of simulation without disturbances.

The derivative ofVwith respect to time is given by

wherek3＞0.

Fig.6.Tracking speed errors of simulation without disturbances.

Fig.7.Control inputs of simulation without disturbances.

According to Eq.(42),asymptotic output tracking error of the AMEHS is achieved after a finite timet0by the proposed BLF-DSC controller.Furthermore,all signals are bounded and the closedloop system is stable.

4.Simulation results

To verify the performance of the proposed BLF-DSC controller for the AMEHS,the co-simulation system based on MATLAB Simulink/AMESim platform is exhibited in Fig.2.Main physical parameters of the system are given in Table 1.

Fig.8.Pressures of simulation without disturbances.

The structure of the BLF-DSC controller shows in Fig.3.The following controller parameters were used:kc1=1°,kc2=60°/s,k1=35,k2=115,k3=850,D2=1000,D3=10,σ2=100,σ3=10,τ2=τ3=1×10-4.

Furthermore,the simulation performance of the BLF-DSC controller was compared with the traditional PID and DSC controllers[13]which are given as follows.And the desired tracking position shows in Fig.4.

1)PID controller

Fig.9.Tracking errors of simulation with disturbances.

Fig.10.Tracking speed errors of simulation with disturbances.

Fig.11.Control inputs of simulation with disturbances.

whereKP=2.75,KI=0.5,KD=0.1.

2)DSC controller

Fig.12.Pressures of simulation with disturbances.

Fig.13.Trajectory of disturbances.

Fig.14.Experimental setup.

Fig.15.Tracking errors of experiment.

Fig.16.Control inputs of experiment.

wherek1=25k2=15k3=120D2=1000D3=10τ2=τ3=1×10-4,sgn(*)is a sign function given as sgn(*)=｛-1if*＜0[-1，+1]if*=0+1if*＞0.

4.1.Simulation results without disturbances

The comparative simulation results of three methods are shown in Figs.5-8.Fig.5 shows the tracking errors of three controllers.The tracking error of the proposed BLF-DSC controller was kept within±0.2°,which is better than the DSC controller with an error of±0.6°and the PID controller with an error of±1.6°.Figs.5 and 6 indicate the simulation results of the BLF-DSC controller satisfied the constraints｜e1｜＜kc1=1°,｜e2｜＜kc2=60°/s.

Fig.17.Pressures of experiment.

Fig.7 shows the control inputs of three controllers.There is a remarkable control input delay of the PID controller comparing with the other two methods.The control input of the DSC controller remains a comparatively smaller delay than the BLF-DSC in Fig.7.The curves in Figs.5 and 7 reveal that the tracking errors of the DSC and the BLF-DSC are nearly equal when the control inputs of the two controllers are equal.Where the control input of the DSC delays more markedly than the BLF-DSC,the tracking error of the DSC is larger.The pressuresP1andP2of the BLF-DSC method are shown in Fig.8,which indicates there was pressure fluctuation in the initial stage and the pressures varied steadily in the rest stage.

4.2.Simulation results with disturbances

The uncertainties of the AMEHS mainly consist ofd2(t)andd3(t).However,d3(t)plays a very small role when ammunition manipulator moves rapidly.To avoid repeatedly complicated description,there only exhibits the simulation results with disturbanced2(t).The comparative simulation results of three methods with disturbanced2(t)in Fig.13 are shown in Figs.9-12.

Fig.9 shows the tracking errors of three controllers with external disturbances.The tracking errors of the BLF-DSC and PID method were still kept within±0.2°and±1.6°respectively,while the tracking errors of the DSC method increased to±0.9°.As shown in Fig.10,the tracking speed errors of the DSC controller trembled evidently during the tracking process because of external disturbances.Comparing Figs.10 and 6,there exist inherent trembles in the DSC controller when the motion direction of the cylinder changes because of the sign function sgn(*),and it’s obvious that the initial motion is a special case of movement direction changing.To eliminate above chatters,the sign function is taken place by the damping termsin the BLF-DSC method.Fig.11 shows the control inputs of three controllers under disturbances.Pressure change of the BLF-DSC controller was shown in Fig.12.

5.Experimental results

The photograph of the experimental setup is shown in Fig.14.The AMEHS was controlled by a high frequency response servoproportional directional valve(Atos:DLHZO)whose response time to±100%step signal is 10 ms with spool position transducer.The positionx1was measured by the 16 bit angle encoder(BMPD39016S)and the statex2was calculated by differentiating the position.The pressures were directly measured by pressure transducers(ISPH-250/I-M-CE)with the measuring range of 0-25 MPa.The control algorithm in the real-time operation was carried out by a high-performance PLC(B&R:X20CP1585)whose main frequency reaches 1 GHz.A/D conversion was handled by X20AI4622 module and D/A conversion was handled by X20AO4632 module.The communication between host computer and PLC was achieved by the Ethernet POWERLINK interface.And PLC communicated with angle encoder by CANopen interface.The sampling rate of the system was 1 kHz.

The experimental position tracking errors of three controllers were shown in Fig.15.The tracking error of the BLF-DSC method was kept within±0.3°,while the tracking errors of the DSC and PID method were kept within±0.7°and±1.6°respectively.In addition,the tracking error of the DSC controller trembled as the cylinder changed the motion direction.Fig.16 shows the control input of the BLF-DSC controller in experiments.The experimental pressures dynamics were shown in Fig.17.There exists pressure oscillation when the motion direction of the cylinder changes.

Comparative analysis of the simulation and experimental results indicate that:Firstly,the BLF-DSC controller holds a smaller tracking error range than the other two controllers.Secondly,the DSC controller is of inherent trembles as the motion direction of the cylinder changes.Last but not the least,the pressures dynamic of two chambers in the cylinder are up to the system states such as uncertain external disturbances,current location of ammunition manipulator,initial values of the system states,leakages and some other umodeled uncertainties.

6.Conclusion

The BLF-DSC strategy was designed to improve the position tracking performance of the AMEHS with its tracking error constraints.Based on nonlinear state space model of the AMEHS established in this paper,the BLF and the DSC were combined to prevent the constraints violation of tracking errors and the explosion of differentiation calculation in backstepping iteration in the meantime.The stability and convergence of the closed-loop overall system were theoretically proved.It was shown that desired position tracking performance was achieved and the proposed method guaranteed control accuracy with tracking error constraints via simulations and experiments.

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.

Acknowledgments

The authors acknowledge the National Natural Science Foundation of China,China;Grant ID:11472137.