Nonlinear robust adaptive control for bidirectional stabilization system of all-electric tank with unknown actuator backlash compensation and disturbance estimation

Shusen Yuan, Wenxiang Deng, Jianyong Yao, Guolai Yang

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

Keywords:Bidirectional stabilization system Robust control Adaptive control Backlash inverse Disturbance estimation

ABSTRACT Since backlash nonlinearity is inevitably existing in actuators for bidirectional stabilization system of allelectric tank, it behaves more drastically in high maneuvering environments.In this work, the accurate tracking control for bidirectional stabilization system of moving all-electric tank with actuator backlash and unmodeled disturbance is solved.By utilizing the smooth adaptive backlash inverse model, a nonlinear robust adaptive feedback control scheme is presented.The unknown parameters and unmodelled disturbance are addressed separately through the derived parametric adaptive function and the continuous nonlinear robust term.Because the unknown backlash parameters are updated via adaptive function and the backlash effect can be suppressed successfully by inverse operation, which ensures the system stability.Meanwhile,the system disturbance in the high maneuverable environment can be estimated with the constructed adaptive law online improving the engineering practicality.Finally, Lyapunov-based analysis proves that the developed controller can ensure the tracking error asymptotically converges to zero even with unmodeled disturbance and unknown actuator backlash.Contrast co-simulations and experiments illustrate the advantages of the proposed approach.

1.Introduction

New generation tank bidirectional stabilization systems are moving toward an all-electric structure, with the horizontal and vertical subsystems driven by servo motor and electric cylinder,respectively [1].However, as the future battlefield environment deteriorates,the inherent electromechanical coupling,nonlinearity and uncertainty of bidirectional stabilization systems in high mobility combat mode will inevitably affect the pointing control of the system [2].Meanwhile, the modern warfare environment is subject to malicious spoofing of sensor signals[3]and cyber attacks[4] all of which pose a hazard to the control system.For the most important subsystem at the end of fire control system, the traditional classical control theory can only ensure the local stability of the system [5-8], which is difficult to meet the increasing requirements of high precision tracking.By this, advanced control strategies for improving the hitting accuracy of tank guns need to be further explored.

In view of this, scholars have made many contributions to achieve improvements in the tracking performance of tank stabilization systems in the past decades.The main research includes two aspects [9,10]: exact modeling and controller design.Specifically,the dynamics of the vertical subsystem is modeled in Refs.[11,12],ignoring the dynamical coupling between the bidirectional stabilization system.The modeling for actuator combined with the design of the control strategy is considered in Refs.[13,14],but only the load of the system is converted to the total rotational inertia of the motor rotor,which essentially controlled the servo motor at the driving end.In Refs.[15-18], the two-axis coupled dynamical equation of bidirectional stabilization system is established based on Lagrange dynamics and the interaction between horizontal and vertical subsystems is analyzed.Frankly, it seems more perfect to further analyze the influence of the actuator servo motor on the kinematics,avoiding the fact that the final controller is based on the torque control of the system.

Moreover, much effort has been devoted to the development with various nonlinear controllers that enhance the tracking performance for tank stabilization system.Typically, adaptive controllers [12,19] can handle uncertain parameters, but cannot suppress the adverse effects of unmodeled disturbances.Therefore,in order to improve the robustness of adaptive controllers,scholars propose the adaptive robust control (ARC) [1,11,20] to deal with both parametric uncertainty and unmodeled disturbance of bidirectional stabilization system.To further actively compensate the unknown dynamics of the system, the method based on the extended state observer (ESO) is skillfully combined with other control strategies applied to bidirectional stabilization system[17,21,22].Unfortunately, they are similarly to ARC only guarantee bounded control performance with time-varying uncertainties.Integrating the adaptive backstepping framework with robust integral of the sign of the error (RISE) feedback method [18], this controller can obtain asymptotic tracking performance even with uncertain nonlinearities in bidirectional stabilization system under continuous control input.However,the control method in Ref.[18]essentially obtains the desired performance by suppressing various uncertainties of the bidirectional stabilization system through high gains.This amplifies the effect of noise on the controller to a certain extent, leading to system instability.Also, the design of the RISE controller requires a strong assumption that the first and second order derivatives of the unknown disturbance are bounded as a precondition,which is difficult to satisfy in practice.Therefore,we need an efficient controller that can obtain asymptotic stability performance even when the bidirectional stabilization system is subject to various complex uncertainties.

Besides the uncertainties mentioned above,the actuators of allelectric tank bidirectional stabilization system are servo motor and electric cylinder[23],and the inherent backlash nonlinearity of the actuators affects the tracking performance of the system [24,25].Currently, all the research on nonlinear control strategies for bidirectional stabilization systems are without consideration of active compensation for backlash nonlinearity.Based on this,this paper is devoted to the backlash compensation of bidirectional stabilization systems in the hope of making a breakthrough.In fact,the backlash effect is an important issue in the field of electromechanical servo control, and its discontinuous and non-smooth property is even a major obstacle to further improvement of the tracking performance of bidirectional stabilization systems.Therefore, it is of practical significance to select an efficient and suitable backlash compensation method.Many rigorous methods have been studied to eliminate the adverse effects of backlash,which can be divided into two categories [26,27].The first approach is to describe the backlash model simply as a mathematical function with a combination of constant or time-varying control coefficients and a bounded disturbance term [1,20,23,28,29].In detail, these methods ultimately unify the bounded disturbance term of the backlash function into a lumped disturbance of the system.Furthermore, the vibration characteristics of bidirectional stabilization system under high maneuverable environment are particularly violent, and the influence of actuator backlash will be dramatically amplified.By simply approximating the backlash effect as lumped disturbance term, this not only increases the burden of the robust control law but also requires stronger robust control gain, which undoubtedly leads to the risk of system instability.The second method to eliminate the backlash effect is to construct a smooth inverse backlash model and realize active compensation through the inverse operation in the controller design process [27,30-32].Its advantage is that the inverse function is cleverly constructed,making the approximation error sufficiently small.Specifically, in Refs.[27,31],the combination of fuzzy control and backlash inverse compensation effectively suppresses the backlash effect and ensures system stability.Based on the adaptive control technique[30,32], the compensation of backlash nonlinearity is obtained by updating the unknown parameters of the backlash inverse function in real time.To the author's knowledge,this compensation method has not been tried in bidirectional stabilization systems,but it has been widely applied in electromechanical servo control due to its availability,such as robot system[27]and vessel-riser system[32].Unfortunately, the above controllers designed with the backlash inverse method cannot obtain perfect steady-state performance and can only ensure that the tracking error of the system is bounded.With the above analysis, a controller that simultaneously handles backlash nonlinearity, unknown system dynamics, strong external disturbances and electromechanical coupling is difficult.Therefore,it is valuable to design a control method considering the unknown actuator backlash compensation and disturbance estimation for bidirectional stabilization system of moving all-electric tank,and obtain excellent asymptotic output tracking performance.

In this paper, the coupled nonlinear mathematical model of bidirectional stabilization system with consideration of actuator dynamics is established.Meanwhile, a new robust adaptive feedback control scheme is developed for precision pointing control of bidirectional stabilization system exposed to system disturbances and backlash nonlinearity.The unfavorable effect of backlash property is eliminated by the smooth backlash inverse model.In addition,the disturbances and unknown parameters of the system are estimated in real time by the derived adaptive laws.Furthermore, the robust feedback control term is designed to address approximation errors for smooth backlash inverse model and unmodeled disturbances.The stability analysis shows that excellent asymptotic tracking performance can be achieved even in the presence of various model uncertainties.

The main contributions are summarized as

(1) It is the first attempt to establish an integrated mechatronic dynamical model for bidirectional stabilization system of moving tank with comprehensive consideration of kinetic coupling, nonlinearity and uncertainty.

(2) A novel adaptive backlash inverse function is employed to realize the active compensation of backlash nonlinearity.

(3) The tracking error is guaranteed to converge asymptotically to zero with the proposed robust adaptive control strategy.

(4) The exact bound of the unknown disturbances need not be known in advance but is estimated online by the synthesized adaptive law.

The structural framework of this paper consists of: Section 2 shows the dynamical model for bidirectional stabilization system of all-electric tank.Section 3 derives the proposed controller.The comparative co-simulation and experiments are presented in Section 4.Section 5 gives some conclusions.

2.Dynamical model for bidirectional stabilization system of moving tank

2.1.Analytical dynamical model of bidirectional stabilization system

The bidirectional stabilization system of moving tank is described in Fig.1 which includes two dimensions for the load of the horizontal subsystem turret and the vertical subsystem barrel.According to the command of the fire control system,the horizontal and vertical subsystems drive the turret and the gun respectively.

Based on the Euler-Lagrange kinetic equation, the coupled dynamical model for bidirectional stabilization system can be described in a mathematically analytical way as [25]

Fig.1.Bidirectional stabilization system of moving tank.

where t∊R is the time, q1∊R is the angular displacement of horizontal subsystem, q2∊R is the angular displacement of vertical subsystem, T1∊R is the driving torque of horizontal subsystem,T2∊R is the driving torque of horizontal subsystem, d1(t)∊R and d2(t)∊R denote the unmodeled disturbance of each subsystem,m1is the mass of horizontal subsystem, m2is the mass of vertical subsystem,L1is the radius of horizontal subsystem,L2is the radius of vertical subsystem, g is the gravitational constant, Tf1∊R and Tf2∊R denote the friction torque of each subsystem.

For addressing the effects of friction nonlinearity in bidirectional stabilization system, a recently proposed continuously differentiable friction model [33] is applied to describe the frictional behavior of the system, which is expressed as follows:

where a11,a21,a31,a12,a22,a32indicate levels with different friction characteristics, c11, c21, c31, c12, c22, c32describe the shape coefficients of friction characteristics.

2.2.Mechatronic analytical dynamical model of horizontal subsystem actuator

The actuator for horizontal subsystem of all-electric tank bidirectional stabilization system drives the turret rotation through a servo motor via a gear reducer.Considering the small armature inductance of servo motor, the armature current dynamics can be neglected, so the dynamic of the horizontal subsystem actuator is expressed as [34]

where Jw1denotes the motor inertia, θw1∊R denotes the angular displacement of the motor, Tw1∊R denotes the electromagnetic torque,Bw1denotes the viscous damping factor,Tg1∊R denotes the gear input torque,Kw1denotes the constant of motor torque,uw1∊R denotes the control input voltage, Rw1denotes the armature resistance of the motor, Ke1denotes the counter-electromotive force.

Because of the gear meshing error in horizontal subsystem,the analytical model between Tg1and T1is represented as

2.3.Mechatronic analytical dynamical model of vertical subsystem actuator

The vertical subsystem of all-electric tank bidirectional stabilization system is driven by electric cylinder, and the transmission principle of electric cylinder is shown in Fig.2.

The dynamics of the electric cylinder is described by[35].

Fig.2.Transmission principle diagram of the electric cylinder.

where Tw2∊R represents the motor electromagnetic torque, Two∊R represents the motor output torque, Jw2represents the motor inertia, θw2∊R represents the angular position signal, Bw2represents the viscous damping factor, Kw2represents the electromagnetic torque coefficient,uw2∊R represents the motor control input,Rw2represents the armature resistance, Tco∊R represents the output torque which is generated by the electric cylinder, Jc2represents the driver rotational inertia, Bc2represents the driver viscous damping coefficient.

The output torque Tcoof the electric cylinder is given by

where Slrepresents the lead of the screw, Fe∊R represents the output thrust, η represents the transmission efficiency, N2represents the transmission ratio, ΔR∊R represents the output displacement of the electric cylinder actuator.

The installation position of electric cylinder is shown in Fig.3.

As observed in Fig.3, it clearly describes the institutional nonlinear relationship with the upper and lower hinge points of the electric cylinder, the gun barrel and the rocker.Rais the distance between the center of the upper hinge point of electric cylinder and the center of trunnion,Rdis the distance between the center of the lower hinge point of the electric cylinder and the center of the trunnion, R0is the initial length of the electric cylinder, α0is the initial vertex angle of the electric cylinder.Hence the corresponding vertex angle α∊R of the electric cylinder shows that α = q2+α0,then the output displacement ΔR is given as

Combining Eqs.(14) and (15), yields

Fig.3.Installation position of electric cylinder.

From Eqs.(10)-(12), we have

Further analysis of Fig.3 indicates that the input torque of the electric cylinder to the vertical subsystem is

3.Controller design for bidirectional stabilization system with backlash compensation and disturbance estimation

3.1.Problem formulation and dynamical model

Considering the complexity of the horizontal and vertical subsystem dynamics models Eqs.(9) and (21), define the following state variables x1= [x11,x12]T= [q1,q2]T, x2= [x21,x22]T=[˙q1, ˙q2]T.Hence, the whole dynamical model for bidirectional stabilization system of the all-electric tank is recharacterized as

where the angle C∊R denotes the angle between the output thrust and the horizontal axis of gun.

According to the geometric relationship in Fig.3,we can obtain

With Eqs.(13), (17) and (19), we have

where Mi(x1), Ki(x1), uw, Ci(x1, x2), Gi(x1), Δi(t) have the following structure

where dw2(t)∊R denotes the unmodeled error of the vertical subsystem actuator.

Taking Eq.(20)into Eq.(2),the comprehensive dynamical model with mechatronics of the vertical subsystem is given by

in which

In Eq.(22),Δi1and Δi2denote external load disturbances,unmodeled friction, unmodeled dynamics of horizontal and vertical subsystem.

It is worth pointing out that the unknown actuator backlash nonlinearity is inevitable because the bidirectional stabilization system is driven by all electricity.Therefore, the backlash nonlinearity of the bidirectional stabilization system is described by Bli(uw)=[Bli(uw1),Bli(uw2)]Tin this paper which reflects the input and output characteristics between the control input uwand the backlash property as

where Bli(uw)denotes the backlash characteristic,the slope matrix b =diag(b1,b2)>O,the vectors sa=[sa1,sa2]T>0,sbj=[sb1,sb2]T<0 in which sa1, sa2, sb1, sb2are constant parameters, the vector Bli(uw(t-))=[Bli(uw1(t-)),Bli(uw2(t-))]Tdenotes the backlash output value in the former period.In Fig.4, the curve of backlash nonlinearity Bli(uwj),j=1,2 for bidirectional stabilization is presented.In addition, j=1 and j=2 represent the relevant parameters of the horizontal and vertical subsystems,respectively.

3.2.Backlash compensation

As given in Fig.5, Blidjdenotes the adaptive backlash inverse model, Blidjdenotes the backlash model with linear parameterization of the bidirectional stabilization system.The final goal is eliminating the negative impact of actuator backlash of the bidirectional stabilization system, this paper proposes a new smooth and continuous inverse function of backlash nonlinearity based on the backlash inverse compensation principle shown in Fig.5 as

Fig.4.Backlash nonlinearity.

Assumption 2.The defined unknown vectors θ and ϑ both have their own upper and lower bounds, i.e.,

where θijmin,θijmax,ϑikminand ϑikmaxare known.Meanwhile,the influence of backlash nonlinearity is considered, assuming b1≠0 and b2≠0.

To ensure that θij,i=1,2,j=1,2 and ϑik,i=1,2,k=1,2,3 are bounded.From Assumption 2, two discontinuous projections can be defined as [36]

Fig.5.Schematic diagram of the RACB controller with backlash inverse.

where Гθand Гϑare positive diagonal adaptation rate matrices,τθand τϑare synthesized adaptive feedback law; for any adaptive feedback laws τθand τϑ, the projection mappings used in Eq.(36)guarantee [36]

Assumption 3.The desired motion trajectory vector x11d(t)∊C2and x12d(t)∊C2.

Remark 2:In fact, the unmodeled error Δi(t) is bounded, and the boundedness of the approximation error db(t) is proved in detail in Ref.[30].Therefore, it can be saw that the Δ(t) has boundedness naturally which shows that Assumption 1 is reasonable.In addition, the controller proposed in this paper without knowing the exact bound of the unknown lumped uncertainties Δ(t) of bidirectional stabilization system, it can be estimated online through the synthesized adaptive law.

3.3.Robust adaptive controller design

The design parallels the recursive backstepping design [37]since the comprehensive dynamical model Eq.(33) includes various uncertainties.Given a desired motion trajectory x1d=[x11d(t),x12d(t)]Tof bidirectional stabilization system,and the ultimate goal of the controller is to synthesize a bounded continuous control input u = [uw1, uw2]Tsuch that the system output x1= [x11, x12]Ttracks the desired motion trajectory with an error tending to zero or within the finite range.

Step 1.Define the following error variable

Based on Eq.(43), an actual control input is constructed to ensure asymptotic stability performance.In fact, the synthesized control input Blidprevents the bidirectional stabilization system from being affected by backlash nonlinearity.Then the Blidcan be derived from Eq.(42) as

in which k2=diag(k21,k22) and k2s=diag(k2s1,k2s2) are positive definite matrixes; ω(t) represents a positive function with the property of being integrable; ^δ1and ^δ2represents the estimated value of δ1and δ2.

As deduced from Eq.(44), Blida=[Blida1,Blida2]Tis a regulable model compensation term obtained by online parameter adaptive with performance enhancement;Blids1=[Blids11,Blids12]Tis a linear robust control law employed to make the nominal model of the bidirectional stabilization system reliable, and Blids2=[Blids21,Blids22]Tis a nonlinear robust feedback control term applied to suppress the various uncertainties.

Combining Eqs.(44) and (43) yields

Remark 3:According to Eq.(44), we can find that the derived nonlinear controller considers the estimation of the bounds of the lumped uncertainties for bidirectional stabilization system.This demonstrates that there is no need for explicit upper and lower bounds on the lumped uncertainties in the designed controller process.Furthermore, the design of the controller makes clever employ of the tanh and ω(t) functions to ensure smoothness and continuity of the developed control method.Therefore, the presented controller becomes more suitable for tank bidirectional stabilization system than the conventional discontinuous controller.

3.4.Main results

Theorem.Noting Assumptions 1-4, four adaptation laws are constructed as

where Гθand Гϑrepresent diagonal matrices, they are positive definite.ε represents a positive adaptation control gain.We also need to utilize the control gains k11,k12,k21and k22so large that the given matrix Eq.(48) is positive definite

To conclude, the proposed controller Eq.(46) can pledge all model signals of bidirectional stabilization system are bounded and obtain the asymptotic output tracking performance, i.e., z1→0 as t→∞.

Proof:See Appendix A.

Remark 4:According to the Theorem analysis,it can be noticed that the designed control strategy achieves a perfect asymptotic stability with the error results converging to zero for bidirectional stabilization system with various uncertainties and actuator backlash.Also, if the control gains k11, k12, k21and k22are gradually increased, the convergence efficiency is enhanced.The asymptotic convergence to zero output tracking results ensure precise pointing control for bidirectional stabilization system of moving all-electric tank in high maneuvering environments, which is of great engineering utility.

4.Comparative verification

4.1.Co-simulation results

In order to complete the validation with the developed nonlinear control scheme of bidirectional stabilization system of moving tank,the co-simulation verification is constructed in Fig.6 which describes its principle structure.The dynamical model of moving tank is modeled through Recurdyn, it has following parameters m1= 5000 kg, L1= 1.1 m, m2= 2000 kg, L2= 5.1 m,Sl=0.016 m,Ra=0.44 m,R0=0.4 m,Rd=0.32 m,α0=arccos(0.48),a11=125, a21=300, a31=1.56,a12=48,a22=99, a32=1.25.The actuators of bidirectional stabilization system are modeled in Simulink and set with the parameters Kw1= 1.89 Nm/A,Rw1=0.607 Ω, Ke1=1.09 V/(rad∙s-1), Bw1=0.0012 Nm/(rad∙s-1),N1=400, Rw2= 0.41 Ω, ke2= 0.89 V/(rad∙s-1), Bw2= 0.0015 Nm/(rad∙s-1),Jw2=0.002 kgm2,Jc2=0.0012 kgm2,Bc2=0.00063 Nm/(rad∙s-1), Kw2=1.54 Nm/A, N2= 5.In Simulink, the Runge-Kutta solver with a fixed step size of 0.5 ms is chosen for simulation,and in Recurdyn the moving environment of the tank is a Class F road condition with a speed of 30 km/h.Given the desired tracking trajectory, the output driving torque of actuator in Simulink is transmitted in real time to the motor output gears of bidirectional stabilization system in Recurdyn through the data interaction interface.Simulink accepts the actual motion trajectories of horizontal and vertical subsystems from Recurdyn to form a closedloop control.

To show the superiority of the control algorithm, we choose b = 1, sa= 0.8, sb= -0.8.Hence, the backlash nonlinearity is expressed by Eq.(28) as

Fig.6.Principle structure of co-simulation.

For convenience,we let the backlash parameter b=1 be known.Three different control methods are utilized to achieve comparative validation.

(1) RACB: The controller is developed in this paper considering unknown actuator compensation and disturbance estimation Eq.(46) for bidirectional stabilization system.In cosimulation, the set control gains for horizontal and vertical subsystems are as follows: k11=50, k21=30, k2s1=0.5 and k12= 50, k22= 30, k2s2= 0.5.Choosing the vector n = 1,ω(t) = 10,000/(1 + t2).Giving the initial estimated values of unknown parameters are ^ϑi,k(0) = 0,i = 1,2,k = 1,2,3,^θi,j(0)=0,i=1,2,j=1,2 and^δi(0) =0,i = 1,2.Parameter adaptation rates are set at Гϑ= diag{10,10,10,10,10,10},Гθ= {2,2,2,2}, ε = 10.

(2) RAC:It is the proposed RACB controller without considering the backlash compensation.The advantages of the smooth backlash inverse operation can be demonstrated by comparing them.For a fair comparison, it has the same control parameters as RACB.

(3) ACB:This is a traditional adaptive control method which uses the inverse function of backlash to eliminate the actuator backlash.The comparison with ACB is chosen to reflect the superiority of the nonlinear robust control law derived in this paper to estimate and suppress system disturbance.For fairness,other design parameters of ACB are chosen as same as that of RACB.

In addition, to assess the effectiveness of each control method more clearly, maximum ME, average AE, and standard deviation SDE of tracking errors are selected as evaluation criteria [37].

Fig.7.Tracking effect of RACB during co-simulation.

Fig.8.Compared tracking results of the horizontal subsystem during co-simulation.

Fig.9.Compared tracking results of the vertical subsystem during co-simulation.

The smooth desired tracking trajectories are chosen of the horizontal and vertical subsystems, respectively: x11d(t) = 0.6(1-exp(-t3)) rad and x12d(t) = 0.1[1-exp(-t3)] rad in co-simulation.The tracking performance results of horizontal and vertical subsystems of the bidirectional stabilization system obtained are shown in Figs.7-12.Meanwhile,the relevant performance indices of the horizontal and vertical subsystems are shown in Tables 1 and 2,respectively.As presented in Fig.7,the presented RACB controller can ensure that the bidirectional stabilization system tracks the specified desired motion commands accurately.Comparing the results of tracking error of the RACB, RAC, and ACB controllers in Figs.8 and 9, it can be noticed that the presented RACB has the optimal tracking control effect for both the horizontal and vertical subsystems.This is made possible by using a smooth and continuous backlash inverse model that eliminates the adverse effects of unknown actuator backlash nonlinearity.The robustness of the RACB is also further enhanced by the design of the accurate disturbance upper bound estimation and the nonlinear robust term.Specifically analyzing Tables 1 and 2,it can be found that the index of AE of the RACB controller for the horizontal and vertical subsystems is improved by 19.1%,93.4%and 15.2%,89.0%compared to the RAC and ACB, respectively.The convergence of parameter estimation for the horizontal and vertical subsystems under RACB control is provided in Figs.10 and 11.The disturbance upper bound estimates δ1and δ2for the horizontal and vertical subsystems of the moving tank are shown in Fig.12.It can be known that the adaptive law of disturbance upper bound estimation designed in this paper has good convergence.

Fig.10.Parameter estimation of horizontal subsystem during co-simulation.

Fig.11.Parameter estimation of vertical subsystem during co-simulation.

Fig.12.Disturbance estimation of horizontal and vertical subsystem during cosimulation.

Table 1 Performance indices of the horizontal subsystem after 5 s during co-simulation.

Table 2 Performance indices of the vertical subsystem after 5 s during co-simulation.

4.2.Experimental results

In order to fully illustrate the validity of RACB,the experimental testbed of tank vertical subsystem is set up, it is shown in Fig.13.The experimental testbed mainly has the mechanical part and the electrical part.The mechanical part mainly contains the barrel,rocker, trunnion, gun breech, electric cylinder, bracket, and angle encoder, etc.The electrical part mainly contains a 150 MHz TMS320F28335 DSP system processing chip.To implement the proposed controller and other comparative controllers, the discretization C++ codes are programmed.The angle encoder provides angle signal of vertical subsystem in real time and forms a closed-loop control with the desired position trajectory in the DSP chip.The sampling time of control process is 2 ms.

Remark 5: Currently, the existing experimental testbed only includes the vertical subsystem for bidirectional stabilization system, and we consider actuator backlash nonlinearity and other uncertainties of the system during the experiment,which can fully verify the superiority of the developed control scheme in this paper.

To ensure the validity of the experimental results, the DSP controller based on the vertical subsystem is constructed with the same artificial backlash as Eq.(49).In addition, the experimental procedure still chooses the above comparative controllers RACB,RAC, and ACB to illustrate the advantages of the proposed control strategy.The nominal parameters of vertical subsystem experimental platform are given as: Ra= 194.3 mm, Rd= 94.2 mm,R0=169.9 mm,m2=71 kg,Kw2=0.195 Nm/A,N2=36,Sl=8 mm,b = 1.The following control parameters are applied: k12= 60,k22= 20, k2s2= 0.5.Choosing the n12=1, ω2(t) = 10,000/(1 + t2)which also conforms to the definition in Assumption 4.In the experiment, the viscous friction of the vertical subsystem is considered,and the viscous friction coefficient a32is assumed to be an unknown parameter of the system,and its initial value is chosen as ^a32(0) = 0.The initial estimates for sa2and sb2are selected by^sa2(0)=0 and ^sb2(0) = 0, and the initial value of upper bound estimate of disturbance is set as^δ2(0) =0.In addition,the adaptive control rates are designed as Гϑ=diag{6},Гθ=diag{0.5,0.5},ε =2.Considering the fairness of the three controller comparisons,the RAC controller does not consider the backlash compensation,and it has the same control parameters as the RACB.For fairness,the other design parameters of ACB are chosen as same as that of RACB,and the comparison between RACB and ACB can verify the effectiveness of disturbance estimation and the superiority of the nonlinear robust feedback law to suppress external disturbances.

Fig.13.Experimental platform of tank vertical subsystem.

Fig.14.Compared tracking results of vertical subsystem during experiment.

Table 3 Performance indices of vertical subsystem after 5 s during experiment.

Fig.15.Parameter estimation and disturbance estimation of RACB during experiment.

The vertical subsystem tracks the smooth desired tracking trajectory: x12d(t) = 0.1[1-exp(-t3)] rad in experiment.In the experiment, we mainly test the stability accuracy of the gun.The comparative experimental errors with the compared three controllers present in Fig.14 and their performance indices show in Table 3.The proposed RACB control strategy achieves better control results in both transient and steady-state tracking performance due to the adoption of smooth backlash inverse compensation and accurate disturbance estimation.The comparison of the tracking results between RACB and RAC indicates the availability of the continuous and smooth adaptive backlash inverse model, and the comparison with ACB demonstrates the strong robust performance of the designed robust feedback control term.Furthermore, the estimation of unknow parameter and disturbance upper bound are collected in Fig.15 with RACB.It can be found that they gradually converge to a constant value because of the parameter adaptive law.As described in Fig.16, the control input of RACB is bounded and continuous.

Fig.16.Control input of RACB during experiment.

5.Conclusions

In this article,a nonlinear robust adaptive control scheme based unknown actuator backlash compensation and disturbance estimation has proposed of bidirectional stabilization system of moving all-electric tank.First, the mechatronic dynamics of bidirectional stabilization system is modeled with coupling, nonlinearities and uncertainties.Meanwhile, the unknown system parameters and backlash parameters are updated through the designed adaptive laws via backstepping and the feedforward compensation is carried out.In addition, the designed controller can update the upper range of unmodeled system disturbance between tank movements in real time.Moreover,the tanh function and the positive integrable function are combined by obtaining a nonlinear robust control term to attenuate the modeling uncertainties.The presented controller realizes effective compensation of actuator backlash for bidirectional stabilization system and the precise estimation of lumped disturbance between moving tank is completed.Based on Lyapunov analysis, the asymptotic stability can be guaranteed even with backlash nonlinearity and various uncertainties.Comparative co-simulation and experiments demonstrate the validity of the presented method.In the future,it will be an interesting research to combine Radial-Basis-Function neural network [38] and prescribed performance constraint[39,40] for precise pointing control of bidirectional stabilization system.

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

This work was supported by the National Natural Science Foundation of China (No.52275062) and (No.52075262).

Appendix A

Completing the proof of Theorem requires the definition of the following Lyapunov formula

Then, according to Assumption 1, we can obtain the upper bound of ˙V as

where λmin(k2s) is the minimal eigenvalue of the matrix k2s.

Considering the matrix Λ is positive definite,we can rewrite Eq.(A4) as

Observing Eq.(A6), it is known that V(t)∊L∞and W∊L2.Therefore,it is concluded that z1,z2,～θ, ～ϑ,～δiare bounded according to the defined Lyapunov function Eq.(A1).Then, ^θ, ^ϑ, ^δiare also bounded.Noting Assumption Eqs.(1)and(3),we easily obtain that the signal x has boundedness.Furthermore, from the definition of Assumption Eq.(4),we can know that the actual control input u is also bounded.That is to say,all signals of the closed-loop system are bounded.Based on error dynamics, it is easy to check that ˙W is bounded and the function W is uniformly continuous.Based on Barbalat,s lemma [41], there is the following conclusion: we have W→0 when t→∞, i.e., we achieve z1→0 when t→∞.Finally, we obtain the proof forTheorem.