Adaptive robust control for four-motor driving servo system with uncertain nonlinearities

Wei ZHAO,Xuemei REN

Key Laboratory of Intelligent Control and Decision of Complex Systems,School of Automation,Beijing Institute of Technology,Beijing,100081,China

Adaptive robust control for four-motor driving servo system with uncertain nonlinearities

Wei ZHAO,Xuemei REN?

Key Laboratory of Intelligent Control and Decision of Complex Systems,School of Automation,Beijing Institute of Technology,Beijing,100081,China

A novel adaptive robust control(ARC)is presented for the four-motor driving servo systems with the uncertain nonlinearities and actuation failures,such that the load tracking control is achieved with the proximate optimal-time.By applying the proposed scheme,several control objectives are achieved.First,the nonlinear synchronization algorithm is presented to maintain the velocity synchronization of each motor,which provides fast convergence without chatting.Moreover,the time-varying bias torque is applied to eliminate the effect of backlash and reduce the waste of energy.Then,the ARC is designed to achieve the proximate optimal-time output tracking with the transient performance inL2norm,where the friction and actuation failures are addressed by the adaptive scheme based on the norm estimation of unknown parameter vector.Finally,the extensive simulated and experimental results validate the effectiveness of the proposed method.

Four-motor driving servo systems,adaptive robust control,velocity synchronization control,friction,backlash,actuation failures

1 Introduction

Multi-motor driving servo systems have the potential for many important practical processes,e.g.,the radar servo systems,the artillery control systems and so on.Thus,the multi-motor systems have attracted various attentions from the control community.For the multimotor driving servo systems,the system performance is affected by the degree of motor velocity synchronization and other nonlinear factors,such as friction and backlash.Therefore,it is necessary to design the tracking controller based on the synchronization scheme and nonlinearities compensation such that the control performance is improved.

In the mechanical systems,the modeling and com-pensation for the friction nonlinearity have been investigated for decades.Many models have been proposed to describe the friction phenomenon,such as Dahl model [1], Stribeck model [2], LuGre model [3] and so on.Moreover,due to the learning capability of neural network(NN),the NN-based compensation has been studied.Huang et al.[4]presented two NNs-based controller to compensate for the effect of nonlinear friction.Incorporated with the radical basis function NN scheme,Xia et al.[5]proposed the energy-based controller to offset the bad effect of friction.Different from the NN compensation,the adaptive technology is designed to compensate for the friction nonlinearity by applying the friction parameters estimation.Misovec[6]studied the adaptive nonlinear friction compensation algorithm with the persistent excitation.Similarly,Li et al.[7]developed the ARC to compensate for the nonlinearly parameterized dynamic friction.In this paper,a continuously differentiable friction model is employed to describe friction and the ARC based on the upper bound of friction is proposed to eliminate its effect in the four-motor driving servo systems.

Backlash is another nonlinearity widely existing in the transmission parts of servo systems.To compensate for the backlash,Selmic and Lewis[8]proposed a NN-based dynamic inversion compensation scheme,where the modified Hebbian algorithm was presented for NN tuning to guarantee the closed-loop system stability.In[9],the adaptive fuzzy algorithm was presented to compensate for the effect of backlash,where the tuning algorithm for fuzzy logic parameters was used to ensure the stable performance.Whereas,the above algorithms[8,9]can not completely eliminate the uncontrollability as the gears separation.To deal with this issue,the bias torque is proposed based on the multi-driver systems,where two groups of drivers are controlled by the equal but opposite torques,respectively.Gawronski et al.[10]modified the traditional bias torque to eliminate the effects of backlash and improve the system performance.In[11],a real-time bias torque controller was designed to completely compensate for the backlash.However,the bias torque needs to be applied on the motors all the while,which may increase the energy consumption of servo systems.In this paper,the time-varying bias torque is presented to compensate for the backlash nonlinearity,which can reduce the waste of energy and minimize its effect on the tracking control as the motors contact to the load.

The actuation failures occur inevitably for the long time operation.Thus,the fault-tolerant control(FTC)should be considered to guarantee the system stability and improve the system performance in fault situation.In[12,13],the integral sliding mode schemes were proposed to design the FTC for the aircraft and spacecraft respectively,which were proven to tolerate the actuator failures and guarantee the closed-loop system stability.Lin et al.[14,15]proposed the Takagi-Sugeno-Kang type fuzzy NN with asymmetric membership function to present the FTC control,such that the system stability was achieved under faulty condition.Furthermore,a novel NN-based active FTC scheme with fault alarm was presented in [16] by using the implicit function theorem,which required no additional fault detection, and had the advantage of passive FTC scheme as well as traditional active FTC.Although the above literatures have obtained some satisfactory results,there exists no research dedicated to the FTC in multi-motor driving servo systems.Thus,this paper employs the passive FTC scheme for the ARC design to further guarantee the performances of four-motor driving servo systems in fault situation.

Synchronization is a basic issue of multi-motor driving servo systems control, which requires that all the motors converge to the same position and velocity.To achieve the synchronization of motors,many researchers have been dedicated to the synchronization algorithm and obtained some important results.For instance,Sun[17]employed the coupling scheme to design an adaptive controller by feeding back the position errors and differential position errors,which guaranteed the asymptotic convergence of synchronization errors.Based on[17],a model-free cross-coupled controller was proposed in[18],where the proportional and derivative-type synchronization algorithms with error feedback were employed for the position synchronization.Similarly,the cross-coupling scheme was incorporated with the robust sliding mode control in[19]to reduce the synchronization error.Besides,the functional link radial basis function network was used in[20]to design the controller for the dual-motor system,such that the velocity synchronization errors can be reduced.Although the aforementioned methods can achieve synchronization,the load tracking was rarely taken into consideration.

This paper focus on the finite-time output tracking control for the four-motor driving servo systems by employing ARC.The main contributions of this paper are summarized as follows:

1)The nonlinear synchronization controller is employed to achieve the velocity synchronization between the motors.This scheme provides much faster convergence without the chatting.

2)The time-varying bias torque is proposed to compensate for the backlash and maintain the controllability as separation of gears,which can also minimize its effect on the tracking control in contact mode.

3)Based on the bound of actuation failures,the novel ARC is proposed to achieve the proximate optimal-time load output tracking in fault situation and compensate for the friction nonlinearity and disturbance without any priori information.

4)The online estimation scheme is proposed based on the norm estimation of unknown parameter vector,which can significantly reduce the number of estimated parameters.

This paper is organized as follows.In Section 2,the mathematical model of the four-motor driving servo systems is described in detail.The nonlinear synchronization scheme,time-varying bias torque and ARC are designed in Section 3.Then,the simulated and experimental results are shown in Section 4.Section 5 concludes this paper.

2 Problem formulation

As shown in Fig.1,the four-motor driving servo system consists of four driving motors,the wheel gears with backlash,the transmission parts with elastic joint and the load.The motors are controlled by the microprocessor-based position controllers.Moreover,the actuation failures occur inevitably in servo systems for the long time operation,which is accordingly considered after the controller.

In the following,we will provide the dynamics of driving system and load system,respectively.

Fig.1 Schematic diagram of four-motor driving servo systems.

2.1 Driving system modeling

From Fig.1,it is obvious that the driving system is mainly composed of four driving motors contacting with gears,which are described by

where θiis the angle position of motori,Jmiis the moment of inertia,and defined asJmi=J,i=1,...,4,bidenotes the viscous friction coefficient.ui(t)is the input control signal to drive each motor,which is given by

whereusiis the synchronization controller to achieve the velocity synchronization of motors and the bias torquewiis used to eliminate the effect of backlash.In the next section,the four motors are divided into two parts to be driven by different bias torqueswi,which guarantees one group of motors in contact with the load as the backlash appears.

Besides,uois the actual input through the actuation failures and defined as

The backlash phenomenon shown in Fig.2 occurs in the transmission parts as the motors change directions.Thus,the transmission torque τiis defined to represent the backlash nonlinearity

with the functionf(zi(t))[21]as

Fig.2 Transmission part with backlash.

2.2 Load system modeling

Generally,the dynamics of load can be described as

where θlandJlare the position and the moment of inertia,flis the friction force,dlis the bounded disturbance,Tlis the torsional torque,which is derived from the control inputuothrough the backlash-type nonlinearity and elastic transmission.

Conventional friction models are discontinuous or piecewise continuous,which may be problematic for the smooth control design.Therefore,the continuous friction torquefl[22,23]is employed in this paper

where β1,β2,β3,a1,a2,a3are the uncertain positive parameters.

2.3 Model transformation

From the above analysis,the whole servo system is represented as a kind of two-mass system,which consists of the driving system with the inputuoand output ω1,and the load with the output θl.

Considering the effect of elastic joint on the torsional torque,the dynamics equations (1) and (6) can be rewritten as the matrix form

Due to the existence of backlash and bias torquewi,the velocity ω1is specified as

From the state equation(8),the transformed model of the four-motor driving servo system is given by

According to(1)and the definition of ω1,it is found that as all the motors contact to the load,(10)is rewritten as

and as only the motors 1 and 3(or motors 2 and 4)contact to the load,it is obtained as

From the above analysis,one obtains that(11)and(12)have the same type as

Remark 1In this paper,all the parameters are completely unknown, which consist of the motor parameters(J,J1andbi),the load system parameters(Jlanddl),the transmission parts(ki,ci,ˉα,ν andSc),the friction parameters(βjandaj),and the actuation failures(ρ andˉρ).Furthermore,the occurring time of the actuation failures is uncertain as well.

Afterwards,the transformed system(13)is studied in detail and the control objective is to determine an adaptive controllerui,such that

1)The velocity synchronization is attained by the nonlinear scheme.Moreover,the effect of backlash is eliminated by the time-varying bias torque.

2)The output signal θltracks a reference trajectoryyd(t)in finite time with the transient performance in term ofL2norm.

3)The effects of friction and actuation failures are all compensated without the prior information and fault diagnosis.

3 Controller design

In this section,the controller design procedure is proposed,which is comprised of the nonlinear synchronization scheme,the bias torque design,and the ARC.

3.1 Synchronization controller

To achieve the velocity synchronization quickly,the effective synchronization scheme needs to be designed.Define the cross-coupling velocity synchronization errors as

which describe the velocity differences between each motor.

Then,the nonlinear controller is proposed as

whereksis the control gain and the function fal(·)[24]is defined as

where α ＞ 0 and δ ＞ 0 are the adjustable parameters.Different from the traditional linear scheme(e.g.,usi=?ksesi),the nonlinear algorithm can attain the velocity synchronization with fast rate.Due to the switching as|esi|≤ δ,the chattering phenomenon near origin,which is derived from the function sgn(·),is completely removed.As an example,the curves of linear control and nonlinear scheme(15)are shown in Fig.3,where the control gain isks=3.From Fig.3,it is obvious that the proposed controller(15)gives faster convergence,which can be increased by a large α.

Fig.3 The comparative curves of linear and nonlinear control.

3.2 Bias torque design

From(5),it is obtained that the system uncontrollability is derived as separation of gears,which may severely affect the system performance. For this purpose, the bias torque is considered to eliminate the effect of backlash in this part.To reduce the waste of energy,the state feedback-based time-varying bias torque is proposed as follows:

Fig.4 The curve of time-varying bias torque.

Remark 2Compared with the conventional bias torques(e.g.,the constant bias torque and the proportional bias torque),the proposed method provides the time-varying torque according to the position difference,which can eliminate the uncontrollability caused by the backlash.Moreover,it is able to reduce the waste of energy and minimize its effect on tracking control.

Remark 3Due to the nonlinear synchronization controlusi,the motor velocity synchronization is maintained with fast convergence,which improves the system performance.Furthermore,the designed bias torquewican keep some motors contacting to the load at any time,which eliminates the system uncontrollability as separation of gears.By the applications ofusiandwi,the issues of velocity synchronization and backlash are both addressed for the four-motor driving servo systems.

3.3 Adaptive robust controller

In this section,the tracking controluis designed to achieve the control objectives proposed in Section 2.To facilitate the tracking controller,the transformed error is proposed as

The functionfp(e)is given by

Moreover,it follows that

Remark 4Inspired by the proximate time optimal servomechanism(PTOS)[25,26],the error transformation(18)is proposed to design the tracking controller.This scheme achieves the suboptimal time convergence as|e|＞eland the asymptotic convergence as|e|≤el,which is able to avoid the chattering and maintain the error convergence with the proximate optimal-time.

The above analysis is summarized as follows.

Lemma 1For the servo systems(13),as the transformed erroret=0 is achieved in finite time Ts,then the tracking erroreconverges to zero with the proximate optimal-time T.

ProofAccording to[25],it is stressed that the time optimal switching curve ofeis defined as

which gives the time optimal convergence ofe.But due to the application of sgn(·),the chattering on the switching curve is inevitable.

Thus,the functionfp(e)is proposed to address the convergence issue.Aset=0 and|e|＞el,(18)is converted into

which can achieve|e|convergence to a region aroundelwith the proximate optimal-time T1.The discount factorais used to reduce the oscillation of convergence curve.

Then,as|e|≤el,it becomes the nonlinear term to achieve the convergence to zero in finite time T2

Thus,it is evident that the tracking erroreconverges to zero in the proximate optimal-time T=Ts+T1+T2.This ends the proof. □

To achieve the erroretconvergence,the ARC is proposed as

where the tracking termutis defined as

and the robust termuris given by

with the positive constantsk1,k2,the robust term gaink3＞0 and sigr(et)=sgnet|et|r.

Before giving the adaptive compensation designuw,we first discuss the boundedness of the lumped nonlinearityTin(13).From the definitions of τi,flandusi,it is able to obtain the useful inequalities as follows:

where all of κijare the positive constants.

Then,combining(27)with(13)and(20)yields that the lumped functionTis bounded and there are some positive constants satisfying

According to(28),there exist fourteen parameters to be estimated online for the nonlinearity compensation,where the number of parameters is too large to increase the computational burden and limit its applanation in practice.To deal with this issue,the adaptive compensation schemeuwis proposed based on the norm estimation of unknown parameter vector as

withkcbeing a positive constant.

Remark 5The adaptive robust tracking controller is composed of the finite-time convergence termut,the robust termurto eliminate the effects of parameter estimation errors,and the adaptive termuwto compensate for the uncertain nonlinearities,the unknown parameters and the actuation failures.Moreover,the robust scheme,rather than other algorithms,such as the intelligent methods and extended state observer,is designed to online compensate for the lumped nonlinearity.This improves the robustness of servo systems essentially.

Remark 6The parameter norm-based robust compensationuwextremely reduces the number of estimated parameters,and simultaneously simplifies the process of parameter estimation.For example,the number of estimated parameters is changed from fourteen to one in this paper.Furthermore,only one adaptive gainkcneeds to be adjusted.Therefore,the proposed scheme is favorable for the application in practice and the computational burden is removed significantly.

Theorem 1Consider the four-motor driving servo systems(13)with the actuation failures(3).If the controllers are presented as(24)with the adaptive law(31),then the following control objectives are achieved.

1)The uncertain nonlinearities and actuation failures are compensated without the fault diagnosis.Furthermore,the tracking erroreis uniformly ultimately bounded(UUB).

2)If the gaink3satisfies

Proof1)Choose the Lyapunov function candidate as

Then,the derivative ofV1along with(18)is given by

where the multiplicative fault ρ and additive faultˉρ are included inGandT,respectively.

Taking the controller(24)into(35)obtains

Taking the condition(30)into(37)obtains

By substituting the control(29)into(38)leads to

Then,the derivative ofV2along with(31)is given by

Combining(39)with(40),the derivative ofVis deduced as

From the following inequality

then(41)is transformed as

From(43),it is evident that both the erroretand the parameter estimation error?ˉcare UUB and finally converge to the bounded regions

where|et|can be made arbitrarily small by a largek2.

2)Furthermore,if there exists a constantk3satisfying(32),then it is obvious that(39)can be transformed as

According to the finite-time theory,it is found thatetconverges to zero in finite time Tsand

Then,from Lemma 1 and(47),one finds that the load output tracking control is achieved with the proximate optimal-time T.

This ends proof of Theorem 1. □

In the following,the transient performance in the sense ofL2norm is studied based on Theorem 1.Furthermore,ask3satisfies the condition(32),then it is found that

Considering the inequality(48),one obtains that

Then it follows that

From the above analysis, by the appropriate choices ofk2andk3,the transient performances of tracking control are maintained in the sense ofL2norm(50).

4 Simulated and experimental validation

4.1 Simulation analysis

In this section,the four-motor driving servo system is considered to illustrate the performance of the proposed ARC.The aim is to guarantee the motor velocity synchronization and make the output θltrack a referenceyd=sin(2/5πt)under the external disturbancedl=0.2cos(3/5πt).Moreover,consider the actuation failures as described by(3),which occur at the timet≥6.The system parameters fori=1,...,4 are listed in Table1,which are supposed to be completely unknown for the controller design.

Table 1 System parameters.

Fig.5 The tracking curves of the proposed ARC.

Moreover,Fig.5 describes the tracking results of ARC,which illustrates that the proposed controller is able to maintain the finite-time output tracking with the bounded error.Although the actuation failures occur at the timet≥6 as seen in Fig.6,the proposed scheme can quickly guarantee the system stability,which further verifies that the controller has the strong robustness to eliminate the effects of failures and disturbance.

Fig.6 The curve of control uo.

Fig.7 gives the synchronization results of the proposed scheme.It is shown that the finite-time motor velocity synchronization is achieved under the different initial values.There exist some errors derived from the actuation failures of tracking control as depicted in Fig.7,but the proposed scheme can guarantee the velocity synchronization with fast convergence.This implies that the achieved synchronization performance is fairly good.

Fig.7 The velocity synchronization curves of the proposed ARC.

4.2 Experimental validation

To validate the effectiveness of the proposed adaptive method,the extensive experiments are conducted on the four-motor driving test rig as shown in Fig.8,which consists of four motors embedded in the motor drive card(Panasonic MCDDT3520),a Pentium 3.0 GHz industrial computer for control and an electromotor(180ST-M 35105)with 64000p/r(i.e.,per rotation)resolution encoder as the load. The proposed algorithms are implemented via a VC++program in CCS5.0.The sampling time ists=0.001s.In the experiments,we will give the extra signal with ρ=0.85 andˉρ=0.25 to represent the actuation failures(3)ast≥6.

Fig.8 Diagram of the four-motor driving servo systems.

In this section,three control methods are compared to verify the proposed control.

2)NPID:The NPID is designed as

3)PID:Similar to NPID,four PID controllers are used to drive four motors with the same parameterskp=80,ki=0.2 andkd=0.

First,the sinusoidal waveyd=35sin(5πt/9)is employed as the reference.Figs.9 and 10 describe the tracking performance and velocity synchronization performance of ARC.It is shown that the satisfactory output tracking control is achieved in finite time.Moreover,the velocity synchronization errors converge to the bounded regions around zero. This indicates that the transient and steady-state performances of output tracking and velocity synchronization are both retained as described in Theorem 1.

Fig.9 Output tracking and tracking errors of ARC.

Fig.10 Velocity synchronization results of ARC.(a)Velocity synchronization.(b)Velocity synchronization errors.

To further evaluate the effectiveness of tracking control,the comparative results are depicted in Fig.11.From Fig.11,one can find that the proposed ARC gives the smallest tracking errors at the point of maximum amplitude and provides the fastest convergence and the smallest steady-state error as described in Fig.11(b).

Fig.11 Comparative results of output tracking control.(a)Output tracking performance.(b)Output tracking errors.

All the aforementioned results of simulations and experiments clearly verify that the proposed schemes can compensate for the backlash and friction.Moreover,it can achieve the motors synchronization and output tracking control in fault situation.

Table 2 Tracking comparison for the different sinusoidal references.

5 Conclusions

The ARC was presented for the four-motors driving servo systems with the unknown nonlinearities and actuation failures.Based on the error transformation,the ARC was proposed to achieve the proximate optimaltime output tracking with the transient performances in term ofL2norm,which can simultaneously address the uncertain nonlinearities and actuation failures without any fault diagnosis.Furthermore,the nonlinear synchronization scheme and state feedback-based bias torque were designed,which attained the velocity synchronization quickly and compensated for the backlash with the small energy consumption,respectively.It was shown from the simulations and experiments that the proposed ARC improved the control performance.

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25 November 2015;revised 24 August 2016;accepted 25 August 2016

DOI10.1007/s11768-017-5120-7

?Corresponding author.

E-mail:xmren@bit.edu.cn.

This work was supported by the National Natural Science Foundation of China(Nos.61433003,61273150),the Foundation for Innovative Research Groups of the National Natural Science Foundation of China(No.61321002)and the Doctoral Program of Higher Education of China(No.20121101110029).

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Wei ZHAOreceived the B.Sc.and M.Sc.degrees from Henan Polytechnic University,Henan,China,in 2009 and 2012,respectively.He is currently pursuing the Ph.D.degree with the School of Automation,Beijing Institute of Technology,Beijing,China.His current research interests include multidrive servo systems,adaptive robust control,neural network control,and sliding mode control.E-mail:zw198603@126.com.

Xuemei RENreceived the B.Sc.degree from Shandong University,Shandong,China,in 1989,and the M.Sc.and Ph.D.degrees in Control Engineering from the Beijing University of Aeronautics and Astronautics,Beijing,China,in 1992 and 1995,respectively.She worked at the School of Automation,Beijing Institute of Technology as a professor from 2002.Her research interests include nonlinear systems,intelligent control,neural network control,adaptive control,multi-drive servo systems and time delay systems.E-mail:xmren@bit.edu.cn.