Zhe SunYunrui BiSongle ChenBing HuFeng XiangYawen Ling and Zhixin Sun

Abstract:For enhancing the control effectiveness,we firstly design a fuzzy logic based sliding mode controller(FSMC)for nonlinear crane systems.On basis of overhead crane dynamic characteristic,the sliding mode function with regard to trolley position and payload angle.Additionally,in order to eliminate the chattering problem of sliding mode control,the fuzzy logic theory is adopted to soften the control performance.Moreover,aiming at the FSMC parameter setting problem,a DE algorithm based optimization scheme is proposed for enhancing the control performance.Finally,by implementing the computer simulation,the DE based FSMC can effectively tackle the overhead crane sway problem and avoid unexpected accident greatly.

Keywords:Sliding mode control,fuzzy logic theory,systems optimization.

1 Introduction

Nonlinear crane systems is a class commonly used lifting appliance for the heavy cargoes transportation in harbours,construction site and industrial factories.According to the control requirement,the payload should rapidly and accurately arrive at the given site,the residual oscillation time and amplitude require shorter and smaller as far as possible to against the unexpected impact.In addition,the control difficulty will greatly increase due to the its underactuation trait.Hence,it is significant to develop the effective control method for solving the overhead crane systems payload swing problem.

Recently,a series of studies are made for overhead crane systems to damp the payload oscillations.In Sun et al.[Sun,Wang,Bi et al.(2015b);Yu,Li and Panuncio(2014)],the optimized PID controller are designed for damping the load vibration.The PID controllers performance are promoted by heuristic algorithm and neural compensation.To compare with nonlinear control method,the linear control method can not effectively tackle the nonlinear feature and easily gives rise to the unsatisfied anti-swing control performance for complex control environment.Aiming at this problem,the nonlinear control methods[Sun,Fang and Chen(2017);Sun and Fang(2014);Sun,Fang,Chen et al.(2014,2016);Sun,Wu,Fang et al.(2017)]are successfully applied in overhead crane systems and get satis fied control performance.

Among aforementioned nonlinear control methods,the sliding mode control method[Cheng(2016);Chwa(2017);Du,Yang,Li et al.(2018);Li,Shi,Yao et al.(2016)]is regarded as a kind of the effective control approach because of the advantages of the easy designing and high robustness,and widely utilized various engineering fields.But it also suffers the chattering problem coming from the discontinuous switching characteristic around the prede fined manifold.Hence,in order to figure out this problem,researchers incorporate the sliding-mode with fuzzy logic uncertainty observer to realize the overhead crane systems efficient control[Park,Chwa and Eom(2014)].In Pezeshki et al.[Pezeshki,Badamchizadeh,Ghiasi et al.(2015)],a T-S fuzzy logic based sliding mode controller is developed to tackle the dynamic performance of overhead crane.But the parameter con figuration will play an important role for control performance.Hence,to design an efficient parameter learning scheme is necessary to enhance the control performance.The optimization approaches are widely utilized in various domain for improving the system performance[Takahashi,Shibata,Motoyama et al.(2017);Li,Niu,Liu et al.(2018);Efe(2018)].To compare with traditional optimization approaches,evolutionary computation begin to get a lot of attentions for system parameter identi fication and optimization because of the outstanding optimization capability[Santucci,Baioletti and Milani(2016);Fan and Yan(2016);Sun,Wang,Srinivasan et al.(2014);Sun,Wang,Bi et al.(2015a);Sabar,Abawajy and Yearwood(2017);Wang and Tang(2016);Suganthi,Devaraj,Ramar et al.(2018)].On basis of the powerful optimization capability,we propose a optimization scheme by incorporating DE algorithm to set the FSMC parameter.

2 Fuzzy sliding mode controller designing

Sliding mode control(SMC)is a kind of commonly used lonlinear control method and has been successfully utilized in complex nonlinear systems.It also is a special kind of nonlinear control which the nonlinearity is expressed as the discontinuity of the control.The control strategy of SMC is different to compare with other control method,in which thesystemstructureispurposefullychangedinthedynamicprocessaccordingtothecurrent state of the system.

Given the typical nonlinear control systems as follows.

The nonlinear functions(FandG),control signal(u)and external disturbance(d(t))construct the nonlinear system.Andare denoted as the system state variables.Here,we denotedxdas reference state track,and the error of system is obtained as follows.

To lets(E)=0,the sliding mode surface function is acquired as follows.

In order to let the E(t)arrive at the sliding mode surface and move to the origin,the control process is separate two stages which are the approaching phase(s(E)/=0)and the sliding phase(s(E)=0).Fors(E)/=0,the control law should satisfy the condition of＜0 so that control law can drive the system error E to the sliding mode surface.Then,the corresponding switching control lawuswcan be depicted as follows.

Notes that,sgn()andu0represent the sign function and control signal initial value.

Fors(E)=0,the equivalent control lawueqis usually adopted to let the dynamic characteristics of the system remain on the sliding surface.The corresponding control forceueqis described to lets(E)=0.

Then,the expression can be acquired as:

In order to figure out the chattering problem,the fuzzy theory is utilized to soft the discontinuous switching around the prede fined manifold.To consider a second-order nonlinear systems:

where,F(x1,x2)andG(x1,x2)are linear and nonlinear functions,uis the control force.Based on the aforementioned description,s(x1,x2)is depicted as follows.

Figure 1:Fuzzification of sliding mode function

In order to soft the control output,we adopt five fuzzy language partitions with the form ofNB,NS,···,PBfor fuzzification,and the corresponding fuzzification of sliding mode function is shown in Fig.2.

3 Fuzzy sliding mode controller optimization

3.1 Differential evolution algorithm

Differential evolution algorithm is a commonly used global real value optimization approach for parameter identification and system optimization.It includes three basic operations which are mutation,crossover and selection.At the beginning evolution stage,NP individuals x={x1,x2,...,xNP}are randomly produce.And the mutation,crossover and selection operations are implemented orderly based on the following scheme.

3.1.1 Mutation

During the mutation process,xr1,xr2,xr3(xr1/=xr2/=xr3)are randomly choice.And these picked out individuals do not conformity.The produced new individual can be obtained according to the following formula.

Notes that,the scaling parameter(Fm)is used for controlling the amplification of vi.

3.1.2 Crossover

In order to incorporate the population information,the new mutant individuals need to perform the crossover operation with origin individuals.The new individual ui=[ui,1,ui,2,...,ui,D]can be produced crossing operation the between mutant viand xi.

Thejth random number(rj)belongs to 0 and 1,and the crossover probability(pc)is a constant among the range of[0,1].The rand number(rand)belongs to[1,D]to ensure thevi,jelement be obtained.

3.1.3 Selection

In order to select the high quality individual from population,the selection operation is employed to choose the excellent individual from uiand xi.That is the individual uishould picked out if the evaluation value of uiis better than the xi,if not,the individual xiis remained.

3.2 Parameter learning scheme

According to DE algorithm optimization process,the parameters of fuzzy sliding mode controller firstly are encoded.These parameter can be divided into two classes which are unknown sliding mode function and fuzz fication parameters.For evaluating the control performance of FSMC,the evaluation function(Fun)is given on basis of control requirement.

Here,the trolley displacement(x)and payload swing angle(θ)are used for evaluating control performance.

The corresponding parameter learning process is described in below.

Step 1:De fine the leaning parameters range and initialize DE algorithm parameters(G=500,NP=30,pc=0.5,Fm=0.7);

Step 2:ProduceNPgroups FSMC parameters and perform control process for each group of parameters;

Step 3:Evaluate the control performance value of each group of parameters;

Step 4:Letiter=iter+1 andi=1;

Step 5:Based on(eq.(10)),do the mutation and acquire new mutant individual vi;

Step 6:Perform the crossover between xiand vito get new crossover individual ui;

Step 7:Leti=i+1 and return to the 5th step untili=NP;

Step 8:Evaluate the newNPgroups parameters by implementing the control process;

Step 9:Choose the best parameter to remain next generation and letiter=iter+1;

Step 10:Go back to the 4th step when the end condition is satis fied,otherwise end the optimization process.

Figure 2:Nonlinear overhead crane systems

4 Description of overhead crane systems and simulation experiment

4.1 Nonlinear overhead crane systems

This section gives the brief description of nonlinear overhead crane,in which the under-actuated feature bring the dif ficulties for controlling payload oscillation.From the Fig.2,we can see that it consists on trolley and payload basic parts.According to the overhead crane dynamic feature,the mechanism model is described as follows.

Here,Mandmare the trolley and load weight,θis used for representing the swing angle,xis the trolley displacement,lis the rope length,Frepresents the driving force from control system.

where,x=[x1,x2,x3,x4]T,x1=x,x2=x,x3=θandx4=θare trolley displacement,trolley velocity,payload swing angle and payload angular velocity;urepresents the control force.f1,f2,g1andg2are described as follows.

4.2 Simulation result discussions

For evaluating the optimal tunning FSMC effectiveness,this section gives the simulation experiment under different operation conditions.According to the description of overhead crane,we construct trolley position and payload angle sliding functions.By incorporating this two sliding functions,the overall sliding function is derived.

whereλ1,λ2,C1,C2are overall sliding mode function coef ficient which are adjustable.

To con firm the proposed method effectiveness,two different operation conditions(Con1:m=3kg,xd=3,xd=9m,Con2:m=9kg,xd=3,xd=9m)simulation are performed and do the comparisons with the optimal PID controllerSun,Wang,Bi et al.(2015b)and sliding mode controller.

Figure 3:The simulation results under the first condition

Figure 4:The simulation results under the second condition

From Fig.3 and Fig.4,we can infer that the three control methods can rapidly and accurately drive the trolley to the given point.Comparing with the optimal PID and SMC,the FSMC doesn’t take much adjustment.For damping the payload oscillation,the performance of the optimal PID controller is unsatisfactory,and the payload vibration time will last for a long time.On the contrary,the SMC and FSMC can effectively eliminate the payload residual vibration.Note that,the FSMC is superior no matter in oscillation amplitude control or eliminate residual vibration.

5 Conclusions

In this paper,we presented a fuzzy logic based sliding mode controller for nonlinear overhead crane systems to solve the payload oscillation problem.In order to reasonably con figure the FSMC parameter,DE algorithm is incorporated for tunning the corresponding FSMC parameters so as to improve the control performance.Finally,for demonstrating the proposed method effectiveness,the corresponding simulation experiment is done at two different operation conditions.By comparing with commonly used methods,the DE based FSMC method shows the excellent anti-swing control performance.This proposed method also can be used in other engineering fields such as robot control,power system control and chemical control.

Acknowledgment

This work is supported by the Natural Science Foundation of Jiangsu Province(No.BK20160913),the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(No.18KJB520035),the High Level Teacher Research Foundation of Nanjing University of Posts and Telecommunications(No.NY2016021),the Incubation Foundation of Nanjing University of Posts and Telecommunications(No.NY217055),Postdoctoral Foundation of Jiangsu Province(No.1701016A),Natural Science Foundation of China(No.61602259,No.61373135 and No.61672299)and National Engineering Laboratory for Logistics Information Technology,YuanTong Express Co.LTD.