基于二阶终端滑模优化的电流环滑模控制
一种基于粒子群优化(PSO)算法的全局快速终端滑模控制方法
一种基于粒子群优化(PSO)算法的全局快速终端滑模控制方法郝春玲【摘要】为了减小六轴机械臂运行时的震动,提高控制系统的响应时间,基于PSO 智能算法对其进行全局优化控制,将机械臂系统离散成6个子系统,分别设计各个子系统,并且基于Lyapunov理论验证控制系统的稳定性.仿真结果表明,采用PSO算法对六轴机械臂进行运行轨迹控制时具有精度高、运行误差小及收敛周期短的优点,切实提高了控制系统的响应速度及控制精度.%In order to eliminate the chattering and improve the response speed of the system. This paper deals with a fast terminal sliding mode control method based on PSO( particle swarm optimization) for the six manipulator control system. Then stability of the system is demonstrated by Lyapunov theory and the optimization control parameters are achieved based on PSO algorithm. The simulation results show that the PSO algorithm has the advantages of high precision,low running error and short convergence period when the trajectory control of the six-axis manipulator is carried out,thus the response speed and control precision of the control system are all improved.【期刊名称】《电子器件》【年(卷),期】2017(040)005【总页数】5页(P1304-1308)【关键词】智能机器人;PSO算法;六轴机械臂;终端滑模控制【作者】郝春玲【作者单位】渤海船舶职业学院机电工程系,辽宁葫芦岛125100【正文语种】中文【中图分类】TP24随着科技的不断进步,以及时下较为流行的工业4.0,机器人逐渐体现了其特有的优势,对其进行智能控制成为了国内外学者的研究热点,文献[1-3]阐述了基于模糊控制策略调整PID值,表现出了较好的鲁棒性。
基于改进分数阶滑模控制的PMLSM直接推力控制
基于改进分数阶滑模控制的PMLSM直接推力控制
李国洪;钱凌志
【期刊名称】《计算机应用与软件》
【年(卷),期】2024(41)3
【摘要】针对分数阶滑模控制在永磁直线同步电机(Permanent Magnet Linear Synchronous Motor, PMLSM)直接推力控制下转速环精度不高、磁链和推力波动大等问题,提出一种改进的分数阶滑模控制算法,即分数阶双幂次指数趋近律、分数阶变速趋近律相结合的组合趋近律,能有效减弱滑模控制引起的系统振荡,加快趋近速率。
仿真结果表明:与分数阶滑模控制相比,在PMLSM直接推力控制系统中采取改进分数阶滑模控制,磁链和推力波动明显减弱,系统响应速度加快,提高了PMLSM推力响应的抗干扰能力。
【总页数】6页(P75-80)
【作者】李国洪;钱凌志
【作者单位】天津理工大学天津市复杂系统控制理论及应用重点实验室
【正文语种】中文
【中图分类】TP273
【相关文献】
1.基于二阶滑模的PMLSM悬浮平台直接解耦控制
2.基于滑模变结构的PMLSM 直接推力控制
3.基于改进分数阶滑模控制的 PMSM直接转矩控制
4.基于改进型滑
模观测器的永磁同步电机分数阶微积分滑模控制5.基于智能分数阶互补滑模的PMLSM直接推力控制
因版权原因，仅展示原文概要，查看原文内容请购买。
26966814_双电机同步消隙伺服系统的鲁棒滑模控制策略
2022年4月第50卷第2期Apr.2022Vol.50No.2现代防御技术MODERN DEFENCE TECHNOLOGY双电机同步消隙伺服系统的鲁棒滑模控制策略*李方俊，王生捷（北京机械设备研究所，北京100854）摘要：针对常见的双电机同步消隙伺服系统电流环、速度环、位置环控制结构，提出一种鲁棒性更强、响应速度更快的简化控制策略。
利用反步递推理论设计了伺服系统的位置控制器，由经指令滤波处理后的参考位置指令直接计算出力矩信号传递至电流环，简化了系统的控制结构；在反步控制器设计过程中引入积分非奇异终端滑模面，克服了线性滑模的抖动问题，并使得系统在有限时间内到达平衡状态；采用扩张状态观测器对总扰动进行补偿，使得系统对内部结构参数变化及外在扰动均具有较强的鲁棒性。
在Matlab/Simulink与Adams中进行了联合仿真实验，仿真结果表明，所研究的新型双电机同步消隙控制方法比传统方案具有更佳的跟踪性能，比普通的扰动补偿方法有更好的抗扰能力。
关键词：双电机同步消隙；指令滤波；反步法；积分非奇异终端滑模面；扩张状态观测器doi：10.3969/j.issn.1009-086x.2022.02.014中图分类号：TM921.2；TJ765；TP391.9文献标志码：A文章编号：1009-086X（2022）-02-0104-09 Control of the Dual-Motor Synchronous Anti-Backlash Servo Systemwith Robust Terminal Sliding Mode ControlLI Fang-jun，WANG Sheng-jie（Beijing Institute of Mechanical Equipment，Beijing100854，China）Abstract：Aiming at the common current loop，speed loop and position loop control structure of dual-motor synchronous anti-backlash servo system，a simplified control strategy with stronger robustness and faster response speed is proposed.The positon controller of the servo system is designed based on the backstepping theory and the torque reference command is directly calculated and transmitted to the cur‑rent loop from the reference position command after command-filter processing,which simplifies the con‑trol structure.The integral nonsingular terminal sliding surface is introduced in the backstepping control‑ler design process which overcomes the chattering problem of the linear sliding mode control and makes the system reach the equilibrium state in finite time.The extended state observer（ESO）is adopted to compensate the external disturbance which makes the servo system more robust to the changes of the in‑ternal structure parameters and external disturbance torque.The co-simulation experiment is carried out in Matlab/Simulink and Adams and the results show that the proposed new dual-motor synchronous anti-backlash method has better tracking performance than tradition scheme and better anti-disturbance abil‑ity than the ordinary disturbance compensation method.☞测试、发射技术☜*收稿日期：2021-09-23;修回日期：2021-12-02第一作者简介：李方俊（1993-），男，湖南常德人。
终端滑模控制方法
终端滑模控制方法1.1终端滑模控制1.1.1基于终端滑模的非线性系统控制[1]控制系统设计的主要需求包括两个主要方面：控制（收敛）性能和控制鲁棒性，前者需要实现有限时间收敛控制，后者需要在不适用高增益开关的条件下实现鲁棒控制。
为提高动态系统的收敛性能，Zak提出了终端吸引子（terminal attractor）[2]的概念，并在神经网络学习中表现出较好的性能，其具有如下三次抛物线型式：(0-1)且平衡点位于原点，对其在初始时刻和平衡时刻间进行积分得到：(0-2) 由此可知，系统(0-1)将在有限时间内收敛到平衡点，收敛时间只取决于系统初始状态。
考虑如下二阶系统(0-3)其中为系统状态，为系统输入，跟踪误差，其中为期望轨迹。
设计如下控制律(0-4)其中，均为正奇数且。
将上式代入式(0-3)得到如下闭环系统：(0-5)并设计滑模面如下(0-6)其中表示初始条件。
那么式(0-5)和(0-6)确保了系统(0-3)在控制律(0-4)下的终端稳定性，定义滑模面为终端滑模子（terminal slider），并定义形如式(0-4)的控制律为终端滑模控制（terminal slider control）。
显然，式(0-4)所示的控制比全状态反馈线性化控制性能优越。
结合式(0-6)(0-4)得到如下控制律(0-7)那么考虑到控制量有界且误差有界，误差的指数必须为正，即(0-8)该条件进一步缩小了参数的设计范围。
但是以上分析设计基础是滑模面初始条件，那么对于不同的期望轨迹其初始值不同（也就是说式(0-6)不一定对仍以期望轨迹均能满足），因此需要对滑模控制器的参数进行重新设计。
传统滑模利用高增益开关切换来迫使系统从任意初始条件均可收敛到滑模面，文献[]提出建立初始条件和滑模面之间的动态系统来解决传统滑模的缺陷。
设计如下滑模控制律(0-9)并将其代入系统(0-3)中得到(0-10)上式表明对于任意初始条件，滑模变量均将在有限时间收敛到稳态值，之后系统跟踪误差将在滑模面(0-6)上有限时间内到达平衡点。
基于二阶滑模观测器的永磁同步电机无位置传感器控制
基于二阶滑模观测器的永磁同步电机无位置传感器控制张晓光;孙力;陈小龙;安群涛【摘要】A kind of second-order sliding mode observer 1s proposed to estimate the rotor position and speed of PMSM(Permanent Magnet Synchronous Motor),which combines the linear sliding mode with the hybrid non-singular terminal sliding mode to avoid the phase lag caused by the low-pass filter in conventional sliding mode observer and to improve the precision of rotor position and speed estimation.The sliding mode control law is designed to restrain the inherent chattering and to guarantee the stability of observer.The tracking algorithm with phase-lock function is employed to demodulate rotor position and speed from the observed back electromotive force.Simulative and experimental results demonstrate the validity of the proposed observer.%为了准确估计永磁同步电机的转子位置与速度,提出一种二阶滑模观测器.该观测器在传统线性滑模面基础上引入了混合非奇异终端滑模面,避免了常规滑模观测器由于低通滤波所产生的相位滞后问题,同时可以提高转子位置与速度的估算精度.为了保证观测器的稳定并抑制滑模固有的抖振现象,设计了滑模控制律.最后,采用具有锁相功能的转子位置与速度跟踪算法从观测的反电动势中解调出转子位置和速度信息.仿真和实验证明了所提观测器的正确性.【期刊名称】《电力自动化设备》【年(卷),期】2013(033)008【总页数】6页(P36-41)【关键词】二阶滑模观测器;滑模控制;锁相环;永磁同步电机;控制【作者】张晓光;孙力;陈小龙;安群涛【作者单位】哈尔滨工业大学电气工程及自动化学院,黑龙江哈尔滨150001;哈尔滨工业大学电气工程及自动化学院,黑龙江哈尔滨150001;哈尔滨工业大学电气工程及自动化学院,黑龙江哈尔滨150001;哈尔滨工业大学电气工程及自动化学院,黑龙江哈尔滨150001【正文语种】中文【中图分类】TM3410 引言永磁同步电机PMSM（Permanent Magnet Synchronous Motor）的转子位置与速度是进行矢量解耦控制的必要条件，通常采用位置传感器进行检测，其中光电编码器、磁编码器以及旋转变压器等最常见。
二阶滑模控制
Wheel Slip Control via Second-OrderSliding-Mode GenerationMatteo Amodeo,Antonella Ferrara,Senior Member,IEEE,Riccardo Terzaghi,and Claudio VecchioAbstract—During skid braking and spin acceleration,the driving force exerted by the tires is reduced considerably,and the vehicle cannot speed up or brake as desired.It may become very difficult to control the vehicle under these conditions.To solve this problem,a second-order sliding-mode traction controller is presented in this paper.The controller design is coupled with the design of a suitable sliding-mode observer to estimate the tire–road adhesion coefficient.The traction control is achieved by maintaining the wheel slip at a desired value.In particular, by controlling the wheel slip at the optimal value,the proposed traction control enables antiskid braking and antispin acceler-ation,thus improving safety in difficult weather conditions,as well as stability during high-performance driving.The choice of second-order sliding-mode control methodology is motivated by its robustness feature with respect to parameter uncertainties and disturbances,which are typical of the automotive context. Moreover,the proposed second-order sliding-mode controller,in contrast to conventional sliding-mode controllers,generates con-tinuous control actions,thus being particularly suitable for appli-cation to automotive systems.Index Terms—Chattering avoidance,higher order sliding modes,robust control,slip control,traction force control.N OMENCLATUREv x Longitudinal velocity(in meters per second).w f Front wheel angular velocity(in radians per second).w r Rear wheel angular velocity(in radians per second).T f Input torque on the front wheel(in newton meter).T r Input torque on the rear wheel(in newton meter).λf Front wheel slip ratio.λr Rear wheel slip ratio.F xf Longitudinal force at the front wheel(in newtons).F xr Longitudinal force at the rear wheel(in newtons).F zf Normal force on the front wheel(in newtons).F zr Normal force on the rear wheel(in newtons).F air Air drag force(in newtons).F roll Rolling resistance force(in newtons).m Vehicle mass(in kilograms).J f Front wheel moment of inertia(in kilograms per square meter).Manuscript received November8,2007;revised August4,2008and May19, 2009.First published November24,2009;current version published March3, 2010.The Associate Editor for this paper was A.Hegyi.M.Amodeo and R.Terzaghi are with Siemens S.p.a.,20128Milano,Italy (e-mail:matteo.amodeo@;riccardo.terzaghi@). A.Ferrara is with the Dipartimento di Informatica e Sistemistica,Universitàdegli studi di Pavia,27100Pavia,Italy(e-mail:antonella.ferrara@unipv.it). C.Vecchio is with the Dipartimento di Informatica e Sistemistica,Universitàdegli studi di Pavia,27100Pavia,Italy,and also with Temis s.r.l.,20011 Corbetta,Italy(e-mail:claudio.vecchio@;claudio.vecchio@ unipv.it).Digital Object Identifier10.1109/TITS.2009.2035438J r Rear wheel moment of inertia(in kilograms per square meter).R f Front wheel radius(in meters).R r Rear wheel radius(in meters).c x Longitudinal wind drag coefficient(in kilograms permeter).f roll Rolling resistance coefficient.l f Distance from the front axle to the center of gravity(in meters).l r Distance from the rear axle to the center of gravity(in meters).l h Height of the center of gravity(in meters).μp Road adhesion coefficient.ˆμp Estimated road adhesion coefficient.I.I NTRODUCTIONI N RECENT years,numerous different vehicle active controlsystems have been investigated and implemented in pro-duction[1].Among them,the traction control of vehicles is becoming increasingly important due to recent research efforts on intelligent transportation systems,particularly on automated highway systems,and on automated driver-assistance systems (see,for instance,[2]–[6]and the references therein).The objective of traction control systems is to prevent the degradation of vehicle performances,which occur during skid braking and spin acceleration.As a result,the vehicle perfor-mance and stability,particularly under adverse external condi-tions such as wet,snowy,or icy roads,are greatly improved. Moreover,the limitation of the slip between the road and the tire significantly reduces the wear of the tires.The traction force produced by a wheel is a function of the wheel slipλ,of the normal force acting on a wheel F z,and of the adhesion coefficientμp between road and tire,which,in turn,depends on road conditions[7],[8].Since the adhesion coefficientμp is unknown and time varying during driving,it is necessary to estimate such a parameter on the basis of the data acquired by the sensors.Because of its direct influence on the vehicle traction force,the wheel slipλis regarded as the controlled variable in the traction force control system.The design of such a control system is based on the assumption that the vehicle velocity and the wheel angular velocities are both available online by direct measurements.As the wheel angular velocity can easily be measured with sensors,only the vehicle velocity is needed to calculate the wheel slipλ. The vehicle longitudinal velocity can be directly measured[9], [10],indirectly measured[11],and/or estimated through the use of observers[12],[13].Since the problem of measuring1524-9050/$26.00©2009IEEEthe longitudinal velocity is out of the scope of this paper,we assume that both the vehicle velocity and the wheel angular velocities are directly measured.The traction control problem is addressed in this paper.The main difficulty arising in the design of a traction force control system is due to the high nonlinearity of the system and the presence of disturbances and parameter uncertainties[6],[14].A robust control methodology needs to be adopted to solve the problem in question.In this paper,we rely on sliding-mode control[15],[16]because of its appreciable properties,which make it particularly suitable to deal with uncertain nonlinear time-varying systems.Different sliding-mode controllers have been proposed in the literature to solve the problem of controlling the wheel slip.For instance,sliding-mode control is used to steer the wheel slip to the optimal value to produce the maximum braking force,and a sliding-mode observer for the longitudinal traction force is proposed in[6].A sliding-mode-based observer for the vehicle speed is proposed in[13].In[5],a sliding-mode control law that uses an online estimation of the tire–road adhesion coefficient is presented.Other different sliding-mode approaches to the traction control problem have been proposed(see,for instance, [17]–[21]and the reference therein).However,the conventional sliding-mode control generates a discontinuous control action that has the drawback of producing high-frequency chattering, with the consequent excessive mechanical wear and passen-gers’discomfort,due to the propagation of vibrations through-out the different subsystems of the controlled vehicle.To reduce the vibrations induced by the controller,a possible solution consists of the approximation of the discontinuous control signals with continuous signals.This is,for instance,the solution adopted in[5]and[14].However,this kind of solution only generates pseudosliding modes[15],[22].This means that the controlled system state evolves in the boundary layer of the ideal sliding subspace and features a dynamical behavior different from that attainable if ideal sliding modes could be generated.Therefore,even,if from a practical viewpoint,this solution can produce acceptable results,the robustness features with respect to matched uncertainties[22]are lost.The idea investigated in this paper to circumvent the incon-venience of the vibrations induced by sliding-mode controllers is to exploit the positive features of second-order sliding-mode control[23].Second-order sliding-mode controllers feature higher accuracy with respect tofirst-order sliding-mode control and generate continuous control actions,since the discontinuity is confined to the derivative of the control signal while keeping the robustness feature typical of conventional sliding-mode controllers[16].Nevertheless,the generated sliding modes are ideal,in contrast to what happens for solutions that rely on con-tinuous approximations of the discontinuous control laws[16]. The particular traction control problem addressed in this paper is the so–called fastest acceleration/deceleration control (FADC)problem.It can be formulated as the problem of maximizing the magnitude of the traction force to produce the maximum acceleration while driving and the smallest stopping distance during braking,even on a possibly slippery road. This is attained by regulating the wheel slip ratio at the value corresponding to the maximum/minimum traction force.Since the reference slip ratios depend on the adhesion coefficientμp, which is unknown and time varying during driving,the con-troller design is coupled with the design of a suitable sliding-mode observer to estimate the tire–road adhesion coefficient. This makes the performance of the proposed control system insensitive to possible variations of the road conditions,since such variations are compensated online by the controller.This paper is organized as follows:Section II introduces the model of the vehicle dynamics,specifies the assumptions, and states the control objectives.The proposed second-order sliding-mode slip controller is presented in Section III.A sliding-mode observer for the tire–road adhesion coefficient is proposed in Section IV.In Section V,the FADC problem is described.Simulation results relevant to the proposed controller are reported in Section VI,whereas somefinal comments are gathered in the last section.II.V EHICLE L ONGITUDINAL D YNAMICSIn this paper,a nonlinear model of the vehicle is adopted[7]. The vehicle is modeled as a rigid body,and only longitudinal motion is considered.The difference between the left and right tires is ignored,making reference to a so-called bicycle model. The lateral,yawing,pitch,and roll dynamics,as well as actuator dynamics,are also neglected.The resulting equations of motion for the vehicle arem˙v x=F xf(λf,F zf)+F xr(λr,F zr)−F loss(v x)(1)J f˙ωf=T f−R f F xf(λf,F zf)(2)J r˙ωr=T r−R r F xr(λr,F zr)(3)F loss(v x)=F air(v x)+F roll=c x v2x·sign(v x)+f roll mg(4)F zf=l r mg−l h m˙v xl f+l r(5)F zr=l f mg+l h m˙v xl f+l r(6)where v x is the longitudinal velocity of the vehicle center of gravity,ωf andωr are the angular velocity of the front and rear wheels,T f and T r are the front and rear input torque,λf and λr are the slip ratio at the front and rear wheel,F xf and F xr are the front and rear longitudinal tire–road contact forces,F zf and F zr are the normal force on the front and rear wheels,F air is the air drag resistance,and F roll is the rolling resistance(see Fig.1).The vehicle parameters are the following:m is the vehicle mass,c x is the longitudinal wind drag coefficient,f roll is the rolling resistance coefficient,J f and J r are the front and rear wheel moments of inertia,R f and R r are the front and rear wheel radius,l f is the distance from the front axle to the center of gravity,l r is the distance from the rear axle to the center of gravity,and l h is the height of the center of gravity(see Fig.1). The normal force calculation method adopted in this paper[see (5)and(6)]is based on a static force model,as described in [8],ignoring the influence of suspension.This method gives a fairly accurate estimate of the normal force,particularly when the road surface is fairly paved and not bumpy.Fig.1.Vehicle model.The longitudinal slip λi ,i ∈{f,r }for a wheel is defined as the relative difference between a driven wheel angular velocity and the vehicle absolute velocity,i.e.,λi =ωi R i −v xωi R i,ωi R i >v x ,ωi =0,acceleration ωi R i −v xv x,ωi R i <v x ,v x =0,braking i ∈{f,r }.(7)The wheel slip dynamics during acceleration can be obtainedby differentiating (7)with respect to time,thus obtaining˙λi =f a i +h a iT i ,i ∈{f,r }(8)wheref a i =−˙v x R i ωi −v x F xiJ i ω2i,i ∈{f,r }(9)h a i =v xJ i R i ω2i,i ∈{f,r }.(10)The dynamics during braking can analogously be obtained bydifferentiating (7)for the brake situation and results in˙λi =f b i +h b iT i ,i ∈{f,r }(11)wheref bi =−R i ωi ˙v x v 2x −R 2iF xi J i v x,i ∈{f,r }(12)h b i =R ii x 2i,i ∈{f,r }.(13)The traction force F xi in the longitudinal direction generatedat each tire is a nonlinear function of the longitudinal slip λi ,of the normal force applied at the tire F zi ,and of the road adhesion coefficient μp [7].Different longitudinal tire–road friction models for vehicle motion control have been proposed in the literature (see [24]).In this paper,the so–called “Magic Formula”tire model developed by Bakker et al.[25]is con-sidered.This model is generally accepted as the most useful and viable model in describing the relationship between the slip ratio and the tire force.The model for the longitudinal force is as follows:F xi =f t (μp ,λi ,F zi ),i ∈{f,r }(14)III.S LIP C ONTROL D ESIGNAs previously mentioned,due to the high nonlinearity of the system and to the presence of time–varying parameters and uncertainties,typical of the automotive context,the control system is designed by relying on a robust control approach, i.e.,second-order sliding-mode control.The main advantage of second-order sliding-mode control[23]with respect to the first-order case[15]is that it features higher accuracy[16]and generates continuous control actions while keeping the same robustness properties with respect to matched uncertainties[22] and a comparable design complexity.As previously discussed,the controlled variable in the pro-posed traction force control system is the slip ratio at a wheel λi,i∈{f,r},because of its strong influence on the traction force.Indeed,it is possible to adjust the traction force produced by a tire F xi,i∈{f,r}to the desired value by controlling the wheel slip.Thus,the control objective of the control sys-tem is to make the actual slip ratioλi track the desired slip ratioλd,i.The sliding variables are chosen as the error between the current slip and the desired slip ratio,i.e.,s i=λei=λi−λd,i,i∈{f,r}.(17) As a consequence,the chosen sliding manifolds are given bys i=λei=λi−λd,i=0,i∈{f,r}(18) and the objective of the control is to design continuous control laws T i,i∈{f,r}that is capable of enforcing sliding modes on the sliding manifolds[see(18)]infinite time.Note that, once the sliding mode is enforced,the actual slip ratio correctlytracks the desired slip ratio since on the sliding manifoldλei =0,and the control objective is attained infinite time.Thefirst and second derivatives of the sliding variable s i in the acceleration case are given by˙s i=f a i+h a i T i−˙λd i,i∈{f,r}¨s i=ϕa i+h a i˙T i,i∈{f,r}(19) whereϕa i andγa i,i∈{f,r}are defined asϕa i=−¨v xR iωi+2˙v x˙ωiR iω2i−2v x˙ω2iR iω3i−¨λdi−v x˙FxiJ iω2i.(20)Note that the quantities h a i,i∈{f,r}are known.From(1)and(15),we get|˙v x|≤2Ψ−F loss(v x)m=f1(v x).(21)Taking into account thefirst time derivative of(1),(16),and (21),one has that|¨v x|≤2Γ−2|˙v x ||v x|m ≤2Γ−2f1(v x)|v x|m=f2(v x).(22)From(2),(3),and(15),it results in|˙ωi|≤Ψ−T iJ i =f3i(T i),i∈{f,r}.(23)Relying on(21)–(23),one has that the quantitiesϕa i,i∈{f,r}are bounded.From a physical viewpoint,this means that,whena constant torque T i,i∈{f,r}is applied,the second timederivative of the slip ratios is bounded.To apply a second-order sliding-mode controller,it is notnecessary for a precise evaluation ofϕa i to be available.In thesequel of this paper,it will only be assumed that suitable boundsΦa i(v x,ωi,T i)ofϕa i,i.e.,|ϕa i|≤Φa i(v x,ωi,T i),i∈{f,r}(24)are known.As for the braking case,the functionsϕb i andγb i can beobtained by following the same procedure previously describedfor the acceleration case.As forϕa i,ϕb i can be regarded asunknown bounded functions with known boundsΦb i(v x,ωi,T i),i.e.,|ϕb i|≤Φb i(v x,ωi,T i),i∈{f,r}.(25)To design a second-order sliding-mode control law,introducethe auxiliary variables y1,i=s i and y2,i=˙s i.Then,system(19)can be rewritten as˙y1,i=y2,i˙y2,i=ϕji+h ji˙Ti,i∈{f,r},j∈{a,b}(26)where˙T i can be regarded as the auxiliary control input[23].Theorem1:Given system(26),whereϕjisatisfies(24)and(25),and y2,i is not measurable,the auxiliary control law is˙Ti=−V i signs i−12s iM,i∈{f,r}(27)where the control gain V i is chosen such thatV i>2Φa i(v x,ωi,T i)/h a i,acceleration case2Φb i(v x,ωi,T i)/h b i,braking casei∈{f,r}(28)and s iM is a piecewise constant function representing the valueof the last singular point of s i(t)[i.e.,s iM is the value of themost recent maximum or minimum of s i(t)]that causes theconvergence of the system trajectory on the sliding manifolds i=˙s i=0infinite time.Proof:The control law[see(27)]is a suboptimal second-order sliding-mode control law.Therefore,by following a the-oretical development as that provided in[26]for the generalcase,it can be proved that the trajectories on the s i O˙s i plane areconfined within limit parabolic arcs,including the origin.Theabsolute values of the coordinates of the trajectory intersectionswith the s i-and˙s i-axes decrease in time.As shown in[26],under condition(28),the following relationships hold:|s i|≤|s iM|,|˙s i|≤|s iM|and the convergence of s iM(t)to zero takes place infinitetime[26].As a consequence,the origin of the plane,i.e.,s i=˙s i=0,is reached infinite since s i and˙s i are both boundedby max(|s iM|,|s iM|).This,in turn,implies that the sliperrorsλei ,i∈{f,r}are steered to zero as required to attainthe objective of the traction control problem.IV.T IRE–R OAD A DHESION C OEFFICIENT E STIMATE To identify theλ−F x curve corresponding to the actual road condition,the tire–road adhesion coefficientμp needs to be estimated.Different estimation techniques for this parameter have been proposed in the literature,and most of them are based on the Bakker–Pacejka Magic Formula model.For instance,in [27],a procedure for the real-time estimation ofμp is presented, whereas in[20],a scheme to identify different classes of roads with a Kalmanfilter and a least-square algorithm is presented. In[5]and[28],a recursive least-square algorithm[29]is adopted to estimate the tire–road adhesion coefficient.A dif-ferent approach is proposed in[30],where an extended Kalman filter is used to estimate the forces produced by the tires.A sliding-mode observer to estimate the longitudinal stiffness for a simplified linear tire–road interaction model was proposed in[6]and[31],while a dynamical tire–road interaction model with a nonlinear observer to estimate the adhesion coefficient has been proposed in[32].In this section,afirst-order sliding-mode observer for the online estimation of the adhesion coefficientμp is designed. The sliding-mode methodology has also been adopted to design the observer since it is applicable to nonlinear systems and has good robustness properties against disturbances,modeling inaccuracy,and parameter uncertainties[15].Following the approach proposed in[5],a simplified tire model is considered instead of(14),i.e.,F xi=μp f t(λi,F zi),i∈{f,r}.(29)To design the sliding-mode observer forμp relying on the so-called equivalent control method[22],introduce the sliding variablesμ=v x−ˆv x(30) whereˆv x is an estimate of the longitudinal velocity v x.The dynamics ofˆv x is chosen as˙ˆv x =1m(Ω−F loss(v x))(31)whereΩ=K sign(sμ)(32) is the control signal of the sliding-mode observer.In the sequel,for notation simplicity,the dependence of the tire force F x on the slip ratioλand the normal force F z has been omitted.By differentiating(30)and substituting(1),one has that˙sμ=˙v x−˙ˆv x=1m(F xf+F xr−K sign(sμ)).(33)From(14),the following relationship holds:F xf+F xr≤F zf+F zr=mg.(34)Relying on(33)and(34),if the gain K in(32)is chosen such thatK>mg≥F xf+F xr(35) then the so-called reaching condition[22]sμ˙sμ≤−η|sμ|,η∈I R+(36) is satisfied,and a sliding mode on the sliding manifold sμ=0 is attained infinite time.The tire–road adhesion coefficientμp can be estimated by taking into account the so-called equivalent controlΩeq,which is defined as the continuous control signal that maintains the system on the sliding surface sμ=0[15].The equivalent control can be calculated by setting the time derivative of the sliding variable˙sμequal to zero,i.e.,˙sμ=1m(F xf+F xr−Ωeq)=0(37) thus the equivalent controlΩeq is given byΩeq=F xf+F xr.(38) If we assume that the front and rear wheels are on the same road surface,which is true for many driving situations,then(38)can be rewritten asΩeq=F xf+F xr=μpf tf(λf,F zf)+f tr(λr,F zr).(39) The equivalent controlΩeq is close to the slow component of the real control and can be obtained byfiltering out the high-frequency component ofΩusing a low-passfilter[15],[22], that isτ˙ˆΩ+ˆΩ=Ω(40)Ωeq≈ˆΩ(41) whereτis thefilter time constant.Thefilter time constant should be chosen sufficiently small to preserve the slow compo-nents of the controlΩundistorted but large enough to eliminate the high-frequency component.From(39)and(41),the estimated tire–road adhesion coeffi-cientˆμp can be calculated asˆμp=ˆΩf tf(λf,F zf)+f tr(λr,F zr).(42) Note that,from(38)and(41),one has thatˆΩ=Fxf+F xr.(43) Thus,ˆΩcan also be regarded as a sliding-mode observer to estimate the total longitudinal force exerted by the vehicle.V.F ASTEST A CCELERATION/D ECELERATIONC ONTROL P ROBLEMThe particular traction-control problem taken into account in this paper is the so-called FADC problem.It can be formulated as the problem of maximizing the magnitude of the tractionforce to produce the maximum acceleration while driving and the smallest stopping distance during braking,even on a possi-bly slippery road.Looking at theλ−F x curve in Fig.2,the maximum ac-celeration can be attained by steering the slipλto the value corresponding to the positive peak of the curve,namely,λMax, i.e.,considering the i th axleλd,i=λMaxi.(44) Beyond this value,the wheels begin to spin,the longitudinal force produced decreases,and the vehicle cannot accelerate as desired.By maximizing the traction force between the tire and the road,the traction controller prevents the wheels from slipping and,at the same time,improves the vehicle’s stability and steerability.Similarly,the target slip to obtain the maximum braking force,i.e.,the minimum braking distance,is determined as the slip value corresponding to the minimum of theλ−F x curve,namely,λMin.Thus,the maximum braking force can be attained by the steering the tire slipλtoλMin,i.e.,considering the i th axleλd,i=λMini.(45)The position ofλMaxi varies,depending on the actualλi−F xicurve considered,and its value is generally unknown duringdriving.The same holds forλMini .As a consequence,the con-trol task has to include the online searching of the peak slip.In the proposed approach,this task is accomplished in two steps.1)The tire–road adhesion coefficientμp is estimated asdescribed in Section IV,and the currentλi−F xi curve is identified.2)For the acceleration case,the desired slip,i.e.,the slipratio corresponding to the maximum of the curve,is calculated by maximizing the functionˆF xi=f ti(ˆμp,λi,F zi)asλd,i=arg minλi −ˆF xi=arg minλi−f ti(ˆμp,λi,F zi).(46)As for the braking case,the desired slip ratio correspond-ing toλMini is calculated by minimizing the functionˆF xi,that isλd,i=arg minλif ti(ˆμp,λi,F zi).(47)Note that the minimum(maximum)of the functionˆF xi can be calculated,for instance,with a minimization algorithm without derivatives[34].Note that different strategies have been proposed in the litera-ture tofind the slip ratio corresponding to the maximum of the λ−F x curve(see,for instance,[3],[5],[6],and[35]).VI.S IMULATION R ESULTSThe traction control presented in this paper has been tested in simulation,considering a scenario with different road con-ditions.The vehicle is travelling at an initial velocity v x(0)= 20m/s,with initial slip ratiosλf(0)=λr(0)=0.02,and theTABLE IS IMULATION PARAMETERSh j i−ηi sign(s i)−f j i+˙λdii∈{f,r},j∈{a,b}(48)whereηi>0.As can be seen,in contrast with the proposed second-order sliding-mode controller,conventional sliding-mode con-trol laws produce discontinuous control inputs that generate high-frequency chattering,with the consequent excessive me-chanical wear and passengers’discomfort.To exploit the robustness feature of the proposed control scheme,the controlled system is tested in simulation in the presence of model uncertainties and disturbances and is com-pared with afirst-order sliding-mode solution,where the sgn(·) function is approximated with the sat(·)function,as in[5]. The nominal model parameters are as in Table I,whereas the real values for the mass,the wheel moment of inertia, and the wheel radius are m=1702kg,J f=J r=1.8kg m2, and R f=R r=0.5m,respectively.Moreover,to model some matched disturbances,the real control input is calculated as T i(t)=¯T i(t)+A sin(t),i∈{f,r}(49) where¯T i is the nominal control input given by(27),and A is the amplitude of the disturbances acting on the control input. Figs.11and12show the simulation results obtained with the proposed second-order sliding-mode control scheme with A= 300in(49).As expected,the proposed control scheme is robust against parameter uncertainties and matched disturbances.One can note that the tire–road adhesion coefficient iscorrectlyTABLE IIP ERFORMANCE I NDEXES[32]C.Canudas-De-Wit and R.Horowitz,“Observers for tire/road contactfriction using only wheel angular velocity information,”in Proc.38th Conf.Decision Control,Phoenix,AZ,1999,pp.3932–3937.[33]R.Marino and P.Tomei,“Global adaptive observer for nonlinear systemsviafiltered transformations,”IEEE Trans.Autom.Control,vol.37,no.8, pp.1239–1245,Aug.1992.[34]R.P.Brent,Algorithms for Minimization Without Derivatives.Englewood Cliffs,NJ:Prentice-Hall,1973.[35]D.Hong,P.Yoon,H.Kang,I.Hwang,and K.Huh,“Wheel slip controlsystems utilizing the estimated tire force,”in Proc.Amer.Control Conf.,Minneapolis,MN,2006,pp.5873–5878.Matteo Amodeo was born in Vizzolo Predabissi, Italy.He received the Master’s degree in computer engineering from the University of Pavia,Pavia, Italy,in2006.Since January2007,he has been with Siemens S.p.a.,Sector BT FSP-DMS,Milano,Italy.His re-search activities are mainly in the area of sliding-mode control applied to automotivecontrol.Antonella Ferrara(S’86–M’88–SM’03)was bornin Genova,Italy.She received the Laurea degreein electronic engineering and the Ph.D.degree incomputer science and electronics from the Universityof Genova in1987and1992,respectively.In1992,she was an Assistant Professor withthe Department of Communication,Computer andSystem Sciences,University of Genova.In1998,she was an Associate Professor of automatic controlwith the Universitàdegli studi di Pavia,Pavia,Italy.Since January2005,she has been a Full Professor of automatic control with the Department of Computer Engineering and Systems Science,Universitàdegli studi di Pavia.She has authored or coauthored more than230papers,including more than70international journal papers. Her research activities are mainly in the area of sliding-mode control with application to automotive control,process control,and robotics.Dr.Ferrara is a Senior Member of the IEEE Control Systems Society and a member of the IEEE Technical Committee on Variable Structure and Sliding-Mode Control,the IEEE Robotics and Automation’s Technical Committee on Autonomous Ground Vehicles and Intelligent Transportation Systems,and the IFAC Technical Committee on Transportation Systems.From2000to2004, she was an Associate Editor of the IEEE T RANSACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY.At present,she is an Associate Editor of the IEEE T RANSACTIONS ON A UTOMATIC C ONTROL.She has been a member of the International Program Committee of numerous international conferences and events.As a student at the Faculty of Engineering,University of Genova,she received the“IEEE North Italy Section Electrical Engineering Student Award”in1986.Riccardo Terzaghi was born in Vizzolo Predabissi, Italy.He received the Master’s degree in computer engineering from the University of Pavia,Pavia, Italy,in2006.Since January2007,he has been with Siemens S.p.a.,Sector BT FSP-DMS,Milano,Italy.His re-search activities are mainly in the area of higher order sliding-mode control and robust control with application to automotivesystems.Claudio Vecchio received the Master’s degree in computer engineering and the Ph.D.degree from the Universitàdegli studi di Pavia,Pavia,Italy,in2005 and2008,respectively.Since November2008,he has been with Temis s.r.l.,Corbetta,Italy.He is also with the Dipartimento di Informatica e Sistemistica,Universitàdegli studi di Pavia.His research interests are mainly in the area of higher order sliding-mode control and robust and nonlinear control,with application to automotive control.。
永磁同步电机电流环PI_控制器参数整定及优化
1. 5R s ,此时电机的电角速度 ω e = 0,通过对式(2) 中
的 d 轴电压方程进行化简得到电阻值,为
第 10 期
刘 林等:永磁同步电机电流环 PI 控制器参数整定及优化
Rs =
2 ud
。
3 id
133
(3)
图 3 电流环控制框图
Fig. 3 Block diagram of current loop control
}
(2)
对永磁同步电机的内部结构进行分析,当给逆
类为基于模型的参数自整定 [5 - 8] ,这类参数自整定
变器施加固定占空比,电机得到固定电压,受电机内
方法为目前主流研究方法,基于模型的参数自整定
部的电感作用,电机的相电流将作用一段时间后快
对建立的数学模型和参数的精度依赖程度比较大;
速趋于稳定状态。 在空载情况下对电机参数进行辨
parameters with the best performance were optimized in the neighborhood of the initial value of PI parame-
ters by the two degree of freedom variable rotation method. The experimental results show that this method
法。 首先采用简单实用的离线参数辨识方法辨识出
电机的电阻和电感参数,然后将得到的电机参数代
入数学模型中整定出一组控制参数,最后以这一组
参数作为基于规则的参数自整定的初值,整定出最
优的控制参数,确保电机可以稳定启动运行。
1 永磁同步电机离线参数辨识
二阶滑模
二阶滑模控制（读书笔记）详细推导一、改进时间最优二阶滑模控制算法1、非线性系统()[100]x Ax B x u Df y x=++= 0()0()B x b x ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦010D ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦u 为系统的控制量输入电压，y 为台车输出转角，f 为转向负载和外界扰动之和，()b x 为系统的非线性控制增益。
2、选取滑模切换函数33222111()()()T d d d d s C x x x x c x x c x x =-=-+-+-采用极点配置或二次型最优法确定矢量C,保证系统进入滑动模态后具有满意的动态特性。
为构造s 的二阶趋近律,令12,y s y s ==，状态方程为122,y s y y s v ====当满足时间最优的目标时，可导出控制量v2222112211sgn ,022sgn(),02m m m m my y y y a y y a a v y y a y y a ⎧⎛⎫-++≠⎪ ⎪⎪⎝⎭=⎨⎪+=⎪⎩ 其轨迹由两段抛物线组成，v 的符号只切换一次，开关线为22102m y y y a +=，m a 为趋近滑模的最大加速度。
3、则 s 的一阶导数 ()[()]()T T d d T T T ds C x x C Ax B x u Df x C Ax b x u C Df C x =-=++-=++-其中12(,,1)T C c c =，0()0()B x b x ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦s 的二阶导数12()()[()]()()()()()(,,)()()(,,,)()T T dT T dT T T T dd d s C Ax b x u b x u C x C A Ax B x u Df b x u b x u C x C AAx C AB x u C ADf b x u b x u C x x x f x u b x u x x f u b x uψψψ=++-=++++-=++++-==++=+则控制量 11ˆˆˆˆˆˆ(())[(,,,)](())[(,,,)]d d u bx s x x f u b x v x x f u ψψ--=-=- 解得0()(0)()tu t u u d ττ=+⎰其中ˆψ是ψ相对应的标称值 把()u t 代入s 1ˆˆˆ(,,,)()(())[(,,,)]()()ˆˆ(,,,)(,,,)ˆˆ()()(,,,)()d d d d d s x x f u b x bx v x x f u b x b x x x f u x x f u v b x b x x x f u x vψψψψφξ-=+-=-+=∆+显然，式中(,,,)()d x x f u x φξ∆和是由外干扰和参数摄动引起的，理 想 情 况 下扰 动为零，可验证(,,,)=0d x x f u φ∆并且根据假设可以推出12(,,,)()d x x f u H r x r φξ∆≤≤≤,其中H 为正实数。
基于滑模观测器的永磁同步电机无传感器控制系统的优化研究
基于滑模观测器的永磁同步电机无传感器控制系统的优化研究李江波;刘述喜;童磊【摘要】为了优化传统基于滑模观测器(SMO)的永磁同步电机(PMSM)无传感器控制系统的性能,分析了传统控制系统中相应模块的数学模型和控制算法存在的问题,建立了优化的基于SMO的PMSM无传感器控制系统模型并进行了仿真.通过分析传统SMO的数学模型,在保证SMO满足Lyapunov稳定条件的前提下,根据电机不同工况对滑模增益进行实时优化,提高了SMO的观测性能.为了提高锁相环(PLL)输出信号的质量,在PLL输入端加上低通滤波器,并对低通滤波环节造成的位置角度偏差进行补偿.通过仿真验证上述所采用优化策略的正确性和可行性,结果表明:优化模型在保证系统动态性能的同时能够获得比传统观察系统质量更高的观测信号.【期刊名称】《电机与控制应用》【年(卷),期】2018(045)011【总页数】6页(P50-55)【关键词】永磁同步电机;滑模观测器;锁相环;角度补偿【作者】李江波;刘述喜;童磊【作者单位】重庆理工大学电气与电子工程学院,重庆400054;重庆理工大学电气与电子工程学院,重庆400054;重庆理工大学电气与电子工程学院,重庆400054【正文语种】中文【中图分类】TM3510 引言永磁同步电机(Permanent Magnet Synchronous Motor,PMSM)具有功率密度大、可靠性高、能量转换效率高等优点[1]，发展前景广阔。
在电机控制过程中，必须利用准确的转子位置与速度信号进行相应的坐标变换[2]。
传统的电机控制系统中大多采用磁编码器、光电编码器、旋转变压器等物理传感器对电机转子速度与位置进行检测。
物理传感器存在着成本高、可靠性低、结构复杂等缺点，阻碍了PMSM的推广。
电机无速度传感器技术由于其自身的优越性，成为目前交流调速的一个重要发展方向。
无传感器控制策略主要包括：高频注入法[3-4]、模型参考自适应观测器法[3，5-6]、卡尔曼滤波器法[3，7]、滑模观测器(Sliding Mode Observer,SMO)法[3，8-13]以及假定坐标法[14]等。
基于自适应快速终端滑模的车轮滑移率跟踪控制
DOI: 10.3785/j.issn.1008-973X.2021.01.020基于自适应快速终端滑模的车轮滑移率跟踪控制李静1，王晨1，张家旭1,2(1. 吉林大学 汽车仿真与控制国家重点实验室，吉林 长春 130011；2. 中国第一汽车集团有限公司智能网联研发院，吉林 长春 130011)摘 要：针对汽车对连续、快速和稳定的车轮滑移率跟踪控制的需求，提出基于自适应快速终端滑模的车轮滑移率跟踪控制策略. 基于Burckhardt 轮胎模型建立车轮滑移率跟踪控制模型，将模型简化过程中的不确定性考虑成复合干扰项，将轮胎侧向力对纵向力的影响考虑成未知参数. 利用双曲正切函数和终端吸引因子设计改进的跟踪微分器，平滑车轮滑移率跟踪误差和估计车轮滑移率跟踪误差的一阶导数. 以车轮滑移率跟踪控制模型和改进的跟踪微分器输出为基础，基于自适应快速终端滑模控制理论，设计对复合干扰项具有强鲁棒性的车轮滑移率跟踪控制律；基于投影算子理论设计自适应律来实时补偿未知参数，利用LaSalle 不变性原理证明了闭环系统的渐近稳定性. 利用车辆动力学软件仿真验证提出的控制律的可行性和有效性. 结果表明，提出的车轮滑移率跟踪控制策略具有精度高和鲁棒性强的优点.关键词： 汽车动力学；车轮滑移率跟踪控制；自适应控制；快速终端滑模控制中图分类号： U 461 文献标志码： A 文章编号： 1008−973X （2021）01−0169−08Wheel slip tracking control of vehicle based on adaptive fastterminal sliding mode control methodLI Jing 1, WANG Chen 1, ZHANG Jia-xu 1,2(1. State Key Laboratory of Automotive Simulation and Control , Jilin University , Changchun 130011, China ;2. Intelligent Network R&D Institute , China FAW Group Limited Company , Changchun 130011, China )Abstract: A novel wheel slip tracking control strategy was proposed based on adaptive fast terminal sliding modelcontrol method aiming at the requirement of the vehicle for the continuous, fast and stable wheel slip tracking control. The wheel slip tracking control model was established based on Burckhardt tire model. The uncertainty during the process of model simplification and the influence of the tire lateral force on the tire longitudinal force in the wheel slip tracking control model were considered as the lumped uncertainty and the unknown parameter,respectively. The modified tracking differentiator was deduced based on hyperbolic tangent function and terminal attraction factor in order to smooth the wheel slip tracking error and estimate the derivative of the wheel slip tracking error. The wheel slip rate tracking control law with strong robustness for the system uncertainty was proposed based on the adaptive fast terminal sliding mode control theory according to the wheel slip tracking control model and the outputs of the modified tracking differentiator. The adaptive law was proposed based on projection operator theory to compensate the unknown parameter. The asymptotic stability of the closed-loop system was proved using the LaSalle invariance principle. The feasibility and effectiveness of the proposed wheel slip tracking control strategy was verified based on vehicle dynamics simulation software. Results show that the proposed wheel slip tracking control strategy has high accuracy and strong robustness.Key words: vehicle dynamics; wheel slip tracking control; adaptive control; fast terminal sliding mode control收稿日期：2019−12−24. 网址：/eng/article/2021/1008-973X/202101020.shtml 基金项目：电动汽车智能辅助驾驶关键技术研究与产品开发项目（2016YFB0101000）.作者简介：李静（1976—），男，教授，博导，从事汽车地面系统分析与控制的研究. /0000-0002-6126-1632. E-mail ：****************通信联系人：张家旭，男，高级工程师. /0000-0001-6159-1965. E-mail ：****************第 55 卷第 1 期 2021 年 1 月浙 江 大 学 学 报(工学版)Journal of Zhejiang University (Engineering Science)Vol.55 No.1Jan. 2021目前，安全、智能和环保汽车已经成为现阶段国内外汽车厂商的主要发展目标之一，连续、快速和稳定的车轮滑移率跟踪控制是汽车实现智能安全避障、制动能量回收等先进控制技术的重要基础，因此深入研究连续、快速和稳定的车轮滑移率跟踪控制方法具有重要的工程实用价值.车轮滑移率跟踪控制早期多应用于汽车的防抱死制动系统（anti-lock braking system，ABS）中，通过控制车轮滑移率使轮胎-地面附着系数最大，进而提高汽车制动稳定性和缩短汽车制动距离.由于车轮滑移率动态在汽车制动过程中呈现出复杂的非线性特点，早期多采用逻辑切换规则的控制方式实现车轮滑移率的强鲁棒跟踪控制[1-5]. Kuo等[1]利用车轮滑移率、角加速度和车轮制动力矩信息，设计ABS工作模式的逻辑切换控制规则，将轮胎-地面附着系数控制到最大值. Tanelli 等[2]将车轮制动力矩和车轮滑移率等信息作为切换控制信号，实现ABS工作模式的切换控制. 马付雷[4]基于车轮角加速度辅助车轮滑移率等信息设计逻辑切换规则，实现ABS工作模式的平滑切换控制. 上述基于逻辑切换规则的控制方式难以对任意目标滑移率进行高精度的稳定跟踪控制，无法满足安全、智能和环保汽车对车轮滑移率跟踪控制的需求.基于动态模型的控制方式更容易实现连续目标滑移率的跟踪控制，易满足安全、智能和环保汽车对车轮滑移率跟踪控制的需求. Pasillas-Lepine 等[6]采用Backstepping方法，设计车轮滑移率跟踪控制律. Shim等[7]基于滑模控制理论，设计车轮滑移率跟踪控制律. Yu等[8]采用最小化目标滑移率跟踪误差准则，设计车轮滑移率跟踪控制律. Mirzaei等[9]采用最优预测控制方法，设计车轮滑移率跟踪控制律. Amodeo等[10]采用二阶滑模控制方法，设计车轮滑移律跟踪控制律. Tanelli等[11]基于李雅普诺夫稳定性理论，推导出可智能区分车轮滑移率稳定和非稳定区域的车轮滑移率跟踪控制律. Johansen等[12]采用线性二次最优控制方法，设计具有实时调度特性的车轮滑移率跟踪控制律. Zhang等[13]设计干扰观测器，在线补偿车轮滑移率闭环系统的不确定性. Zhang等[14]基于耗散性理论，设计车轮滑移率跟踪控制律. He等[15]利用障碍李雅普诺夫函数，设计车轮滑移率跟踪控制律. He等[16]利用时变非对称障碍函数，设计车轮滑移率跟踪控制律. 上述基于动态模型的控制方式通常需要依靠简化的控制模型才能得到有效的控制方案，简化后的控制模型无法精确地表征车轮滑移率状态在汽车制动过程中呈现出的复杂非线性. 在基于动态模型的控制方式中，采用简化的控制模型设计控制方案之前需要解决简化的控制模型无法精确表征系统状态的问题.本文基于Burckhardt轮胎模型建立车轮滑移率跟踪控制模型，将模型简化过程中的不确定性考虑成系统的复合干扰项，同时将轮胎侧向力对纵向力的影响考虑成未知参数，从而解耦控制模型中的轮胎纵向力和侧向力. 利用双曲正切函数和终端吸引因子设计改进的跟踪微分器，平滑车轮滑移率跟踪误差和估计车轮滑移率跟踪误差的一阶导数，分别基于自适应快速终端滑模控制理论和投影算子理论设计车轮滑移率跟踪控制律和未知参数的自适应律. 仿真结果表明，所设计的车轮滑移率跟踪控制律具有控制精度高、鲁棒性强的优点，可以有效地抑制滑模控制固有的“抖振”现象.1 车轮滑移率跟踪控制模型如图1所示的车轮滑移率动态模型，基于Burckhardt轮胎模型，建立关于车轮角速度和汽车纵向速度的动力学方程：JωrF wl F wsT bT fm v xα式中：为车轮转动惯量，为车轮角速度，为车轮有效滚动半径，、分别为Burckhardt轮胎模型描述的轮胎纵向力和侧向力，为车轮制动力矩，为轮胎滚动阻力矩和机械摩擦阻力矩之和，为整车质量的1/4，为汽车纵向速度，为轮胎侧偏角.为了简化式（1）描述的关于车轮角速度和汽车纵向速度的动力学方程，将包含轮胎侧向力的因子与轮胎滚动阻力矩和机械摩擦阻力矩之和考虑成式（1）的扰动项，则式（1）修正为图 1 车轮滑移率动态模型Fig.1 Wheel slip dynamic model170浙江大学学报（工学版）第 55 卷d 1=−rF ws sin α−T f d 2=F ws sin α式中：，.Burckhardt 轮胎模型描述的轮胎纵向力可以表示为F z λl λres µres 式中：为车轮垂向载荷；为车轮纵向滑移率；为车轮合成滑移率；为轮胎-地面附着系数，可以表示为[17]ϑ1ϑ2ϑ3其中、和反映了轮胎的合成滑移率对轮胎-地面附着系数的影响.若用未知参数描述车轮侧向滑移率对轮胎纵向力的影响，则式（3）修正为cos α≈1假设，将式（5）代入式（2），可得Burckhardt 轮胎模型在制动工况下定义的车轮滑移率可以简化为对式（7）求二阶微分运算，可得对式（6）求一阶微分运算，可得x 1=λl −λd x 2=˙x1u =˙T b 由于车身惯量远大于车轮惯量，车轮滑移率的变化快于车身运动状态的变化. 将式（6）、（9）代入式（8），令，，，可得车轮滑移率跟踪控制模型为λd u 式中：为车轮目标滑移率；为控制输入；f (x )=v −1x {2F z µres (x 1+λd )(x 2+˙λd )/m −((1−x 1−λd )/m +r 2/J )F z ˙µres (x 1+λd )},x =[x 1,x 2]TG =r /(Jv x )d s =v x −1(d 2˙λl /m −˙d 2(1−λl )/m −r ˙d1/J )−¨λd |d s |⩽B d 其中为系统的状态向量；；d s 为系统的复合干扰项，，假设满足. 2 车轮滑移率跟踪控制器考虑式（10）描述的车轮滑移率跟踪控制模型，目标是设计改进的跟踪微分器、车轮滑移率跟踪控制律和未知参数自适应律，使得闭环系统可以高精度地跟踪车轮目标滑移率，对系统的复合干扰项具有较强的鲁棒性. 系统的总体架构如图2所示，改进的跟踪微分器用于平滑车轮滑移率跟踪误差，估计车轮滑移率跟踪误差的一阶导数. 车轮滑移率跟踪控制律基于改进的跟踪微分器的输出，计算车轮滑移率动态系统高精度的跟踪车轮目标滑移率所需要的车轮制动力矩变化率；利用未知参数自适应律实时估计和补偿车轮滑移率跟踪控制律中的未知参数，以提高车轮滑移率跟踪控制律对未建模动态的自适应能力. 车轮滑移率跟踪控制律的输出通过积分运算后作用于车轮滑移率动态系统，可以抑制系统的“抖振”现象.图 2 系统总体架构Fig.2 System overall architecture2.1 改进的跟踪微分器利用跟踪微分器可以准确地计算信号的一阶导数，为全状态反馈控制律设计奠定基础. 申帅等[18]利用跟踪微分器计算航空光电稳定平台的加速度信号，实现航空光电稳定平台的高增益加速度反馈控制. 谭诗利等[19]利用跟踪微分器计算高超声速飞行器反演控制律中的虚拟控制量，有效解决了高超声速飞行器反演控制律的微分膨胀问题. 为了提高跟踪微分器的收敛速度，利用双曲正切函数和终端吸引因子设计改进的跟踪微分器来平滑车轮滑移率跟踪误差，估计车轮滑移率跟踪误差的一阶导数，以避免利用传统的一阶惯性环节求解信号微分过程中产生的噪声放大效应. 基于双曲正切函数和终端吸引子，设计定理1描述的具有渐近稳定平衡原点的动态系统. 基于该动态系统，构建改进的跟踪微分器.(0,0)定理1 假设原点为如下系统的平衡点：第 1 期李静, 等：基于自适应快速终端滑模的车轮滑移率跟踪控制 [J]. 浙江大学学报：工学版,2021, 55(1): 169–176.171tanh(·)a 0>0a 1>0a 2>0b 0>00<p <1q 式中：为双曲正切函数. 若系统的匹配参数满足、、、、和为正奇数，则系统的平衡点是渐近稳定的.证明：定义Lyapunov 候选函数为z 1 0z 1tanh (b 0z 1)>0由双曲正切函数为奇函数，且时有可知，式（12）定义的Lyapunov 候选函数满足对式（12）求导，可得将式（11）代入式（13），可得定义集合z 2=0z 1=0N {z 1=0,z 2=0}由式（11）可知，当时，必有，因此集合的最大不变集为. 由LaSalle 不变性原理[20]可知，式（11）描述的系统平衡原点是渐近稳定的.根据文献[21]可知，基于式（11）构建改进的跟踪微分器为W >0ˆx 1ˆx 2x 1x 2W →∞ˆx 1→x1ˆx 2→x 2式中：W 为改进跟踪微分器的匹配参数，；和为车轮滑移率跟踪控制模型状态变量和的估计，且时，有和.2.2 车轮滑移率跟踪控制律基于改进跟踪微分器的输出，将式（10）描述的车轮滑移率跟踪控制模型转化为以式（18）为基础，设计车轮滑移率跟踪控制律，使得闭环系统可以高精度地跟踪车轮目标滑移率，且对系统的复合干扰项具有较强的鲁棒性.定义车轮滑移率跟踪误差滑模面为κ0κ0>0式中：为设计参数，.基于式（19），定义如下快速终端滑模面：κ1、κ2、、γκ1>0κ2>01.0<β<2.0γ>β式中：β为设计参数，，，，.由文献[22]可知，式（20）描述的快速终端滑模面可以在有限时间收敛到平衡点，收敛时间可以表示为F (·)式中：为高斯超几何函数，定义参见文献[23].定理2 考虑式（18）描述的车轮滑移率跟踪η1>0η2>0ˆθθ式中：η1、η2为设计参数，，；为未知参数的估计值，采用如下自适应律更新：θmin θmax κ3>0Proj θ(·)θ式中：和分别为未知参数的最小值和最大值；为待设计的自适应增益系数；为投影算子，可以保证未知参数的估计值满足如η1⩾B d 若设计参数，则车轮滑移率跟踪闭环系统的平衡原点是渐近稳定的.证明：定义Lyapunov 候选函数为˜θθ˜θ=θ−ˆθ式中：为未知参数的估计误差，定义为.对式（25）求导，可得将式（23）代入式（27），可得172浙 江 大 学 学 报（工学版）第 55 卷由投影算子的性质[24]，可知η1⩾B d 若设计参数，则有将式（29）、（30）代入式（28），可得由式（30）可以定义如下不变集：˙s=0当时，根据式（19），可知˙s=0因此，当时，车轮滑移率跟踪闭环系统的状态轨迹将渐近收敛到平衡原点.σ=0˙s =0当时，根据式（20）可知，车轮滑移率跟踪闭环系统的状态轨迹将在有限时间内收敛到. 由式（33）可知，车轮滑移率跟踪闭环系统的状态轨迹将进一步渐近收敛到平衡原点.综上所述，由LaSalle 不变性原理[20]可知，闭环系统是渐近稳定的.˙s=0注1：在系统的复合干扰存在条件下，车轮滑移率跟踪闭环系统的状态轨迹可以渐近收敛到平衡原点，特别是在快速终端滑模面上的状态轨迹可以在有限时间内收敛到，说明由式（22）、（23）构成的车轮滑移率跟踪闭环系统具有较强的鲁棒性和良好的控制精度.注2：由式（22）可知，利用高阶滑模控制思想，将引起的“抖振”现象的不连续控制输入项包含于控制指令的一阶导数中；通过积分运算求解控制指令，从而有效抑制不连续控制输入项对系统性能的影响.3 仿真结果及分析a 0=10a 1=10a 2=10p =0.9q =3W =200κ0=3κ1=260κ2=150κ3=35γ=3β=1.5η1=2η2=12m =1416kg J =0.9kg ·m 2r=0.31m 基于车辆动力学仿真软件CarSim ，对设计的车轮滑移率跟踪控制律进行仿真验证，选择车辆前左轮和后右轮的仿真结果进行分析. 设计的改进跟踪微分器参数设置如下：，，，，，. 设计的车轮滑移率跟踪控制律参数设置如下：，，，，，，，. 仿真车辆参数设置如下：，，，车辆质心点到前轴和后轴的距离分别为1.016和1.562 m.3.1 斜坡信号工况在斜坡信号仿真工况中，车速和路面附着系数分别设置为33.34 m/s 和1，斜坡信号幅值和变化速率分别设置为0.1和0.03. 为了避免车速为零时控制律产生奇异性，仿真截止车速设置为5 m/s.仿真结果如图3、4所示. 图中，e fl 、e rr 分别为左前图 3 斜坡信号工况的跟踪误差及误差变化率Fig.3 Tracking error and its rate of ramp signal maneuver第 1 期李静, 等：基于自适应快速终端滑模的车轮滑移率跟踪控制 [J]. 浙江大学学报：工学版,2021, 55(1): 169–176.173前车轮和右后车轮滑移率跟踪误差变化率.从图3可知，设计的改进跟踪微分器可以较好地平滑车轮滑移率跟踪误差，且可以准确地估计车轮滑移率跟踪误差变化率. 从图4（a ）~（c ）、4（e ）可知，车辆左前轮和右后轮轮速无明显波动，左前轮和右后轮的滑移率在系统存在复合干扰时可以快速、稳定地跟踪斜坡信号，且轮胎-地面附着系数可以跟随车轮滑移率平稳过渡到稳态阶段. 从图4（d ）可知，在设计过程中利用积分运算求解车轮制动力矩，有效地抑制了滑模控制的“抖振”现象. 综上所述，所设计的车轮滑移率跟踪控制律可以在车辆垂向载荷动态变化过程中保证闭环系统具有良好的鲁棒性和控制精度.3.2 正弦信号工况在正弦信号仿真工况中，路面峰值附着系数设置为1，初始车速设置为33.34 m/s ，正弦信号幅值、偏移量和频率分别设置为0.045、0.055和9.42 rad/s.为了避免车速为零时控制律产生奇异性，仿真截止车速设置为5 m/s. 仿真结果如图5、6所示.从图5可知，所设计的改进跟踪微分器可以较好地平滑车轮滑移率跟踪误差，且可以无噪声放大效应地估计车轮滑移率跟踪误差变化率. 从图6（a ）~（c ）、（e ）可知，车辆左前轮和右后轮轮速跟随正弦信号的周期性波动，左前轮和右后轮的滑移率在系统存在复合干扰时可以快速、稳定地跟踪正弦信号，无明显的相位滞后，且轮胎-地面附着系数可以跟随车轮滑移率周期性波动. 从图6（d ）可知，利用积分运算求解车轮制动力矩可以有效地抑制滑模控制的“抖振”现象. 综上所述，所设计的车轮滑移率跟踪控制律可以在车辆垂向载荷动态变化过程中保证闭环系统具有良好的动态过程品质.4 结 论（1）基于Burckhardt 轮胎模型建立车轮滑移率跟踪控制模型，将模型简化过程中的不确定性考虑成系统复合干扰项，将轮胎侧向力对纵向力的影响考虑成未知参数，解耦控制模型中的轮胎纵向力和侧向力.（2）利用双曲正切函数和终端吸引因子设计改进的跟踪微分器，平滑车轮滑移率跟踪误差和估计车轮滑移率跟踪误差的一阶导数. 以车轮滑移率跟踪控制模型和改进的跟踪微分器输出为基础，基于自适应快速终端滑模控制理论和投影图 4 斜坡信号工况的车辆状态信息Fig.4 Vehicle state information of ramp signal maneuver174浙 江 大 学 学 报（工学版）第 55 卷算子理论推导出未知参数自适应律，对系统复合干扰项具有强鲁棒性的车轮滑移率跟踪控制律；利用LaSalle 不变性原理，证明了闭环系统的渐近稳定性.（3）利用车辆动力学软件仿真，验证所提出图 6 正弦信号工况的车辆状态信息Fig.6 Vehicle state information of sine signal maneuver图 5 正弦信号工况的跟踪误差及误差变化率Fig.5 Tracking error and its rate of sine signal maneuver第 1 期李静, 等：基于自适应快速终端滑模的车轮滑移率跟踪控制 [J]. 浙江大学学报：工学版,2021, 55(1): 169–176.175的控制律的可行性和有效性. 结果表明，设计的改进跟踪微分器可以较好地平滑车轮滑移率跟踪误差，且可以无噪声地估计车轮滑移率跟踪误差变化率；提出的车轮滑移率跟踪控制律具有良好的控制品质，可以有效地抑制滑模控制中固有的“抖振”现象.参考文献(References):KUO C Y, YEH E C. A four-phase control scheme of an anti-skid brake system for all road conditions [J]. Proceeding of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 1992, 206(44): 275–283.[1]TANELLI M, OSORIO G, BERNARDO M D, et al. Existence, stability and robustness analysis of limit cycles in hybrid anti-lock braking systems [J]. International Journal of Control, 2009, 82(4): 659–678.[2]JING H H, LIU Z Y, CHEN H. A switched control strategy for antilock braking system with on/off valves [J]. IEEE Transactions on Vehicular Technology, 2011, 60(4): 1470–1484.[3]马付雷. 汽车ABS逻辑门限值控制算法研究与实现[D]. 重庆:重庆邮电大学, 2011.[4]王伟达, 丁能根, 张为, 等. ABS逻辑门限值自调整控制方法研究与试验验证[J].机械工程学报, 2010, 46(22): 90–104.WANG Wei-da, DING Neng-gen, ZHANG Wei, et al. Research and verification of the logic threshold self-adjusting control method for ABS [J]. Journal of Mechanical Engineering, 2010, 46(22): 90–104.[5]PASILLAS-LEPINE W, LORIA A, GERARD M. Design and experimental validation of a nonlinear wheel slip control algorithm [J]. Automatica, 2012, 48: 1852–1859.[6]SHIM T, CHANG S, LEE S. Investigation of sliding-surface design on the performance of sliding mode controller in antilock braking systems [J]. IEEE Transactions on Vehicular Technology, 2008, 57(2): 747–759.[7]YU H X, QI Z Q, DUAN J M, et al. Multiple model adaptive backstepping control for antilock braking system based on LuGre dynamic tyre model [J]. International Journal of Vehicle Design, 2015, 69(1-4): 168–184.[8]MIRZAEI M, MIRZAEINEJAD H. Optimal design of a non-linear controller for anti-lock braking system [J]. Transportation Research Part C, 2012, 24: 19–35.[9]AMODEO M, FERRARA A, TERZAGHI R, et al. Wheel slip control via second-order sliding-mode generation [J]. IEEE Transactions on Intelligent Transportation Systems, 2010, 11(1): 122–131.[10]TANELLI M, ASTOLFI A, SAVARESI S M. Robust nonlinear [11]output feedback control for brake by wire control systems [J].Automatica, 2008, 44: 1078–1083.JOHANSEN T A, PETERSEN I, KALKKUHL J, et al. Gain-scheduled wheel slip control in automotive brake systems [J].IEEE Transactions on Control Systems Technology, 2003, 11(6): 799–811.[12]ZHANG J X, LI J. Adaptive backstepping sliding mode control for wheel slip tracking of vehicle with uncertainty observer [J].Measurement and Control, 2018, 51(9/10): 396–405.[13]ZHANG J X, LI J. Robust backstepping sliding mode control with L2-gain performance for reference input wheel slip tracking of vehicle [J]. Information Technology and Control, 2019, 48(4): 660–672.[14]HE Y G, LU C D, SHEN J, et al. Design and analysis of output feedback constraint control for antilock braking system with time-varying slip ratio [J]. Mathematical Problems in Engineering, 2019, 2019: 1–11.[15]HE Y G, LU C D, SHEN J, et al. Design and analysis of output feedback constraint control for antilock braking system based on Burckhardt's model [J]. Assembly Automation, 2019, 39(4): 497–513.[16]KIENCKE U, NIELSEN L. Automotive control systems[M].Berlin: Springer, 2000.[17]申帅, 张葆, 李贤涛, 等. 基于跟踪微分器的加速度反馈控制[J]. 吉林大学学报: 工学版, 2017, 47(4): 1217–1224.SHEN Shuai, ZHANG Bao, LI Xian-tao, et al. Acceleration feedback control based on tracking differentiator [J]. Journal of Jilin University: Engineering and Technology Edition, 2017, 47(4): 1217–1224.[18]谭诗利, 雷虎民, 王鹏飞. 应用跟踪微分器的高超声速飞行器的反演控制[J]. 宇航学报, 2019, 40(6): 673–683.TAN Shi-li, LEI Hu-min, WANG Peng-fei. Backstepping control for hypersonic vehicle with a novel tracking differentiator [J].Journal of Astronautics, 2019, 40(6): 673–683.[19]姜长生, 吴庆宪, 费树岷, 等. 现代非线性系统鲁棒控制基础[M]. 哈尔滨: 哈尔滨工业大学出版社, 2012.[20]HAN J Q, WANG W. Nonlinear tracking differentiator [J].Journal of Systems Science and Mathematical Sciences, 1994, 14(2): 177–183.[21]YANG L, YANG J Y. Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems [J]. International Journal of Robust and Nonlinear Control, 2011, 21: 1865–1879.[22]ABRAMOWITZ M, STEGUN I A. Handbook of mathematical functions: with formulas, graphs, and mathematical tables [M]. New York: Dover, 1972.[23]YAO B, TOMIZUKA M. Adaptive robust control of MIMO nonlinear systems in semi-strict feedback form [J]. Automatica, 2001, 37(9): 1305–1321.[24]176浙江大学学报（工学版）第 55 卷。
二阶系统 滑模控制
二阶系统滑模控制
二阶系统是指系统的动态方程具有二阶微分方程形式的系统。
在控制领域中，二阶系统是比较常见的一种系统类型。
滑模控制是一种针对非线性系统的控制方法，其主要思想是通过引入一个滑动面来控制系统的状态。
在二阶系统中，可以通过滑模控制来实现对系统状态的控制。
滑模控制的核心是滑动面的设计。
对于二阶系统，通常选择一个二次函数形式的滑动面。
通过选择合适的滑动面以及设计相应的控制律，可以实现对二阶系统的稳定控制。
在实际应用中，二阶系统滑模控制常常被用于控制机械臂、飞行器等系统。
它具有控制精度高、稳定性好等优点，是一种非常有效的控制方法。
- 1 -。
基于双模态有限时间滑模的永磁同步电机抗扰动控制
包装工程第45卷第5期·188·PACKAGING ENGINEERING2024年3月基于双模态有限时间滑模的永磁同步电机抗扰动控制张会林，王帅，张建平（上海理工大学机械工程学院，上海200093）摘要：目的解决包装机工作过程中，因永磁同步电机的动态响应慢和抗扰动能力弱导致包装机精度不高的问题。
方法设计一种双模态有限时间滑模控制器，实现系统有限时间收敛。
将双模态函数引入趋近律增益，不仅能实现“大误差大增益，小误差小增益”，而且趋近律增益切换为小增益的时间可调，从而使电机获得更快的响应速度和更小的抖振。
同时，设计有限时间扰动观测器对扰动进行观测，并进行前馈补偿，以此来提高系统的抗扰性能。
结果实验结果表明，文中方法相较于另外2种对照方法，可以使电机的动态响应分别提升27%、37%，控制性能分别提升40%、70%，相较于超螺旋扰动观测器，可以使电机的抗扰性能提升58%。
结论所提控制策略可以明显提高系统的动态响应、控制性能、抗扰性能，使得永磁同步电机更符合包装机的要求。
关键词：永磁同步电机；动态响应；抗扰动；双模态有限时间滑模控制器；有限时间扰动观测器中图分类号：TP13；TB486 文献标志码：A 文章编号：1001-3563(2024)05-0188-09DOI：10.19554/ki.1001-3563.2024.05.023Disturbance Rejection Control of Permanent Magnet Synchronous Motor Based onDual-mode Finite Time Sliding ModeZHANG Huilin, WANG Shuai, ZHANG Jianping(School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China)ABSTRACT: The work aims to solve the problem of low precision in packaging machines in work due to the slow dynamic response and weak disturbance rejection ability of permanent magnet synchronous motors. A dual-mode finite time sliding mode controller was designed to realize finite time convergence of the system. By adding a dual-mode function to the reaching law, it could make the sliding mode gain achieve "large error and large gain, small error and small gain", and the time of the sliding mode gain switched to small gain was adjustable, so that the motor could achieve faster response speed and less chattering. At the same time, a finite time disturbance observer was designed to observe the disturbance, and feed-forward compensation was performed to improve the disturbances rejection performance of the motor. The experimental results showed that if the method proposed was used to control the motor, compared with the control groups, the dynamic response was improved by 27% and 37% and the control performance was improved by 40% and 70%. Compared with the super-spiral disturbance observer, the disturbance rejection performance of the motor was improved by 58%. Therefore, the control strategy proposed can significantly improve the dynamic response, control performance, and disturbance rejection ability of the system, which will let permanent magnet synchronous motor more suitable for the needs of packaging machines.KEY WORDS: permanent magnet synchronous motor; dynamic response; disturbance rejection; dual-mode finite time sliding mode controller; finite time disturbance observer收稿日期：2023-05-17基金项目：国家自然科学基金（12172228,11572187）；上海市自然科学基金（22ZR144400）第45卷第5期张会林，等：基于双模态有限时间滑模的永磁同步电机抗扰动控制·189·近年来，随着人们生活水平的不断提高，包装行业迎来很好的发展机遇，同时也对包装产业提出了更高的要求。
基于RBF网络最小参数学习法的机械手终端滑模控制
基于RBF网络最小参数学习法的机械手终端滑模控制刘昕明;吕东东【摘要】针对机械手轨迹跟踪控制算法的问题研究,该文提出了基于RBF神经网络最小参数学习法的终端滑模控制(TSMC)方案.终端滑模控制算法解决了线性滑模控制算法不能在有限时间收敛到系统滑模面的问题,并保持了其对被控系统不确定性的鲁棒性.采用RBF神经网络逼近系统中的不确定项,用单个参数代替神经网络中的权值,从而简化自适应算法,增强了实时控制的要求.同时,用一个鲁棒控制项来抑制神经网络的建模误差和估计误差.Lyapunov理论保证闭环系统的有限时间收敛性和稳定性.最后,以两关节机械手作为被控对象,实验结果证实该控制方案的有效性.【期刊名称】《自动化与仪表》【年(卷),期】2018(033)006【总页数】6页(P28-33)【关键词】滑模控制;机械手;轨迹跟踪;RBF神经网络;终端滑模;Matlab【作者】刘昕明;吕东东【作者单位】辽宁工程技术大学电气工程与控制工程学院,葫芦岛 125105;辽宁工程技术大学电气工程与控制工程学院,葫芦岛 125105【正文语种】中文【中图分类】TP241.3滑模变结构控制以其对系统参数变化和外部扰动的鲁棒性而著称，线性滑模控制（LSMC）是系统到达滑模面后，跟踪误差渐进收敛至零[1]，并且可以通过选择滑动模态参数来调整渐进收敛的速度，但是无论如何调整，状态跟踪误差都不能在有限时间内收敛至零。
为了解决无限时间收敛问题，采用终端滑模控制，通过在线性滑模面中引入非线性函数项和适当设计控制器，使得跟踪误差在有限时间内收敛到零，并且相对于线性滑模控制鲁棒性更强。
由于TSMC自身也存在缺点会出现奇异问题[2-3]，为了避免奇异问题的出现，对终端滑模面进行了改进[4]。
由于要设计的控制器依赖于机械手的精确数学模型，而机械手的某些项是不能确定的，可以采用RBF神经网络去逼近不确定项。
RBF神经网络是一种非线性模型，具有收敛速度快、全局逼近能力强等优点，对于复杂不确定问题具有自适应能力和自学习能力，可以应用于非线性和不确定系统的控制器设计中[5]。
基于终端滑模控制器的高精度伺服系统设计
基于终端滑模控制器的高精度伺服系统设计Design of high precision servo system based on terminal sliding mode controller王朝阳，潘松峰WANG Zhao-yang, PAN Song-feng（青岛大学 自动化学院，青岛 266000）摘 要：随着工业机器人与精密机床的发展，传统的三环控制策略无法满足伺服系统对动态响应速度、位置跟踪精度及超调量越来越高的要求。
针对这一问题，结合已有的研究成果，提出微分前馈控制与终端滑模结合的控制策略，在负反馈基础上，添加位置前馈控制，实现位置信号的快速跟踪，将非奇异快速终端滑模控制方法应用到位置环与速度环，提高电机位置跟踪精度以及在电机出现较大位置误差时减小超调量。
理论分析和仿真验证算法的可行性和有效性。
关键词：永磁同步电机；位置伺服系统；终端滑模控制；前馈控制中图分类号：TP273 文献标识码：A 文章编号：1009-0134(2021)01-0113-05收稿日期：2019-10-20作者简介：王朝阳（1996 -），男，山东德州人，硕士研究生，研究方向为电机控制。
0 引言在工业机器人以及精密机床中的伺服系统要求较高的位置跟踪精度，同时位置伺服系统经常会出现阶跃的位置信号或者发生较大的位置误差，出现这种瞬时脉冲信号时无法避免出现较大的超调量，而位置环存在超调量的一部分原因是速度环抑制速度波动以及抗负载变化的能力弱。
这使传统的控制方法难以同时满足这两个要求。
文献[1～3]将滑模控制应用于伺服系统位置和速度环，通过对伺服系统滑模面及控制律的合理设计，较好地实现了位置精确定位；文献[4]将终端滑模控制方法应用于PMSM 的调速系统中，但用于位置伺服系统中时，系统位置跟踪精度较差需要改进位置环，同时添加位置前馈控制时，接收前馈补偿信号的速度环在采用滑模控制时也需要进行改进；文献[5]针对伺服系统的高精度位置控制问题，提出了基于自适应模糊补偿器的终端滑模控制方法；文献[6]将终端滑模控制应用到伺服系统的电流环、速度环和位置环中，系统具有较高的跟踪精度和较小的超调量；文献[7,8]将前馈控制应用于PMSM 位置伺服系统，将位置前馈补偿信号作用在速度环，在速度环使用PI 控制，系统跟踪误差小，但抑制速度波动以及抗负载变化的能力弱，超调量较大。
基于扰动观测器和改进自适应二阶快速终端滑模的PMLSM_伺服系统控制
第28卷㊀第4期2024年4月㊀电㊀机㊀与㊀控㊀制㊀学㊀报Electri c ㊀Machines ㊀and ㊀Control㊀Vol.28No.4Apr.2024㊀㊀㊀㊀㊀㊀基于扰动观测器和改进自适应二阶快速终端滑模的PMLSM 伺服系统控制赵希梅,㊀李德豪,㊀金鸿雁(沈阳工业大学电气工程学院,辽宁沈阳110870)摘㊀要:针对永磁直线同步电动机(PMLSM )在运行时易受到摩擦力㊁负载扰动及环境变化等不确定影响,设计一种基于扰动观测器和自适应二阶快速终端滑模控制方法㊂首先,该方法在高阶滑模的基础上结合了快速终端滑模控制,有效加快了系统响应速度㊂其次,在二阶滑模面的基础上引入条件积分项,该项仅作用于边界层内部,能够有效避免滑模到达阶段由于积分饱和现象导致的超调量过大及响应时间过慢的问题㊂此外,设计自适应切换控制律,使切换增益在合理的区间内,并根据系统状态进行实时更新,从而削弱抖振现象㊂最后,为进一步提升系统的鲁棒性,采用扰动观测器对PMLSM 系统的集总不确定性进行补偿㊂经实验验证,该控制方法具有良好的位置跟踪精度和强鲁棒性,且对系统抖振现象有明显削弱作用㊂关键词:永磁直线同步电动机;二阶滑模控制;终端滑模控制;条件积分项;自适应控制;扰动观测器DOI :10.15938/j.emc.2024.04.005中图分类号:TM351;TP273文献标志码:A文章编号:1007-449X(2024)04-0041-09㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀收稿日期:2022-11-29基金项目:辽宁省博士科研启动基金计划项目(2022-BS -177);辽宁省科技厅揭榜挂帅项目(2023JI1/11100004)作者简介:赵希梅(1979 ),女,博士,教授,研究方向为直线伺服㊁智能控制㊁鲁棒控制㊁机器人控制等;李德豪(1999 ),男,硕士研究生,研究方向为直线电机及其控制;金鸿雁(1993 ),女,博士,讲师,研究方向为直线伺服㊁智能控制等㊂通信作者:赵希梅PMLSM servo system control based on disturbance observer and improved adaptive second order fast terminal sliding modeZHAO Ximei,㊀LI Dehao,㊀JIN Hongyan(School of Electrical Engineering,Shenyang University of Technology,Shenyang 110870,China)Abstract :Permanent magnet linear synchronous motor (PMLSM)is vulnerable to uncertain influences such as friction,load disturbance and environmental changes during operation.A fast terminal sliding mode control method based on disturbance observer and adaptive second-order was designed.Firstly,the method combined the fast terminal sliding mode control on the basis of high-order sliding mode,which ef-fectively accelerated the system response speed.Secondly,a conditional integral term was introduced on the basis of the second-order sliding surface,which only acted inside the boundary layer,and could ef-fectively avoid the problem of excessive overshoot and too slow response time caused by integral saturation in the sliding mode arrival stage.In addition,an adaptive switching control law was designed to make the switching gain within a reasonable interval and update in real time according to the system state,thereby weakening the chattering phenomenon.Finally,in order to further improve robustness of the system,the disturbance observer was used to compensate the lumped uncertainty of the PMLSM system.The experi-mental results show that the control method has good position tracking accuracy and strong robustness,and can significantly weaken the chattering phenomenon of the system.Keywords:permanent magnet linear synchronous motor;second-order sliding mode control;terminal sliding mode control;conditional integral term;adaptive control;disturbance observer0㊀引㊀言永磁同步直线电动机(permanent magnet linear synchronous motor,PMLSM)与传统旋转电机相比,具有定位精度高㊁响应速度快㊁功率密度大㊁无齿隙等优点,在工业得到广泛应用㊂然而,PMLSM是一个多变量㊁强耦合的复杂非线性系统,存在负载扰动㊁非线性摩擦扰动㊁参数变化等不确定性因素㊂因此,如何对系统的集总不确定性进行抑制和补偿,从而满足高精度位置跟踪控制成为研究的热点问题㊂滑模控制(sliding mode control,SMC)由于具有响应速度快㊁鲁棒性强等优点被广泛应用于PMLSM 伺服系统中[1-3]㊂但传统SMC存在严重的抖振现象,而终端滑模控制(terminal sliding mode control, TSMC)因具有更快的收敛速度且能在有限时间内收敛至滑模面上,获得广泛应用㊂文献[4]采用非奇异快速TSMC方法提高位置跟踪精度,并采用径向基神经网络(radial basis function neural network, RBFNN)对集总不确定性进行估计,然而神经网络中神经元个数和权值等参数的选取困难,对估计精度影响较大,收敛的快速性难以保证㊂文献[5]设计了递归非奇异TSMC方法,采用非奇异终端滑模面和递归积分终端滑模面结合组成双层滑模面结构,可保证两滑模面依次㊁连续到达,保证了有限时间收敛,并在RBFNN的作用下有效地削弱抖振㊂文献[6]将分数阶算子引入到TSMC的滑模面中,其特有的记忆特性可有效地削弱抖振并提升系统的鲁棒性㊂高阶滑模控制(high order sliding mode con-trol,HOSMC)是对传统SMC的扩展,将不连续的状态变量作用到滑模状态的高阶导数中,在保留了传统SMC优点的同时,削弱了抖振现象并提高控制精度[7-9]㊂HOSMC常与自适应控制㊁模糊控制㊁神经网络控制等算法相结合,广泛地应用于高精度伺服控制领域中㊂文献[10-11]引入低通滤波器进一步削弱HOSMC的抖振现象,然而滤波器存在滞后现象,会使响应时间变慢,跟踪精度变差㊂文献[12]采用离散自适应HOSMC,将HOSMC从连续域拓展至离散域,丰富了HOSMC的应用范围㊂文献[13]将HOSMC与积分超螺旋算法相结合构造切换控制律,积分超螺旋算法较传统控制律能有效地抑制抖振㊂文献[14]将HOSMC与时变SMC相结合,能够保证控制变量始终处于滑模面上,然而时变项的选取比较困难,容易对跟踪精度产生较大影响㊂文献[15]针对HOSMC提出自适应切换控制律,使系统具有强鲁棒性和快速收敛性,然而控制律中存在开关函数,不连续项会加剧切换处的抖振现象㊂因此,本文提出基于扰动观测器和改进自适应二阶快速终端滑模控制(adaptive second order fastterminal sliding mode control,ASOFTSMC)方法㊂相较于传统二阶快速TSMC,ASOFTSMC在滑模面的设计中引入条件积分项,能够有效避免滑模到达阶段的积分饱和现象,具有响应速度快㊁鲁棒性强㊁跟踪精度高的特点㊂此外,引入自适应切换控制律,使切换增益保持在合理区间,提高快速性的同时避免增益高估所引起的抖振㊂为进一步提高系统的鲁棒性,针对系统的集总不确定性采用扰动观测器(dis-turbance observer,DOB)进行补偿㊂最后,通过系统仿真和实验证明该控制方法能有效地提高PMLSM 伺服系统位置跟踪精度和鲁棒性能㊂1㊀PMLSM数学模型在忽略涡流㊁磁滞损耗且不计初级齿槽效应的前提下,对PMLSM采用基于i d=0的矢量控制,在同步旋转坐标系下,电机的电磁推力方程表示为F e=3πp n2τ[ψf i q+(L d-L q)i d i q]㊂(1)式中:p n表示极对数;τ表示极距;ψf表示永磁体基波磁链;i d㊁i q和L d㊁L q分别表示d㊁q轴电流和电感㊂针对表贴式PMLSM,有L d=L q=L s㊂(2)式中L s为等效电感㊂此时,式(1)可简化为:F e=k f i q;k f=3πp nψf2τ㊂}(3)式中k f为电磁推力系数㊂PMLSM的机械运动方程可表示为:x㊃=010-B Méëêêêùûúúúx+k fMéëêêêùûúúúu-1Méëêêêùûúúúd;y=[10]x㊂üþýïïïï(4)24电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第28卷㊀式中:x 为电机的状态变量,定义电机的位置为x 1㊁速度为x 2;y 为电机的位置输出;B 为粘滞摩擦系数;M 为动子的总质量;u 为电机的输入信号,即q 轴电流;d 为电机的集总不确定性,可表示为d =d r +d f +d L +d 0㊂(5)式中:d r 为推力波动,为一频率已知㊁幅值和相位未知的周期性正弦扰动;d f 为摩擦扰动;d L 为负载扰动;d 0为其他扰动,为非周期性扰动㊂d 存在一不确定性上界H ,满足:H ȡ|d |,H >0㊂(6)2㊀PMLSM 位置伺服系统设计为满足PMLSM 伺服系统的高精度位置跟踪性能及鲁棒性要求,设计一种包含条件积分项的ASOFTSMC㊂同时,为进一步提升系统的鲁棒性,引入扰动观测器进行估计和补偿㊂基于扰动观测器和ASOFTSMC 的PMLSM 伺服系统位置控制框图如图1所示㊂图1㊀基于扰动观测器和ASOFTSMC 的PMLSM 伺服系统位置控制框图Fig.1㊀Position control block diagram of PMLSM servo system based on ASOFTSMC and disturbance observer2.1㊀ASOFTSMC 设计定义动子位置误差e 1㊁动子速度误差e 2分别为:e 1=x ∗1-x 1;e 2=e ㊃1㊂}(7)传统二阶快速TSMC 滑模面设计为s 1=k 1|e 1|α1sgn(s 1)+k 2|e 1|α2sgn(s 1)+k 3e 2㊂(8)式中:0<α1<1;1<α2<2;k 1㊁k 2㊁k 3为待设计正参数㊂当处于滑模到达阶段时,k 2|e 1|α2sgn(s 1)起主要作用,当处于稳定阶段时,k 1|e 1|α1sgn(s 1)起主要作用,k 1㊁k 2的大小决定了两个阶段对于滑模面的权重大小,通常情况下选取较大的k 2值以提高快速性,k 3则保证了e 1的变化趋势趋于0,但过大的k 3值可能会造成系统失稳,故选取一较小正常数较为适宜㊂然而符号函数sgn(s )是非连续函数,会在切换点上产生抖振现象,为此引入饱和函数sat(s )替代传统符号函数㊂相较于传统的符号函数,sat(s )在切换处的曲线更加平滑,可有效地削弱抖振㊂定义边界层厚度为μ,sat(s )表示为sat(s )=-1,s <-μ;s μ,-μ<s <μ;1,s >μ㊂ìîíïïïï(9)积分滑模面相较于线性滑模面,在滑模面中引入了积分环节,可提高跟踪精度并增强系统鲁棒性㊂在传统二阶TSMC 滑模面中引入积分环节,并采用饱和函数代替符号函数,可得到二阶积分FTSMC 的滑模面为s =k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2+k ʏe 1d t ㊂(10)然而,在滑模到达阶段,积分项可能会引起响应的振荡,当控制饱和时,积分器堆积,会造成较大的超调量和较慢的响应时间㊂故引入条件积分项以代替传统的积分项,该项只作用于边界层内部,可有效避免积分环节在滑模到达阶段对位置跟踪带来的负面效果㊂设计条件积分项为:s 2=k 4σ;σ㊃=-σ0+μsat(s )㊂}(11)式中:k 4为待设计的正常数;σ为中间变量;σ0为34第4期赵希梅等:基于扰动观测器和改进自适应二阶快速终端滑模的PMLSM 伺服系统控制数值很小的正常数㊂将式(9)代入式(11)可得σ㊃=-σ0+μ,s >|μ|;-σ+s ,s ɤ|μ|㊂{(12)由式(12)可知,当滑模状态处于边界层外部时,有s 2=0㊂故条件积分项仅作用于边界层内部,有s 2=k 4ʏ(σ0+s )d t ʈk 4ʏ[k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2]d t ㊂(13)由式(9)和式(13)可得改进二阶FTSMC 滑模面为s =k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2+k 4ʏs 2d t ㊂(14)设状态变量在边界层内部,对式(14)求导得s ㊃=k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3e ㊃2+k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2)㊂(15)当不考虑外部扰动,即d =0时,令s ㊃=0可得到等效控制律u eq ㊂对式(15)进行整理,有㊀s ㊃=k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3(x㊃㊃∗1-x ㊃㊃1)+k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2)=k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3(x㊃㊃∗1-(-BM x 2+k f u eq ))+k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2)=k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3x㊃㊃∗1+k 3BM x 2-k 3k f Mu eq +k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2)㊂(16)令式(16)等于0,可得u eq =Mk 3k f (k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3x㊃㊃∗1+k 3BMx 2+k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2))㊂(17)在等效控制律式(17)中,包含参数k 4的项为条件积分项,当状态变量处于边界层外部时,该项为0;其余项则保证状态变量从初态收敛至边界层内部,k 4可选取一相对较大值以提升系统鲁棒性㊂为满足PMLSM 高精度位置伺服控制的强鲁棒性要求,考虑推力波动和其他周期性扰动对于控制系统的影响,引入自适应切换控制律u asw ,表达式为u asw =Mk 3k fK sat(s )㊂(18)式中K 为自适应参数,K 值越大,响应速度越快,但同时抖振越剧烈㊂通过自适应律的设计可防止因切换增益K 过高而引起的过度抖振现象,自适应律设计为K ㊃=κs sat(|s |-μ),K >H ;χ,K <H ㊂{(19)式中:κ>0,为切换增益的衰减速率;χ为正实数,表示切换增益的提高速率;μ为边界层厚度,其厚度越小,跟踪精度越高,但收敛速度则响应越慢㊂通过式(19)可知,当切换增益K 小于H 时,K ㊃>0,K 值增加以提高收敛速度;当切换增益K 大于H时,若滑模状态变量大于边界层厚度μ,K 值缓慢增加以提高跟踪精度,反之,则K 值减小以抑制抖振㊂切换增益K 存在上界K ∗,满足K <K ∗㊂由式(17)和式(18)可得总控制律为u 0=u eq +u asw =Mk 3k f (k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3x ㊃㊃∗1+k 3BM x 2+k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2)-K sat(s ))㊂(20)定义Lyapunov 函数:V =12s 2+12γ(K -K ∗)2㊂(21)式中γ为一正常数㊂对式(21)求导可得V ㊃=ss ㊃+1γ(K -K ∗)κs sat(|s |-μ)[]=s (k 1α1|e 2|α1-1+k 2α2|e 2|α2-1+k 3x ㊃㊃∗1+k 3BMx 2-k 3k f (u eq +u asw )+k 4(k 1|e 1|α1sat(s )+k 2|e 1|α2sat(s )+k 3e 2)-d )+η=s (-K sat(s )-d )+η=-K |s |-ds +η㊂(22)在式(22)中,定义η=1γ(K -K ∗)κs sat(|s |-μ)[]㊂(23)整理式(22)可得V ㊃=-K |s |-ds +1γ(K -K ∗)κs sat(|s |-μ)[]㊂(24)根据自适应律式(19)可知,切换增益K 满足维持系统稳定的最小值,即K ȡH ㊂故此时有关系K ȡH >d 成立㊂故在式(24)中,有44电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第28卷㊀-K |s |-ds =(K -d )s ɤ0,s ɤ0;-(K +d )s ɤ0,s >0㊂{(25)改写式(25)可得-K |s |-ds ң-ζA |s |㊂(26)式中ζA >0㊂引入ζB >0,代入式(24)可得V ㊃=-ζA |s |+(K -K ∗)κγ|s |sat(|s |-μ)[]+(ζB |K -K ∗|-ζB |K -K ∗|)=(-ζA |s |-ζB |K -K ∗|)-(κγ|s |sat(|s |-μ)-ζB )|K -K ∗|㊂(27)在式(27)中,定义ξ=κγ|s |sat(|s |-μ)-ζB[]|K -K ∗|㊂(28)整理式(27),有V ㊃=-ζA |s |-ζB |K -K ∗|-ξ=-ζA 2|s |2-ζB 2γ|K -K ∗|2γ-ξɤ-min{ζA 2,ζB 2γ}(|s |2+|K -K ∗|2γ)-ξɤ-ζV 12-ξ㊂(29)式中ζ=2min{ζA ,ζB γ}㊂现分别从以下2种情况分析控制器的稳定性㊂情况1):|s |>μ㊂此时,若ξ>0,则需满足κγ|s |-ζB >0ңγ<κ|s |ζB㊂(30)代入式(29)可得V ㊃ɤ-ζV 12-ξɤ-ζV 12㊂(31)故在式(27)的前提下,在任何初始状态下均可在有限时间内收敛至|s |ɤμ㊂情况2):|s |ɤμ㊂当|s |ɤμ时,在式(30)的前提下,ξ不能保证恒大于0,故不可使用上述推导结果证明控制器的稳定性㊂此时,假设系统处于不稳定阶段:s (0)=μ+(从μ的右侧趋向于μ),K 0为一数值很小的正常数成立㊂根据式(19)和式(25)可得:s ㊃=-d -K sat(s );K ㊃=κ|s |㊂}(32)对式(32)进行求解,可得s (t )=s (0)cos(κsat(s )t )+-d -K 0sat(s )κsat(s )t=s (0)2+-d -K 0sat(s )2κsat(s )sin(κsat(s )t +υ)㊂(33)式中υ为相位值㊂根据式(33),当s (0)=μ+ңμ,有s max =μ2+d 2+K 20+2dK 0κsat(s )ʈμ2+d 2κsat(s )㊂(34)此时,若κ足够大,使得d 2κsat(s )ʈ0,则可称系统状态变量在任何初始状态下都能在有限时间内收敛至|s |ɤμ的区域内,控制器满足渐近稳定条件㊂2.2㊀扰动观测器设计为进一步提高系统的鲁棒性,针对PMLSM 运行过程中的集总不确定性d 采用基于DOB 的方法对其进行补偿㊂在式(4)中,做如下定义:A 0=010-B M éëêêêùûúúú;B 0=0k f M éëêêêùûúúú;Θ=01M éëêêêùûúúú㊂üþýïïïïïïïïïïï(35)式中:A 0为系统状态矩阵;B 0为输入矩阵;Θ为扰动输入矩阵㊂根据式(4)和传统状态观测器理论,针对一存在扰动的系统,扰动量d 可通过DOB 进行估计:d ^=L (x ㊃-A 0x -B 0u +Θd ^)㊂(36)式中:d ^为扰动的观测值;L 为观测器的增益矩阵㊂然而,式(36)中包含了对状态变量的导数项,在存在不确定性扰动的复杂运行状况下,导数项会对观测的精确度造成较大的影响㊂为此,引入中间变量ω为ω㊃=-LB 0(ω+L x )-L (A 0x +Θu )㊂(37)通过引入中间变量ω,可以避免导数项对观测结果的影响㊂扰动观测器设计如下:54第4期赵希梅等:基于扰动观测器和改进自适应二阶快速终端滑模的PMLSM 伺服系统控制ω㊃=-LB 0(ω+L x )-L (A 0x +Θu );d ^=ω+L x ㊂}(38)定义观测误差e d 为e d =d ~=d ^-d ㊂(39)整理式(37)~式(39)可得e ㊃d=-LΘe d -d ㊃㊂(40)假设系统集总不确定性扰动d 为一缓慢变化的量㊂此时,满足d ㊃ʈ0,则式(40)可改写为e ㊃d=-LΘe d ㊂(41)根据Hurwitz 稳定性判据,当-LΘ满足Hur-witz,即-LΘ>0时,由式(38)设计的扰动观测器收敛,满足Hurwitz 稳定条件㊂3㊀实验验证为验证方法的有效性,将基于DOB-ASOFTSMC(方法一)㊁改进二阶FTSMC (方法二)㊁传统二阶FTSMC(方法三)3种方法应用在PMLSM 位置伺服系统中,并进行半实物仿真实验,半实物仿真平台图和实验系统框图分别如图2和图3所示㊂图2㊀实验平台图Fig.2㊀Diagram of experimentalplatform图3㊀实验系统框图Fig.3㊀Block diagram of experimental system实验中采用PMLSM 参数如下:M =3.1kg,B =0.002N㊃s /m,τ=0.052m,ψf =0.536Wb,p n =2,L s =0.0504mH㊂为保证参数选取的一致性,3种方法中k 1㊁k 2㊁k 3㊁k 4的参数取值相同㊂控制器参数选取如下:1)DOB-ASOFTSMC:k 1=10,k 2=15,k 3=1,k 4=50,μ0=0.01,χ=0.2,κ=20,K =100,α1=0.5,α2=1.8;2)改进二阶FTSMC:k 1=10,k 2=15,k 3=1,k 4=50;3)传统二阶FTSMC:k 1=10,k 2=15,k 3=1,k 4=50㊂1)实验Ⅰ:方波信号给定下的跟踪效果㊂位置给定设置为一方波信号,时长为6s,分别在0㊁1.5㊁3和4.5s 施加大小分别为3㊁5㊁-1和-2μm 的位置信号,同时在0.3~1.5s 的区间内施加如图4所示的变负载扰动㊂基于3种不同方法的位置跟踪曲线如图5所示㊂图4㊀变负载扰动曲线Fig 4㊀Variable load disturbancecurve图5㊀基于3种方法的位置跟踪曲线Fig.5㊀Position tracking curves based on three methodsgiven in the square waves由图5可知,方法一较另外2种方法同样拥有更快的响应速度㊂从1.55~1.8s 的跟踪局部放大图中可以更直观地看出,3种方法在1.5s 施加5μm 位置信号下的响应时间分别为1.68㊁1.73和1.78s;同时,3种方法均具有较高的位置跟踪精度,并且在施加变负载扰动时跟踪信号无明显波动,有较强的鲁棒性㊂为体现仿真结果和半实物仿真实验结果的一致64电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第28卷㊀性,基于3种方法的仿真和实验位置跟踪误差曲线分别如图6~图8所示㊂图6㊀方波下基于方法一的位置跟踪误差曲线Fig.6㊀Position tracking error curves based on method 1given in squarewave图7㊀方波下基于方法二的位置跟踪误差曲线Fig.7㊀Position tracking error curves based on method 2given in square wave观察图6~图8可以发现,仿真曲线和实验曲线在结果上具有一致性,说明方案具有可行性㊂进一步地,针对实验曲线进行分析,可以得出方法一具有更快的响应速度,位置误差曲线也相对更为平滑;处于跟踪稳态时,方法一的跟踪误差处于ʃ1μm 的区间内,具有良好的跟踪性能,方法二和方法三则分别处于ʃ2和ʃ3.5μm 的区间内,且抖振更剧烈,这是由于自适应切换增益可根据系统状态进行动态调整,可防止因切换增益过大而导致的抖振现象;施加变负载扰动时,方法一有0.5μm 左右的位置偏差,方法二和方法三则分别有2和4μm 左右的位置偏差,可见DOB 对于扰动具有强鲁棒性㊂针对图6~图8中的变负载(0.1s <t <1.5s)区域分别从极大值(max)㊁极小值(min)和均方根(RMS)三方面进行统计学分析㊂均方根相较于平均数,由于防止了误差正负抵消所带来的统计学偏差,更能反映出实际的位置跟踪精度㊂分析结果如表1所示㊂图8㊀方波给定下基于方法三的位置跟踪误差曲线Fig.8㊀Position tracking error curves based on method 3given in square wave表1㊀方波给定下的位置误差统计学分析Table 1㊀Statistical analysis of position error given insquare wave单位:μm方法误差(仿真结果/实验结果)max min RMS 一 1.02/1.22-0.78/-0.940.25/0.52二4.21/4.86-0.85/-1.62 1.12/1.53三4.33/6.524-1.07/-1.71.86/2.642)实验Ⅱ:余弦信号给定下的跟踪效果㊂为验证在不同给定信号下3种控制方法的位置跟踪效果,给定一幅值为3mm㊁频率为10Hz 的余弦信号,时长为1s㊂基于3种控制方法的位置跟踪曲线如图9所示㊂74第4期赵希梅等:基于扰动观测器和改进自适应二阶快速终端滑模的PMLSM 伺服系统控制图9㊀余弦给定下基于3种方法的位置跟踪曲线Fig.9㊀Position tracking curves based on the three methods given in the cosine根据图9可以直观地看出,方法一具有更快的响应时间,3种方法的响应时间分别为0.13㊁0.18和0.22s;在进入跟踪稳态后,3种方法均可稳定跟踪,具有良好的动态跟踪性能㊂为进一步分析3种方法的跟踪性能,基于3种控制方法在余弦给定下的位置跟踪误差曲线如图10~图12所示㊂图10㊀余弦给定下基于方法一的位置跟踪误差曲线Fig.10㊀Position tracking error curves based on method 1given in cosine与实验Ⅰ类似,3种方法在仿真和实验结果上具有一致性,说明在不同参考信号下,提出方法均具有可行性㊂通过观察图10~图12可以直观地看出,对于更加平滑的余弦信号,3种控制方法相较于存在突变的方波信号有着更好的控制性能,分别体现在响应时间更快且跟踪误差更小,方法一的位置跟踪误差在ʃ1.5μm 的区间内,另外2种方法的位置跟踪误差则分别在ʃ2和ʃ2.5μm 的区间内㊂与实验一类似,针对图10~图12的稳态误差(0.25s <t <1s)区域从max㊁min㊁RMS 三方面进行统计学分析,分析结果如表2所示㊂图11㊀余弦给定下基于方法二的位置跟踪误差曲线Fig.11㊀Position tracking error curves based on method 2given incosine图12㊀余弦给定下基于方法三的位置跟踪误差曲线Fig.12㊀Position tracking error curves based on method 3given in cosine表2㊀余弦给定下的位置误差统计学分析Table 2㊀Statistical analysis of position error givenin cosine单位:μm方法误差(仿真结果/实验结果)max min RMS 一0.68/1.34-0.71/-1.320.18/0.35二 1.10/1.82-1.21/-2.010.40/0.66三1.84/2.84-1.86/-2.320.59/0.8684电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第28卷㊀㊀㊀通过两组系统实验可知,在不同给定信号作用下,提出的基于DOB的ASOFTSMC方法在仿真和实验环境下均具有可靠性且对于PMLSM有着良好的位置跟踪性能和强鲁棒性㊂4㊀结㊀论针对PMLSM系统运行时易受到不确定性影响的问题,设计一种基于DOB的ASOFTSMC方法㊂该方法使用条件积分项代替传统积分项,解决了滑模到达阶段可能产生的积分饱和现象,提高了响应时间㊂同时设计自适应切换控制律,结合边界层法可有效地削弱抖振㊂采用DOB进一步提升系统的鲁棒性,对不确定扰动进行补偿㊂实验结果表明,在不同的信号给定下,该控制方法具有高跟踪精度㊁强鲁棒性和更快的响应速度并且有效地削弱了抖振㊂在施加变负载扰动时,由于DOB可对系统扰动进行补偿,故较传统方法有更强的鲁棒性㊂参考文献:[1]㊀张康,王丽梅.基于周期性扰动学习的永磁直线电机自适应滑模位置控制[J].电机与控制学报,2021,25(8):132.ZHANG Kang,WANG Limei.Adaptive sliding mode position con-trol for permanent magnet linear motor based on periodic disturb-ance learning[J].Electric Machines and Control,2021,25(8): 132.[2]㊀沈燚明,卢琴芬.初级励磁型永磁直线电机研究现状与展望[J].电工技术学报,2021,36(11):2325.SHEN Yiming,LU Qinfen.Overview of permanent magnet linear machines with primary excitation[J].Transactions of China Elec-trotechnical Society,2021,36(11):2325.[3]㊀吕刚.直线电机在轨道交通中的应用与关键技术综述[J].中国电机工程学报,2020,40(17):5665.LÜGang.Review of the application and key technology in the lin-ear motor for the rail transit[J].Proceedings of the CSEE,2020, 40(17):5665.[4]㊀DINH T X,AHN K K.Radial basis function neural network basedadaptive fast nonsingular terminal sliding mode controller for piezo positioning stage[J].International Journal of Control Automation and Systems,2017,15(7):2892.[5]㊀徐驰,赵希梅.永磁直线同步电动机智能递归非奇异终端滑模控制[J].控制理论与应用,2022,39(7):1242.XU Chi,ZHAO Ximei.Intelligent recursive nonsingular terminal sliding mode control of permanent magnet linear synchronous motor [J].Control Theory&Applications,2022,39(7):1242. [6]㊀高洁,王华宇,徐萌.开关磁阻电机的分数阶终端滑模控制[J].电机与控制学报,2022,26(2):142.GAO Jie,WANG Huayu,XU Meng.Improved terminal sliding mode control strategy of switched reluctance motors based on frac-tional calculus[J].Electric Machines and Control,2022,26(2):142.[7]㊀WANG Liang,SHENG Yongzhi,LIU Xiangdong.A novel adap-tive high-order sliding mode control based on integral sliding mode [J].International Journal of Control,Automation and Systems, 2014,12:459.[8]㊀RABIEE H,ATAEI M,EKRAMIAN M.Continuous nonsingularterminal sliding mode control based on adaptive sliding mode dis-turbance observer for uncertain nonlinear systems[J].Automati-ca,2019,109:108515.[9]㊀UTKIN V.Discussion aspects of high-order sliding mode control[J].IEEE Transactions on Automatic Control,2016,61(3):829.[10]㊀KOMMURI S K,RATH J J,VELUVOLU K C,et al.Decoupledcurrent control and sensor fault detection with second-order slid-ing mode for induction motor[J].IET Control Theory&Applica-tions,2015,9(4):608.[11]㊀SONG Zhuoyue,DUAN Chao,WANG Jianan,et al.Chattering-free full-order recursive sliding mode control for finite-time atti-tude synchronization of rigid spacecraft[J].Journal of the Frank-lin Institute,2018,356(2):998.[12]㊀XIANG Xianbo,LIU Chao,SU Housheng,et al.On decentral-ized adaptive full-order sliding mode control of multiple UAVs[J].ISA Transactions,2017,71(2):196.[13]㊀SANJOY M,JAWHAR G,MAAROUF S.Homogeneous finite-time consensus control for higher-order multi-agent systems by fullorder sliding mode[J].Journal of Systems Science and Complex-ity,2018,31(5):1186.[14]㊀CONG Binglong,CHEN Zhen,LIU Xiangdong.On adaptivesliding mode control without switching gain-over estimation[J].International Journal of Robust and Nonlinear Control,2014,24(3):515.[15]㊀GUO Fuhui,LU Pingli.Fast self-adapting high-order slidingmode control for a class of uncertain nonlinear systems[J].Jour-nal of Systems Engineering and Electronics,2021,32(3):690.(编辑:邱赫男)94第4期赵希梅等:基于扰动观测器和改进自适应二阶快速终端滑模的PMLSM伺服系统控制。
基于终端滑模的变参数血管介入手术机器人力反馈控制
基于终端滑模的变参数血管介入手术机器人力反馈控制
孙昊悦;胡陟
【期刊名称】《上海工程技术大学学报》
【年(卷),期】2024(38)1
【摘要】针对主从式血管介入手术机器人力反馈控制系统存在的变参数及外部扰动问题,提出一种含有连续函数特性的终端滑模控制方法。
首先分别基于计算和力测量的方法对血管介入手术力反馈控制系统参数及其变化进行分析。
然后在终端滑模控制方法的基础上,设计一种包含连续特性函数的终端滑模控制器,并使用Lyapunov稳定性分析方法,证明了控制器的稳定性。
仿真验证表明,从跟踪效果和控制器输出两个角度分析,改进后的终端滑模控制方法不仅有效地抑制了抖振,还提高了信号的跟踪精度,证明了该策略在提升变参数系统稳定性和提升控制精度方面具有重要的应用价值。
【总页数】7页(P63-69)
【作者】孙昊悦;胡陟
【作者单位】上海工程技术大学电子电气工程学院
【正文语种】中文
【中图分类】TP242.3
【相关文献】
1.血管介入手术主从导管机器人滑模控制
2.新型变倾斜参数S型函数的积分滑模水下机器人控制方法研究
3.基于滑模变结构控制滤波的水下机器人水平面轨迹跟踪
控制4.机器人的终端滑模变结构控制研究5.机器人血管介入手术自适应滑模预测控制
因版权原因，仅展示原文概要，查看原文内容请购买。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
基于二阶终端滑模优化的电流环滑模控制黄宴委;刘喆怡;熊少华;陈少斌【摘要】In order to suppress uncertainties caused by parameteric variations and disturbances,a sliding mode control(SMC)optimized by a second order terminal sliding mode control(STSMC)was derived to control current loops in a permanent magnet synchronous motor(PMSM).The equivalent control quantity for the current loop is easily obtained by SMC.The switching control for uncertainties was deduced by STSMC integrated with an adaptive exponential function law.Moreover, the stability of STSMC system was analyzed for current loops,and the gain of the switching function in STSMC can be set small,even to be set theoretic zero.STSMC is effective to suppress the"chattering"and retains the system robustness. The comparative results of simulations and experiments indicate that the current loop control system by STSMC improves the steady state accuracy of the current with the result in a better performance of speed regulation for PMSM,and is easy to implement.%针对永磁同步电机电流环存在参数摄动、干扰等不确定因素,提出了一种基于二阶终端滑模优化的电流滑模控制.借助常规滑模控制的设计简单、容易实现等优点,计算得到电机电流环的等效控制量.利用二阶终端滑模设计不确定量的控制量,并引入自适应指数趋近律抑制系统参数摄动和干扰等不确定量.所提出的电流环滑模控制系统进行了稳定性分析,理论上可获得自适应零增益切换函数控制,达到有效抑制"抖振",具有很好的鲁棒性.对仿真和实验波形的计算结果表明:所提出的控制方法提高了电流稳态精度,获得更好的调速性能,且设计简单.【期刊名称】《电机与控制学报》【年(卷),期】2018(022)003【总页数】8页(P74-81)【关键词】永磁同步电机;电流环;滑模控制;二阶终端滑模;自适应增益【作者】黄宴委;刘喆怡;熊少华;陈少斌【作者单位】福州大学电气工程与自动化学院,福建福州350116;福州大学电气工程与自动化学院,福建福州350116;福州大学电气工程与自动化学院,福建福州350116;福州大学电气工程与自动化学院,福建福州350116【正文语种】中文【中图分类】TM341;TP273永磁同步电机(permanent magnet synchronous motor，PMSM)伺服系统利用矢量控制技术使得定子电流与电机输出转矩之间成线性关系，使其在数控、运动伺服系统得到了广泛应用[1]。
定子电流性能将直接影响PMSM输出转矩的性能，也使得定子电流控制成为矢量控制的关键技术之一[2-3]。
但是，PMSM模型存在着d-q轴电流耦合、反电动势等非线性因素，且依赖于电机参数和工作点[4]，难以获得准确的PMSM数学模型，使得单纯比例积分控制器很难获得高动态和稳态性能的电流，进而难以获得准确快速的输出转矩，限制了PMSM在高精度高性能工业场合的应用[5]。
近年来，一些先进的控制方法被广泛地应用于PMSM电流控制。
比如，在比例积分控制器的基础上引入谐波调节器作为前馈补偿来抑制电流谐波[6]；在PMSM电流预测控制模型基础上，引入d轴电流误差积分控制并动态调节磁链参数来消除由于电机模型参数不准确引起的电流静差[7-8]；改进迟滞电流控制[9]、预置电流控制[10]、无源电流控制[11]、鲁棒滑模电流控制[12]等等。
在这些先进电流控制方法中，滑模控制(sliding mode control，SMC)是一种结构简单、响应快的非线性控制，对外界干扰和参数摄动等不确定性具有很强的鲁棒性，被广泛应用于运动伺服控制系统中[13]。
SMC利用非连续控制逻辑之间的切换来迫使控制系统在滑模面上运动，保持控制系统的鲁棒性，进而抑制系统的不确定因素，缺点是切换运动导致了控制系统的高频“抖振”。
同时，由于实际系统的不确定量和干扰往往是未知的，导致非连续控制逻辑的增益难以准确设定。
为了保证系统的稳定性，通常采用大增益控制，会加剧SMC系统的“抖振”[14]。
为了消除高频“抖振”现象，文献[15]利用平滑函数估计代替符号函数使得系统进入滑模面后不再进行切换运动，虽然减轻了“抖振”现象，但也弱化的系统的鲁棒性。
文献[12]提出在滑模电流控制的基础上，设计一个简单观测器估计参数摄动量和干扰作为前馈补偿量，减小符号函数的增益，减轻控制系统的“抖振”现象，提高了电流稳态精度。
这种利用观测器实现自适应增益的方法可有效减轻“抖振”现象，又能保持较好的鲁棒性，但是系统设计比较复杂，稳定性难以分析与保证。
目前，高阶滑模控制既能有效抑制“抖振”现象又能保持鲁棒性。
其中，二阶滑模控制将不连续控制作用在滑模量的二阶导数上，保证滑模量及其一阶微分在有限时间内收敛到零，有效消除“抖振”现象。
相对其它高阶滑模控制而言，二阶滑模以设计简单且对高阶信息需求量小而被广泛应用于工业场合[16]。
但是，基于线性滑模面的二阶滑模虽然能够很好地抑制“抖振”现象，但是很难在有限时间内令滑模量收敛到零[17]。
文献[18]为了提高最优控制方法的鲁棒性，提出一种基于两种滑模面的二阶滑模控制技术，特点是设计高阶滑模面可保证滑模量从任意初始状态收敛于高阶滑模面，再趋近线性滑模面，实现改进型二阶滑模控制有限时间内收敛。
本文提出一种基于二阶终端(second-order terminal，ST)滑模优化的PMSM电流环滑模控制(second-order terminal sliding mode control，STSMC)。
考虑系统参数摄动和干扰等不确定因素条件下，建立PMSM的电流环数学模型。
借助SMC的设计简单、容易实现等优点来获得PMSM电流环的等效控制量。
然后，针对系统参数摄动和干扰等不确定量，利用二阶终端滑模设计不确定量的控制量，即二阶终端滑模趋近律的控制量，引入自适应指数趋近律[19]，理论上可获得自适应零增益切换函数控制，达到有效减小“抖振”，能很好地抑制系统不确定量。
同时，对PMSM的电流环STSMC系统进行了稳定性分析，通过仿真与实验说明STSMC的控制性能。
在d-q轴正交坐标系下，矢量变换的PMSM非线性数学模型为[20]：式中：we为电角速度；Rs为定子阻抗；Ld和Lq分别为d-q轴电感；φ为永磁体与定子交链的磁链；id和iq分别为d-q轴定子电流；ud和uq为d-q轴定子电压；J为转子转动惯量；B为转子粘性摩擦系数；TL为负载转矩；Te为电机转矩；w为转子角速度；p为磁极对数，we=pw。
由式(1)和式(2)可见模型的准确度依赖于系统参数。
当实际系统发生可加性的参数摄动和干扰时，电流环模型可重新表示为：式中：dq和dd表示系统的不确定量；εq和εd为不可测干扰；ΔLq=Lq-Lqo；ΔLd=Ld-Ldo；ΔRs=Rs-Rso和Δφ=φ-φo，分别表示系统参数实际值Lq，Ld，Rs和φ与额定值Lqo，Ldo，Rso和φo之间的偏差量。
令x=[iq, id]T，u=[uq, ud]T，式(4)～式(7)的状态方程为其中A=[-Rso/Lqo, 0;0, -Rso/Ldo]，B=[1/Lqo, 0;0, 1/Ldo]，C=[-Ldoweid/Lqo-weφo/Lqo,Lqoweiq/Ldo]T为耦合量和反电动势，D=[-dq/Lqo, -dd/Ldo]T为系统不确定量。
定义电流的跟踪误差为其中和分别为参考输入信号。
同时定义一阶比例积分滑模面为式中：S为滑模面矢量，S=[sq,sd]T；k为正实数。
假设控制系统可得到适当的控制律，能保证在有限时间内到达滑模段，则滑模面有由电机状态模型式(8)中，在不考虑不确定量D时，由式(11)可得到电压等效控制律为式中：k=diag(k, k)。
式(12)可得到电机电流环的电压等效控制律u*。
但是，仅靠等效控制律式(12)是无法抑制不确定因素D而获得良好的电流性能。
为抑制不确定量D，需要设计不确定量的控制量Δu。
由式(9)求导和式(8)得到误差导数为重新定义一阶比例积分滑模面S为式中则滑模面S的一阶微分为由式(13)、式(14)和式(15)可得到考虑系统状态偏离滑模面时的可到达性，针对式(16)在满足前提下，设计各种基于趋近律的控制量Δu。
比如，采用指数趋近律S=-λS-ηsgn(S)，则Δu=B-1(λS+ηsgn(S)-D)。
在电机滑模控制系统中，参数摄动和干扰D是未知且非线性的，为了保证电流环控制系统的稳定，需要设置足够大的趋近律增益η作为不确定量的上界值：η≥|D|，则Δu=B-1(λS+ηs gn(S))，导致电流环高频抖振严重，降低了电流稳态精度[12]。
高阶滑模控制能很好地抑制滑模控制高频抖动和提高到达段性能。
其中二阶滑模以其设计简单，且对高阶信息依赖性少，被广泛应用于工业场合。
二阶滑模优点是将不连续控制作用在滑模面的二阶导数上，保证滑模面及其一阶导数能够在有限时间内收敛到0，有效消除了抖振。
但是，在滑模面式(10)和式(14)可知，控制量Δu不需要式(10)的积分初始条件，而控制系统的初始状态往往处于滑模面之外，导致收敛慢。
终端滑模控制却能保证在有限时间内从任意初始状态收敛到滑模面。
因此，在式(14)的基础上，构造二阶非奇异终端滑模面ξ[18]有其中ξ=[ξq, ξd]T，γ=diag(γq, γd)>0，α和β为终端滑模面参量，满足如下条件：由式(17)得到终端滑模面的一阶微分为由于α，β满足式(18)，则可得式(19)中γα/βα/β-1由常数ρ>0代替，则由文献[19]取滑模面ξ的自适应指数趋近律，其中λ，η>0，σ>0。