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
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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
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3 id
133
(3)
图 3 电流环控制框图
Fig. 3 Block diagram of current loop control
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(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。
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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。