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
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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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。

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.。

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。

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. 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