电机转速环节Super-Twisting算法二阶滑模控制律设计与研究

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基于super-twisting 二阶滑模算法的作业型rov路径跟踪控制方法

基于super-twisting 二阶滑模算法的作业型rov路
径跟踪控制方法
随着水下作业机器人(ROV, remotely operated vehicle) 技术的发展和应用，如何有效地设计控制系统以提高机器人的实际定位和导航性能也成为ROV研究的重要内容。

本研究将从滑模控制技术角度出发，探究基于Super-Twisting 二阶滑模算法的作业型ROV路径跟踪控制方法，并运用虚拟三维水下智能体框架MATLAB/SIMULINK 平台模拟仿真，基于算法设计性能指标如追踪误差、响应时间优势，进行系统设计参数优化，从而提高其路径跟踪控制效果及稳定性。

本研究可以为ROV研发和应用奠定一定的技术基础，为水下作业机器人的发展和应用提供支持。

本研究将由以下几部分组成：
首先，介绍虚拟水下智能体框架MATLAB/SIMULINK 平台，以及Six-DOF 机械臂系统仿真模型构建和参数优化方法，确立水下作业机器人模型及参数；
其次，研究Super-Twisting 二阶滑模算法的原理和波特兰跟踪控制器的设计，并与PID控制器做对比，探究其优势；
然后，利用MATLAB/SIMULINK 平台进行仿真，并在仿真的基础上，运用算法设计性能指标（如追踪误差、响应时间等指标），对波特兰跟踪控制器进行参数
优化，进而提高控制效果和稳定性
最后，结合自然环境下实际ROV路径跟踪控制实验结果，比较Matlab/Simulink 模拟结果和实验结果，得出仿真与实验结果是否一致，从而证明本研究设计策略的有效性。

二阶滑模控制方法及其对环形倒立摆的控制应用

的ｒ阶滑动集描述形式如下：＝＝＝… ＝ｓ＝０。
２．２超螺旋（Ｓｕｐｅｒ－Ｔｗｉｓｔｉｎｇ）算法
超螺旋算法与其他二阶滑模算法相比存在着特殊之
高阶滑模的定义是由Ｌｅｖａｎｔ和Ｆｉｒｄｍａｎ在１９６６年给
二阶滑模控制方法及其对环形倒立摆的控制应用
Ｓｅｃｏｎｄ－ｏｒｄｅｒｓｌｉｄｉｎｇｍｏｄｅｃｏｎｔｒｏ－ａｎｄｉｔｓａｐｐｌｉｃａｔｉｏｎｔｏｒｏｔａｒｙｉｎｖｅｒｔｅｄｐｅｎｄｕｌｕｍ
孙灵芳，邢宇，李斌
滑模控制存在 “ 抖振 ”现象，这在实际工作中具有很大
的危害．近几十年来，科研工作者们一直在寻找抑制 “ 抖振 ”现象的方法，如 “ 边界层 ”法、滤波方法、高为炳提出的 “ 趋近律 ”法等。在此期间，由Ｌｅｖａｎｔ￣出的高阶滑模控制方法得到了广泛的关注．高阶滑模不但可以保
出的，如下：
处，它仅仅需要滑模变量Ｓ的信息，而且当控制系统的
持传统滑模具有的优点，而且可以有效地抑制了传统滑模
控制产生的 “ 抖振 ”，从而提高了控制的精度翻。
１高阶滑模的定义
首先介绍滑动阶（ＳｌｉｄｉｎｇＯｒｄｅｒ）的定义】：
１＝（ｆ，）
其中ｅＲ “ 为系统状态量，ＵＥＲ为控制输入，ｆ（ｔ，ｘ）＊Ｉｇ（ｔ，ｘ）为不确定光滑函数，ｓ＝ｓ（ｔ，ｘ）为滑模面。二阶滑模控制的目标是使系统状态在有限的时间内到达滑模面ｓ＝ｓ（ｔ，ｘ）并且具有二阶滑动模态。

基于super-twisting滑模的永磁同步电机转矩环控制器设计

基于super-twisting滑模的永磁同步电机转矩环控制器设计葛如愿;邓福军
【期刊名称】《微电机》
【年(卷),期】2018(051)002
【摘要】针对永磁同步电机的直接转矩控制系统中因扰动引起转矩脉动及系统鲁棒性差的问题,本文采用一种基于Super-twisting滑模的转矩环控制方法,设计转矩和磁链控制器.根据滑模变结构控制的特点来抑制系统中的扰动,进而减小转矩脉动并增加系统的鲁棒性;同时在分析Super-twisting滑模的基础上,将其中的开关函数用双曲正切函数替换,系统在高频运动下没有明显抖振,效果良好.与常规的滞缓控制相比,Super-twisting滑模控制的转矩脉动明显减少,且对电机的扰动具有更强的鲁棒性,仿真结果验证了该法的有效性.
【总页数】6页(P51-55,68)
【作者】葛如愿;邓福军
【作者单位】大连交通大学,大连116028;大连交通大学,大连116028
【正文语种】中文
【中图分类】TM351;TM341
【相关文献】
1.基于永磁同步电机直接转矩复合型滑模控制器的设计 [J], 姜文;贺昱曜;刘伟超;周淳
2.基于Super-twisting滑模永磁同步电机驱动的转速和转矩控制 [J], 万东灵;赵
朝会;王飞宇;孙强
3.基于Super-twisting滑模控制的PMSM直接转矩控制中速度控制器研究 [J], 张惠智;王英
4.基于Super-twisting算法的永磁同步电机自适应滑模速度控制 [J], 汤成;胡继胜
5.基于Super-twisting算法的永磁同步电机直接转矩控制 [J], 李少朋;谢源;张凯;贺耀庭
因版权原因，仅展示原文概要，查看原文内容请购买。

基于二阶滑模与定子电阻自适应的转子磁链观测器及其无速度传感器应用

基于二阶滑模与定子电阻自适应的转子磁链观测器及其无速度传感器应用黄进;赵力航;刘赫【摘要】针对感应电机高性能矢量控制需求,设计一种基于Super-Twisting二阶滑模理论的转子磁链观测器,并提出以其观测结果作为参考模型的无速度传感器控制方案.该观测器属于非线性滑模观测器,充分利用了辅助滑模面,因而对电机转子电阻变化以及外部扰动具有良好的鲁棒性,且反应速度优于传统电压型转子磁链观测器.而Super-Twisting理论无法处理的定子侧参数变化与扰动问题,则由并行定子电阻辨识机构进行修正.实验证明,转子磁链观测结果的相位和幅值较为准确.整套控制方案的有效性也经由仿真和实验得到验证.【期刊名称】《电工技术学报》【年(卷),期】2013(028)011【总页数】8页(P54-61)【关键词】感应电机;模型参考自适应;无传感器控制;二阶滑模;Super-Twisting 【作者】黄进;赵力航;刘赫【作者单位】浙江大学电气工程学院杭州 310027;浙江大学电气工程学院杭州310027;浙江大学电气工程学院杭州 310027【正文语种】中文【中图分类】TM3511 引言自20 世纪70 年代发展至今，三相异步电机矢量控制技术已趋成熟，能够满足大部分工业需求。

而精确的转子磁场定向和准确的转速观测，是保证矢量控制效果的两个最重要的因素。

为了能通过电机外部量“观测”电机内部的磁通，大部分控制方案采用的是基于数学模型的间接磁场定向矢量控制。

该方法的控制性能受数学模型准确性和电机参数稳定性的影响，其中定子电阻与转子时间常数影响最大。

另一方面，转速是非常重要的反馈量。

传统的机械式速度传感器会带来额外的成本和安装维护方面的问题，因此无速度传感器技术一直是电机控制领域的重要研究方向。

转子磁链和转速的观测可以在只测得定子侧电信号的前提下实现[1]。

将定子电流、转子磁链作为状态变量写出异步电机数学模型，在此基础上即可构建转子磁链观测器，并实现转速辨识。

Supertwisting

A Super-Twisting Algorithm for Systems of Relative Degree MoreThan OnePost ASCC2013paperAbstract—This paper presents a data-driven homogeneous continuous super-twisting algorithm for systems with relative de-gree more than one.The conditions ofﬁnite-time convergence to an equilibrium are obtained demonstrating that the equilibrium can be moved as close to the origin as necessary,increasing a value of the control gain.The paper concludes with numerical simulations illustrating performance of the designed algorithms.Index Terms—Sliding mode control,super-twisting,systems of relative degree more than oneI.I NTRODUCTIONIt is well known that the classical discontinuous sliding mode control providesﬁnite-time convergence for a system of relative degree one[1].Aﬁnite-time stabilizing control for a system of relative degree two is realized using the twisting algorithm[2],where the second order sliding mode control is also discontinuous.Both algorithms are robust with respect to bounded disturbances.On the other hand,using a continuous second-order sliding mode super-twisting algorithm[3],a state of a relative degree one system can be stabilized along with itsﬁrst derivative.The super-twisting algorithm is robust with respect to unbounded disturbances satisfying a Lipschitz con-dition.Theﬁnite-time convergence of the designed algorithms is conventionally established using geometrical techniques[2], [3],direct Lyapunov method[4],[5],[6],or homogeneity approach[7],[8].The explicit Lyapunov functions for their second-order super-twisting algorithms can be found in[6]. The homogeneity approach,mentioned even in the classical book[9],was consistently developed in the mentioned papers and applied to the observer design in[10].Various modiﬁcations of the sliding mode technique have always been actively used in industrial applications([11], [12],[13],[14],[15],[16],[17]),including data-driven ones ([18],[19],[20],[21]).This paper presents a data-driven homogeneous continuous super-twisting algorithm for systems with relative degree more than one.First,the case of relative degree two is addressed.The conditions ofﬁnite-time conver-gence to an equilibrium are obtained in demonstrating that the equilibrium can be moved as close to the origin as necessary, increasing a value of the control gain.Then,a modiﬁcation of the designed algorithm enablingﬁnite-time convergence to the origin is proposed,based on the knowledge of the equilibrium from the previous run.Finally,the robustness of the designed data-driven algorithm is discussed.Similar results are then obtained for systems of relative degree more then two. The paper concludes with numerical simulations illustrating performance of the designed algorithms.The simulation results are discussed and demonstrated in a number ofﬁgures.The paper is organized as follows.The problem statement is given in Section2.A super-twisting-like control algorithm for systems of relative degree two is designed in Section3.The corresponding examples are provided in Section4.A super-twisting-like control algorithm for systems of relative degree more than two is presented in Section5and illustrated by examples in Section6.The proofs of all theorems are given in Appendix.A brief conference version of this paper was presented in[22].II.C ONTROL P ROBLEM S TATEMENT Consider a conventional dynamic system of relative degree two˙x1(t)=x2(t),x1(t0)=x10,(1)˙x2(t)=u(t),x1(t0)=x20,where x(t)=[x1(t),x2(t)]∈R2is the system state and u(t)∈R is the control input.In the classical second-order sliding mode control theory, aﬁnite-time stabilizing control for the system(1)is designed using the twisting algorithm[2]in the formu(t)=−k1sign(x1(t))−k2sign(x2(t)),(2) where k1,k2>0are certain positive constants,and the signum function of a scalar x is deﬁned as sign(x)=1,if x>0, sign(x)=0,if x=0,and sign(x)=−1,if x<0([9]).On the other hand,if a scalar dynamic system is of relative degree one˙x(t)=u(t),x(t0)=x0,(3) a continuousﬁnite-time stabilizing control for the system(3) can be designed using the super-twisting algorithm[3]as followsu(t)=−λ|x(t)|1/2sign(x(t))−α∫tt0sign(x(s))ds,(4)whereλ>0,α>0are certain positive constants.Note that applying the continuous control(4)to the system(3) results in a second-order sliding mode,i.e.,both x(t)and ˙x(t)converge to zero for aﬁnite time.In other words,the continuous control(4)yieldsﬁnite-time convergence similar to that produced by a classical discontinuous sliding mode control u(t)=−Ksign(x(t)),where K>0is sufﬁciently large, for the system(3).In this paper,we propose a homogeneous super-twisting-like continuous modiﬁcation of the twisting control algorithm (2)as followsu(t)=−λ1|x1(t)|1/3sign(x1(t))−−λ2|x2(t)|1/2sign(x2(t))−α∫tt0sign(x2(s))ds,(5)where λ1,λ2>0,α>0are certain positive constants.It would be demonstrated that the designed continuous control (5)works similarly to the twisting control (2),i.e.,results in ﬁnite-time convergence of both states x 1(t )and x 2(t )of the system (1).The announced result is formalized in the next section and then proved in Appendix.III.S UPER -T WISTING A LGORITHM FOR R ELATIVED EGREE T WO S YSTEMS The result for the control law (5)is given as follows.Theorem 1.Consider a dynamic system (1)of relative degree two.Then,the modiﬁed super-twisting control law (5)yields ﬁnite-time convergence of both states x 1(t )and x 2(t )to a point [x 1f ,0].Proofs of all the theorems are given in Appendix.Remark 1.Theorem 1ensures ﬁnite-time stability of the system (1)with the control law (5)with respect to an equilibrium point [x 1f ,0]located in the manifold x 2=0,which is however different from the origin.Nonetheless,the equilibrium point [x 1f ,0]could be moved as close to the origin as necessary,increasing a value of the control gain λ1>0,inview of the inequality |x 1f |1/3≤λ−11αT 1,where T 1is the ﬁnite convergence time.Although Remark 1underlines that the control law (5)cannot lead both states of the system (1)to the origin,this problem can be solved using the control law proposed in the following theorem.Theorem 2.Consider a dynamic system (1)of relative degree two.If upon applying the control law (5),the system (1)was ﬁnite-time stabilized at a point [x 1f ,0],it can be stabilized at the origin from the same initial condition [x 10,x 20]using the control lawu (t )=−λ1|x 1(t )+x 1f |1/3sign (x 1(t )+x 1f )−−λ2|x 2(t )|1/2sign (x 2(t ))−α∫tt 0sign (x 2(s ))ds .(6)Consider now a system (1)in presence of a disturbance:˙x 1(t )=x 2(t ),x 1(t 0)=x 10,(7)˙x 2(t )=u (t )+ξ(t ),x 1(t 0)=x 20,where ξ(t )satisﬁes the Lipschitz condition with a constant L .The system (7)can still be stabilized at a point [x 1f ,0]in view of the following theorem.Theorem 3.Consider a dynamic system (7)of relative degree two in presence of a disturbance ξ(t )satisfying the Lipschitz condition with a constant L .Then,the modiﬁed super-twisting control law (5)yields ﬁnite-time convergence of both states x 1(t )and x 2(t )to a point [x 1f ,0],provided that the following conditions hold for control gains:α>L ,λ22>2(α+L )2/(α−L ).IV.E XAMPLES :I.R ELATIVE D EGREE T WOThis section presents examples of designing a ﬁnite-time stabilizing regulator for a dynamic system (1)of relative degree two,based on the modiﬁed super-twisting regulator (5)in Theorems 1–3.1.Consider a linear system (1).The modiﬁed super-twisting regulator (4)is applied with the control gains selected as λ1=20,λ2=10,α=1.The initial conditions are assigned as x 10=1000,x 20=1000.The obtained results are shown in Fig.1.The ﬁnal value of x 1(t )is equal to x 1f =0.0254.The initial conditions x 10=1000,x 20=−1000yield x 1f =−0.0004.The obtained results are shown in Fig.2.2.Consider the linear system from the previous example,assigning the same initial conditions,x 10=1000,x 20=1000and x 10=1000,x 20=−1000,and applying the control law (6),with λ1=20,λ2=10,α=1to both the considered cases,assuming x 1f =0.0254and x 1f =−0.0004.It can be observed from Figs.3and 4and their ampliﬁcations around the ﬁnal time,Figs.5and 6,respectively,that the system is stabilized at the origin in each case.3.Consider a linear system (7)with disturbance ξ(t )=sin (1000t ).Again,the modiﬁed super-twisting regulator is applied with the control gains selected as λ1=20,λ2=10,α=1.The initial conditions are assigned as x 10=1000,x 20=1000.The obtained results are shown in Fig.7.The ﬁnal value of x 1(t )is equal to x 1f =0.0244.The initial conditions x 10=1000,x 20=−1000yield x 1f =−0.0007.The obtained results are shown in Fig.8.This example clearly demonstrates that the sufﬁcient condi-tions for the control gains in Theorem 3are too conservative,and the ﬁnite-time convergence takes place with much relaxed values.In particular,the value of constant L in this example is equal to 1000,due to high-frequency sinusoidal oscillations sin (1000t ).V.S UPER -T WISTING A LGORITHM FOR R ELATIVE D EGREEM ORE T HAN T WO S YSTEMS The main result can be generalized as follows.Consider a dynamic system of relative degree n >2˙x 1(t )=x 2(t ),x 1(t 0)=x 10,(8)˙x 2(t )=x 3(t ),x 2(t 0)=x 20,···˙x n (t )=u (t ),x n (t 0)=x n 0,using the notation for the system (1).We propose a gener-alization the super-twisting-like continuous control algorithm (5)as followsu (t )=−v 1(t )−v 2(t )−...−v n (t )+v n +1(t ),(9)where v i (t )=λi |x i (t )|γi sign (x i (t )),i =1,...,n ,v n +1(t )=|(s (t ))|γ/(1−γ)sign (s (t )),s (t )=−α∫tt 0sign (x n (s ))ds ,and certain positive constants λ1,...,λn >0,α>0and exponents γi ,i =1,...,n ,are assigned according to [7]to yield homogeneous ﬁnite-time convergence of all the states of the closed-loop system (8),(9).Namely,γi ∈(0,1),i =1,...,n satisfy the recurrent relations γi −1=γi γi +1/(2γi +1−γi ),i =2,...,n ,γn +1=1,and γn =γ,where γbelongs to an interval(1−ε,1),ε>0.Theorem8.1in[7]establishes that there exists such anε>0that the homogeneous(in view of deﬁnition ofγi)closed-loop system(8),(9)is globallyﬁnite-time stable at the origin.It would be demonstrated that the designed continuous con-trol(7)works similarly to the twisting control(5),i.e.,results inﬁnite-time convergence of the states x1(t),x2(t),...,x n(t)of the system(6).The announced result is formalized in the next theorem and then proved in Appendix.Theorem 4.Consider a dynamic system(8)of rela-tive degree n>2.Then,the modiﬁed super-twisting con-trol law(9)yieldsﬁnite-time convergence of the states x1(t),...,x n−1(t),x n(t)to a point[x1f,...,x(n−1)f,0]. Remark2.Theorem4ensuresﬁnite-time stability of the system(8)with the control law(9)with respect to an equilib-rium point[x1f,...,x(n−1)f,0]located in the manifold x n=0, which is however different from the origin.Then,it follows from the equations(8)that x2f=x3f=...=x(n−1)f=0. Thus,the equilibrium point is given by[x1f,0...,0,0]and located in the manifold x2=x3=...=x n=0.Nonetheless, the equilibrium point[x1f,0...,0,0]could be moved as close to the origin as necessary,increasing a value of the control gainλ1>0in such a way thatλ1,...,λn still correspond to a Hurwitz polynomial s n+λn s n−1+...+λ2s+λ1(see[7]), in view of the inequality|x1f|1/3≤λ−11αT1,where T1is the ﬁnite convergence time.Although Remark2underlines that the control law(9) cannot lead all states of the system(1)to the origin,this problem can be solved using the control law proposed in the following theorem.Theorem 5.Consider a dynamic system(8)of relative degree n>2.If upon applying the control law(9),the system (8)wasﬁnite-time stabilized at a point[x1f,...,x(n−1)f,0],it can be stabilized at the origin from the same initial condition [x10,...,x(n−1)0,x n0]using the control lawu(t)=−w1(t)−w2(t)−...−w n−1(t)−v n(t)+v n+1(t), where w i(t)=λi|x i(t)+x i f|γi sign(x i(t)+x i f),i=1,...,n−1.Consider now a system(8)in presence of a disturbance:˙x1(t)=x2(t),x1(t0)=x10,(10)˙x2(t)=x3(t),x2(t0)=x20,···˙x n(t)=u(t)+ξ(t),x n(t0)=x n0,whereξ(t)satisﬁes the Lipschitz condition with a con-stant L.The system(10)can still be stabilized at a point [x1f,...,x(n−1)f,0]in view of the following theorem. Theorem6.Consider a dynamic system(10)of relative degree n>2in presence of a disturbanceξ(t)satisfying the Lipschitz condition with a constant L.Then,the modiﬁed super-twisting control law(9)yieldsﬁnite-time convergence of the states x1(t),...,x n−1(t),x n(t)to a point[x1f,...,x(n−1)f,0], provided that the following conditions hold for control gains:α>L,λ2n>2(α+L)2/(α−L).VI.E XAMPLES:II.R ELATIVE D EGREE M ORE T HAN T WO This section presents examples of designing aﬁnite-time stabilizing regulator for a dynamic system(8)of a relative degree more than two,based on the modiﬁed super-twisting regulator(9)in Theorems4–6.4.Consider a linear3D system˙x1(t)=x2(t),x1(t0)=x10,(11)˙x2(t)=x3(t),x2(t0)=x20,˙x3(t)=u(t),x3(t0)=x30,The modiﬁed super-twisting regulator(9)u(t)=−λ1|x1(t)|1/4sign(x1(t))−λ2|x2(t)|1/3sign(x2(t))−λ3|x3(t)|1/2sign(x3(t))−α∫tt0sign(x3(s))ds,(12)is applied with the control gains selected asλ1=λ2=20,λ3=10,α=1.The initial conditions are assigned as x10= x20=x30=1000.The obtained results are shown in Fig.9. Theﬁnal value of x1(t)is equal to x1f=−6.03×10−9.The initial conditions x10=x30=1000,x20=−1000yield x1f=−0.00017.The obtained results are shown in Fig.10.5.Consider a linear system(11)with disturbanceξ(t)= sin(1000t).Again,the modiﬁed super-twisting regulator is applied with the control gains selected asλ1=λ2=20,λ3=10,α=1.The initial conditions are assigned as x10= x20=x30=1000.The obtained results are shown in Fig.11.Theﬁnal value of x1(t)is equal to x1f=−1.38×10−7. The initial conditions x10=x30=1000,x20=−1000yield x1f=−1.13×10−5.The obtained results are shown in Fig.12.This example again demonstrates that the sufﬁcient condi-tions for the control gains in Theorem6are too conservative, and theﬁnite-time convergence takes place with much relaxed values.In particular,the value of constant L in this example is equal to1000,due to high-frequency sinusoidal oscillations sin(1000t).VII.A PPENDIXA.Proof of Theorem1.The system(1),(5)can be recast in the time-invariant form˙x1(t)=x2(t),x1(t0)=x10,(13)˙x2(t)=−λ1|x1(t)|1/3sign(x1(t))−−λ2|x2(t)|1/2sign(x2(t))+x3(t),x2(t0)=x20,˙x3(t)=−αsign(x2(t)),x3(t0)=0.The vectorﬁeld f in the right-hand side of(13)can be represented as the sum of two homogeneous vectorﬁelds, f=g1+g2,where g1=[x2,−λ1|x1(t)|1/3sign(x1(t))−ρλ2|x2(t)|1/2sign(x2(t)),0],ρ∈(0,1),and g2=[0,−(1−ρ)λ2|x2(t)|1/2sign(x2(t))+x3(t),−αsign(x2(t))]of homo-geneity degree m2=−1.The homogeneity degree m1for g1 can be selected as m1=−2<m2.Theﬁeld g1provides theﬁnite-time stability at a point[0,0,x3(t0)]in view of its homo-geneity and Lyapunov function V(x1,x2)=λ1(3/4)|x1(t)|4/3 +(1/2)|x2(t)|2.Theﬁeld g2corresponds to a super-twisting algorithm[3],which converges to a point[x1f,0,0]for aﬁnite time.The theorem assertion now follows from Theorem7.4in [7],taking into account that the Lyapunov function for super-twisting has a continuous total derivative in time along the trajectory,so the results of Theorem6.2,Lemma4.2and the inequalities(34)-(36)from[7]hold.B.Proof of Theorem2.The theorem assertion follows from the fact that the equilib-rium[x1f,0,x3f]of the time-invariant representation(13)for the system(1),(5)can be moved to a point[0,0,x3f]using the coordinate change x1−x1f,x2,x3.C.Proof of Theorem3.The system(7)can be recast in the time-invariant form˙x1(t)=x2(t),x1(t0)=x10,(14)˙x2(t)=−λ1|x1(t)|1/3sign(x1(t))−−λ2|x2(t)|1/2sign(x2(t))+x3(t),x2(t0)=x20,˙x3(t)=−αsign(x2(t))+˙ξ(t),x3(t0)=0. where˙ξ(t)exists and is bounded for almost all t≥t0.The the-orem assertion follows from Theorem1and the convergence conditions for a super-twisting algorithm[3].D.Proof of Theorem4.The system(8),(9)can be recast in the time-invariant form˙x1(t)=x2(t),x1(t0)=x10,(15)˙x2(t)=x3(t),x2(t0)=x20,···˙x n−1(t)=x n(t),x n−1(t0)=x(n−1)0,˙x n(t)=−v1(t)−v2(t)−...−v n(t)++|x n+1(t)|γ/(1−γ)sign(x n+1(t)),x n(t0)=x n0,˙x n+1(t)=−αsign(x n(t)),x n+1(t0)=0. Similarly to(13),the vectorﬁeld f in the right-hand side of(13)can be represented as the sum of two homogeneous vectorﬁelds,f=g1+g2,where g1=[x2,x3,...,x n,−v1(t)−v2(t)−...v n−1(t)−ρv n(t),0],ρ∈(0,1),of homogeneity de-gree m1=(γ−1)/γ<0,and g2=[0,0,...,0,−(1−ρ)v n(t)+ v n+1(t),−αsign(x n(t))].The homogeneity degree m2for g2 can always be selected greater than m1,m2>m1=(γ−1)/γ, by virtue of exponentγ/(1−γ)for v n+1,making the entire system(15)homogeneous.Theﬁeld g1provides theﬁnite-time stability at a point[0,...,0,x n+1(t0)]in view of Theorem8.1in [7].Theﬁeld g2introduces a modiﬁcation of a super-twisting algorithm[3],which converges to a point[x1f,...,x(n−1)f,0,0] for aﬁnite time,in view of Lyapunov function V(x n,x n+1)=α|x n(t)|+(1−γ)|x n+1(t)|1/(1−γ).The theorem assertion now follows from Theorem7.4in[7],taking into account that V(x n,x n+1)has a continuous total derivative in time along the trajectory,so the results of Theorem6.2,Lemma4.2and the inequalities(34)-(36)from[7]hold. E.Proof of Theorem5.The theorem assertion follows from the fact that the equi-librium[x1f,...,x(n−1)f,0,x(n+1)f]of the time-invariant rep-resentation(15)for the system(8),(9)can be moved to a point[0,...,0,0,x(n+1)f]origin using the coordinate change x1−x1f,...,x n−1−x(n−1)f,x n,x n+1.F.Proof of Theorem6.The system(10)can be recast in the time-invariant form˙x1(t)=x2(t),x1(t0)=x10,˙x2(t)=x3(t),x2(t0)=x20,···˙x n−1(t)=x n(t),x n−1(t0)=x(n−1)0,˙x n(t)=−v1(t)−v2(t)−...−v n(t)++|x n+1(t)|γ/(1−γ)sign(x n+1(t)),x n(t0)=x n0,˙x n+1(t)=−αsign(x n(t))+˙ξ(t),x n+1(t0)=0. where˙ξ(t)exists and is bounded for almost all t≥t0.The the-orem assertion follows from Theorem4and the convergence conditions for a super-twisting algorithm[3].Fig.1.Graphs of the linear system(1)upon applying the control law(4) with initial conditions x10=1000,x20=1000.Theﬁnal value of x1(t)is equal to x1f=0.0254.R EFERENCES[1]V.I.Utkin,Sliding Modes in Control and Optimization,Springer,1992.[2] A.Levant,“Sliding mode and sliding accuracy in sliding mode control,”International Journal of Control,V ol.58,1993,pp.1247–1263. [3] A.Levant,“Robust exact differentiation via sliding mode technique,”Automatica,V ol.34,1998,pp.379–384.[4] A.Polyakov and A.Poznyak,“Lyapunov function design forﬁnite-timeconvergence analysis:Twisting controller for second-order sliding mode realization,”Automatica,V ol.45,2009,pp.444–448.Fig.2.Graphs of the linear system(1)upon applying the control law(4) with initial conditions x10=1000,x20=−1000.Theﬁnal value of x1(t)is equal to x1f=−0.0004.Fig.3.Graphs of the linear system(1)upon applying the control law(6) with initial conditions x10=1000,x20=1000and x1f=0.0254.[5] A.Polyakov and A.Poznyak,“Reaching time estimation for super-twisting second order sliding mode controller via Lyapunov function design,”IEEE Trans.on Automatic Control,V ol.54,2009,pp.1951–1955.[6]J.A.Moreno and M.Osorio,“Strict Lyapunov functions for the super-twisting algorithm,”IEEE Trans.on Automatic Control,V ol.57,2012, pp.1035–1040.[7]S.P.Bhat and D.S.Bernstein,“Geometric homogeneity with appli-cations toﬁnite-time stability,”Mathematics of Control,Signals,and Systems,V ol.17,2005,pp.101–127.[8] A.Levant,“Homogeneity approach to high-order sliding mode design,”Automatica,V ol.41,2005,pp.823–830.[9] A.F.Filippov,Differential Equations with Discontinuous RighthandSides,Kluwer,1988.[10]W.Perruquetti,T.Floquet,and E.Moulay,“Finite-time observers:appli-cations to secure communication,”IEEE Trans.on Automatic Control,Fig.4.Graphs of the linear system(1)upon applying the control law(6) with initial conditions x10=1000,x20=−1000and x1f=−0.0004.Fig.5.Graphs of the linear system(1)upon applying the control law(6) with initial conditions x10=1000,x20=1000and x1f=0.0254.(Ampliﬁed) V ol.53,2008,pp.356–360.[11] E.Kayacan,Y.Oniz,and O.Kaynak,“A grey system modeling approachfor sliding-mode control of antilock braking system,”IEEE Trans.on Industrial Electronics,V ol.56,2009,pp.3244–3252.[12]Y.Xia,Z.Zhu,M.Fu,and S.Wang,“Attitude tracking of rigid spacecraftwith bounded disturbances,”IEEE Trans.on Industrial Electronics,V ol.57,2010,pp.647–659.[13]Y.Xia,Z.Zhu,and M.Fu,“Back-stepping sliding mode control formissile systems based on extended state observer,IET Control Theory and Applications,V ol.5,2011,pp.93–102.[14] E.Kayacan,O.Cigdem,and O.Kaynak,“Sliding mode control approachfor online learning as applied to type-2fuzzy neural networks and its experimental evaluation,”IEEE Trans.on Industrial Electronics,V ol.59,2012,pp.3510–3520.[15]H.Gao,W.Zhan,H.R.Karimi,X.Yang,and S.Yin,“Allocation of ac-tuators and sensors for coupled-adjacent-building vibration attenuation,”IEEE Trans.on Industrial Electronics,V ol.60,2013,pp.5792–5801.Fig.6.Graphs of the linear system(1)upon applying the control law(6)with initial conditions x10=1000,x20=−1000and x1f=−0.0004.(Ampliﬁed)Fig.7.Graphs of the linear system(7)with disturbanceξ(t)=sin(1000t) upon applying the control law(4)with initial conditions x10=1000,x20= 1000.Theﬁnal value of x1(t)is equal to x1f=−0.0244.[16]Z.Zhu,D.Xu,J.Liu,and Y.Xia,“Missile guidance law based onextended state observer,”IEEE Trans.on Industrial Electronics,V ol.60,2013,pp.5882–5891.[17]H.Li,X.Jing,H.R.Karimi,“Output-feedback-based control for vehiclesuspension systems with control delay,”IEEE Trans.on Industrial Electronics,V ol.61,2014,pp.436–446.[18]L.Wu,P.Shi,and H.Gao,“State estimation and sliding-mode control ofMarkovian jump singular systems,IEEE Trans.on Automatic Control, V ol.55,2010,pp.1213–1219.[19]L.Ma,Z.Wang,Y.Bo,and Z.Guo,“Robust H∞sliding mode control fornonlinear stochastic systems with multiple data packet losses,”Intern.Journal of Robust and Nonlinear Control,V ol.22,2012,pp.473–491.[20]L.Wu,X.Su,and P.Shi,“Sliding mode control with bounded L2gain performance of Markovian jump singular time-delay systems,”Automatica,V ol.48,2012,pp.1929–1933.[21]J.Hu,Z.Wang,H.Gao,and L.K.Stergioulas,“Robust sliding modeFig.8.Graphs of the linear system(7)with disturbanceξ(t)=sin(1000t) upon applying the control law(4)with initial conditions x10=1000,x20=−1000.Theﬁnal value of x1(t)is equal to x1f=−0.0007.Fig.9.Graphs of the linear system(11)upon applying the control law(9)with initial conditions x10=x20=x30=1000.Theﬁnal value of x1(t)is equalto x1f=−6.03×10−9.control for discrete stochastic systems with mixed time delays,randomlyoccurring uncertainties,and randomly occurring nonlinearities,”IEEETrans.on Industrial Electronics,V ol.59,2012,pp.3008–3015. [22]M.V.Basin and P.Rodriguez-Ramirez,“A super-twisting algorithm forsystems of relative degree more than one,”Proc.9th Asian ControlConference,Istanbul,Turkey,2013.Fig.10.Graphs of the linear system(11)upon applying the control law (9)with initial conditions x10=x30=1000,x20=−1000.Theﬁnal value of x1(t)is equal to x1f=−0.00017.Fig.11.Graphs of the linear system(11)with disturbanceξ(t)=sin(1000t) upon applying the control law(9)with initial conditions x10=x20=x30= 1000.Theﬁnal value of x1(t)is equal to x1f=−1.38×10−7.Fig.12.Graphs of the linear system(11)with disturbanceξ(t)=sin(1000t) upon applying the control law(9)with initial conditions x10=x30=1000, x20=−1000.Theﬁnal value of x1(t)is equal to x1f=−1.13×10−5.。

基于PMSM的二阶滑模无位置传感器控制

乡驱动控制rie and c ontrl--飆蒔电力□2021年第49卷第1期基于PMSM的二阶滑模无位置传感器控制蔡军，李鹏泽，黄袁园(重庆邮电大学自动化学院，重庆400065)摘要:根据Super-twisting算法设计了二阶STASMO无位置传感器控制方案，该方案不仅充分地抑制了抖振现象，而且取消了低通滤波器的使用。

当电机运行时，定子电阻会随着温度的升高而变化，研究了旋转坐标系下的定子电阻观测器方案来实时观测定子电阻，避免了定子电阻变化对电机位置或速度估计精度的影响。

仿真分析表明该方案对电机位置或速度有较高的估计精度。

关键词:永磁同步电机;超螺旋算法滑模观测器;无位置传感器控制;定子电阻观测器中图分类号：TM351,TM464文献标志码:A文章编号:1004-7018(2021)01-0032-05PMSM Based Second-Order Sliding Mode Position Sensorless ControlCAI Jun,LI Peng-ze,HUANG Yuan-yuan(College of Automation,Chongqing University of Posts and Telecommunications,Chongqing400065,China) Abstract:The designs a second-order STASMO position sensorless control scheme based on the Super-twisting algorithm,which not only sufficiently suppresses chattering,but also eliminates the use of low-pass filters.When the motor is running,the stator resistance will change as the temperature rises.The proposes a stator resistance observer scheme in the rotating coordinate system to observe the stator resistance in real time,the influence of the change of the stator resistance on the accuracy of the position or speed estimation of the motor is avoided.The simulation analysis of the scheme proposed,it is proved that the scheme proposed has higher estimation accuracy for the motor position or speed.Key words:permanent magnet synchronous motor(PMSM),super-twisting algorithm based sliding-mode observer ( STASMO),position sensorless control,stator resistance observer羅M ■•咖轉中PMSMEI |盒班擲朮归鋼迪巒噩理軽铝0引言永磁同步电机（以下简称PMSM）因为具有功率密度高、转动惯量小和动态性能好等优势而被广泛应用于众多传动系统中。

基于super-twisting滑模的永磁同步电机转矩环控制器设计

=>引>言
由于永磁电机 9'-' 具有高效率高功率密度动态响应快结构简单等优点 9'-' 的驱动控制成为了一个热门的研究课题# 然而永磁同步电机驱动的状态估计和控制比感应电机更具挑战性主要是因为用于永磁同步电动机的有效方程数目少数学模型单一
直接转矩控制 c,) 是感应电机驱动的一种先进控制策略后来被扩展应用到永磁同步电机传
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23598136_基于super-twisting_二阶滑模算法的作业型ROV_路径跟踪控制方法

第29卷第1期水下无人系统学报 Vol.29No.12021年2月JOURNAL OF UNMANNED UNDERSEA SYSTEMS Feb. 2021收稿日期: 2020-03-13; 修回日期: 2020-04-23.基金项目: 国家重点研发计划项目资助(2017YFC0306704).作者简介: 黄博伦(1989-), 男, 在读博士, 主要研究方向为水下机器人控制技术.[引用格式] 黄博伦, 杨启. 基于super-twisting 二阶滑模算法的作业型ROV 路径跟踪控制方法[J]. 水下无人系统学报, 2021,29(1): 14-22.基于super-twisting 二阶滑模算法的作业型ROV 路径跟踪控制方法黄博伦1, 杨启1,2(1. 上海交通大学高新船舶与海洋开发装备协同创新中心海洋工程国家重点实验室, 上海, 200240; 2. 上海交通大学海洋水下工程科学研究院有限公司, 上海, 200231)摘要: 作业型遥控无人水下航行器(ROV)的运动存在时变外界干扰和系统不确定性, 利用常规滑模方法设计其运动控制器会产生抖振现象, 而常用的饱和函数联合边界层法(SatSMC)在消除抖振的同时无法保证控制精度。

针对上述问题, 文中设计了super-twisting 二阶滑模控制器(STSMC)来实现作业型ROV 的空间路径跟踪。

利用Lyapunov 方法分析了系统的稳定性, 并证明该方法能够保证跟踪误差在有限时间内收敛。

将提出的STSMC 与SatSMC 及比例-积分-微分法进行了仿真试验对比, 结果表明: STSMC 能够使ROV 完成对既定路径的跟踪, 并具有更好的鲁棒性、快速性和控制精度, 同时产生的抖振也明显小于SatSMC, 控制参数也未增加, 更适于ROV 的实际使用。

关键词: 遥控无人水下航行器; super-twisting 算法; 滑模控制; 路径跟踪中图分类号: TJ630; TB53 文献标识码: A 文章编号: 2096-3920(2021)01-0014-09 DOI: 10.11993/j.issn.2096-3920.2021.01.003Trajectory Tracking Control Method of a Work-class ROV Based on aSuper-twisting Second-order Sliding Mode ControllerHUANG Bo-lun 1, YANG Qi 1,2(1. Collaborative Innovation Center for Advanced Ship and Deep-sea Exploration, State Key laboratory of Ocean Engi-neering, Shanghai Jiao Tong University, Shanghai 200240, China; 2. Shanghai Jiaotong University Underwater Engineering Institute Co. Ltd .Shanghai 200231. China)Abstract: Time-varying external disturbances and system uncertainties affect the motion of work-class remote-operated vehicles(ROVs). The conventional sliding mode method for ROV motion control has the drawback of a chattering phe-nomenon, whereas the common method for eliminating chattering, namely, the saturation function combined with a boundary layer sliding mode controller(SatSMC), cannot guarantee control accuracy. To address these problems, a su-per-twisting second-order sliding mode controller(STSMC) is proposed to realize trajectory tracking of a work-class ROV. The Lyapunov method is used to analyze the stability of the system. It is proved that the proposed controller can ensure the convergence of a tracking error in finite time. A simulation experiment of the proposed STSMC and SatSMC methods and the proportional integral derivative(PID) control are compared. Results show that the STSMC method ena-bles the ROV to complete the tracking of a predetermined path. This method also has stronger robustness, rapidity and accuracy. The chattering of the STSMC is also significantly reduced compared to that of the SatSMC. In addition, the2021年2月黄博伦, 等: 基于super-twisting二阶滑模算法的作业型ROV路径跟踪控制方法第1期control parameters are not increased, making the STSMC more suitable for actual use with ROVs.Keywords: remote-operated vehicle; super-twisting algorithm; sliding mode control; trajectory tracking0 引言遥控无人水下航行器(remote operated vehicle, ROV)已经广泛应用在深海探测、海底管线维修、深海采矿和水底搜救等深海任务中, 成为人类探索海洋、开发海洋、保护海洋不可或缺的工具。

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电机转速环节Super-Twisting算法二阶滑模控制律设计与研究张庆超;马瑞卿;皇甫宜耿;王姣【摘要】Aiming at the speed outer close⁃loop control for the motors, the design method of the second⁃order slid⁃ing mode control law based on the Super⁃Twisting algorithm( STA) is researched, which doesn′t need the derivative or the extreme value of the speed. The stability is proved by using Lyapunov function. And the influence for the sys⁃tem stability and the rapidity of convergence by the proportional of the sliding mode variable is analyzed, also the commonality and the differences of the various types proportional of the sliding mode variable. Simulation results show that the introduced proportional of the sliding mode variable in STA can improve the convergence speed of the system regardless of the running state of the motor, and the STA second⁃order sliding mode features of the motor system is not changed. Due to the unified rotor motion equation, the analyzed result of this paper is general to the speed loop second⁃order sliding mode control for all kinds of motors%针对电机转速环节，研究了以Super⁃Twisting算法（ STA）为基础的转速闭环二阶滑模控制律设计方法，无需对转速求导或求解转速极值。

利用Lyapunov函数证明了系统的稳定性，并分析了系统中比例滑模项对系统收敛的稳定性和快速性的影响，以及不同类型比例滑模项的共同点与区别。

仿真结果表明，在STA基础上人为增加比例滑模项可以提高系统的收敛速度，而不受系统工作状态影响，并且不改变系统的STA二阶滑模特性。

由于各种电机都具有统一的转子运动方程，因此，研究结果对各种电机的转速环节二阶滑模闭环控制具有通用性。

【期刊名称】《西北工业大学学报》【年(卷),期】2016(034)004【总页数】8页(P668-675)【关键词】电机;转速闭环;二阶滑模;Super-Twisting算法;稳定性;收敛性【作者】张庆超;马瑞卿;皇甫宜耿;王姣【作者单位】西北工业大学自动化学院，陕西西安 710129;西北工业大学自动化学院，陕西西安 710129;西北工业大学自动化学院，陕西西安 710129;中国兵器第203研究所，陕西西安 710065【正文语种】中文【中图分类】TM351;TP273滑模控制由于对系统匹配的有界不确定扰动具有强鲁棒性，在各种电机控制系统中得到了广泛关注与研究[1-6]。

在高性能转速、电流/转矩双闭环电机调速系统结构，转速外环控制律输出就是内环的给定，要求是连续控制指令。

但传统滑模控制是一种非连续控制方法，输出控制信号存在离散化的高频抖振，导致传统滑模不适合用于多环级联系统的外环控制。

针对传统滑模控制中的抖振问题，众多学者提出了多种抖振抑制方法，例如趋近律法、边界层法、高阶滑模等，并将其用于各种电机控制系统中[1-2,5-6]。

在这些方法中，高阶滑模控制是一种既能够保留传统滑模强鲁棒性，又能够有效抑制抖振的新型滑模控制方法，而考虑到电机系统阶数以及控制律复杂性，电机调速系统通常采用二阶滑模。

常见的二阶滑模控制算法有Twisting算法、Super-Twisting算法、Sub-Optimal算法、Drift算法和Prescribed Convergence Law算法等[7-9]。

其中，Super-Twisting算法(Super-Twisting Algorithm,STA)由于无需求取滑模变量的导数或极值，且输出控制信号连续无抖振，因此，被广泛应用于各种电机调速控制中[3-4,10-12]。

文献[3]将STA用于开关磁阻电机调速控制；文献[4]将STA用于风力发电系统中绕线转子感应发电机的转矩控制；文献[10]将STA用于有刷直流电机调速控制；文献[11]将STA用于BLDCM的高性能转速控制；文献[12]将STA用于永磁同步直线电机的速度控制。

但上述方法都属于电机的单转速闭环二阶滑模控制，无法控制电流，并且需要对转速求导，增加了系统引入干扰、噪声等风险。

对于电机的双闭环滑模控制，文献[13]对采用指数趋近律法的BLDCM转速、电流双闭环滑模控制进行了仿真研究；文献[14]提出一种基于高阶非奇异终端滑模的控制方法，用于永磁同步电机的转速、电流双闭环控制；文献[15]将二阶滑模Sub-Optimal算法用于永磁直流电机的转速、电流双闭环控制；文献[16]将二阶积分滑模用于永磁同步电机的转速外环控制。

上述双闭环滑模控制方法，也都需要对转速求导或求导数为零处的滑模变量极值。

本文针对电机双闭环系统中转速外环的二阶滑模控制，基于STA设计了二阶滑模控制律，无需对转速求导或求极值；利用Lyapunov 函数证明了系统的稳定性，并分析了比例滑模项对系统收敛的稳定性和快速性的影响，在此基础上，设计了带有比例滑模项的电机转速外环STA 二阶滑模控制律；最后，根据系统特点，结合仿真分析了不同类型的比例滑模项的共同点与区别。

对于各种旋转电机，例如无刷直流电机、永磁同步电机、开关磁阻电机、异步电机、直流电机等，其转子运动方程都可以统一表示为式中,Te为电磁转矩;Tl为负载转矩;J为电机等效转动惯量;ω为转子角速度;β为黏滞摩擦系数。

设ω*为转速给定,定义转速误差状态变量x=ω*-ω,根据(1)式,电机转速环节可表示为如(2)式所示的一阶误差状态方程,并将Te改写为,用以表示Te同时作为转速外环控制变量和转矩内环给定指令2个含义。

式中,表示由电机的参数摄动带来的匹配性扰动。

2.1 控制律设计二阶滑模STA的具体表达式可写为[7]式中,s为滑模变量,而控制参数α>0,λ>0。

STA是一种针对相关度为1的系统提出的二阶滑模控制算法,而显然(2)式所示的电机转速误差状态方程的相关度为1,因此,设计滑模面为针对(2)式,可设计基于STA的直接控制律为将(5)式带入(2)式,并对(4)式沿(2)式所示系统轨迹求导可得在(6)式所示的闭环系统中,λ′=bλ,α′=bα,γ=β/J=-a。

显然,γ为电机参数,并且γ>0。

在控制律的参数调节过程中,可将系统参数b融入控制参数λ′和α′整体调节,而无需关注b的实际准确值。

这样,即使在不知道电机转速环节确切的模型参数的条件下,对于(6)式所示系统,仍然只有2个控制参数需要调节λ和α。

2.2 稳定性分析对于STA的稳定性分析,文献[18-19]提出了一种相对简单的利用类二次型Lyapunov函数的证明方法,并分析了常值扰动与时变扰动2种情况下的STA收敛性;而实际上,常值扰动可以作为时变扰动的一个特例,因此,这里仅从时变扰动角度,借鉴文献[18-19]构造的Lyapunov函数,对(6)式所示系统进行稳定性证明。

令φ=f(t),并采用变量代换则,(6)式可以改写为至此,(5)式所示控制律设计的稳定性分析问题就转化为(8)式所示系统的平衡点镇定问题。

针对(8)式所示系统,选取Lyapunov函数式中,ξT=[ξ1,ξ2,ξ3]=[|z1|1/2sign(z1),z1,z2],Π为实对称正定矩阵,并取。

(9)式所示V(z)是连续正定函数,除集合外{z1=s=0},V处处可微,对V沿系统轨迹求导有式中因为是标量,而CTΠξ和ξTΠC也为标量,并且Π=ΠT,因此令m=CTΠξ=ξTΠC,则m2=ξTΠCCTΠξ,根据不等式(m-ρ)2=m2-2mρ+ρ2≥0,∀m,ρ∈R,可以推出假设φ是Lipschitz连续的,∀t≥0,有,再根据,可知,又因为,其中D=[1 0 0],于是可以推出结合(10)式可得式中对于1,当Q1为正定矩阵时有而根据Schur补定理,可推出Q1为正定矩阵的一个充分条件为对于2,分析Q2可知,由于γ实际为系统参数,并且>0,当参数满足(17)式时Q2正定此时对于3,计算并展开3可得式中对于(20)式,由于>0,那么根据STA中α′>0的要求,以及假设条件进一步分析可知,在所有和种条件下,只要,就有,即在的条件下,ξ1ξ3<0,ξ2ξ3<0。

从而可知,当参数满足(21)式时根据(19)式可推出综合(15)式～(22)式可知,系统控制参数应满足(23)式所示条件在此条件下,对(9)式所示的连续正定候选Lyapunov函数V=ξTΠξ的沿系统轨迹求导有2.3 比例滑模项对系统稳定性及收敛速度影响1) 稳定性影响分析当对(2)式所示的电机转速误差方程直接采用STA时,如(6)式所示,相比文献[17-20]中分析的纯STA系统多出一个关于滑模面的比例项-γs。

而根据(7)式～(24)式的稳定性分析过程可知,只要保证γ>0,-γs就不影响系统的稳定性。

而对于γ=β/J,其中β和J都是电机的实际物理参数,并且β>0,J>0,所以γ>0是客观参数条件。

2) 控制律改进与收敛速度影响分析既然比例滑模项不影响电机转速环节二阶滑模STA闭环控制的稳定性,那么可通过人为地增加比例滑模项ks(k>0)把控制律(5)式改为将(25)式带入(2)式,并对(4)式沿(2)式所示系统轨迹求导可得式中,k′=bk。

对于(26)式所示的系统,当把k′+γ视为一个整体参数时,参照(7)式～(24)式的稳定性分析过程,结合上述分析,可知在原STA控制律中加入比例项ks,亦不影响原系统的稳定性对于控制律(25)式,相比(5)式所示的纯STA控制律中的开方滑模比例项——λ|s|1/2sign(s),只要合理设置k、ks的引入可以增加系统在远离平衡点处的滑模变量增益,增大控制指令输出,保证鲁棒性,并提高收敛速度;在到达平衡点附近时,STA起到主导作用,保证系统在平衡点邻域内的稳定性和鲁棒性。