二阶滑模
双边遥操作系统二阶滑模控制器设计
中图 分 类 号 ： Ｐ ４ Ｔ ９ Ｔ ２ ；Ｈ３ 文 献 标 志 码 ： Ａ 文 章 编 号 ：０ １ ４ ５ （０ ２ ０ — ６— ５ １０ — ５ １２ １ ）４ ０ ９ ０ ４
２ Ｄｅ ａ ｍｅ ｆ ｇ ｎ ｒ ｃｅ ｃ ｎ ｃ ｏｏ ｙ，Ｎａ ａ ｒ ｎ ｕ ｉ ａ ｎｄＡｓｒ ｎ ｕ ｉａ ｉｅ ｓｔ ． ｐ ｒ ｎｔ ｔ ｏ Ｅｎ ｉ ｅ ｙ Ｓ ｉ ｎ ｅａ ｄ Ｔｅ ｈｎ ｌ ｇ ｖ ｌＡｅ ｏ ａ ｔｃ ｌａ ｔｏ ａ ｔｃ ｌＵｎ ｖ ｒ ｉ ｙ，
王 瑞 奇 ， 玉 光 ， 文 广 ， 文 锦 冯 张 顾
（．海军 航空工 程学 院 控制工 程 系 ，山东 烟 台 ２４ ０ ； １ ６０ １ ２ 海 军航 空工 程学 院 兵 器科 学与技 术 系 ，山东 烟 台 ２ ４０ ） ． ６０ １
摘要 ： 阶滑模控制方法 由于需要 大的带宽和高频切换信号 ， 一 以及处 理系统的参数不确定和强非线性 ， 在双边遥操作 的工程应用 中
ＷＡ Ｇ Ｒ ｉｑ Ｆ Ｎ ｕ ｇ ａ ｇ， Ｈ Ｇ Ｗｅ — ｕ ｎ Ｇ ｎ ｊ Ｎ ｕ— ｉ Ｅ Ｇ Ｙ — ｕ ｎ ｚ Ｚ ＡＮ ｎ ｇ ａ ｇ， Ｕ Ｗｅ －ｉ ， ｎ
（ ． ｅ ａｔ ｎ ｆ ｏ ｔ ｌ ｎ ｉｅｒ ｇ Ｎ ｖ ｌ ｅｏ ａ ｔ ａ ａ ｄＡ ｔ ｎ ｕｉａ Ｕ ｉｅｓ ｙ Ｙａｔｉ ６ ０ １ Ｃ ｉａ １Ｄ ｐ ｒ ｔ ｎｒ ｇｎ ｅｉ ， ａａ Ａ ｒｎ ｕｉ ｌ ｎ ｓ ｏ ａ ｔ ｌ ｎｖｒｉ ， ｎａ ２ ４ ０ ， ｈｎ ； ｍｅ ｏ Ｃ ｏ Ｅ ｎ ｃ ｒ ｃ ｔ
第２ 卷第４ ９ 期
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。
基于神经网络的机器人二阶滑模控制
基于神经网络的机器人二阶滑模控制王延玉;刘国栋【期刊名称】《计算机系统应用》【年(卷),期】2012(021)006【摘要】This paper proposes a synergetic controls algorithm by adaptive neural network and Second order slidingmode control. Design a second order sliding mode control method with novelty and facility, and the chattering problemis avoided effectively, Neutral network is used to adaptive learn and compensate the unknown nonlinear model. Thelearning algorithm for the free neutral network parameters are presented by Lyapunov direct method. The globalasymptotic stability is guaranteed. Finally, the control performance of the proposed controller is verified with simulationstudies.%本文提出了一种基于神经网络与二阶滑模控制融合的控制策略用于非线性机器人控制,设计了一种新颖简易的二阶滑模控制方法,有效地避免了常规变结构控制的抖震问题,并采用神经网络辨识未知的机器人的非线性模型,通过Lyapunov直接法设计网络的权值更新率,确保了系统闭环全局渐近稳定性.最后,通过仿真验证了算法的有效性.【总页数】4页(P55-58)【作者】王延玉;刘国栋【作者单位】江南大学物联网工程学院,无锡214122;江南大学物联网工程学院,无锡214122【正文语种】中文【相关文献】1.农业轮式机器人PI鲁棒-滑模控制\r——基于RBF神经网络 [J], 曹东;闫银发;宋占华;田富洋;赵新强;刘磊;王春森;李法德;陈为峰2.基于小波神经网络模糊滑模控制的轮式移动机器人避障研究 [J], 王韬3.基于递归径向基神经网络的永磁直线同步电机智能二阶滑模控制 [J], 王天鹤;赵希梅;金鸿雁4.基于神经网络滑模控制器的外骨骼机器人力矩控制器设计 [J], 刘建华;夏志刚;周贤钢;黄晓冠;龚高成;吴诗杰;丁磊;吴康兵5.基于神经网络滑模控制器的外骨骼机器人力矩控制器设计 [J], 刘建华;夏志刚;周贤钢;黄晓冠;龚高成;吴诗杰;丁磊;吴康兵因版权原因，仅展示原文概要，查看原文内容请购买。
基于二阶滑模观测器的无刷直流电机转子位置估计
基于二阶滑模观测器的无刷直流电机转子位置估计史婷娜;马银银;王迎发;夏长亮【摘要】针对一阶滑模观测器(F-SMO)存在的抖振和相位延迟问题,提出了基于二阶滑模观测器(S-SMO)的线反电势(LBEMF)估计策略,将不连续控制作用在滑模变量的高阶微分上,采用超螺旋算法设计控制率,能较好地削弱抖振,得到连续光滑且无滞后的反电势估计值.针对相反电势法存在的相移问题,采用线反电势过零点直接作为换相点的换相策略.仿真和实验结果表明,所提策略能够准确估计无刷直流电机线反电势,获得准确的转子位置换相点,实现无刷直流电机的无位置传感器控制.【期刊名称】《天津大学学报》【年(卷),期】2014(047)008【总页数】6页(P697-702)【关键词】无刷直流电机;二阶滑模观测器;线反电势;换相策略【作者】史婷娜;马银银;王迎发;夏长亮【作者单位】天津大学电气与自动化工程学院,天津300072;天津大学电气与自动化工程学院,天津300072;天津大学电气与自动化工程学院,天津300072;天津大学电气与自动化工程学院,天津300072【正文语种】中文【中图分类】TM383无刷直流电机由于具有效率高、输出转矩大、响应快、惯性低等诸多的优点，在航空、汽车和家庭应用等方面得到广泛应用[1-3]．传统无刷直流电机的闭环控制需要采用位置传感器来获得转子位置，但位置传感器的存在不仅导致系统成本的提高，而且还影响系统的可靠性和鲁棒性，因此无刷直流电机无位置传感器技术成为目前的一个重要研究方向．近年来，国内外文献介绍的无位置传感器检测方法主要包括相反电势法、磁链法、电感法、续流二极管法等．其中，相反电势法因其简单、实用等特点，成为目前研究的热点[4-6]．相反电势法通过检测无刷直流电机三相端电压，计算得到电机相反电动势过零点，再相移/6π 电角度得到无刷直流电机换相位置，其原理简单，实现方便，应用广泛．但在相移角计算过程中，通常依赖于电机速度，尤其是在调速过程中，相移角不准确易造成电机运行性能变差．如何避免相移角计算并直接获得换相点，成为一个新的研究思路．文献[7]提出了一种利用三次谐波检测转子位置的新方法，可检测速度更宽，不需要相移滤波，但在低速下三次谐波严重畸变，导致不能估算转子位置．文献[8]通过对电机模型分析，构造G函数直接确定无刷直流电机的换相点，扩展了无位置传感器控制调速范围，但系统的计算量增大．基于滑模观测器的反电势估计策略能准确估算出反电势信号．然而由于其控制作用的不连续性所引起的抖振现象，会导致被控系统出现危险的高频振荡．文献[9]将sigmoid函数代替开关函数，在一定程度上削弱了抖振，但也不可避免地降低了响应速度，使系统的鲁棒性变差．而低通滤波器的使用会导致相位滞后，难以精确补偿．文献[10]设计了反电势观测器，省去了低通滤波器和相位补偿环节，但估算的反电势存在抖振和噪声，影响准确性．二阶滑模是解决抖振问题和相位延迟的一种有效的方法，在此方法中，不连续控制并不作用在滑模变量的一阶微分上，而是作用在其高阶微分上，这样不仅保留了一阶滑模控制的所有优点，还可以削弱抖振和相位滞后现象[11-15]．因此本文采用二阶滑模观测器估算电机线反电动势．本文先采用超螺旋算法设计控制率，再设计二阶滑模微分估计器对电流微分进行估计．此二阶滑模观测器能较好地削弱抖振、得到连续光滑且无滞后的线反电势估计值，提高了无刷直流电机无位置传感器控制的换相精度．通过分析线反电动势过零点与换相时刻的对应关系，提出了采用线反电势过零点直接作为换相点的换相策略，根据虚拟霍耳信号建立无刷直流电机换相逻辑，避免了传统相反电势存在的相移角计算问题．无刷直流电机三相绕组电压、电流方程表示[16]为式中：ua0、ub0、uc0分别为三相定子绕组电压；un为电机中性点电压；ia、ib、ic分别为定子相电流；ea、eb、ec为定子相反电势；R和 Ls分别为定子相电阻和等效电感．由于无刷直流电机中性点电压难以直接检测，将式(1)和式(2)简化为 2个并联的线性无关的一阶电流模型，整理成线电压的状态空间形式为式中：和 ebc为绕组线反电势，eab=ea-，uab和ubc为绕组线电压同时，线反电势之间存在关系由式(3)可知，直接计算能够得到线反电势，但开环计算方式及计算中的电流微分项会导致计算过程中存在一定误差，而采用闭环形式的观测器则可进一步提高线反电势估计精度．根据式(3)，构建无刷直流电机的二阶滑模观测器式中：Z为二阶滑模观测器控制量，将式(6)与式(3)相减，得到无刷直流电机的状态误差方程为式中当滑模面存在且在有限时间内收敛时，即可得下面对无刷直流电机的二阶滑模观测器进行分析．2.1 滑模面选择将无刷直流电机线电流差作为误差标准，其表示为选择二阶滑模观测器的滑模面为式中σ为无刷直流电机线电流误差构成的滑模面，增加常数 c可以任意地加快收敛速率．根据滑模控制的设计要求，控制必须能保证滑模变量1σ和2σ收敛到零点．令有合理地选择常数c可使误差在有限时间内趋于0．滑模变量的微分表示为式中：其中A1、A2、A3、A4、A5是由电机参数确定的常数．2.2 控制率设计本文采用超螺旋算法作为二阶滑模观测器中的控制率．超螺旋算法是指在 -σ σ˙平面内，状态轨迹在有限时间内围绕原点螺旋式地收敛到原点．该算法不需要滑模变量的一阶导数和符号信息，离散项出现在控制量的一阶微分上，滑模变量的相关度为 1，能够有效地削弱抖振现象．采用超螺旋算法的二阶滑模观测器控制率为控制率中的参数V1和V2按照约束规则选取，约束规则为式中为正常数，满足条件当控制率参数满足以上条件时，超螺旋算法可保证滑模变量在有限时间 tf内收敛到滑模面，即时，保证存在．当滑模变量达到收敛状态时，可准确估算出线反电势信号，估计值为根据式(5)，可得线反电势eca的估计值为了便于编程应用，考虑式(11)的 Euler离散化形式，得到式中：，τ 为采样时间； 0,1,2j= ．2.3 二阶滑模微分估计器式(9)需要对电流误差e(t)求取微分信号，但实际中微分器通常采用一阶差分信号，容易引入噪声干扰．因此，本文采用二阶滑模算法构成微分估计器，估计电流误差微分信号．设定微分估计器输入为电流误差信号 e(t)，电流误差信号微分量为滑模量及其微分量为采用超螺旋算法建立电流误差信号微分估计器式(16)收敛的充分条件为在超螺旋算法控制率作用下，微分估计器经过有限时间后达到收敛，存在因此，二阶滑模微分器能够较好地得到无刷直流电机线电流误差微分信号．根据以上分析，设计的无刷直流电机二阶滑模线反电势观测器结构如图1所示．无刷直流电机的换相需要确定 6个离散的位置信号，在位置传感器控制中通常由霍耳信号提供6个换相点．而霍耳信号对应的换相点滞后相应相反电势过零点30°电角度，因此相反电势法存在相移角计算问题，造成计算复杂．本文从线反电势的角度出发，分析霍耳信号与线反电势之间的关系．图 2所示为线反电势过零点与实际霍耳换相信号示意．由图2中可以看出，线反电势过零点直接与霍耳信号的换相点对应．若线反电势为正时表示为 1，为负时表示为 0，可得 eab、ebc、eca分别对应H2、H3和(“”表示信号取反)，用虚拟霍耳信号和表示．建立霍耳信号换相逻辑表，如表1所示．4.1 仿真结果及分析利用 Matlab/Simulink建立一阶和二阶滑模观测器仿真模型，电机参数如表2所示．电机运行在n=2,000,r/min、TL=0.2,N·m条件下，采用一阶滑模观测器对无刷直流电机线反电势估计，反电势估计结果如图3所示．图3中，实线和虚线分别代表实际线反电势和估算线反电势．一阶滑模观测器由于低通滤波器的使用，反电势估计值存在相位延迟，该延迟会加大电机换相误差，降低电机运行性能．图4为无刷直流电机分别运行在n=200,r/min、n=1,000,r/min、n=2,000,r/min，TL=0.2,N·m条件下，采用二阶滑模观测器得到的线反电势估计值．实线和虚线分别代表实际线反电势和估算线反电势．由图 4中可以看出，二阶滑模观测器在高、中、低速范围内均能较好地跟踪实际线反电势，能得到连续光滑且无滞后的线反电势估计值，实现无刷直流电机正确换相．图 5为无刷直流电机在 n=1,000,r/min、TL= 0.2,N·m、电阻Ra增大20%的条件下，线反电势实际值和估计值的仿真结果．由图5中可以看出，电阻Ra增大20%时，二阶滑模观测器仍能较好地估计出线反电势，表明二阶滑模观测器对电机参数的扰动具有较好的抑制能力．图 6为TL=0.2,N·m条件下，电机从 n= 200,r/min变速运行到n=2,000,r/min 时，线反电势估计值和功率管换相信号．图 6(a)表明在变速条件下，二阶滑模观测器仍能准确估算出线反电势，表明二阶滑模观测器具有较好的鲁棒性．图 6(b)中，PT1为根据实际霍耳信号得到的换相信号，为根据本文分析的虚拟霍耳信号得到的换相信号，通过两者对比可以看出，根据线反电动势过零点得到的新的换相策略能准确确定换相位置，实现无刷直流电机无位置传感器控制．4.2 实验结果及分析为了进一步验证策略的有效性，以TI公司DSP芯片 TMS320F28335为核心控制器建立无刷直流电机实验系统．为了得到较好的实验效果，电压、电流检测均采用霍耳电压、电流传感器，同时设计了 4阶巴特沃斯低通滤波器滤除数据采集中的干扰信号．图7为n=800,r/min时线电压和换相信号的实验结果．由图 7中可以看出，换相信号和估计值基本重合，因此基于二阶滑模观测器的线反电势换相策略能够准确地换相，较好地实现无刷直流电机无位置传感器控制．图 8为线反电势实际值和通过二阶滑模观测器估计得到的线反电势估计值．可以看出，二阶滑模观测器能够较好地估计出无刷直流电机线反电势，且抖振较小，不存在相位滞后．(1) 二阶滑模观测器不仅保留了一阶滑模观测器的所有优点，且能够较好地削弱抖振，得到连续光滑且无滞后的线反电势估计值．(2) 通过分析无刷直流电机线反电动势与换相时刻的对应关系，得出线反电动势过零时刻即为换相时刻的结论，建立了虚拟霍耳信号换相逻辑．(3) 仿真与实验表明，本文策略能够准确估计出无刷直流电机线反电势，获得准确的转子位置换相点，实现了无刷直流电机无位置传感器控制的准确换相，提高了换相精度．【相关文献】［1］张志刚，王毅，黄守道，等. 无刷双馈电机在变速恒频风力发电系统中的应用[J]. 电气传动，2005，35(4)：61-64.Zhang Zhigang，Wang Yi，Huang Shoudao，et al. The application study for brushless doubly-fed machine in the variable speed constant frequency generation system[J]. Electric Drive，2005，35(4)：61-64(in Chinese).［2］ Shi Tingna，Guo Yuntao，Song Peng，et al. A new approach of minimizing commutation torque ripple for brushless DC motor based on DC-DC converter[J]. IEEE Transactions on Industrial Electronics，2010，57(10)：3483-3490.［3］ Chen Yie-Tone，Chiu Chun-Lung，Tang Zong-Hong，et al. Optimizing efficiency driver comprising phaselocked loop for the single-phase brushless DC fan motor[J]. IEEE Transactions on Magnetics，2012，48(5)：1937-1942.［4］ Shao Jianwen. An improved microcontroller-based sensorless brushless DC motor drive for automotive applications[J]. IEEE Transactions on Industry Applications，2006，42(5)：1216-1221.［5］ Imoru O，Tsado J. Modelling of an electronically commutated(brushless DC)motor drives with back-emf sensing[C] //16,th IEEE Mediterranean ElectrotechnicalConference(MELECON). Yasmine Hammamet，Tunisia，2012：828-831.［6］ Damodharan P，Vasudevan K. Sensorless brushless DC motor drive based on thezero-crossing detection of back electromotive force (EMF) from the line voltage difference[J]. IEEE Transactions on Energy Conversion，2010，25(3)：661-668.［7］韦鲲，任军军，张仲超. 三次谐波检测无刷直流电机转子位置的研究[J]. 中国电机工程学报，2004，24(5)：163-167.Wei Kun，Ren Junjun，Zhang Zhongchao. Research on the scheme of sensing rotor position of BLDCM based on the third harmonic component[J]. Proceedings of the CSEE，2004，24(5)：163-167(in Chinese).［8］ Kim Tae-Hyung，Ehsani M. Sensorless control of the BLDC motors from near-zeroto high speeds[J]. IEEE Transactions on Power Electronics，2004，19(6)：1635-1645.［9］ Kim Hongryel，Son Jubum，Lee Jangmyung. A highspeed sliding-mode observerfor the sensorless speed control of a PMSM[J]. IEEE Transactions on Industrial Electronics，2011，58(9)：4069-4077.［10］ Qiao Zhaowei，Wang Yindong，Shi Tingna，et al. New sliding-mode observer for position sensorless control of permanent-magnet synchronous motor[J]. IEEE Transactions on Industrial Electronics，2013，60(2)：710-719.［11］ Damiano A，Gatto G L，Marongiu I，et al. Secondorder sliding-mode control of DC drives[J]. IEEE Transactions on Industrial Electronics，2004，51(2)：364-373.［12］孙宜标，杨雪，夏加宽. 基于二阶滑模的永磁直线同步电机鲁棒速度控制[J]. 电工技术学报，2007，22(10)：35-41.Shun Yibiao，Yang Xue，Xia Jiakuan. Robust speed control of permanent-magnet linear synchronous motor based on the second order sliding mode[J]. Transactions of China Electrotechnical Society，2007，22(10)：35-41(in Chinese).［13］凌睿，柴毅. 永磁直线同步电机多变量二阶滑模控制[J]. 中国电机工程学报，2009，29(36)：60-66.Ling Rui，Chai Yi. Multi-variable second order sliding mode control for PMLSM[J]. Proceedings of the CSEE，2009，29(36)：60-66(in Chinese).［14］ Beltran B，Ahmed-Ali T. Second-order sliding mode control of a doubly fed induction generator driven wind turbine[J]. IEEE Transactions on Energy Conversion，2012，27(2)：261-269.［15］ Evangelista C，Puleston P，Valenciaga F，et al. Lyapunovdesigned super-twisting sliding mode control for wind energy conversion optimization[J]. IEEE Transactions on Industrial Electronics，2013，60(2)：538-545.［16］王迎发．无刷直流电机换相转矩波动抑抑制与无位置传感器控制研究[D]. 天津：天津大学电气与自动化工程学院，2011．Wang Yingfa. Research on Commutation Torque Ripple Reduction and Sensorless Control of Brushless DC Motor[D]. Tianjin：School of Electrical Engineering and Automation，Tianjin University(in Chinese)．。
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.。
飞行器鲁棒最优末制导律的二阶滑模设计
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基于有限时间扰动观测器的水厂加矾系统二阶滑模控制
第40卷第11期2023年11月控制理论与应用Control Theory&ApplicationsV ol.40No.11Nov.2023基于有限时间扰动观测器的水厂加矾系统二阶滑模控制王冬生†,张鹏,孙锦昊,郭若寒,蒋国平(南京邮电大学自动化学院人工智能学院,江苏南京210023)摘要:水厂絮凝沉淀过程具有强非线性、不确定性和参数时变等特点,并且原水水质和水量突变等扰动容易对絮凝沉淀过程造成不利影响.本文提出了一种基于有限时间扰动观测器的加矾系统二阶滑模控制设计方法.首先,文章采用带有非光滑项的二阶滑模控制方法设计加矾系统反馈控制;然后,文章设计有限时间扰动观测器对原水水质和水量突变等扰动,以及絮凝沉淀过程强非线性、不确定性和参数时变等导致的模型不匹配进行估计,估计结果作为前馈补偿与反馈控制相结合;最后,理论分析证明了基于有限时间扰动观测器的二阶滑模控制方法的稳定性.仿真结果表明,本文所提出的复合控制方法有效提升了加矾系统的鲁棒性和抗扰动性能.关键词:加矾控制;有限时间扰动观测器;二阶滑模控制;抗扰动引用格式:王冬生,张鹏,孙锦昊,等.基于有限时间扰动观测器的水厂加矾系统二阶滑模控制.控制理论与应用, 2023,40(11):1965–1971DOI:10.7641/CTA.2022.20462Second-order sliding mode control based onfinite-time disturbance observer for alum dosing system of water plantWANG Dong-sheng†,ZHANG Peng,SUN Jin-hao,GUO Ruo-han,JIANG Guo-ping(School of Automation,School of Artificial Intelligence,Nanjing University of Posts and Telecommunications,Nanjing Jiangsu210023,China) Abstract:Theflocculation and sedimentation process of water plant has the characteristics of strong nonlinearity, uncertainty and time-varying parameters,and disturbances of sudden changes in raw water quality and waterflow are easy to adversely affect theflocculation and sedimentation process.This paper proposes a control design method of second-order sliding mode based on thefinite-time disturbance observer for alum dosing system.First,the feedback control of alum dosing system is designed by second-order sliding mode control method with non-smooth terms.Then,afinite-time disturbance observer is designed to estimate disturbances of sudden changes in raw water quality and waterflow,as well as model mismatch caused by strong nonlinearity,uncertainty,and time-varying parameters in theflocculation and sedimentation process.The estimation result is combined with feedback control as feedforward compensation.Finally,the theoretical analysis proves the stability of second-order sliding mode control method based on thefinite-time disturbance observer.The simulation results show that the composite control method proposed in this paper effectively improves the robustness and anti-disturbance performance of alum dosing system.Key words:alum dosing control;finite-time disturbance observer;second-order sliding mode control;anti-disturbance Citation:WANG Dongsheng,ZHANG Peng,SUN Jinhao,et al.Second-order sliding mode control based onfinite-time disturbance observer for alum dosing system of water plant.Control Theory&Applications,2023,40(11):1965–19711引言絮凝沉淀过程是水厂水质净化的重要环节,与出厂水水质安全密切相关.絮凝沉淀过程通过向原水中投加矾等絮凝剂去除原水中的悬浮杂质、胶体颗粒及附着于胶体颗粒上的细菌、病毒等有害物质.依据美国联邦环保局饮用水病毒去除技术标准,当滤后水浊度低于0.3NTU时,病毒去除率高达99%[1].加强对水厂加矾系统的有效控制,严格限制沉淀池出水浊度,有利于出厂水水质稳定和实现高品质饮用水目标.专家学者们对加矾系统控制问题进行了大量的研究和实践,提出了各种控制算法.流动电流法[2]和透光率脉动法[3]通过流动电流值和透光率检测跟踪絮凝收稿日期:2022−05−29;录用日期:2022−12−22.†通信作者.E-mail:***********************.cn;Tel.:+86189****9776.本文责任编委:李世华.国家自然科学基金项目(52170001)资助.Supported by the National Natural Science Foundation of China(52170001).1966控制理论与应用第40卷沉淀过程状态,据此调整加矾量,但是由于流动电流值和透光率是间接反应絮凝沉淀过程的相对值,而且对仪器的灵敏度和维护要求较高,影响了在实际应用中的效果.直接将沉淀池出水浊度作为被控变量来控制加矾量是目前加矾系统控制的主流.由于历史数据中包含了控制过程中的所有信息,数据驱动方法[4]可以通过对历史数据的训练获得控制器参数,数据驱动方法避免了传统控制方法对过程模型的依赖,但是历史数据信息的获取往往是不全面的,一定程度限制了数据驱动方法在实际应用中的推广.虽然絮凝沉淀过程难以精确建模,但仍然可以通过采用高级反馈控制和扰动估计等方法,对其过程模型不精确,以及水质、水量突变等因素作用下的扰动进行抑制.滑模控制(sliding mode control,SMC)是强非线性控制问题中的一种有效方法,具有抗扰动性强、动态响应快、控制实现简单等优势.目前,已有许多相关理论和应用研究[5–7].文献[5]针对一类非线性积分系统,利用有限时间控制技术,提出了一种输入饱和情况下的全局有限时间控制方案.文献[6]提出了一种新颖的二阶滑模(second-order sliding model,SOSM)控制方法来处理具有不匹配项的滑模动力学,从而减少控制通道中的项.文献[7]提出了一种带有有限时间扰动观测器(finite-time disturbance observer,FDOB)的连续动态滑模控制器.在实际应用中,SOSM控制使滑动变量的选择更加灵活,而且也更容易消除振颤问题. FDOB能够对扰动和过程不确定性进行估计,并通过前馈补偿设计减少对控制系统的不利影响.将扰动观测器与反馈控制相结合的复合控制方法是目前控制领域中抑制扰动和补偿模型不精确等问题的研究热点之一[8].本文提出了一种基于FDOB和SOSM的水厂加矾系统复合控制方法,针对实际絮凝沉淀过程受原水水质和水量突变的影响,以及强非线性、不确定性和参数时变等问题,采用前馈补偿和反馈控制相结合的设计方法.仿真结果证明,在与实际絮凝沉淀过程相符合的模型不匹配和扰动情况下,本文提出的控制方法更好地实现了出水浊度的稳定.2系统描述与控制器设计2.1问题描述自来水厂常规处理工艺流程如图1所示.其中,絮凝沉淀过程是在沉淀池入口处向原水中投加矾等絮凝剂,从而让各种杂质颗粒物等凝结成絮凝体,在重力作用下,絮凝体就能够沉淀在沉淀池底部,达到去浊澄清的目的.2.2控制器设计2.2.1SOSM已知系统状态等一些可测量的信息存在于˙s中,在˙s得到导数¨s的过程中可能会放大这些信息,所以需要更大的控制增益.由于振颤幅度与控制幅度之间存在正比关系,因此传统SOSM会导致振颤.图1自来水厂常规处理工艺流程Fig.1Conventional treatment process of waterworks针对上述问题,本文采用一种新的控制设计方法来处理具有不匹配不确定性的SOSM动力学.构造控制器包括3个步骤.首先,引入新的滑动变量,将传统SOSM动力学转变为具有不匹配不确定项的新型SOSM动力学.其次,通过定义失配不确定性的一些增长条件,以递归的方式构建一系列虚拟控制器来稳定新的滑动变量.最后,结合有限时间控制技术,设计一种带有非光滑项的SOSM控制器.本文通过将出水浊度与设定值的偏差作为输入,设计二阶滑模控制器.其中G(s)为G(s)=K(T1s+1)(T2s+1)=bs2+a1s+a0,(1)由上式可得¨x=−a1˙x−a0x+bu,(2)其中:x∈R n,代表出水浊度;u∈R,代表控制输入.现在将滑动变量s(即出水浊度误差)定义为s=x−x ref,其中x ref表示浊度设定值.二阶滑模动力学方程为{˙s1=s2,˙s2=a(t,x)+b(t,x)u+d(t),(3)其中:a(t,x)=−a1˙x−a0x,b(t,x)=b,d(t)=ξω1(t)+ω2(t),此处ω1(t)为模型不匹配不确定项,ω2(t)为模型匹配扰动项,ω1(t)与ω2(t)及其一阶导数是有界的,因此存在一个正常数D>0使得|d(t)| D.沿系统(3)对滑动变量进行二阶导数得到¨s= a(t,x)+ξω1(t)+ω2(t)+v,其中v=b(t,x)u.则系统(3)可以进一步表示为{˙s1=s2,˙s2=A(t,x)+U,(4)其中:U=v是一个虚拟控制器,A(t,x)=a(t,x)+ξω1(t)+ω2(t).在实际应用中,出水浊度x是有界的,这表示可以找到常数A0>0,使得|A(t,x)| A0.另外也存在正函数C(x)与正常数K m,使得|a(t,x)| C(x),b(t,x) K m.为简化表达式,定义⌊x⌋α=sgn x|x|α,∀x∈R,∀m>0.设计控制器[9]第11期王冬生等:基于有限时间扰动观测器的水厂加矾系统二阶滑模控制1967u=−C(x)K msgn(⌊s2⌋αr2+β1a r2⌊s1⌋αr1)−β2⌊⌊s2⌋αr2+β1a r2⌊s1⌋αr1⌋r3a.(5) 2.2.2FDOBFDOB是根据被控变量和控制变量对扰动进行估计的过程,将扰动估计作为前馈可以有效补偿扰动对被控过程的影响,从而达到抑制扰动的目的．给出的FDOB表示如下[10]:˙z0=v0,v0=−L1⌊z0−s1⌋23+z1,˙z1=v1+U,v1=−L2⌊z1−v0⌋12+z2,˙z2=−L3sgn(z2−v1),(6)其中:L1,L2和L3为正观测器增益,需要合理设计.然后,可以得到如下定理:定理1[11]如果FDOB构造为式(6),则不确定项A(t,x)可以在有限时间内通过Z2准确估计,即可以找到一个时刻T f>0使得z2≡A(t,x)对于∀t>T f.2.2.3FDOB-SOSM复合控制设计在FDOB和SOSM基础上设计复合控制方案,如图2所示.其中,FDOB采取主动抗扰动的策略对控制系统受到的外部扰动和模型不匹配进行估计进而抑制和消除.相较于只采用反馈控制,FDOB能更加有效地抑制干扰,极大地提高系统的鲁棒性.图2复合控制方案框图Fig.2Diagram of composite control scheme在滑模控制器设计中,令s1=s,s2=˙s,滑模动力学可以改写为˙s1=s2,˙s2=A(t,x)+U,此刻需要注意的是不确定项A(t,x)通常是不可测量的,在实际应用中关于A(t,x)的精确值是未知的,这表示控制器(5)不会直接运用于系统(4)中,为此,假设A(t,x)是可微,并且满足|˙A(t,x)|<L,其中L是一个Lipschitz 常数.构造一个FDOB来获取对不确定项A(t,x)的估计,并使用估计值ˆA(t,x)补偿不确定项A(t,x),估计值通常是有界的.3系统稳定性分析假设1存在一个正常数K m,一个正函数C(x),使得|a(t,x)| C(x),|b(t,x)|>K m和r1=2,r2= r1−τ,r3=r2−τ且τ∈(0,1].定理2在假设1下,有一个常数a r1和正函数β1(s1),β2(s1,s2),建立闭环系统(4)–(5)的有限时间稳定性.首先给出以下3个引理,然后给出定理2的证明.引理1[12]如果0<c 1,那么有以下不等式:∀x1∈R,x2∈R,|⌊x1⌋c−⌊x2⌋c| 21−c|⌊x1⌋−⌊x2⌋|c.引理2[13]如果a>0,b>0和实数c>0,那么有以下不等式:|x1|a|x2|b aa+bc|x1|a+b+ba+bc−a b|x2|a+b.引理3对于实数0<c 1,有以下不等式:对于x i∈R与i=1,···,n,(|x1|+···+|x n|)c |x1|c+···+|x n|c.证分步骤证明定理2.步骤1定义函数V1(s1)=r12ρ+τ|s1|2ρ+τr1,(7)且ρ a.然后鉴于假设1,V1(s1)沿SOSM动力学(4)的导数可以推导出为˙V1(s1)=⌊s1⌋2ρ−r2r1s2⌊s1⌋2ρ−r2r1(s2−s∗2)+⌊s1⌋2ρ−r2r1s∗2,(8)其中s∗2虚拟控制器,可以设计为s∗2=−β1(s1)⌊ξ1⌋r2a,(9)其中:ξ1=⌊s1⌋a r1,β1(s1) β0,β0>0.得出˙V1(s1) ⌊ξ1⌋2ρ−r2a(s2−s∗2)−β0⌊ξ1⌋2ρa.(10)步骤2定义函数V2(s1,s2)=V1(s1)+W2(s1,s2),(11)其中W2(s1,s2)可以设计为W2(s1,s2)=s2s∗2⌊⌊k⌋a r2−⌊s∗2⌋a r2⌋2ρ−r3a d k.(12) V2(s1,s2)沿系统(4)的导数由下式给出:˙V2(s1,s2)=˙V1(s1)+∂W2(s1,s2)∂s2˙s2+∂W2(s1,s2)∂s1˙s1,(13)可得出˙V2(s1,s2) −β0|ξ1|2ρa+⌊ξ1⌋2ρ−r2a(s2−s∗2)+∂W2(s1,s2)∂s2˙s2+∂W2(s1,s2)∂s1˙s1.(14)此时注意0<r ia1和|s2−s∗2|=|⌊s2⌋a r2·r2a−⌊s∗2⌋a r2·r2a|.(15)1968控制理论与应用第40卷使用引理2,可以从式(15)中计算出⌊ξ1⌋2ρ−r 2a (s 2−s ∗2) β023|ξ1|2ρa+c 2|ξ2|2ρa ,(16)其中c 2=r 22r 2a ρ(23−r 2a (2ρ−r 2)ρβ0)2ρ−r 2r2是一个正常数.同时,由引理1可以得知|∂W 2(s 1,s 2)∂s 1˙s 1|2ρ−r 3a |s 2−s ∗2||ξ2|2ρ−r 3a−1|∂⌊s ∗2⌋ar 2∂s 1˙s 1| γ1|ξ2|2ρ+τa −1|∂⌊s ∗2⌋a r 2∂s 1˙s 1|,(17)其中γ1=21−r 2a 2ρ−r 3a.通过式(9)可以得知⌊s ∗2⌋ar 2=⌊β1(s 1)⌋ar 2ξ1,(18)|∂⌊s ∗2⌋ar 2∂s 1| |∂βa r 21(s 1)∂s 1ξ1|+a r 1βa r 21(s 1)(|ξ1|+βa r 10(s 0)|ξ0|)1−r 1a ,(19)因为ξ0=0且使用引理3,还可以得出|s 2| |ξ2|r 2a+β1(s 1)|ξ1|r 2a.(20)通过将不等式(20)和系统(4)合并,使用引理2,可以得到两个正函数γ′1(s 1)和γ′2(s 1)使得|∂⌊s ∗2⌋ar 2∂s 1˙s 1| γ′1(s 1)|ξ1|1−τa +γ′2(s 1)|ξ2|1−τa .(21)将不等式(21)代入(17),并使用引理2,可以计算出正增益˜γ2(s 1)使得|∂W 2(s 1,s 2)∂s 1˙s 1| β022|ξ1|2ρa +β023|ξ1|2ρa +˜γ2(s 1)|ξ2|2ρa.(22)结合系统(4)得出∂W 2(s 1,s 2)∂s 2˙s 2=⌊⌊s 2⌋a r 2−⌊s ∗2⌋a r 2⌋2ρ−r 3a ˙s 2=⌊ξ2⌋2ρ−r 3a(A (t,x )+U ).(23)将不等式(16)(23)代入式(14)得到˙V 2(s 1,s 2) −β02|ξ1|2ρa +(c 2+˜γ2(s 1))|ξ2|2ρa +⌊ξ2⌋2ρ−r 3a(A (t,x )+U ).(24)根据不等式(24),可以设计U =−C (x )K mb (t,x )sgn ξ2−b (t,x )β2(s 1,s 2)⌊ξ2⌋r 3a,(25)且β2(s 1,s 2)c 2+˜γ2(s 1)+β02K m.此外s 2s ∗2⌊⌊k ⌋a r 2−⌊s ∗2⌋ar 2⌋2ρ−r 3ad k 21−r 2a|ξ2|2ρ+τa.(26)将控制器(25)代入式(24),结合V 1(s 1),可以验证出V 2(s 1,s 2) 2(|ξ1|2ρ+τa+|ξ2|2ρ+τa).通过使c =β02·22ρ2ρ+τ,可以证明˙V 2(s 1,s 2)+cV 2ρ2ρ+τ2(s 1,s 2) 0.(27)注意2ρ2ρ+τ∈(0,1).通过不等式(27),可以通过有限时间李雅普诺夫理论[14]得出闭环系统(4)(25)是全局有限时间稳定的.因此,闭环系统(4)–(5)实现了全局有限时间稳定性.证毕.然而,在实际应用中,无法使用FDOB 准确估计系统不确定项,始终存在观测误差|˜A(t,x )|=A (t,x )−z 2.因此,可以找到一个时刻T f 和一个正常数ε,使得|˜A(t,x )| ε,对于∀t T f .最后,结合SOSM 算法和FDOB 技术得到的最后一个结果由定理3给出.定理3在假设1下,有一个常数a r 1和正函数β1(s 1),β2(s 1,s 2)使得下面的SOSM 控制律成立:u =−C (x )K msgn(⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋a r 1)−β2(s 1,s 2)⌊⌊s 2⌋ar 2+βar 21(s 1)⌊s 1⌋ar 1⌋r 3a −z 2,(28)其中z 2是FDOB(6)给出的不确定项A (t,x )的估计,建立闭环系统(4)–(5)的有限时间稳定性.证根据U =v 和v =b (t,x )u 的定义,得到U =−C (x )K mb (t,x )sgn(⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1)−b (t,x )β2(s 1,s 2)×⌊⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1⌋r 3a −z 2,(29)将控制器(29)放入系统(4)中,可以得到{˙s 1=s 2,˙s 2=A (t,x )−z 2+U s ,(30)U s =−C (x )K mb (t,x )sgn(⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1)−b (t,x )β2(s 1,s 2)×⌊⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1⌋r 3a .(31)第11期王冬生等:基于有限时间扰动观测器的水厂加矾系统二阶滑模控制1969因为系统(30)与系统(4)结构相似,则系统(30)在控制器U s下将有限时间收敛到原点.由此进一步验证控制器不会在T f之前发散到无穷大.选择一个有限时间有限函数V(s1,s2)=12s21+12s22.(32)由于系统(30)中的不确定项A(t,x)总是有界的,因此可以很容易地得到˜A(t,x)=A(t,x)−z2也是有界的.因此,可以找到一个正常数Υ使得|˙s2| |˜A(t,x)|+|U s| Υ,(33)˙V(s1,s2)=s1s2+s2(A(t,x)−z2+U s)2V(s1,s2)+12Υ2,(34)之后可以得出结论V(s1,s2)=(V(s1(0),s2(0))+14Υ2)e2t−14γ2.(35)这意味着系统(30)的状态s1和s2在时间间隔(0, T f]内是有界的.此外,可以得出结论,系统(30)可以通过复合控制器(28)在有限域内稳定到原点.因此,滑动变量s可以在有限时间内稳定为零.证毕.4仿真验证加矾系统随着实际工况的变化而不同,本文采用在线辨识方法对加矾系统进行建模,即G(S)=1050s2+15s+1.(36)本文在MATLAB环境下进行控制仿真.模拟在0∼60min期间将出水浊度设定值保持在2NTU,在60∼120min期间将出水浊度设定值保持在1NTU.选择超调量、调节时间(∆=0.02min)和绝对误差积分(integral absolute error,IAE)作为量化指标来评估控制方案,即IAE(t)=1NN∑t=1|y r(t)−y(t)|,(37)其中:y r(t)是参考值,y(t)是实际过程输出.为了设计加矾系统的二阶滑模控制器,首先要选择一个滑动变量.将滑动变量s(即浊度误差)定义为s=y−y ref,(38)式中:y表示出水浊度,y ref表示出水浊度设定值,得到滑动变量s的动力学方程{˙s1=s2,˙s2=−0.3s2−0.02s1+0.2u.(39) 4.1模型不匹配情况在加矾系统中,由于天气恶劣或水源受到污染,原水水质有时会发生突变.这导致沉淀池的原水水质超出正常范围,并且建立的模型过程与实际过程不匹配.为了证实所提出的控制方案的鲁棒性,在模型不匹配的情况下,K和T提高20%,从而得到了传递函数G(S)=14.472s2+18s+1,(40)因此,滑动变量s的动力学方程{˙s1=s2,˙s2=−0.25s2−0.014s1+0.2u.(41)通过取−C(x)K m=−(1.5|s2|+0.1|s1|),α=2, r1=2,r2=1.6,r3=1.2,β1=0.4,β2=6控制器可以设计为u=−(1.5|s2|+0.1|s1|)sgn(⌊s2⌋21.6+0.421.6⌊s1⌋1)−6⌊⌊s2⌋21.6+0.421.6⌊s1⌋1⌋1.22.(42)为了更好展示FDOB-SOSM复合控制器的性能,仿真中将工业系统中广泛运用的比例–积分–微分(pr-oportional integral derivative,PID)控制器,SOSM控制器,FDOB-PID复合控制器加入对照实验中,仿真结果如图3和表1所示.由图3可以看出在0∼60min和60∼120min,本文提出的FDOB-SOSM复合控制,能够更好地跟踪出水浊度设定值(reference,REF)的变化;由表1可知FDOB-SOSM复合控制下的系统稳定时间最少,绝对误差积分最小,整体性能要优于其他控制器.3.02.52.01.51.00.50.0≤⍺/NTU020406080100120U / minFDOB + SOSMSOSMFDOB + PIDPIDREF图3模型不匹配情况仿真结果Fig.3Simulation results of model mismatch4.2受扰动情况在加矾系统中,由于原水水质和水量变化、以及传感器信号波动等原因会导致对加矾系统产生一定的扰动.因此考虑受扰动情况,由传递函数(36),得到滑动变量s的动力学方程{˙s1=s2,˙s2=−0.3s2−0.02s1+0.2u(t)+d(t),(43)1970控制理论与应用第40卷式中d (t )为扰动,仿真中取的是幅度0.05,频率为0.1的正弦信号.控制器采用式(42).表1模型不匹配情况控制性能指标Table 1Control performance index of model mismatch0∼60min60∼120min控制方案超调量/稳定绝对误超调量/稳定绝对误%时间/差积分/%时间/差积分/min NTU min NTUFDOB-SOSM 060.0994012.50.2045SOSM 114.50.11410160.2582FDOB-PID 6130.090336410.3036PID 27.535.50.18425049.50.4480仿真结果如图4和表2所示.由图4可以看出在0∼60min,只有FDOB-SOSM 控制方案很好的跟踪设定值.可以看出基于FDOB 的扰动估计补偿,使FDOB-SOSM 复合控制具有更好的抗扰动能力;同时,由表2可知FDOB-SOSM 复合控制下的系统稳定时间最少,绝对误差积分也最小.2.52.01.51.00.50.0≤⍺ / N T U020406080100120U / minFDOB + SOSM SOSMFDOB + PID PID REF图4受扰动情况仿真结果Fig.4Simulation results under disturbance表2受扰动情况控制性能指标Table 2Control performance index under disturbance0∼60min60∼120min控制方案超调量/稳定绝对误超调量/稳定绝对误%时间/差积分/%时间/差积分/min NTU min NTU FDOB-SOSM011.50.11020120.2144SOSM 8.5110.1153–>600.3158FDOB-PID –>600.1602–>600.3377PID–>600.1798–>600.41614.3模型不匹配受扰动情况为了进一步对比FDOB-SOSM 控制方案的性能,在模型不匹配且同时遭受扰动的情况下,由传递函数(40),得到滑动变量s 的动力学方程{˙s 1=s 2,˙s 2=−0.25s 2−0.014s 1+0.2u (t )+d (t ),(44)式中d (t )为扰动,仿真中取的是幅度0.05,频率为0.1的正弦信号,控制器采用式(42).仿真结果如图5和表3所示.由图5可以看出在0∼60min 和60∼120min,只有FDOB-SOSM 控制方案很好的跟踪设定值.可以看出模型不匹配和外部扰动时,基于FDOB 的扰动估计补偿,使FDOB-SOSM 复合控制具有更好的设定值跟踪和抗扰动能力.由表3可知,FDOB-SOSM 控制方案具有更好的鲁棒性、更快的响应和更小的超调.3.02.52.01.51.00.50.0≤⍺ / N T U020406080100120U / minFDOB + SOSM SOSMFDOB + PID PID REF图5模型不匹配受扰动情况仿真结果Fig.5Simulation results of model mismatch underdisturbance表3模型不匹配受扰动情况控制性能指标Table 3Control performance index of model mismatchunder disturbance0∼60min60∼120min控制方案超调量/稳定绝对误超调量/稳定绝对误%时间/差积分/%时间/差积分/min NTU min NTU FDOB-SOSM790.1018017.50.2332SOSM –>600.1623–>600.432FDOB-PID–>600.1498–>600.3833PID–>600.3109–>600.58185结论本文提出了一种水厂加矾系统的FDOB-SOSM 复合控制方案,采用了一种改进的带有非光滑项的SOSM 控制方法实现加矾反馈控制;FDOB 用于估计模型不匹配和扰动,并应用估计值作为前馈补偿削弱模型不匹配和扰动带来的不利影响.采用李亚普诺夫函数证明了系统的稳定性.在实际工程中存在的水第11期王冬生等:基于有限时间扰动观测器的水厂加矾系统二阶滑模控制1971质、水量突变等影响下造成的模型不匹配与扰动分别进行了仿真.仿真结果证明了控制方法的有效性.参考文献:[1]RATNAY AKA D D,BRANDT M J,JOHNSON M K.CHAPTER8-Water Filtration Granular Media Filtration.Oxford:Butterworth-Heinemann,2009.[2]CUI Fuyi,LI Guibai.Coagulation control technology offlowing cur-rent method.Water Supply and Drainage in China,1991,7(6):36–40.(崔福义,李圭白.流动电流法混凝控制技术.中国给水排水,1991, 7(6):36–40.)[3]LIU Qianjun,BAI Hua,LI Guibai.Intelligent control of light trans-mittance pulsatingflocculation dosing system.Water Supply and Drainage in China,2003,19(8):52–53.(刘前军,白桦,李圭白.透光率脉动絮凝投药系统的智能控制.中国给水排水,2003,19(8):52–53.)[4]AI Wei,ZHU Xuefeng.Data-driven direct control method for largelag process offlocculation and dosing in water plant.Control Theory &Applications,2011,28(3):335–342.(哀微,朱学峰.水厂絮凝投药大滞后过程的数据驱动直接控制方法.控制理论与应用,2011,28(3):335–342.)[5]DING Shihong,LI Shihua.Global Finite-time stabilization of nonlin-ear integral systems under input saturation.Acta Automatica Sinica, 2011,37(10):1222–1231.(丁世宏,李世华.输入饱和下的非线性积分系统的全局有限时间镇定.自动化学报,2011,37(10):1222–1231.)[6]DING S,LI S.Second-order sliding mode controller design subjectto mismatched term.Automatica,2017,77:388–392.[7]RAUF A,LI S,MADONSKI R,et al.Continuous dynamic slidingmode control of converter-fed DC motor system with high order mis-matched disturbance compensation.Transactions of the Institute of Measurement and Control,2020,42(14):2812–2821.[8]YANG Bo,SHU Hongchun,ZHU Dena,et al.Maximum powertracking sliding mode control of permanent magnet synchronous gen-erator based on disturbance observer.Control Theory&Applications,2019,36(2):207–219.(杨博,束洪春,朱德娜,等.基于扰动观测器的永磁同步发电机最大功率跟踪滑模控制.控制理论与应用,2019,36(2):207–219.)[9]LIU L,ZHENG W X,DING S.High-order sliding mode controllerdesign subject to lower-triangular nonlinearity and its application to robotic system.Journal of the Franklin Institute,2020,357(15): 10367–10386.[10]CHEN D,SD B,XW A,et posite SOSM controller for pathtracking control of agricultural tractors subject to wheel slip.ISA Transactions,2022,130:389–398.[11] A.LEV ANT.Higher-order sliding modes,differentiation and output-feedback control.International Journal of Control,2003:76(9/10): 924–941.[12]QIAN C,WEI L.A continuous feedback approach to global strongstabilization of nonlinear systems.IEEE Transactions on Automatic Control,2001,46(7):1061–1079.[13]BHAT S P,BERNSTEIN D S.Finite-time stability of continuousautonomous systems.SIAM Journal on Control and Optimization, 2000,38(3):751–766.[14]ZHU J,YU X,ZHANG T,et al.Sliding mode control of MIMOMarkovian jump systems.Automatica,2016,68:286–293.作者简介:王冬生博士,副教授,研究方向为人工智能、大数据处理及智能控制在水处理过程中的应用,E-mail:***********************.cn;张鹏硕士研究生,研究方向为智能控制在水处理过程中的应用,E-mail:*****************;孙锦昊本科生,研究方向为智能控制在水处理过程中的应用, E-mail:*****************;郭若寒本科生,研究方向为智能控制在水处理过程中的应用, E-mail:****************;蒋国平博士,教授,研究方向为复杂网络、复杂系统控制,E-mail: *****************.cn.。
双Buck型逆变器高阶系统二阶滑模控制
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ｓｅ ｄ —ｔ ｔ ｒ ｃｓｏ ｎｄ ｃ ｒｅ ｔｔａ ｉ ｎ ｅｆ ｒ ｎ ｅ ｏ ｌｄ ｎ ｄ ｃ ｎｔｏ ，ａ ｎ ｗ ｈ ｏ ｒｈ—ｒ ｅ ｔ ａ ｙ ｓａｅ ｐ ｅ ｉｉｎ ａ ｕ ｒ ｎ ｒ ｎｓｅ ｔｐ ｒｏ ｍａ ｃ ｆｓｉ ｉ ｇ ｍｏ ｅ ｏ ｒ ｌ ｅ ｔ ｅ ｆ ｕ ｔ ｏ ｄ ｒ ｓ ｓｅ ｗｉｈ ｓ ｃ ｎ ｏ ｄ ｄ．Ｔｈ ｉｃ ｉ ｄ ｎ ｍｉ ｄ ｌ ｐ ｒ ｔ ｙ ｔｍ ｔ ｅ ｏ ｄ・ ｒ ｅ ｌ ｎ ｄｅ ｃ ｎ ｒ ｌｓｒｔ ｇ ｓ ｐ ｏ ｏ ｅ ｄ ｅ ｃｒ ｕ ｔ ｙ ａ ｃ ｍｏ ｅ ，ｏ ｅ ａ— ｉ ｇ ｐ ｉ ｃｐ ｅ ａ ｄ ｃ ｎ ｒ ｌｅ ｅ ｉ ｎ ｏ ｅ ｓｒ ｔ ｇ ｒ ｉｅ ｎ ｔ ｅ ｐ ｐｅ ．Ｔ ｅ ｅ ｐｅｉ ｎ ｅ ｕ ｔ ｈ ｗ ｈ ｔ ｎ — ｒｎ ｉ ｌ ｎ ｏ ｔｏｌｒｄ ｓｇ ｆｔ ｔａｅ ｙ ａ ｅ ｇｖ ｎ ｉ ｈ ａ ｒ ｈ ｘ ｒｍｅ ｔｒ ｓ ｌ ｓ ｏ ｔ ａ ｈ ｓ
永磁同步电机调速系统二阶滑模控制器的设计
微电机MICROMOTORS第54卷第2期2021年 2月Vol. 54. No. 2Feb.2021永磁同步电机调速系统二阶滑模控制器的设计黄鹤松1,王 芮1,宋承林2,张鸿波2（1.山东科技大学电气与自动化工程学院，山266590；2.青岛中加特电气股份有限公司，山东青岛266000）摘 要：在表贴式永磁同步电机调速系统中，针对经典PI 控制中超调大、鲁棒性差、易受负载扰动等问题，设计一种基于 滑模 的 器，采用 型滑模面，通过 数 了其在有限时间 ， 应用于节，对参数变化 感的优点；并针对 扰动问题， 一种 矩 器，矩值补偿到电流中，减了 扰动对系统的 $仿真和实验结 ， 永磁 电机的调速性能，响应 快 超调，系统鲁棒性强。
关键词：永磁 电机； 滑模 ； 矩观测器；负载扰动中图分类号：TM351； TM341: TP273 文献标志码：A文章编号：1001-6848（2021）02-0055-06Design of Second Order Sliting Mode Controller Based on PMSMSpeeS Regulation SystemHUANG Hesong 1，WANG Rui 1，SONG Chenlin 2，ZHANG Hongbo 2(1. Collegc cf Electrical Engineering anC AutoEatioo ， ShanCong University qf Science anC Tchnology ，QingCao ShanCong 266590， China ； 2. QingCao CCS Electric Co.，LtO.， QingCaoShanCong 266000， Chyna )Abstract : In the surface-mounted permanent maanel synchronous motor speed control system ，the problems of targe overshoot ，poor vbustnrs and susceptibgity to toad disturbance in tassic PI control are addressed. A controller based on the second-order sliding mode algorithm was designed. The inUgral sliding mode sur face was adopted. The convergence in finite tima was proved by Lyapunov functUn , and it was appied to the speed control. It is insensitivv to the change of intemal parameters aiming at the problem of load disturb ance ，a load torque observer was designed to compensate the observed vvlua of the load torque into the cur rent to reduce the inauenco of the load disturbance on the system. Simulation and expe/mental results showthat the control method improvvs the speed re-ulation system of the permanent maanel synchronous motor sig-nificontly ，con respond quickly to a givvn speed without ovvrshoot ，and it has strong vbustnrs.Key words : permanent maanel synchronous motor ； second-order sliding mode ； load torque obsrver ； load disturbance0引言贴式永磁电机因其体积小、效率高、功率密度大、等点，在、扌造、备等领域 广泛应用。
二阶系统 滑模控制
二阶系统滑模控制
二阶系统是指系统的动态方程具有二阶微分方程形式的系统。
在控制领域中，二阶系统是比较常见的一种系统类型。
滑模控制是一种针对非线性系统的控制方法，其主要思想是通过引入一个滑动面来控制系统的状态。
在二阶系统中，可以通过滑模控制来实现对系统状态的控制。
滑模控制的核心是滑动面的设计。
对于二阶系统，通常选择一个二次函数形式的滑动面。
通过选择合适的滑动面以及设计相应的控制律，可以实现对二阶系统的稳定控制。
在实际应用中，二阶系统滑模控制常常被用于控制机械臂、飞行器等系统。
它具有控制精度高、稳定性好等优点，是一种非常有效的控制方法。
- 1 -。
基于二阶滑模算法的永磁电机直接转矩控制
[ 7 * 二阶滑模控制算法对
永磁直线电机
统进行设计,保证了电机
出现
与阻力扰动时转子速度能够严
格
信号,消除了磁阻作用和推力波
动的影响， 削弱了永磁直线电机的抖振现
象&
[ 8 *提出一种针对电机速度控制的二
阶滑模算法， 对实际
信号进行检测来
实际速度值,提高了
统的稳定性,但
该方法 量大、控制特性
&
[9* :
、效率高、
率密度高、控制简单等，应于工业自
动化、航空、电动等 [1]& PMSM
控
制性能直接影响电机的输出特性，
提高
PMSM 控制精度一直是电机控制 的研究
热点。
高精度快等特点[2]，传统
进行定子电压矢量输出，
电与转矩
脉动大、 器开关频率不恒定、电机振动 「
大等问题。为了
DTC系统中建立了四维开关表， 于每个有
矢量的长度相等,方向固定，使得DTC方法对
输出矢量的控制能力受到严重限制,容易在稳
态运行时出现不理想的转矩与磁链波动&文献
[10 *将转矩与磁链误差作为输入， 2 PI
控制器输出定
电压,
SVPWM逆
器的开关状态，但PI系数对电机 与工作
特性敏感,无法保证系统的 性&
Shenyang 110870，China)
Abstract: Aiming ai the problems of low precision control accuracy，large torque ripplo and unstable switching frequency in traditional direct torque control ( DTC) method， a DTC metCod based on second order sliding mode control for permanent maanet motor was proposed. Based on the second order sliding mode controi principle， traditionai flux controneo and torque controneo were replaced by sliding mode controneo. Vector-modulating method was used to modulato the space voltaae and the stability of switching frequency was improved. Also，dynamic and output performances of motor were improved. The simulation and experimentai results showed that the controi method cou8d eoecrieeaoeducerhecu oenroipp eand rooqueoipp e.Theanri-inreooeoenceabiiraooconroo8eaerem wae enhanced. The fast dynamio response and strong robust performancc of motor were realized.
基于PMSM的二阶滑模无位置传感器控制
微电机MICROMOTORS第53卷第-期2020年 9月Vol.53. No.—Dec. 2022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440065）摘 要：在永磁同步电机传统滑模观测器（SMO ）无位置传感器控制方案中，针对其因符号函数带来的抖振现象以及因一阶低通滤波器带来的相位滞后问题。
根据SuperOwi S Ung 算法设计了二阶滑模观测器（STASMO -无位置传感器控 制方案，该方案不仅有效地抑制了抖振现象，而且取消了一阶低通滤波器的使用。
当电机运行时，定子电阻会随着电机内部温度的升高而改变，故设计了合理的定子电阻观测器来实时观测定子电阻，从而避免了定子电阻对无位置 传感器控制方案估计精度的影响。
最后通过对所提方案进行系统模型搭建与仿真分析，从而证明了所提方案对电机 位置和转速具有较高的估计精度。
关键词：永磁同步电机；二阶滑模观测器；无位置传感器控制；定子电阻观测器中图分类号：TM341 ； TM351 ； TP073 文献标志码：A 文章编号：—0—6848（2020 ） 9-0083-06PMSM Based Second-ordeo Slicing Mode Position Seysorlest ControlCAL Jun , L i Pengza , HUANG Ynvyyad(College ef Automation , Chongginh University of Posti ang Telecommunicationi , Chongqing 440265 , Chinn -Abstroct : U tha traditional sliding moVa oVse/vn (SMO) position sensorless covt/l schema of permanentmaynei syych/vovs motors , tha chattan phenomenon cansed by tha sign function and tha pPaso lay p/Vlem cansed by tha first-vrUan low-pass filtan are addressed . A secovd-vrUan sliding moVa oVse/vn (STASMO) po sition sensorless covt/l schema was d/igded accorUing to tha sup/Cwisting alnorithm. This schema not ovty Xfectivety supp/ssed chatteOng , but also eliminated tha use of a first-vrUan low-pass filten. When tha motonis running , the staton resistance wilt change with the increase of the inteoial temperature of the moton, so a/asonable staton resistance oVse/en was designed to oVse/e the staton resistance in real hme , the/by avot- ding the 1110x 6/00 of the staton resistance on the estimation acchracy of the sensorless control s cheme. Final-ty , th/nph the system moVel budding and simulation analysis of the p/posed scheme , d is proved that thep/posed scheme has a high estimation acchracy fon the moton position and speed.Key wrrdt : permanent maynei syych/vovs moton ； second-vrUen sliding moVe oVse/en ； position sensorlesscontrol ； Staton resistance oVse/eno 引言永磁同步电机（PMSM ）因其具有结构紧凑、转 矩惯量比高、功率密度大和动态特性好等优势而被广泛应用于众多电力传动系统中。
滑模控制 二阶倒立摆 matlab
滑模控制二阶倒立摆 matlab滑模控制是一种常用的控制方法，在控制二阶倒立摆中也可以得到很好的应用。
通过加入滑模控制器可以提高控制系统的稳定性和鲁棒性。
在 Matlab 中，可以使用 Simulink 来进行二阶倒立摆的仿真和控制器设计。
具体步骤如下：1. 搭建二阶倒立摆的模型，包括小车、摆杆和配重块等组成部分。
2. 设计 PID 控制器，作为基准控制器用于比较滑模控制器的性能；3. 按照滑模控制器设计的思路，搭建滑模控制器模型，其中包括滑模面、滑模控制律等组成部分。
4. 将滑模控制器与二阶倒立摆模型进行连接，并进行仿真。
实现过程中的代码如下：1. 建立模型：使用 Simulink 中的组件、信号源、仿真器等构建二阶倒立摆控制系统模型。
2. PID 控制器设计：```matlabKp = 1.5;Ki = 0.01;Kd = 0.2;pid_controller = pid(Kp, Ki, Kd);```3. 滑模控制器设计：```matlabs = 0.1;r = 0.1;a = sqrt(2 * s * r);s_function = @(s_, r_) sign(s_) * a * tanh(abs(s_ / a) ^ (1 / 2)) - r_ * sign(s_);fcn = @(s_, r_) [s_function(s_(1), r_(1)), s_function(s_(2), r_(2))];smc_controller = @(s_, r_) - fcn(s_, r_);```4. 连接模型和控制器，进行仿真：```matlabmodel = 'inverted_pendulum';load_system(model);set_param(model, 'StopTime', '20');sim(model);% 绘制结果显示figure;subplot(2,1,1);plot(tout, theta, 'r', tout, theta_pid,'b');grid on;title('角度反馈');legend('smc', 'pid');xlabel('时间(s)');ylabel('角度(弧度)');subplot(2,1,2);plot(tout, x, 'r', tout, x_pid, 'b');gridon;title('位置反馈');legend('smc', 'pid');xlabel('时间(s)');ylabel('位置(m)');```在运行成功后，就可以看到二阶倒立摆的仿真结果，包括位置和角度等方面的变化情况，可以通过比较 PID 控制器和滑模控制器的性能表现来验证滑模控制器的优势。
基于二阶滑模观测器的永磁同步电动机伺服控制
Ｓ ｃ ｎ ｄ ｒ Ｓ ｉ ｉ ｇ Ｍ ｏ ｅ Ｏｂ ｅ ｖ ｒ Ｄｅ ｉ ｎ ｆ ｒ ＰＭ Ｓ ｅ ｖ ｎ ｒ ｌ ｅ ｏ ｄ Ｏｒ ｅ ｌ ｎ ｄ ｓ ｒ ｅ ｓｇ ｏ ｄ Ｍ Ｓ ｒ ｏ Ｃｏ ｔｏ
ＨＵＡＮＧＦＵ Ｙｔ—ｇ ｎ Ｌ Ｕ ｅ ｇ， Ｉ Ｗ 一ｇｕ ＭＡ Ｒｕ —ｑｎ ｏ， ｉ ｉ ｇ
ｓ ｐ ｒ—ｔ ｉｔ ｇ ａｇ ｒ ｈ ｗ ｓｕ ｉ ｚ ｄ ｆｒｇ ｎ ｒ ｔ ｇｔ ｅｓ ｅ ｎ ｒ ｅ ｄ ｈ ｂ ｅ ｖ ｒｄ ｎ ｍｉ ａ ｙ ｔｍ．Ｔ ｅｍａ ｎａ － ｕｅ ｗ ｓ ｎ ｌｏ ｔ ｍ ａ ｔ ｉ ０ ｅ ｅ ａｉ ｈ ｅ ｏ ｄ ｏｄ ｒｍｏ ｅ ｉ ｔ ｅ ｏ ｓ ｒｅ ｙ ａ ｃ ｌｓ ｓ ｉ ｉ ｌｅ ｎ ｎ ｅ ｈ ｉ ｄ ｖ ｎ ａ ｅ ｏ ｅｕ ｅ ｏ ｉｈ ｏ ｄ ｒｓｉ ｉｇ ｍｏ ｅ ｔ ｃ ｎ ｑ ｅ ｓｔ ａ ｖ ｉ ｅ ｈ ｈ ｔ ｒｎ ｈ ｎ ｍｅ ｏ ｈ ｃ ａ ｎ ｅｅ ｔ ａ ｔ ｆｔ ｓ ｆｈ ｇ ｒ ｅ ｌ ｎ ｄ ｅ ｈ ｉ ｕ ｓｗａ ｈ ｔｉ ａ ｏｄ ｄ ｔ ｅ ｃ ａｔｉ ｇ ｐ ｅ ｏ ｎ ｎ ｗ ｉｈ ｗ ｓ ｉｈ ｒ ｎ ｇ ｈ ｄ ｔ ｅ ｔ ｔ ｅｃ ａ ｓｃ ｌ ｒｔｏｄ ｒｓｉ ｉｇ ｍｏ ｅ ｏ ｓ ｒ ｅ ｓａ ｄ ｃ ｎｒ ｌ ｒ ．Ｔ ｅ ｅｆ ｉｎ ｙ ｏ ｅｐ ｏ ｏ ｅ ｂ ｅ ｅ ｓ ｉ ｕ ｔａｅ ｎ ｏ ｈ ｌ ｓｉａ ｓ ｒ ｅ ｌ ｎ ｄ ｂ ｅ ｖ ｒ ｎ ｏ ｔ ｌ ｓ ｈ ｆ ｃｅ ｃ ｆｔ ｒ ｐ ｓ ｄ ｏ ｓ ｒ ｒ ｗａ ｌ ｓｒｔ ｄ ｏ ｉ ｆ ｄ ｏｅ ｉ ｈ ｖ ｓ ｌ ａ Ｍａｌ ｂ Ｓ ｍｕ ｉ ｋ ｔ ／ ｉ ｌ ． ａ ｎ
基
基 于二 阶滑模 观 测 器 的永磁 同步 电动 机 伺 服控 制
基于二阶滑模的永磁同步电机直接转矩控制
ｐｏ ｅ ． Ｓ ｍｕ ａｉｎ ｒ ｓ ｌｓ ｓ ｏ ｔ ａ ， ｃ ｍｐａ ｅ ｔ ｈ ｃ ｅ ａ ｅ ｎ ｔａ ｉｉｎａ ｏ ｔｏ ， ｔ ｅ ｓ ｅ ｄ ｓｄ ｉ ｌｔｏ ｅ ｕ ｔ ｈ ｗ ｈ ｔ ｏ ｒ ｄ ｗｉ ｔ ｅ ｓ ｈ ｍｅ ｂ ｓ ｄ ｏ ｒ ｄ ｔｏ ｌＰＩｃ ｎ ｒ ｌ ｈ ｐ ｅ ｈ ｃ ｎ ｒ ｌｅ ｆｔ ｒ ｅ ｌ ｉ ｇ ｍｏ ｅ ｈ ｓ ｓｒ ｎ ｅ ｏ ｓ ｏ ｍｏｏ ａ ａ ｔ ｒａ ｉｔ ｒ ａ ｃ ｏ ｔｏｌｒｏ ｗｏ ｏ ｄ ｒｓｉ ｎ ｄ ａ ｔｏ ｇ ｒ ｒｂｕ ｔｔ ｔ ｒｐ ｒ ｍｅｅ ｎｄ ｄ ｓｕ ｂ ｎ ｅ， ａ ｄ ａ ｔｎ ａ ｅ ｄ ｎ ｔｅ ｕ ｔｄ
０ 引 言
交流 永磁 同 步 电动 机 （ ＭＳ 以 结 构 简 单 、运 Ｐ Ｍ）
也 得 到 了很 大 改 善 。
关键 词 ：二 阶滑 模 ；直 接 转 矩 控 制 ；永 磁 同步 电机
中 图分 类 号 ：Ｔ ４ ；Ｔ ３ １ Ｍ３ １ Ｍ ５ 文 献 标 志 码 ：Ａ 文 章 编 号 ：１０ —８ ８ ２ １ ） ００ ３ —３ ０ １６４ （ ００ １ —０ ３０
ｓｅｄｉ ｔｅｃｎｅ ｔ ｎ ｌ ｉ ｃ ｔｒｕ ｏ ｔ ｌ Ｄ Ｃ）ｏ ｅｍ ｎｎ ａｎ ｔ ｙｃ ｒｎ ｕ ｃ ｉｅ（ Ｍ Ｍ） ｐｅ ｏ ｖｎｉ ａ ｄｒ ｔｏｑ ｅ ｎｒ （ Ｔ ｎｈ ｏ ｅ ｃ ｏ ｆ ｒ ａ ｅｔ ｇ ｅ ｓｎ ｈｏ ｏ ｓ ｈｎ Ｐ Ｓ ， ｐ ｍ ｍａ
ｂ ｓ ｄ ｏ ｈ ｄ ｌｏ ｅ ｍａ ｎ ｇ ｅ ｙ ｃ ｒ ｎ ｕ ｔ ｒ ｃ ｍｂ ｎ ｄ ｔ ｒ ｅ ｌｄｎ ｄ ｔ ｒ ｃ ａ ｅ ｎ ｔ ｅ ｍｏ ｅ ｆｐ ｒ ｎｅ ｔｍａ ｎ ｔｓ ｎ ｈ ｏ ｏ ｓ ｍｏｏ ， ｏ ｉ ｅ ｗｏ ｏ ｄ ｒ ｓｉ ｉｇ ｍｏ ｅ ｗｉ ｄｉｅ ｔ ｈ
速度矢量场二阶滑模无人艇引导律
速度矢量场二阶滑模无人艇引导律作者：温锦元黄宴委来源：《华侨大学学报·哲学社会科学版》2024年第03期摘要：针对航速和航道未知扰动等因素，提出一种速度矢量场二阶滑模无人水面艇（USV）引导律。
首先，建立无人艇运动学和航向角动力学模型；其次构造路径误差（y e）模型，设计基于航速（V g）的路径误差矢量场，速度越大，航向角变化越小；再结合二阶滑模面设计一种速度矢量场二阶滑模无人艇引导律，并考虑未知扰动因素Δ分析速度矢量场二阶滑模无人艇引导律的稳定性。
仿真结果表明：相比于经典矢量场，速度矢量场有效实现航速V g越快，航向角变化率越小，矢量场越平缓，提高了USV航行安全性和稳定性；基于速度矢量场二阶滑模无人艇引导律的路径跟踪控制系统鲁棒性更强，路径跟踪准确度更高，能够较好地完成路径跟踪。
关键词：无人水面艇；矢量场引导律；路径跟踪；滑模控制中图分类号：TP 273文献标志码：A文章编号：1000-5013（2024）03-0324-08Second-Order Sliding Mode Guidance Law in Velocity Vector Field for Unmanned Surface VesselWEN Jinyuan，HUANG Yanwei（College of Electrical Engineering and Automation，Fuzhou University，Fuzhou 350116，China）Abstract：Aiming at factors such as unknown disturbances of course speed and course path，a second-order sliding mode guidance law in velocity vector field fof unmanned surface vessel（ USV） is proposed in the paper. Firstly，kinematics and course angle dynamics models of USV are established. Secondly，the path error （y e）model is constructed，and the path error vector field based on course speed （V g）is designed. The greater the speed，the smaller the change of course angle. Recombined with the second-order sliding mode surface，a velocity vector field second-order sliding mode USV guidance law is designed，and the stability of velocity vector field second-order sliding mode USV guidance law is analyzed considering the unknown disturbance factor Δ. The simulation results show that compared with the classical vector field，the velocity vector field achieves faster course speed V g with smaller course angle change rate and smoother vector field，which improves the navigation safety and stability of USV. The path tracking control system based on the second-order sliding mode guidance law in a velocity vector field exhibits enhanced robustness and higher accuracy in path following， which achieves path tracking with remarkable precision.Keywords：unmanned surface vessel；vector field guidance law；path following；sliding mode control無人水面艇（unmanned surface vessel，USV）是一种用于水上的独立可控的无人驾驶平台[1]。
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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v
2222112211sgn ,022sgn(),02m m m m m
y 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 d
s 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 d
T T d
T T T T d
d 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 u
ψψψ=++-=++++-=++++-==++=+
则控制量 11ˆˆˆˆˆˆ(())[(,,,)](())[(,,,)]d d u b
x s x x f u b x v x x f u ψψ--=-=- 解得0()(0)()t
u 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 b
x 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 v
ψψψψφξ-=+-=-+=∆+
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。
4、这样对应的二阶滑模应重新描述为
122====(,,,)+()d s y y s y x x f u x v
φξ⎧⎨∆⎩ 首先分析理想状态下的系统轨迹，假设12(0)(0)0y y >（即(0)(0)0s s >），则12y y 、（s s 、）在v 作用下，从初始点沿第一段抛物线移动，在1y 轴（s 轴）到达极值即抛物线顶点max y ，当1max 0.5y y =时，到达两段抛物线的交点，()v t 发生切换，并沿第二段抛物线（开关线）到达零点。
存在扰动时，切换发生在开关线附近的某一点，可能需要多次切换才能到达零点，扰动较大时甚至会导致系统不稳定。
考虑简化并改进算法，将开关线22
102m y y y a +=换成max 102
y y -=，对控制量()v t 乘以系数α，通过改变α和m a 约束来抑制扰动，改进算法如定理1所述。
5、定理1：改进的时间最优控制律1max 1()sgn[()]2
m v t a y t y α=--
其中11max max 21(0,),[,]21,r w w y y y r α⎧∈∈⎪=⎨⎪⎩
，其他，112max(/,2/())m a H wr H r wr >-
max y 是1y 的极值，初值取1(0)y ，迭代过程中满足2()0y t =（即()0s t =）时，取max 1()y y t =。
当满足1()y t ε<（ε是很小的正数），取()0v t =。
6、求控制量()u t 需要先求()v t ，可以
2222112211sgn ,022sgn(),02m m m m m
y y y y a y y a a v y y a y y a ⎧⎛⎫-++≠⎪ ⎪⎪⎝⎭=⎨⎪+=⎪⎩
也可以采用改进的()v t 将()v t 代入u 可求出u （含有v s =待求）
下面求(0)u
10(0)(0)T y s C x ==
取1()()s sng s sng y ζζ==，ζ是很小的正常数。
则1()()[()]
()T T d d T T T d s sng y C x x C Ax B x u Df x C Ax b x u C Df C x ζ==-=++-=++-
110ˆˆ(0)(())[((0))]T T T d
u b x sng y C Ax C Df C x ζ-=--+
二、削抖措施
二阶滑模将不连续控制量经过积分变为连续量,有效地抑制 了高频抖动。
本算法中,抖动的幅度和有关。
1、在下两式1max 1()sgn[()]2
m v t a y t y α=-- 11max max 21(0,),[,]21,r w w y y y r α⎧∈∈⎪=⎨⎪⎩
，其他
中的m a 约束条件112max(/,2/())m a H wr H r wr >-是为了保证系统的鲁棒性,根据扰动的上界选取。
接近滑模面时扰动φ将逐渐减小,因此可考虑用连续变化量替换固定不变的m a 。
2、把(,,,)d x x f u φ∆分解为
12111222(,,,)(,,)()()ˆ(,,)ˆ()()ˆ()ˆ()
d d d x x f u x x f x u
b x x x f b x b x x b x φφφφψψφψψ∆=∆+∆=-=-其中
(,,,)d x x f u φ∆的最大值为H ,重新表示为0()H t H u β=+，其中0H β和分别是1(,,)d x x f φ∆和2()x φ的上界。
m a 以u 为自变量，表示为如下形式0m m a a u λ=+，
其中000111
2max[,]m H H a wr r wr >-，1122max[,]wr r wr ββλ>- 三、基于Lyapunov 函数的二阶滑模控制 1、选取Lyapunov 函数2211(,)22V s s s s s ρρε=
++（注意sgn()s s s =），其中ρ和ε为正常数。
2、Lyapunov 函数的时间为 (,)sgn()
(sgn())V s s ss ss s s s s s s ρρερρε=++=++
为满足稳定性条件，选择sgn())s ks s s ρε=---
则(,)0V s s ≤。
3、按照sgn())s ks s s ρε=---设计趋近律，可使s 和s 在有限时间内趋近于平衡点。
则控制律1ˆˆˆ(())[sgn()(,,,)]d u b
x ks s s x x f u ρεψ-=---- 参考文献
[1] 李运华. 电液伺服系统的二阶滑模控制算法研究.机械工程学报，第41卷第3期。