基于滑模控制方法的故障容错控制系统研究
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摘要
本论文论述了以网络控制系统为背景的容错控制系统设计问题,前面部分旨在通过忽略网络环境来表现容错系统的方法,也就是在不考虑网络环境的情况下进行设计。接下来,论文介绍了在网络环境下,当网络受到延时、丢包、通讯干扰等影响的情况下,设计控制系统的问题。本文还研究了新的解决容错系统的联合技术,采用RBF-ARX模型和滑模控制相结合进行系统辨识,并采用特征值分配技术设计滑动面。本文还提出了一种由解耦滑模控制组成的处理子系统而不是整个系统的方法,这一方法所采用的策略是基于滑动曲面分区的选择。
第一章首先从错误和故障的概念入手,简要的讨论了执行器和传感器可能发生的故障类型。这一章对容错控制的概念进行了介绍,并对该研究领域的不同FTC进行了简要概述。同时,还介绍了不同FTC策略的基本分类,并讨论了它们各自的优缺点。此外,本章还给出了一个将显式MPC(模型预测控制)控制器应用于水箱水位的仿真样例。本章论述了SMC(滑模控制)理论的概要,基本原理和关键性技术问题,以及滑模控制的简单组合。
本文提出了滑动面的不同设计方法。
第二章提出了一种利用所谓的分散控制理论和解耦滑模控制以解决容错控制系统的方法,用来处理子系统而不是整个系统。介于某些情况,将一个复杂的问题分为一些更为简单的子系统,而每个子系统又单独受控。因此,待解决的不是一个复杂的问题,而是若干个更为简单的小问题。就作者所知,尚未出现这种分散算法的先例。
这一方法在滑动曲面分段设计以应用分段线性工具的基础上,提出了一种解耦策略,其目的是划定滑模出现的区域,这样我们就可以在划定区域内将复杂系统线性化为一个简单的线性模式,并尝试将这一策略应用于三容水箱水位系统。这一方法的新颖之处是如下的滑动曲面分段设计：可推广为：Si=Gi(x1x21)T,i∈N (2)当Gi=(ci1-ciz)T,i∈N时,q就是我们的控制器分区设计,该分区随着i∈N变化。如第二章2.2节中所说,当o 分别按照文中式(2.6)、(2.7)和(2.11)类似的结构,我们能在第二个滑动面S2上得到控制律,但是,为较好地构成分散方法的方式,我们选择处理一个滑动面S1；因为该系统分为两个子系统,并且我们可以控制第一个子系统,而把第二个子系统用作向第一个子系统输入信息的系统,所以这两个作业面可能通过变量z进行连接；由于S1和S2都是类推的,并且其中之一可以由式(2.4)中的(-c1z)来增强,其中z可能是有界值。备注2.2：
c1和c2都对系统的状态有很大影响。为实现良好的瞬态响应,有必要选择适当的滑动因子。
控制输入端是所选择的子系统的滑模控制端；因为在滑模控制理论中,假定u=u1,那么要控制整个系统,x1的边界可以由0 因此,z是一个衰减信号。根据滑模控制理论,我们想要通过使x1和x2等于零由控制行动实现这一目标,并且通过只由一个滑动面S1(2.4)设计的相同的控制输入,我们能得到S1和S2这两个滑动面,而且这两个滑动面倾向于零,其中包含本章中下文会给出的转移变量z；这个控制输入将由两个滑动面S1和S2注入到这两个子系统中,这是其特点；所以,x1应趋向于z,使S1趋于零,并且z应该趋向于零,使S1模拟为S2。这通过式(2.6)采用一种类似于减少z的输入控制信号。
在第二章中提到的详细设想前提下实现控制器设计的简要步骤大致如下：
首先进行初始化。指定一个与之相适应的线性系统下的初始状态。然后,设定q区,以表示滑动曲面的非唯一性。接着,利用分段性线性工具计算出选定范围,在存在有滑动模态的情况下,得到一个名为稳定矩阵的非空矩阵P。这一设计过程可以简要表达为如下流程图：
若计算得出一个空矩阵,即表示无滑动模态,在这种情况下,我们则回头重新设定另一区域。
稳定性问题由一个分段的二次李雅普诺夫函数来保证,它可以写成：1是各区域的索引集。其中,则V(x)=xT Pix,x∈Xt,Pi是正定矩阵。
为遵从或符合备注2.2所引述的说明,z的设计就是从滑动面S2到S1中取一个转移值,z可作为：其中ψ是S2的边界层。
在设计中,为使z更小并使收敛接近于零,选择一个足够更大的ψ,会导致形成z的无穷小变化。
关于这一方案是否能处理拥有大量已知或未知参数的高度复杂系统,仍存在疑问。为此,我们求助于基于RBF-ARX模型的另一方法,这一方法将在下一章进行介绍。
我们主要的研究兴趣是“滑模”领域,该领域的早期研究主要是在控制环境中探讨滑模。其研究方法包括采用非线性控制策略迫使闭环系统轨道随状态空间曲面而演变。适当的曲面选择决定闭环系统性能,而迫使曲面状态停留的控制规律会保证鲁棒性。理论上讲,滑模控制器可以完全排除一类被称为匹配不确定性因素的影响。这种独特的鲁棒性特性带动了该领域二十多年的研究工作。
过去的滑模控制理论发展都是基于一个假设,即系统所有状态在控制规律下可用。但从工程角度上讲,这个假想是不成立的。在实际系统中,只有该信息的子集可通过正确安装的传感器得到测量值。滑模理论的发展研究了仅有测量信息可用的情况。这是一个更加困难的设计问题,但是重要的理论成果已经证实,在滑模控制方案中可以考虑参数综合的问题。
其他一系列工作探讨了滑模在故障检测与隔离中的使用。该方法的新颖性在于滑模有通过适当缩放和滤波所谓的“等价输出误差引入”可以重建不可测量的信号的能力。这种方法在故障探测时与典型的误差产生方法有很大的不同。重建方法试图捕捉故障大小和“形状”,因此可以同时处理故障检测和隔离工作。事实上,即使出现故障,滑模输出依然有可能很好的跟踪对象的输出。
滑模控制(SMC)中,为保证滑动运动,其设计流程主要包括基于SMC两个主要阶段的两大步骤：
1)到达阶段：系统状态可在限定时间内从任何初始状态到达切换流形(预期的滑模)
2)滑模阶段：系统被纳入切换流形的滑动运动,即切换流形变成一个吸引子。
与以上两阶段相对应的是以下两大设计步骤：
Ⅰ)切换流形选择：根据指定的理想动力学特性选择一组切换流形。常见的流形就是线性超平面。
Ⅱ)不连续控制设计：采取不连续控制策略以保证切换流形在限定时间内的可达性。根据具体的控制要求,控制器可以是局部或全局的。
有几个主要问题被视为影响SMC普及使用的障碍,而它们却推动了软计算(SC)(如神经网络(NNs),模糊逻辑(FL)等)的广泛应用。以下列出了其中一些问题。
1)抖振：在上文中提到SMC是一种特殊的VSSs,其中,滑模由具分裂性的控制作用所引导。尽管拥有简单和鲁棒性的优势,SMC通常会遇到一些众所周知的问题,如：抖振,这是一种在预定义的切换流形周围振荡的运动。
主要归因于以下两点。
Ⅰ)寄生动力学与控制系统关联的出现,导致了小振幅高频率的振荡。这种寄生动力学代表在控制设计中通常被忽视的快速的执行器和传感器动力学。
Ⅱ)切换可引起高频振荡。其中可能包括短时间延迟,和／或控制计算所需要的执行时间延迟及最近在网络控制系统中的传输延迟。
2)匹配和非匹配不确定性：据了解,SMC以其对匹配不确定性的低灵敏度著称。但由于还不清楚SMC的鲁棒性的情况,如果不满足匹配条件,那么滑模运动就要取决于不太理想的不确定性。在这一方面的研究工作主要集中在将不确定性的影响限定在预期边界内。
3)未建模动态：从数学角度,不可能建立一个完整的实用系统。因为总是存在一些未建模动态。如果未建模动态包含可由SMC高频控制切换激活的高频振荡动态,则情况可能会更加严重。
正如上文所述,SMC和软计算(SC)的集成主要表现在以下两个方面。首先是将SC技术应用至SMC可让其更加“智能”,其次是采用SMC以增强SC能力。例如：NNs提供了一种无需建模的方案来从基本动态例子中学习。传统的NNs通过反向传播(BP)学习。此种学习过程通常十分费时。不同于传统的常数参数或函数的学习,动态环境下的学习速度是非常重要的。NNs的应用已经在动态学习环境中成功了,它更注重实现稳定和快速收敛,而不是减少学习误差。
带有SMC的模糊系统基本应用原理与NNs相似,例如减缓与抖振相关的问题,匹配和非匹配的不确定性,以及未建模动态。其主要优势包括被专家经验支持而受到的启发：降低(或减轻)抖振,基于他们多年的经验形成的部分知识为还没有建模的不确定性建模,并且通过对使用FL的集成线性模型进行建模的复杂系统进行控制。
在第三章提出的方法中,根本策略在于结合DTSMC(离散滑模控制)和RBF-ARX以解决传感器容错控制系统。在这一章中,协同仿真器作为一个补充工具用以实施该方法,系统行为将被RBF-ARX模型(3)所描述。RBF-ARX模型是具有高斯径向基函数神经网络的伪ARX,其系数取决于作用点。
滑动曲面的设计表明了其对于控制定律的独立性,可见控制器不会受到输入延误的影响。RBF-ARX模型的优势在于利用结构化的参数优化方法离线估计模型参数,在利用LMM(列文伯格-马夸尔特法)优化非线性模型参数的同时利用LSM(最小二乘法)和SVD(奇异值分解)对线性参数进行估计。
模型(3)的各个变量或参数在第三章中都进行了定义,如ARX模型[189]所述,RBF-ARX模型可以用故障模式进行估计。yF(t)=φF(θN/F,ΘF,(t-1))TθL/F+ξF(t)(4)(4)中的F是指故障大小。
滑动曲面的设计显示了其对于控制律的独立性,使得控制器不会受到输入延误的影响。这一设计依赖于特征值配置,将RBF-ARX模型改写为状态矢量空间的形式：
关于系统(5)的原理以及将(3)转化为(5)的设想在第三章中有详细的叙述。
设计滑动曲面的关键在于在选定的滑动曲面中找到G。St=GXt (6)
其中Xt是跟踪误差,且Xt=Xt.d-Xt,其中Xt,d是期望矢量,Xt则是状态估计。
经过计算和重新排列,我们得出：
即动力学状态反馈闭环形式At-BF,其中
寻找G取决于寻找F,第三章3.6节中给出了计算F的完整程序。这个控制器是基于特征值进行分配,如公式(3.36)所示,对于给定的F,我们可能通过把F分解成G1和G2计算滑动面增益G,然后通过使用变换矩阵来计算G。
在方程(3.16)中,对于给定的G,控制律的设计将取决于具有设计参数Φ和η的开关函数,这两个设计参数的选择方式是公式(3.24)中所示的这个词的特征值使用计算出的G保留在单位圆内。
前面所述协同仿真器带有/AMESim(自适应建模环境仿真),它能提供完整的系统工程平台,能进行复杂多领域系统的建模,模拟的运行以及执行深入的分析。AMESim平台使工程师能够使用一种友好的、面向应用的方法来研究任何组件或系统的静态和动态行为。这使得AMESim平台成为了多领域系统的首选,包括液体、机械、热学、机电和汽车、非公路、宇航或大型工业设备的电动机械元件或控制元件。它被广泛的应用于许多知名公司,如福特、通用、博世等。
一个传感器渗漏导致测量不精确的电动液压系统作为应用的实例。当模型在Matlab的Simulink环境下运行时,这一系统所用的接口可以执行许多AMESim设备。特别的是,AMESim模型实际上以某种方法转变成了S函数,其参数可以在AMESim之内正常变化。通常,AMESim与Simulink同时运行。
在使用带有一些假设的基于滑模控制的RBF-ARX模型和控制设计及基于状态空间RBF-ARX时,考虑到电液伺服系统的驱动回路可能遭受泄漏等故障,采用此策略,在故障的情况下,可提高对系统的识别能力。
在较低的维度可使人能够简化控制设计。因此,我们将滑模控制方法应用于降阶的运动方程中,并且我们通过一个例子,即所谓的块控制原则(Drakunov等人1990年)将原始设计问题降为一组具有较低维度的独立问题。
第四章以带有线性二次型调节器的倒立摆系统为例,将输入输出线性化和滑模控制进行了全面的对比,试图掌握滑模控制的总体意图。关于滑模控制仍然有待进一步研究。
控制某个系统的一个传统方法,就是利用动态系统在工作点附近局部线性化的一阶近似来计算线性控制器。非线性控制设计的反馈线性化方法是将非线性动力系统到完全或部分线性动力系统的一种代数转化,以便线性控制技术可以得到应用。计算一个系统的输入输出反馈所用到的一些数学工具包括：李导数,李氏括号和对合条件,并根据以下步骤：先计算状态反馈律,然后利用具有二态性的非线性转换函数进行处理,最终得到易于实施滑模控制的系统规范形式。
实际上,倒立摆系统并不是反馈线性化,因此我们计算出一个输出函数h(x),以达到系统中相对最大程度：经过线性化后,一旦我们得到系统的规范形式,我们选定的滑动曲面则为：控制器的形式则是：
例子中使用了输入输出线性化和滑模控制,并与线性二次型调节器做了详细对比。这个案例表明,线性二次型调节器和滑模控制都能在鲁棒条件下控制倒立摆。但滑模控制比线性二次型调节器具有更好的抗干扰(或其他突发异常现象)能力,同时,相比于线性二次型调节器,滑模控制能使摆杆状态更为准确的到达期望的参考状态。但滑模控制在接近期望参考状态时也需要更多的控制器动作。滑模控制所表现出来的这些特质,使我们尝试通过网络来完成控制系统,以考察它是否能处理网络控制系统中的各种不同的问题。
第五章简单回顾了网络控制系统(NCS),提出了网络环境中可能面临的不同挑战。网络时延可能会影响到系统运作,降低稳定性,或导致整个闭环系统的不稳定。网络控制引起了控制界极大的兴趣。众所周知,在许多实际系统中,电厂机械设备、控制器、传感器和制动器很难被安装于同一位置,因此信号必须从一处传送到另一处。在现代工业系统中,通常利用网络媒介将这些组件连接起来(典型的是数字有限频宽串行通信),从而产生了所谓的网络控制系统。因此,这些系统为交互式动态程序的控制带来了巨大挑战。因而NCS必须能够容错,这就意味着它必须能够处理错误事件并且进行积极重构。过去十年中,控制理论和计算机技术的进步促进了控制技术的不断改进。同时控制系统也变得日益精良与复杂。为了能长时间地成功运行,这些复杂的系统要求高度的可靠性,可维护性且能进行故障容错。
第六章描述了Matlab中Simulink实时仿真模块的用法,它促进了在实时核、网络传输和连续动态设备中执行控制器任务的协同仿真。本章详细说明了利用网络构成控制系统的各个模块。
一个典型的控制系统包括了工业控制器,控制过程,以及某种输入/输出通道,这种通道通常是通讯网络。在标准情况下,仿真中通常不包括任何网络模式,模型被简化且假设网络具有足够强大的计算机能力和快捷的通讯。这一解决方案并未考虑控制过程中的任何网络影响,但这种影响可能会导致不可预测的系统动作。当利用网络在硬实时系统下交换数据时,我们同样需要在仿真中考虑一个网络模型以得到正确的结果。已有的网络类型会定义特殊的通讯规则并导致数据流的独特反应,这种独特反应能大大改变过程控制中的输出结果。时延、数据丢失或网络负荷会影响实时控制回路中数据交换。这就是为什么我们要在复杂仿真模式下包括网络模型或使用实时嵌入系统以改进控制器的稳定性和鲁棒性的主要原因。
数据流模型,皮特里网或Matlab SimEvents工具箱中的定制方案通常可以用来模拟网络行为。在本文中,Matlab的实时工具箱被用做一个了解网络类别的简易方法,它是通过C++MEX语言编写并使用了事件模拟和外部中断。
本章还介绍了时延如何影响非线性系统控制,并研究了时延对控制回路性能的影响。这一章还讨论了滑模控制的实现以及在面临出现故障、干扰、网络延时、交通干扰等必将导致出现任意的时延问题时其性能的保持问题。可见,按计划执行的任务会随着这些额外的时延或通讯干扰所带来的变化而变化。为了检验我们所做工作的效率,我们和一些其它的工作做了比较,这些工作是其他作者使用和我们相同的系统所做的。
在文献[215]中,在结合了LQR(线性二次型调节器)技术、使用Petri网络模型仿真技术的网络自动化和控制系统的Markov时滞特性建模中,其建模的方法是对网络自动化系统的反应时间的建模,而不是估计传感器到控制器或控制器到执行器(表6.6)的时滞的概率密度分布,但是它并不像我们一样强调信息包丢失问题,因为我们尝试了丢包率达到了60%的情况。
文献[216]处理时变延迟和随机发生的丢包现象,不考虑由网络所导致的干扰,其主导思想是通过在倒立摆中运用蒙特卡洛算法对未知的将来控制输入进行建模。
该技术称为基于序列方式的控制传递,它不仅是传递一个输入,而是传递将来时刻的合理的整个输入序列,本指导思想利用了现代通讯网络的性能,其数据是在具有时间戳的大信息包中传递的。
此过程的主要的问题是控制器必须知道执行器过去和未来的控制输入,以确定当前最优的控制输入序列。
而且此程序只限于考虑控制器与执行器间的通讯网络,系统的状态完全可以被传感器检测,且传感器和控制器间的连结完美,只有很小的系统噪音。
把此方法与标准的NCS方法相比较,其结果可接受,然而与LQR相比效果更好(表6.6)。
在我们的实例中,设备与控制器之间、传感器到控制器之间存在着网络；而执行器到控制器之间没有线路连接,采样时间与作者所运用的方法相同。
在文献[217]中,作者在时滞和包丢失率很低的条件下工作,这是由于他提供更多的干扰,这就意味着仔细检查可用带宽,这可以通过增加更多设备以得到此要素(表6.6)(有信息包丢失情况),并且控制作者使用LQR控制的系统。
作者改变缓冲区大小,以在系统仍然处于稳定的区域的条件下,使其获得不同的阈值时滞,但是作者必须每次都要改变采样时间,并且在系统仍然处于稳定的区域的条件下,获得他所述的最小和最大时滞。在此项研究中,系统的稳定区域可以处于他的论文表6.6所述的边界上。
在我们的实例中,如作者所建议的那样,实时平台提供了一种调节干扰比而不用增加设备数量的方法,因此用此方法可使我们的缓冲区大小固定,我们的时滞测试丢包率在10%到60%的范围,如果丢包率小于此范围,则认为是完美的网络。那么,在我们的实例中时滞可以达到70ms并保持系统的稳定性,而在本论文中,却保持着8.4%的包丢失,时滞为18.23ms。与此实例研究相比,在我们的实例中也没有多速率采样时间。
与[218]中关于单摆系统的研究实例比较,我们主要考虑了3种情况：
第一种情况是采样时间固定,没有延迟或者数据包丢失的系统。[218]中当外力输入从2.25N下降到0.25N时,系统大约在10s内达到稳定的状态。而我们的方法,系统在5s内便达到稳定状态。此外,如果我们不考虑在5s后有意施加的干扰,则其运动所需要的力和0.25N相比完全是可以忽略的。
第二种情况是采样时间可变,有时滞和无数据包丢失的系统。考虑到系统的延迟时间应该小于总取样时间的最大值,故系统的延迟时间的可能范围只能是0.135s到0.02s。[218]提出的方法在延迟时间小于0.02s和位移0.17m的情况下,需要有1.25N到3N的外力输入,系统经过约10s的时间才能回到了稳定状态。而我们的方法,在相同的情况下,0.07s的系统时滞时,只需要17.8N的外力作用、或者0.05s的系统时滞时,只需要11.5N的外力作用,系统便到达其稳定状态。而且,此输入外力的大小取决于其位移(此处约为2m)和系统尺寸。
第三种情况是存在随机延迟和数据包丢失的系统。在[218]的算法(使用TDNN(时滞NN)和LQR控制器)中,当信息包没有达到并且取样时间结束时,其算法指示控制器和执行器使用最后可用信息(在缓存或队列中存贮的)。实施此算法仿真的前提条件是假设取样时间是0.03s,并且网络中有30%的包丢失,传感器到控制器间的时滞上限为0.09s下限为0.01s,而控制器到执行器间的时滞是0.03s到0.005s之间。这么长的时滞是可能发生的时滞,但是作者没有试验和测定系统在固定时滞条件下是如何动作的,因此此时滞可能发生,也可能不发生。系统在0.055m的位移和2N到3N的外力输入的情况下,其达稳定状态的时间大约是7s。当系统超过了一定的延迟时间和数据包丢失率的情况下,此研究方法就无法应对系统不稳定的情况,因此,作者提出：在NCS设计中,我们应当考虑网络时滞的特点。将我们的方法和作者的方法比较。首先,在取样时间为0.01s的情形下,本方法在数据包丢失率60%和延迟时间为0.01s的情况下,系统经过5s便达到了可接受的稳定状态。其次,为了更加接近实际系统,我们还考虑了一些干扰因素。结合应用近似反馈线性化的滑模控制对上述问题的处理似乎已达到某种极限,但迄今仍是可接受的。
第七章对整篇论文的工作进行了总结,并提出展望。指出MACs的出现是由于通信媒介仅能使有限的同步媒介接入通道对用户可用的结果。因此,在网络控制系统中,只有少量的传感器和制动器可以随时与控制器进行沟通。一旦网络超负荷,传输延迟和数据丢包率就开始显著增长,使得许多网络传输无法进行从而导致数据率超标。网络控制系统的鲁棒性还必须考虑到通过一个或多个MACs的网络进行控制。另外MAC概念的合并也导致了互联动态过程控制存在巨大挑战。最后指出,利用具有自适应容错的RBF-ARX模型来建立一个系统将是未来工作的第二大任务。
Our main research interest is in the area of'sliding modes'. The earliest work in this field of research considered sliding modes in the context of control. The approach involves forcing the closed-loop system trajectories to evolve along a surface in the state-space by means of a nonlinear control strategy. The closed-loop system performance is specified by appropriate surface selection, and robustness is ensured via a control law which forces the states to remain on the surface. Theoretically sliding mode controllers are able to completely reject the effect of a class of uncertainties-known as matched uncertainties. This unique robustness property has stimulated research in this area for over two decades.
Our main goal is to seek for method to handle the issues mentioned in the literature and add new contributions with sliding mode combined to other techniques to make any lack of efficiency avoidable, and apply to systems in connected to other component in classical way or over network.
This dissertation presents, networked control system based fault tolerant control system, firstly aims to show the fault tolerant control system method over ignored possible network i.e. working with a structure without going into network's issue, and then gives the control system in the presence of the network influenced by time delay, packed dropout, traffic disturbance. The dissertation also conducts new combination techniques dealing with fault tolerant control system starting by system identification, using RBF-ARX model combined to sliding mode control, where it's sliding surface design, lie to Eigenvalues assignmentand shows its independence from control law. The advantage of the RBF-ARX model, is its joy of an off-line nonlinear model parameter optimization, using LMM (Levenberg-Marquardt method), and a linear parameter estimation using LSM (Least Squares method) with SVD (Singular value decomposition). The example used for the application of this approach is an electro-hydraulic system with leakage issue where the measurement could not be accurate.
This system uses the interface which could implement many of the AMESim facilities while the model is running in the Simulink of Matlab. In particular, the parameters of the AMESim model could be changed within AMESim in the normal way, actually the model in AMESIM, somehow is transformed into S-function. Normally AMESim and Simulink run simultaneously. In other part of this dissertation, a new approach is given, which consists on decoupling fashion sliding mode control, dealing with subsystems instead of whole system. This approach presents a strategy based on the selection of piecewise sliding surface partition, and we apply the PwLTool which has as purpose to delimit regions where sliding mode occurs, then we may linearize the complex system into a simple linearized model in the delimited regions. An attempt to apply this strategy to the3water tank level system as example is given. The novelty of our approach is the design of sliding surface in piecewise style.
The dissertation carries out also a comprehensive comparison to an input-output linearization and sliding mode control using an inverted pendulum system as an example with linear quadratic regulation, attempting to capture the general intent of SMC, where there is still a great deal to be done in SMC.
A classical way to control a system is to compute a linear controller using the first order approximation of the system dynamics around the origin which gives a local linear approximation of the system.
The feedback linearization approach to non-linear control design is an algebraic transformation of non-linear system dynamics into a fully or partially linear one, so that linear control techniques can be applied.
The method to calculate the input-output feedback of a system uses some mathematical tools to mention here, lie-derivative, lie-bracket and involutive condition, this example shows that both the LQR and the SMC were able to control the inverted pendulum in a robust fashion. The SMC had better disturbance (or any other abrupt anomalies) rejection qualities than the LQR.
A typical control system contains an industrial controller, controlled process and some kind input/output channel, usually in a communication network, where its behavior usually, can be simulated with state flow models, Petri nets, or in a custom made schemes in a MatlabSimEvents toolbox.In this dissertation, we use True-Time toolbox (for Matlab) as a simple and easy way to realize several network types and it is written in C++MEX language, to investigate how the performance of a control loop could be influenced by a delay, and implement the approximated feedback linearization sliding mode controller and how far it performance could stand in the presence of multi-issues, like presence of fault, disturbance, network induced delay, traffic interference disturbance which provoke certainly a random time delay. It showed also that the scheduling run tasks changes with the change in the additional delay or traffic disturbance.
Further more we have carried out a serious comparaison with recent researches in the same field, and we find out that the sliding mode controller with approximate feedback linearization implemented seems coping positively with the issues mentioned above to certain limit but so far acceptable.
第一章首先从错误和故障的概念入手,简要的讨论了执行器和传感器可能发生的故障类型。这一章对容错控制的概念进行了介绍,并对该研究领域的不同FTC进行了简要概述。同时,还介绍了不同FTC策略的基本分类,并讨论了它们各自的优缺点。此外,本章还给出了一个将显式MPC(模型预测控制)控制器应用于水箱水位的仿真样例。本章论述了SMC(滑模控制)理论的概要,基本原理和关键性技术问题,以及滑模控制的简单组合。
本文提出了滑动面的不同设计方法。
第二章提出了一种利用所谓的分散控制理论和解耦滑模控制以解决容错控制系统的方法,用来处理子系统而不是整个系统。介于某些情况,将一个复杂的问题分为一些更为简单的子系统,而每个子系统又单独受控。因此,待解决的不是一个复杂的问题,而是若干个更为简单的小问题。就作者所知,尚未出现这种分散算法的先例。
这一方法在滑动曲面分段设计以应用分段线性工具的基础上,提出了一种解耦策略,其目的是划定滑模出现的区域,这样我们就可以在划定区域内将复杂系统线性化为一个简单的线性模式,并尝试将这一策略应用于三容水箱水位系统。这一方法的新颖之处是如下的滑动曲面分段设计：可推广为：Si=Gi(x1x21)T,i∈N (2)当Gi=(ci1-ciz)T,i∈N时,q就是我们的控制器分区设计,该分区随着i∈N变化。如第二章2.2节中所说,当o
c1和c2都对系统的状态有很大影响。为实现良好的瞬态响应,有必要选择适当的滑动因子。
控制输入端是所选择的子系统的滑模控制端；因为在滑模控制理论中,假定u=u1,那么要控制整个系统,x1的边界可以由0
在第二章中提到的详细设想前提下实现控制器设计的简要步骤大致如下：
首先进行初始化。指定一个与之相适应的线性系统下的初始状态。然后,设定q区,以表示滑动曲面的非唯一性。接着,利用分段性线性工具计算出选定范围,在存在有滑动模态的情况下,得到一个名为稳定矩阵的非空矩阵P。这一设计过程可以简要表达为如下流程图：
若计算得出一个空矩阵,即表示无滑动模态,在这种情况下,我们则回头重新设定另一区域。
稳定性问题由一个分段的二次李雅普诺夫函数来保证,它可以写成：1是各区域的索引集。其中,则V(x)=xT Pix,x∈Xt,Pi是正定矩阵。
为遵从或符合备注2.2所引述的说明,z的设计就是从滑动面S2到S1中取一个转移值,z可作为：其中ψ是S2的边界层。
在设计中,为使z更小并使收敛接近于零,选择一个足够更大的ψ,会导致形成z的无穷小变化。
关于这一方案是否能处理拥有大量已知或未知参数的高度复杂系统,仍存在疑问。为此,我们求助于基于RBF-ARX模型的另一方法,这一方法将在下一章进行介绍。
我们主要的研究兴趣是“滑模”领域,该领域的早期研究主要是在控制环境中探讨滑模。其研究方法包括采用非线性控制策略迫使闭环系统轨道随状态空间曲面而演变。适当的曲面选择决定闭环系统性能,而迫使曲面状态停留的控制规律会保证鲁棒性。理论上讲,滑模控制器可以完全排除一类被称为匹配不确定性因素的影响。这种独特的鲁棒性特性带动了该领域二十多年的研究工作。
过去的滑模控制理论发展都是基于一个假设,即系统所有状态在控制规律下可用。但从工程角度上讲,这个假想是不成立的。在实际系统中,只有该信息的子集可通过正确安装的传感器得到测量值。滑模理论的发展研究了仅有测量信息可用的情况。这是一个更加困难的设计问题,但是重要的理论成果已经证实,在滑模控制方案中可以考虑参数综合的问题。
其他一系列工作探讨了滑模在故障检测与隔离中的使用。该方法的新颖性在于滑模有通过适当缩放和滤波所谓的“等价输出误差引入”可以重建不可测量的信号的能力。这种方法在故障探测时与典型的误差产生方法有很大的不同。重建方法试图捕捉故障大小和“形状”,因此可以同时处理故障检测和隔离工作。事实上,即使出现故障,滑模输出依然有可能很好的跟踪对象的输出。
滑模控制(SMC)中,为保证滑动运动,其设计流程主要包括基于SMC两个主要阶段的两大步骤：
1)到达阶段：系统状态可在限定时间内从任何初始状态到达切换流形(预期的滑模)
2)滑模阶段：系统被纳入切换流形的滑动运动,即切换流形变成一个吸引子。
与以上两阶段相对应的是以下两大设计步骤：
Ⅰ)切换流形选择：根据指定的理想动力学特性选择一组切换流形。常见的流形就是线性超平面。
Ⅱ)不连续控制设计：采取不连续控制策略以保证切换流形在限定时间内的可达性。根据具体的控制要求,控制器可以是局部或全局的。
有几个主要问题被视为影响SMC普及使用的障碍,而它们却推动了软计算(SC)(如神经网络(NNs),模糊逻辑(FL)等)的广泛应用。以下列出了其中一些问题。
1)抖振：在上文中提到SMC是一种特殊的VSSs,其中,滑模由具分裂性的控制作用所引导。尽管拥有简单和鲁棒性的优势,SMC通常会遇到一些众所周知的问题,如：抖振,这是一种在预定义的切换流形周围振荡的运动。
主要归因于以下两点。
Ⅰ)寄生动力学与控制系统关联的出现,导致了小振幅高频率的振荡。这种寄生动力学代表在控制设计中通常被忽视的快速的执行器和传感器动力学。
Ⅱ)切换可引起高频振荡。其中可能包括短时间延迟,和／或控制计算所需要的执行时间延迟及最近在网络控制系统中的传输延迟。
2)匹配和非匹配不确定性：据了解,SMC以其对匹配不确定性的低灵敏度著称。但由于还不清楚SMC的鲁棒性的情况,如果不满足匹配条件,那么滑模运动就要取决于不太理想的不确定性。在这一方面的研究工作主要集中在将不确定性的影响限定在预期边界内。
3)未建模动态：从数学角度,不可能建立一个完整的实用系统。因为总是存在一些未建模动态。如果未建模动态包含可由SMC高频控制切换激活的高频振荡动态,则情况可能会更加严重。
正如上文所述,SMC和软计算(SC)的集成主要表现在以下两个方面。首先是将SC技术应用至SMC可让其更加“智能”,其次是采用SMC以增强SC能力。例如：NNs提供了一种无需建模的方案来从基本动态例子中学习。传统的NNs通过反向传播(BP)学习。此种学习过程通常十分费时。不同于传统的常数参数或函数的学习,动态环境下的学习速度是非常重要的。NNs的应用已经在动态学习环境中成功了,它更注重实现稳定和快速收敛,而不是减少学习误差。
带有SMC的模糊系统基本应用原理与NNs相似,例如减缓与抖振相关的问题,匹配和非匹配的不确定性,以及未建模动态。其主要优势包括被专家经验支持而受到的启发：降低(或减轻)抖振,基于他们多年的经验形成的部分知识为还没有建模的不确定性建模,并且通过对使用FL的集成线性模型进行建模的复杂系统进行控制。
在第三章提出的方法中,根本策略在于结合DTSMC(离散滑模控制)和RBF-ARX以解决传感器容错控制系统。在这一章中,协同仿真器作为一个补充工具用以实施该方法,系统行为将被RBF-ARX模型(3)所描述。RBF-ARX模型是具有高斯径向基函数神经网络的伪ARX,其系数取决于作用点。
滑动曲面的设计表明了其对于控制定律的独立性,可见控制器不会受到输入延误的影响。RBF-ARX模型的优势在于利用结构化的参数优化方法离线估计模型参数,在利用LMM(列文伯格-马夸尔特法)优化非线性模型参数的同时利用LSM(最小二乘法)和SVD(奇异值分解)对线性参数进行估计。
模型(3)的各个变量或参数在第三章中都进行了定义,如ARX模型[189]所述,RBF-ARX模型可以用故障模式进行估计。yF(t)=φF(θN/F,ΘF,(t-1))TθL/F+ξF(t)(4)(4)中的F是指故障大小。
滑动曲面的设计显示了其对于控制律的独立性,使得控制器不会受到输入延误的影响。这一设计依赖于特征值配置,将RBF-ARX模型改写为状态矢量空间的形式：
关于系统(5)的原理以及将(3)转化为(5)的设想在第三章中有详细的叙述。
设计滑动曲面的关键在于在选定的滑动曲面中找到G。St=GXt (6)
其中Xt是跟踪误差,且Xt=Xt.d-Xt,其中Xt,d是期望矢量,Xt则是状态估计。
经过计算和重新排列,我们得出：
即动力学状态反馈闭环形式At-BF,其中
寻找G取决于寻找F,第三章3.6节中给出了计算F的完整程序。这个控制器是基于特征值进行分配,如公式(3.36)所示,对于给定的F,我们可能通过把F分解成G1和G2计算滑动面增益G,然后通过使用变换矩阵来计算G。
在方程(3.16)中,对于给定的G,控制律的设计将取决于具有设计参数Φ和η的开关函数,这两个设计参数的选择方式是公式(3.24)中所示的这个词的特征值使用计算出的G保留在单位圆内。
前面所述协同仿真器带有/AMESim(自适应建模环境仿真),它能提供完整的系统工程平台,能进行复杂多领域系统的建模,模拟的运行以及执行深入的分析。AMESim平台使工程师能够使用一种友好的、面向应用的方法来研究任何组件或系统的静态和动态行为。这使得AMESim平台成为了多领域系统的首选,包括液体、机械、热学、机电和汽车、非公路、宇航或大型工业设备的电动机械元件或控制元件。它被广泛的应用于许多知名公司,如福特、通用、博世等。
一个传感器渗漏导致测量不精确的电动液压系统作为应用的实例。当模型在Matlab的Simulink环境下运行时,这一系统所用的接口可以执行许多AMESim设备。特别的是,AMESim模型实际上以某种方法转变成了S函数,其参数可以在AMESim之内正常变化。通常,AMESim与Simulink同时运行。
在使用带有一些假设的基于滑模控制的RBF-ARX模型和控制设计及基于状态空间RBF-ARX时,考虑到电液伺服系统的驱动回路可能遭受泄漏等故障,采用此策略,在故障的情况下,可提高对系统的识别能力。
在较低的维度可使人能够简化控制设计。因此,我们将滑模控制方法应用于降阶的运动方程中,并且我们通过一个例子,即所谓的块控制原则(Drakunov等人1990年)将原始设计问题降为一组具有较低维度的独立问题。
第四章以带有线性二次型调节器的倒立摆系统为例,将输入输出线性化和滑模控制进行了全面的对比,试图掌握滑模控制的总体意图。关于滑模控制仍然有待进一步研究。
控制某个系统的一个传统方法,就是利用动态系统在工作点附近局部线性化的一阶近似来计算线性控制器。非线性控制设计的反馈线性化方法是将非线性动力系统到完全或部分线性动力系统的一种代数转化,以便线性控制技术可以得到应用。计算一个系统的输入输出反馈所用到的一些数学工具包括：李导数,李氏括号和对合条件,并根据以下步骤：先计算状态反馈律,然后利用具有二态性的非线性转换函数进行处理,最终得到易于实施滑模控制的系统规范形式。
实际上,倒立摆系统并不是反馈线性化,因此我们计算出一个输出函数h(x),以达到系统中相对最大程度：经过线性化后,一旦我们得到系统的规范形式,我们选定的滑动曲面则为：控制器的形式则是：
例子中使用了输入输出线性化和滑模控制,并与线性二次型调节器做了详细对比。这个案例表明,线性二次型调节器和滑模控制都能在鲁棒条件下控制倒立摆。但滑模控制比线性二次型调节器具有更好的抗干扰(或其他突发异常现象)能力,同时,相比于线性二次型调节器,滑模控制能使摆杆状态更为准确的到达期望的参考状态。但滑模控制在接近期望参考状态时也需要更多的控制器动作。滑模控制所表现出来的这些特质,使我们尝试通过网络来完成控制系统,以考察它是否能处理网络控制系统中的各种不同的问题。
第五章简单回顾了网络控制系统(NCS),提出了网络环境中可能面临的不同挑战。网络时延可能会影响到系统运作,降低稳定性,或导致整个闭环系统的不稳定。网络控制引起了控制界极大的兴趣。众所周知,在许多实际系统中,电厂机械设备、控制器、传感器和制动器很难被安装于同一位置,因此信号必须从一处传送到另一处。在现代工业系统中,通常利用网络媒介将这些组件连接起来(典型的是数字有限频宽串行通信),从而产生了所谓的网络控制系统。因此,这些系统为交互式动态程序的控制带来了巨大挑战。因而NCS必须能够容错,这就意味着它必须能够处理错误事件并且进行积极重构。过去十年中,控制理论和计算机技术的进步促进了控制技术的不断改进。同时控制系统也变得日益精良与复杂。为了能长时间地成功运行,这些复杂的系统要求高度的可靠性,可维护性且能进行故障容错。
第六章描述了Matlab中Simulink实时仿真模块的用法,它促进了在实时核、网络传输和连续动态设备中执行控制器任务的协同仿真。本章详细说明了利用网络构成控制系统的各个模块。
一个典型的控制系统包括了工业控制器,控制过程,以及某种输入/输出通道,这种通道通常是通讯网络。在标准情况下,仿真中通常不包括任何网络模式,模型被简化且假设网络具有足够强大的计算机能力和快捷的通讯。这一解决方案并未考虑控制过程中的任何网络影响,但这种影响可能会导致不可预测的系统动作。当利用网络在硬实时系统下交换数据时,我们同样需要在仿真中考虑一个网络模型以得到正确的结果。已有的网络类型会定义特殊的通讯规则并导致数据流的独特反应,这种独特反应能大大改变过程控制中的输出结果。时延、数据丢失或网络负荷会影响实时控制回路中数据交换。这就是为什么我们要在复杂仿真模式下包括网络模型或使用实时嵌入系统以改进控制器的稳定性和鲁棒性的主要原因。
数据流模型,皮特里网或Matlab SimEvents工具箱中的定制方案通常可以用来模拟网络行为。在本文中,Matlab的实时工具箱被用做一个了解网络类别的简易方法,它是通过C++MEX语言编写并使用了事件模拟和外部中断。
本章还介绍了时延如何影响非线性系统控制,并研究了时延对控制回路性能的影响。这一章还讨论了滑模控制的实现以及在面临出现故障、干扰、网络延时、交通干扰等必将导致出现任意的时延问题时其性能的保持问题。可见,按计划执行的任务会随着这些额外的时延或通讯干扰所带来的变化而变化。为了检验我们所做工作的效率,我们和一些其它的工作做了比较,这些工作是其他作者使用和我们相同的系统所做的。
在文献[215]中,在结合了LQR(线性二次型调节器)技术、使用Petri网络模型仿真技术的网络自动化和控制系统的Markov时滞特性建模中,其建模的方法是对网络自动化系统的反应时间的建模,而不是估计传感器到控制器或控制器到执行器(表6.6)的时滞的概率密度分布,但是它并不像我们一样强调信息包丢失问题,因为我们尝试了丢包率达到了60%的情况。
文献[216]处理时变延迟和随机发生的丢包现象,不考虑由网络所导致的干扰,其主导思想是通过在倒立摆中运用蒙特卡洛算法对未知的将来控制输入进行建模。
该技术称为基于序列方式的控制传递,它不仅是传递一个输入,而是传递将来时刻的合理的整个输入序列,本指导思想利用了现代通讯网络的性能,其数据是在具有时间戳的大信息包中传递的。
此过程的主要的问题是控制器必须知道执行器过去和未来的控制输入,以确定当前最优的控制输入序列。
而且此程序只限于考虑控制器与执行器间的通讯网络,系统的状态完全可以被传感器检测,且传感器和控制器间的连结完美,只有很小的系统噪音。
把此方法与标准的NCS方法相比较,其结果可接受,然而与LQR相比效果更好(表6.6)。
在我们的实例中,设备与控制器之间、传感器到控制器之间存在着网络；而执行器到控制器之间没有线路连接,采样时间与作者所运用的方法相同。
在文献[217]中,作者在时滞和包丢失率很低的条件下工作,这是由于他提供更多的干扰,这就意味着仔细检查可用带宽,这可以通过增加更多设备以得到此要素(表6.6)(有信息包丢失情况),并且控制作者使用LQR控制的系统。
作者改变缓冲区大小,以在系统仍然处于稳定的区域的条件下,使其获得不同的阈值时滞,但是作者必须每次都要改变采样时间,并且在系统仍然处于稳定的区域的条件下,获得他所述的最小和最大时滞。在此项研究中,系统的稳定区域可以处于他的论文表6.6所述的边界上。
在我们的实例中,如作者所建议的那样,实时平台提供了一种调节干扰比而不用增加设备数量的方法,因此用此方法可使我们的缓冲区大小固定,我们的时滞测试丢包率在10%到60%的范围,如果丢包率小于此范围,则认为是完美的网络。那么,在我们的实例中时滞可以达到70ms并保持系统的稳定性,而在本论文中,却保持着8.4%的包丢失,时滞为18.23ms。与此实例研究相比,在我们的实例中也没有多速率采样时间。
与[218]中关于单摆系统的研究实例比较,我们主要考虑了3种情况：
第一种情况是采样时间固定,没有延迟或者数据包丢失的系统。[218]中当外力输入从2.25N下降到0.25N时,系统大约在10s内达到稳定的状态。而我们的方法,系统在5s内便达到稳定状态。此外,如果我们不考虑在5s后有意施加的干扰,则其运动所需要的力和0.25N相比完全是可以忽略的。
第二种情况是采样时间可变,有时滞和无数据包丢失的系统。考虑到系统的延迟时间应该小于总取样时间的最大值,故系统的延迟时间的可能范围只能是0.135s到0.02s。[218]提出的方法在延迟时间小于0.02s和位移0.17m的情况下,需要有1.25N到3N的外力输入,系统经过约10s的时间才能回到了稳定状态。而我们的方法,在相同的情况下,0.07s的系统时滞时,只需要17.8N的外力作用、或者0.05s的系统时滞时,只需要11.5N的外力作用,系统便到达其稳定状态。而且,此输入外力的大小取决于其位移(此处约为2m)和系统尺寸。
第三种情况是存在随机延迟和数据包丢失的系统。在[218]的算法(使用TDNN(时滞NN)和LQR控制器)中,当信息包没有达到并且取样时间结束时,其算法指示控制器和执行器使用最后可用信息(在缓存或队列中存贮的)。实施此算法仿真的前提条件是假设取样时间是0.03s,并且网络中有30%的包丢失,传感器到控制器间的时滞上限为0.09s下限为0.01s,而控制器到执行器间的时滞是0.03s到0.005s之间。这么长的时滞是可能发生的时滞,但是作者没有试验和测定系统在固定时滞条件下是如何动作的,因此此时滞可能发生,也可能不发生。系统在0.055m的位移和2N到3N的外力输入的情况下,其达稳定状态的时间大约是7s。当系统超过了一定的延迟时间和数据包丢失率的情况下,此研究方法就无法应对系统不稳定的情况,因此,作者提出：在NCS设计中,我们应当考虑网络时滞的特点。将我们的方法和作者的方法比较。首先,在取样时间为0.01s的情形下,本方法在数据包丢失率60%和延迟时间为0.01s的情况下,系统经过5s便达到了可接受的稳定状态。其次,为了更加接近实际系统,我们还考虑了一些干扰因素。结合应用近似反馈线性化的滑模控制对上述问题的处理似乎已达到某种极限,但迄今仍是可接受的。
第七章对整篇论文的工作进行了总结,并提出展望。指出MACs的出现是由于通信媒介仅能使有限的同步媒介接入通道对用户可用的结果。因此,在网络控制系统中,只有少量的传感器和制动器可以随时与控制器进行沟通。一旦网络超负荷,传输延迟和数据丢包率就开始显著增长,使得许多网络传输无法进行从而导致数据率超标。网络控制系统的鲁棒性还必须考虑到通过一个或多个MACs的网络进行控制。另外MAC概念的合并也导致了互联动态过程控制存在巨大挑战。最后指出,利用具有自适应容错的RBF-ARX模型来建立一个系统将是未来工作的第二大任务。
Our main research interest is in the area of'sliding modes'. The earliest work in this field of research considered sliding modes in the context of control. The approach involves forcing the closed-loop system trajectories to evolve along a surface in the state-space by means of a nonlinear control strategy. The closed-loop system performance is specified by appropriate surface selection, and robustness is ensured via a control law which forces the states to remain on the surface. Theoretically sliding mode controllers are able to completely reject the effect of a class of uncertainties-known as matched uncertainties. This unique robustness property has stimulated research in this area for over two decades.
Our main goal is to seek for method to handle the issues mentioned in the literature and add new contributions with sliding mode combined to other techniques to make any lack of efficiency avoidable, and apply to systems in connected to other component in classical way or over network.
This dissertation presents, networked control system based fault tolerant control system, firstly aims to show the fault tolerant control system method over ignored possible network i.e. working with a structure without going into network's issue, and then gives the control system in the presence of the network influenced by time delay, packed dropout, traffic disturbance. The dissertation also conducts new combination techniques dealing with fault tolerant control system starting by system identification, using RBF-ARX model combined to sliding mode control, where it's sliding surface design, lie to Eigenvalues assignmentand shows its independence from control law. The advantage of the RBF-ARX model, is its joy of an off-line nonlinear model parameter optimization, using LMM (Levenberg-Marquardt method), and a linear parameter estimation using LSM (Least Squares method) with SVD (Singular value decomposition). The example used for the application of this approach is an electro-hydraulic system with leakage issue where the measurement could not be accurate.
This system uses the interface which could implement many of the AMESim facilities while the model is running in the Simulink of Matlab. In particular, the parameters of the AMESim model could be changed within AMESim in the normal way, actually the model in AMESIM, somehow is transformed into S-function. Normally AMESim and Simulink run simultaneously. In other part of this dissertation, a new approach is given, which consists on decoupling fashion sliding mode control, dealing with subsystems instead of whole system. This approach presents a strategy based on the selection of piecewise sliding surface partition, and we apply the PwLTool which has as purpose to delimit regions where sliding mode occurs, then we may linearize the complex system into a simple linearized model in the delimited regions. An attempt to apply this strategy to the3water tank level system as example is given. The novelty of our approach is the design of sliding surface in piecewise style.
The dissertation carries out also a comprehensive comparison to an input-output linearization and sliding mode control using an inverted pendulum system as an example with linear quadratic regulation, attempting to capture the general intent of SMC, where there is still a great deal to be done in SMC.
A classical way to control a system is to compute a linear controller using the first order approximation of the system dynamics around the origin which gives a local linear approximation of the system.
The feedback linearization approach to non-linear control design is an algebraic transformation of non-linear system dynamics into a fully or partially linear one, so that linear control techniques can be applied.
The method to calculate the input-output feedback of a system uses some mathematical tools to mention here, lie-derivative, lie-bracket and involutive condition, this example shows that both the LQR and the SMC were able to control the inverted pendulum in a robust fashion. The SMC had better disturbance (or any other abrupt anomalies) rejection qualities than the LQR.
A typical control system contains an industrial controller, controlled process and some kind input/output channel, usually in a communication network, where its behavior usually, can be simulated with state flow models, Petri nets, or in a custom made schemes in a MatlabSimEvents toolbox.In this dissertation, we use True-Time toolbox (for Matlab) as a simple and easy way to realize several network types and it is written in C++MEX language, to investigate how the performance of a control loop could be influenced by a delay, and implement the approximated feedback linearization sliding mode controller and how far it performance could stand in the presence of multi-issues, like presence of fault, disturbance, network induced delay, traffic interference disturbance which provoke certainly a random time delay. It showed also that the scheduling run tasks changes with the change in the additional delay or traffic disturbance.
Further more we have carried out a serious comparaison with recent researches in the same field, and we find out that the sliding mode controller with approximate feedback linearization implemented seems coping positively with the issues mentioned above to certain limit but so far acceptable.
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