Homogeneity is a key symmetry feature of dynamical systems, which provides an interrelation between the solutions with different initial conditions, and that global behavior of trajectories follows the local one. Existence of Lyapunov functions in the same class of homogeneous maps hints a way for their selection. In general, the homogeneous dynamical systems take an intermediate place between linear and nonlinear dynamics, they demonstrate a lot of features common for the linear framework while dealing with strongly nonlinear cases. The application of the theory of homogeneity also ensures the control systems and observers with such useful properties as
- finite‐time or nearly fixed‐time convergence, whose introduction allows the peaking/overshooting transients to be attenuated and the separation principle to be extended to the nonlinear framework;
- bi‐limit homogeneity properties of the systems allowing constructing controllers and observers with fixed‐time convergence independent from the initial conditions;
- in the case of homogeneous sliding mode controllers/observers, theoretically an exact compensation of states‐ and time‐dependent uncertainties/perturbations is permitted;
- inherent robustness to unmatched disturbances, unmodeled dynamics, and delays;
- a theoretical precision during adequate discretization, corresponding to the homogeneity weights of variables.
Moreover, for analysis of a wide class of linear and nonlinear systems, a concept of homogeneous domination (i.e., local or limit homogeneity) can be applied, extending to such systems, the above‐mentioned properties of homogeneous dynamics.
Unfortunately, the finite‐/fixed‐time convergence in homogeneous systems is achieved at the expense of unbounded gains of the controller at zero or infinity. It causes the following main disadvantages of homogeneous controllers or observers:
- (i) Chattering, or high frequency input/output oscillations, caused by parasitic dynamics, time‐delays, conventional discretization tools or noises.
- (ii) To be applied to the real systems homogeneous controllers need to be saturated.
- (iii) For discrete implementation, the special numerical methods are required, because generally the standard Euler discretization is suitable for Lipschitz systems only.
- (iv) A complementary discretization strategy may be needed for the implementation of the controllers with positive homogeneity degrees providing the finite‐time convergence from infinity.
As it often happens in nonlinear setting, the problems of constructive tuning of controllers/observers and nonconservative evaluation of stability and convergence characteristics are of great importance and form a part of the price of having finite‐/fixed‐time decay rates.
By preparing this Special Issue, we have tried to join the leading experts in order to explore the newest ideas in the field. Finally, we are happy to present 25 selected papers written by the authors from 10 countries covering the hottest topics in homogeneous and sliding mode controllers' design.
The selected papers can be separated in 10 topics.
- The first topic presents the methodologies of the homogeneous controllers' design.
E. Cruz‐Zavala and J.A. Moreno in their work “High‐Order Sliding‐Mode Control Design Homogeneous in the Bi‐limit” suggested a new Lyapunov‐based design of high‐order sliding‐mode controllers, providing a fixed‐time convergence for a class of single‐input‐single‐output uncertain nonlinear systems.
The paper “MIMO Homogeneous Integral Control Design using the Implicit Lyapunov Function Approach” by A. Mercado‐Uribe, J.A. Moreno, A. Polyakov, D. Efimov shows how the implicit Lyapunov Function approach can be used for MIMO plants presenting the option of homogeneous integral control design.
- 2. The second topic joins the methods for PI and PID like homogeneous controllers.
The paper “Design of a PID‐like continuous sliding mode controller” by U. Pérez‐Ventura, J. Mendoza‐Avila and L. Fridman proposes two sets of the PID‐like continuous sliding mode controller gains using Harmonic Balance methodology by minimizing the amplitude or energy consumption needed to maintain the system in a real sliding mode in the presence of parasitic dynamics. The authors also presented a Lyapunov analysis ensuring the stability of PID‐like algorithms with proposed gains for the system without unmodeled dynamics.
The work “Generalized Homogenization of Linear Control Theory and Experiment” by S. Wang, A. Polyakov, G. Zheng suggested a methodology for homogenization of linear PID controllers by keeping the design simplicity of PIDs, while introducing the advantageous properties of homogeneous models.
The idea of the contribution “An Integral Extension Technique for Continuous Homogeneous State‐Feedback Control Laws Preserving Nominal Performance” by R. Seeber and J.A. Moreno suggests a homogeneous integral extension of arbitrary‐order homogeneous state‐feedback controllers updated as PI‐like controllers, for which the super‐twisting controller is a particular case. It is shown that appropriate tuning allows to obtain discontinuous integral feedback which, similar to the super‐twisting controller, can reject Lipschitz continuous perturbations also in the higher order case.
- The next topic deals with constrained homogeneous controllers.
The paper “Control of Systems with Time Varying Unilateral Constraints Using Robust and Optimal Controllers: A Homogeneity Framework” by H.B. Oza and Y. Orlov studies the tracking problem, where a time‐varying unilateral constraint and disturbances are presented. An extension to the method of nonsmooth state transformation is developed. The proposed framework enables synthesis and analysis of control of a rigid body colliding with a time‐varying unilateral constraint achieving finite‐time tracking for a class of unilaterally constrained systems, where the system dynamics with impacts are converted into one without impacts.
M.A. Golkani, R. Seeber, M. Reichhartinger and M. Horn in their paper “Lyapunov‐based Saturated Continuous Twisting Algorithm” present a Lyapunov approach for using the homogeneous Continuous Twisting Algorithm (CTA) with a saturated actuator. By combining both discontinuous and continuous twisting algorithms, the approach allows to improve the performance and guarantee stability of the CTA under saturation condition.
- The fourth topic studies a discrete implementation of the homogeneous sliding mode controllers.
The work “Digital implementation of sliding‐mode control via the implicit method: A tutorial” by B. Brogliato and A. Polyakov presents an introduction to the implicit discretization for homogeneous sliding mode controllers.
The paper “Lyapunov‐based Consistent Discretization of Stable Homogeneous Systems” by T. Sanchez, A. Polyakov, D. Efimov proposes an explicit discretization scheme for asymptotically stable homogeneous systems preserving asymptotic stability, the convergence rates, and the Lyapunov function of the original continuous‐time system.
J.E. Carvajal‐Rubio, J.D. Sánchez‐Torres, M. Defoort, M. Djemai and A.G. Loukianov present a work on “Implicit and Explicit Discrete‐Time Realizations of Homogeneous Differentiators.” The authors consider two discrete‐time realizations of the homogeneous differentiator. They based on the methodology used to obtain an exact discretization of linear systems with a zero‐order hold. It is shown that the error dynamics of both discrete‐time differentiators are homogeneous to their respective transformations, and they preserve the accuracy of their continuous‐time counterparts after a finite time. An implementation strategy was proposed for the implicit discrete‐time realization, which is nonanticipative and includes a root‐finding approach based on Halley's method.
- The other topic reflected in the Special Issue joined the papers devoted to Homogeneity based Differentiators, Observers and Feedback Control.
V. Andrieu, D. Astolfi and P. Bernard present “Observer design via interconnections of second‐order mixed sliding‐mode/linear differentiators”. By combining, the global convergence properties of high‐gain observers and the semi‐global finite‐time theoretically exact convergence of super‐twisting differentiators, for lower‐triangular nonlinear dynamics the proposed design allows a global finite time convergent observer to be obtained when the nonlinearities are linearly bounded.
A. Jbara, A. Levant, A. Hanan in their paper “Filtering Homogeneous Observers in Control of Integrator Chains” propose a robust homogeneous filtering observer for any disturbed integrator chains. Designed differentiators demonstrate an accurate estimation of derivatives even in the case of extremely large noises.
The work “Homogeneous Output‐Feedback Control with Disturbance‐Observer for a class of Nonlinear Systems” by T. Sanchez, J.A. Moreno proposes the control scheme allowing finite‐time theoretically exact convergence to the origin despite the uncertain nonlinear terms. The dynamic part of the controller consists of an extended order observer, which is based on the higher order sliding‐mode exact differentiator, estimating the states of the system and uncertain nonlinear terms in finite‐time. Such an estimation is used in the suggested homogeneous controllers.
The paper “Numerical Design of Lyapunov Functions for a Class of Homogeneous Discontinuous Systems” by J. Mendoza‐Avila, D. Efimov, R. Ushirobira and J.A. Moreno proposes the analytical and numerical design of a Lyapunov function for homogeneous and discontinuous systems. Two expressions of homogeneous and locally Lipschitz continuous Lyapunov functions are provided, and a methodology for their numerical construction is realized. These results are applied to the numerical design of a Lyapunov function for some Higher‐Order Sliding Mode algorithms.
- The sixth topic covered in the Special Issue combines the papers devoted to analysis of the homogeneous and the time‐delay systems.
D. Efimov and A. Aleksandrov in their paper “Analysis of robustness of homogeneous systems with time delays using Lyapunov‐Krasovskii functionals” propose stability analysis of homogeneous time‐delay systems applying the Lyapunov–Krasovskii theory, and a generic structure of the functional is given that suits for any homogeneous system of nonzero degree (and can also be used for any dynamics admitting a homogeneous approximation).
J. Xu, L. Fridman, E. Fridman and Y. Niu propose an “Output‐Feedback Lyapunov Redesign of Uncertain Systems with Delayed Measurements”. Instead of the traditional observer/differentiator‐based output‐feedback design, a static state estimator is constructed by the Taylor expansion of delayed measurements with the integral remainders. Then, a sliding variable is constructed according to the nominal Lyapunov function. A Lyapunov redesign approach allows to keep the system trajectory in predefined vicinity of the origin, even subject to approximation errors and exogenous disturbances.
- The Special Issue tackles the analysis of homogeneous systems in the frequency domain.
The work by I. Boiko “On Phase Deficit of Homogeneous Sliding Mode Control” suggests a necessary condition for finite‐time convergence, infinite‐time convergence and finite‐frequency oscillations. The condition is formulated in terms of phase deficit.
The paper “Chattering analysis for Lipschitz continuous sliding‐mode controllers” by C.A. Martínez‐Fuentes, U. Pérez‐Ventura and L. Fridman presents an analysis of chattering in systems driven by Lipschitz continuous sliding‐mode controllers (LCSMC) using the describing function approach. Predictions of amplitude, frequency, and average power of self‐excited oscillations, are used to compare such LCSMC with the super‐twisting controller in systems with fast actuators.
- The Special Issue includes some papers about adaptation methods for homogeneity based controllers and observers.
H. Obeid, S. Laghrouche and L. Fridman present a “Dual layer barrier functions based adaptive higher order sliding mode control” for a disturbed chain of integrators of order
n. The proposed strategy ensures a finite‐time convergence of the sliding variable and its (
n – 1) derivatives to zero without using any information on the bounds of the disturbance or its derivative. A proposed dual layer scheme is based on the barrier function super‐twisting compensation of the Lipschitz disturbances and the adaptation of the discontinuous high‐order sliding mode control, which just counteracts the error of this compensation.
R. Franco, H. Ríos, D. Efimov, W. Perruquetti contribute with the paper “Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding‐mode observer approach,” where adaptation of a nonlinear sliding mode observer based on a nonlinear time‐varying parameter identification algorithm is presented, for uncertain nonlinear systems with measurement noise. Proposed observer can be synthesized for the class of nonlinear systems with unknown time‐varying parameters and measurement noise, when a parameter distribution matrix and nonlinear terms depend on the full state and the input. For the disturbed case, the nonlinear time‐varying parameter identification algorithm provides a fixed‐time rate of convergence while the sliding‐mode observer ensures ultimate boundedness for the state estimation error attenuating the effects of the external disturbances.
- The Special Issue also covers some applications of homogeneity‐based controllers and observers to some problem of control theory.
F.J. Bejarano and M. Mera in their paper “Continuous State Observability and Mode Reconstructability of Switched Nonlinear Systems with Unknown Switching Function” propose the reconstruction procedure that involves the use of finite‐time homogeneous sliding mode differentiator without the requirement that an original system has to be in any specific form.
J. Dávila and A. Pisano present a research “On the fixed‐time consensus problem for nonlinear uncertain multiagent systems under switching topology” suggested local interaction rule providing leader‐following and leader‐less consensus in a network of nonlinear uncertain first‐order agents communicating through a connected and switching graph topology. The proposed interaction rule guarantees that consensus is achieved after a fixed‐time.
A. Taoufik, M. Defoort, M. Djemai, K. Busawon, J.D. Sánchez‐Torres contribute with the paper “Distributed global fault detection scheme in multi‐agent systems with chained‐form dynamics”. The proposed scheme consists of cascades of predefined‐time sliding mode observers providing exact estimate of the global system state, whereby the settling time is a parameter defined in advance, which does not depend on the initial conditions of the system.
- Homogeneous controllers for some mechanical systems.
E. Cruz‐Zavala, E. Nuño and J.A. Moreno apply homogeneous controllers for “Robust Trajectory‐Tracking in Finite‐Time for Robot Manipulators using Nonlinear PD Control plus Feed‐Forward Compensation”. They proposed two kinds of controllers, a discontinuous and its continuous version. On the one hand, the discontinuous PD term is designed like a twisting sliding‐mode controller ensuring robust global uniform finite‐time stabilization of the tracking error, despite model uncertainties and nonvanishing perturbations. On the other hand, the continuous case only ensures practical stability in the presence of model uncertainties and nonvanishing perturbations, but it decreases the chattering phenomenon of the discontinuous case.
D. Gutiérrez‐Oribio, J.A. Mercado‐Uribe, J.A. Moreno, L. Fridman propose an algorithm for “Robust global stabilization of a class of underactuated mechanical systems of two degrees of freedom” generating a continuous homogeneous controller that ensures a theoretically exact finite‐time convergence to the third order sliding mode despite of the presence of Lipschitz disturbances and/or uncertainties, and uncertain control coefficient in the model. The efficacy of the proposed controllers is illustrated via experiments for swinging up the Reaction Wheel Pendulum.
Enjoy the reading!
同质性是动力学系统的关键对称特征，它提供了具有不同初始条件的解之间的相互关系，并且轨迹的整体行为遵循局部行为。Lyapunov函数在同一类齐次映射中的存在为它们的选择提供了一种途径。通常，齐次动力学系统在线性动力学和非线性动力学之间处于中间位置，它们在处理强非线性情况时展示了线性框架的许多共同特征。同质性理论的应用还确保控制系统和观察者具有如下有用的特性：
- 有限时间或接近固定时间的收敛，其引入允许衰减峰值/过冲瞬变，并将分离原理扩展到非线性框架；
- 系统的双极限同质性，可以构造具有固定时间收敛性的控制器和观测器，而不受初始条件的影响；
- 在均质滑模控制器/观测器的情况下，理论上允许对状态和时间相关的不确定性/扰动进行精确补偿；
- 对无与伦比的干扰，无模型的动力学和延迟的固有鲁棒性；
- 充分离散化期间的理论精度，对应于变量的同质权重。
此外，为了分析各种各样的线性和非线性系统，可以应用均质支配（即局部或极限均质）的概念，并将其扩展到此类系统，即均质动力学的上述特性。
不幸的是，同类系统中的有限/固定时间收敛是通过以零或无穷大获得控制器无穷大的增益为代价的。它导致同类控制器或观察者的以下主要缺点：
- （i）由寄生动力学，时间延迟，常规离散化工具或噪声引起的颤动或高频输入/输出振荡。
- （ii）要应用于实际系统，均质控制器必须处于饱和状态。
- （iii）对于离散实现，需要特殊的数值方法，因为通常标准的Euler离散化仅适用于Lipschitz系统。
- （iv）可能需要一种互补的离散化策略，以实现具有正均匀度的控制器，从而提供从无穷远开始的有限时间收敛。
正如在非线性环境中经常发生的那样，对控制器/观测器进行构造性调整以及对稳定性和收敛性进行非保守评估的问题非常重要，并成为具有有限/固定时间衰减率的价格的一部分。
通过准备本期特刊，我们试图与领先的专家一起探索该领域的最新想法。最后，我们很高兴介绍来自10个国家的作者撰写的25篇论文，涉及均匀和滑模控制器设计中的最热门主题。
所选论文可分为10个主题。
- 第一个主题介绍了同类控制器的设计方法。
E. Cruz-Zavala和JA Moreno在他们的工作“双极限同质的高阶滑模控制设计”中提出了一种基于Lyapunov的新型高阶滑模控制器设计，为一类单输入单输出不确定非线性系统。
A. Mercado-Uribe，JA Moreno，A。Polyakov，D。Efimov撰写的论文“使用隐式Lyapunov函数方法的MIMO均质积分控制设计”展示了隐式Lyapunov函数方法可用于MIMO植物，并提供了均质选项整体控制设计。
- 2.第二个主题加入了PI和PID的方法，例如同类控制器。
U.Pérez-Ventura，J。Mendoza-Avila和L.Fridman的论文“ PID型连续滑模控制器的设计”通过使用谐波平衡方法，通过最小化两个PID型连续滑模控制器增益，提出了两组增益。在存在寄生动力学的情况下，将系统保持在实际滑动模式下所需的振幅或能量消耗。作者还提出了Lyapunov分析，以确保类似PID的算法的稳定性，并为系统提供了拟议的收益，而没有未建模的动力学。
S. Wang，A. Polyakov，G. Zheng的著作“线性控制理论和实验的广义均质化”提出了一种线性PID控制器均质化的方法，该方法通过保持PID的设计简单性，同时介绍了均质模型的优势。
R. Seeber和JA Moreno的贡献“一种用于保持均匀性能的连续均质状态反馈控制律的积分扩展技术”的想法提出了将任意阶均质状态反馈控制器更新为PI控制器的均质积分扩展，对于这种情况，超级加捻控制器是一种特殊情况。结果表明，适当的调谐可以获取不连续的积分反馈，这与超扭曲控制器类似，在较高阶情况下也可以拒绝Lipschitz的连续扰动。
- 下一个主题涉及受约束的齐次控制器。
HB Oza和Y. Orlov撰写的论文“使用鲁棒且最优的控制器控制具有时变单边约束的系统：同质性框架”研究了跟踪问题，其中提出了时变单边约束和扰动。开发了非光滑状态转换方法的扩展。所提出的框架能够对刚体的控制进行综合和分析，并与时变单边约束相撞，从而实现对一类单边约束系统的有限时间跟踪，在该系统中，具有影响的系统动力学被转换为没有影响的系统动力学。
MA Golkani，R。Seeber，M。Reichhartinger和M. Horn在他们的论文“基于Lyapunov的饱和连续扭曲算法”中提出了一种Lyapunov方法，该方法可将均质连续扭曲算法（CTA）与饱和执行器一起使用。通过结合不连续和连续扭曲算法，该方法可以提高性能并保证饱和条件下CTA的稳定性。
- 第四个主题研究了均质滑模控制器的离散实现。
B. Brogliato和A.Polyakov撰写的“通过隐式方法进行滑模控制的数字实现：教程”一书介绍了均质滑模控制器的隐式离散化。
T. Sanchez，A. Polyakov，D. Efimov的论文“稳定均质系统的基于Lyapunov的一致离散化”为渐近稳定的齐次系统提出了一个显式离散化方案，该方案保留了渐近稳定性，收敛速度和原始Lyapunov函数连续时间系统。
JE Carvajal-Rubio，JDSánchez-Torres，M。Defoort，M。Djemai和AG Loukianov提出了一项关于“同质微分的隐式和显式离散时间实现”的工作。作者考虑了齐次微分器的两个离散时间实现。他们基于用于获得零阶保持线性系统精确离散的方法。结果表明，两个离散时间微分器的误差动力学对其各自的变换都是同构的，并且它们在有限的时间后仍保持其连续时间微分器的精度。提出了一种隐式离散时间实现的实现策略，该实现策略是非预期的，并且包括基于哈雷方法的寻根方法。
- 特刊中反映的另一个主题与致力于基于同质性的鉴别器，观察者和反馈控制的论文有关。
V. Andrieu，D。Astolfi和P. Bernard提出“通过二阶混合滑模/线性微分器的互连实现观察者设计”。通过将高增益观测器的全局收敛性与超扭曲微分器的半全局有限时间理论上精确收敛相结合，对于较低三角形的非线性动力学，该建议的设计允许在以下情况下获得全局有限时间收敛的观测器：非线性是线性有界的。
A. Jbara，A。Levant，A。Hanan在他们的论文“对积分器链的控制中过滤均质观测器”中提出了一个鲁棒的均质滤波观测器，用于观察任何受干扰的积分器链。设计的微分器即使在噪声非常大的情况下也能证明对导数的准确估计。
JA Moreno的T.Sanchez的著作“一类非线性系统的具有扰动-观测器的同质输出反馈控制”，提出了一种控制方案，尽管存在不确定的非线性项，但仍允许理论上精确地收敛到原点。控制器的动态部分由扩展阶观测器组成，该观测器基于高阶滑模精确微分器，可在有限时间内估算系统状态和不确定的非线性项。在建议的同类控制器中使用了这种估计。
J. Mendoza-Avila，D。Efimov，R.Ushirobira和JA Moreno的论文“一类非连续不连续系统的Lyapunov函数的数值设计”提出了齐次和不连续系统Lyapunov函数的解析和数值设计。提供了齐次和局部Lipschitz连续Lyapunov函数的两个表达式，并实现了其数值构造的方法。这些结果应用于一些高阶滑模算法的Lyapunov函数的数值设计。
- 本期特刊涵盖的第六个主题结合了专门分析齐次系统和时滞系统的论文。
D. Efimov和A. Aleksandrov在他们的论文“使用Lyapunov-Krasovskii泛函分析具有时滞的齐次系统的鲁棒性”中提出了应用Lyapunov-Krasovskii理论的齐次时滞系统的稳定性分析，该泛函的通用结构为假定适用于任何非零度的齐次系统（也可以用于任何接受齐次逼近的动力学）。
J. Xu，L。Fridman，E。Fridman和Y. Niu提出了“不确定时滞系统的输出反馈Lyapunov重新设计”。代替传统的基于观测器/微分器的输出反馈设计，静态估计器由延迟测量的泰勒展开式和积分余数构造而成。然后，根据名义Lyapunov函数构造一个滑动变量。Lyapunov重新设计方法允许将系统轨迹保持在原点的预定范围内，甚至会受到近似误差和外源性干扰的影响。
- 特刊探讨了频域中同类系统的分析。
博伊科一世的著作《关于均匀滑模控制的相位缺陷》为有限时间收敛，无限时间收敛和有限频率振荡提出了必要条件。该条件是根据相位不足制定的。
CAMartínez-Fuentes，U。Pérez-Ventura和L.Fridman撰写的论文“ Lipschitz连续滑模控制器的抖动分析”使用描述函数对由Lipschitz连续滑模控制器（LCSMC）驱动的系统中的抖动进行了分析。方法。自激振荡的幅度，频率和平均功率的预测被用于在具有快速执行器的系统中将此类LCSMC与超扭曲控制器进行比较。
- 特刊包括有关基于同质性的控制器和观察者的适应方法的一些论文。
H. Obeid，S。Laghrouche和L. Fridman为扰动的
n阶积分器提出了“基于双层势垒函数的自适应高阶滑模控制” 。所提出的策略可确保滑动变量及其（
n – 1）导数在零时间内进行有限收敛，而无需使用有关扰动或其导数范围的任何信息。提出的双层方案基于Lipschitz扰动的势垒函数超扭曲补偿和不连续的高阶滑模控制的自适应，这恰好抵消了该补偿的误差。
R. Franco，H.Ríos，D.Efimov，W.Perruquetti撰写了论文“带有测量噪声的不确定非线性系统的自适应估计：滑模观测器方法”，其中基于非线性对非线性滑模观测器进行了自适应针对具有测量噪声的不确定非线性系统，提出了时变参数辨识算法。当参数分布矩阵和非线性项取决于完整状态和输入时，可以为未知时变参数和测量噪声的非线性系统类综合拟议的观测器。对于不安的情况
- 特刊还涵盖了基于同质性的控制器和观察者在控制理论上的某些应用。
FJ Bejarano和M. Mera在他们的论文“具有未知切换功能的切换非线性系统的连续状态可观察性和模式可重构性”中提出了一种重建程序，该程序涉及使用有限时间齐次滑模微分器，而无需原始系统必须采用任何特定形式。
J.Dávila和A. Pisano提出了一项研究“关于切换拓扑下的非线性不确定多主体系统的固定时间共识问题”，建议使用局部交互规则在非线性不确定一阶代理网络中提供领导者跟随和无领导者共识通过连接的和切换的图拓扑进行通信。提议的交互规则可确保在固定时间后达成共识。
A. Taoufik，M。Defoort，M。Djemai，K.Busawon，JDSánchez-Torres撰写了论文“具有链式动力学的多代理系统中的分布式全局故障检测方案”。所提出的方案由预定义时间滑模观察器的级联组成，这些级联提供了对全局系统状态的精确估计，其中建立时间是预先定义的参数，它不依赖于系统的初始条件。
- 用于某些机械系统的均质控制器。
E. Cruz-Zavala，E。Nuño和JA Moreno将同类控制器用于“使用非线性PD控制加前馈补偿的机器人机械手有限时间内的稳健轨迹跟踪”。他们提出了两种控制器，一种是不连续的，另一种是连续的。一方面，不连续的PD项的设计类似于扭曲的滑模控制器，尽管存在模型不确定性和无扰动，但仍可确保跟踪误差的鲁棒全局统一有限时间稳定。另一方面，连续情况仅在存在模型不确定性和不消失的扰动的情况下才能确保实用的稳定性，但它减少了不连续情况的颤动现象。
D.Gutiérrez-Oribio，JA Mercado-Uribe，JA Moreno，L。Fridman提出了一种算法，用于“对一类具有两个自由度的欠驱动机械系统进行鲁棒全局稳定”，从而生成一个连续的均质控制器，该控制器确保理论上精确的有限-尽管存在Lipschitz扰动和/或不确定性，并且模型中的控制系数不确定，但时间收敛到三阶滑模。通过摆起反作用轮摆锤的实验说明了所提出控制器的功效。
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