On the second-order sliding mode control of nonlinear systems with uncertain control direction

This note deals with the implementation of a second-order sliding mode control algorithm for a class of nonlinear systems in which the sign of the high-frequency gain, though constant, is unknown. A specific second-order sliding mode control algorithm, the “Suboptimal” algorithm, is properly modified in order to face the uncertainty in the control direction. It is shown that after a finite time the uncertain sign is identified and the standard finite time convergence takes place from that time on. Simulation results are provided.     

Output tracking control of uncertain nonlinear second-order systems

The solution of a tracking problem for a secondorder nonlinear system with uncertain dynamics and incomplete state measurement is obtained by means of a procedure directly inspired by the solution of the classical minimum-time optimal control problem. Two different types of uncertainty are considered in the paper: in the first case a constant bound on the uncertain dynamics is assumed to be known; in the second case, the bound is a function of both the measurable and the unmeasurable state variable of the system. In both cases, the possibility of applying the proposed control algorithms is proved to be determined by a proper choice of the control signal features. The resulting system is characterized by a suitable feedback switching logic and the convergence of the system trajectory to the desired one (or to a δ-vicinity of this latter) is proved also in the uncertain case.     

Nonsmooth finite-time control of uncertain second-order nonlinear systems

Nonsmooth finite-time stabilizing control laws have been developed for the double integrator system. The objective of this paper is to further explore the finite-time tracking control problem of a general form of uncertain second-order affine nonlinear system with the new forms of terminal sliding mode (TSM). Discontinuous and continuous finite-time controllers are also developed respectively without the singularity problem. Complete robustness can be acquired with the former, and enhanced robustness compared with the conventional boundary layer method can be expressed as explicit bounded function with the latter. Simulation results on the stabilizing and tracking problems are presented to demonstrate the effectiveness of the control algorithms.     

Nonsmooth finite-time control of uncertain second-order nonlinear systems

Nonsmooth finite-time stabilizing control laws have been developed for the double integrator system. The objective of this paper is to further explore the finite-time tracking control problem of a general form of uncertain second-order affine nonlinear system with the new forms of terminal sliding mode (TSM). Discontinuous and continuous finite-time controllers are also developed respectively without the singularity problem. Complete robustness can be acquired with the former, and enhanced robustness compared with the conventional boundary layer method can be expressed as explicit bounded function with the latter. Simulation results on the stabilizing and tracking problems are presented to demonstrate the effectiveness of the control algorithms.     

On design of time-varying feedback control for a class of second-order nonlinear systems

This paper solves the tracking problem by designing the time-varying control for a class of nonlinear pendulum system. First, by analyzing the pendulum dynamics, we get a second-order nonlinear strict-feedback system, and the tracking problem is converted into a stabilizing problem of this nonlinear system. Then, we consider a candidate output feedback control with time-varying parameters, and analyze the system dynamics to get the stabilizing condition through utilizing the backstepping approach. Although this condition is nonlinear, which is not easy to be solved, under some special conditions, the parameters can be calculated. Finally, the simulation on the tracking problem for the pendulum system is presented, which verifies our results.     

Terminal sliding mode control of second-order nonlinear uncertain systems

In this paper, a terminal sliding mode control scheme is proposed for second-order nonlinear uncertain systems. By using a function augmented sliding hyperplane, it is guaranteed that the output tracking error converges to zero in finite time which can be set arbitrarily. In addition, the proposed scheme eliminates the reaching phase problem so that the closed-loop system always shows the invariance property to parameter uncertainties. Copyright © 1999 John Wiley & Sons, Ltd.     

非线性奇异系统非交互控制的反馈实现

研究非线性奇异系统的非交互控制的反馈实现问题．首先给出了非线性奇异系统的向量相对阶与其非交互控制实现的关系；然后对正则非线性奇异系统的反馈控制系统的非交互控制的实现问题，给出了一种可使系统实现非交互控制的反馈律的构造方法。     

具有非线性扰动的广义系统的鲁棒H∞控制

研究具有非线性结构扰动广义系统的鲁棒H∞控制和鲁棒H∞保性能控制问题,该不确定性为时间和状态的函数.且满足Lipschitz条件.目的是分别设计系统的鲁棒H∞控制器和鲁棒H∞保性能控制器.应用线性矩阵不等式方法,分别给出了系统的鲁棒H∞控制器和鲁棒H∞保性能控制器存在的充分条件.当这些条件可解时,分别给出了鲁棒H∞控制器和鲁棒H∞保性能控制器的表达式.最后通过一个仿真算例说明了所给出方法的应用.     

Containment control for second-order nonlinear multi-agent systems with intermittent communications

This article investigates the containment control problem for a class of second-order multi-agent systems with inherent nonlinear dynamics, under the common assumption that each agent can only obtain the relative information of its neighbours intermittently. A kind of distributed protocol based only on the relative local intermittent measurements of neighbouring agents is designed for containment control under fixed directed topology. In the absence of delays, based on the Lyapunov function technology and the intermittent control method, some sufficient conditions are presented to guarantee the intermittent containment control of second-order nonlinear multi-agent systems. In the presence of delays, some containment conditions are also obtained for a second-order multi-agent systems with inherent delayed nonlinear dynamics and intermittent communications. Moreover, the similar results are obtained for second-order nonlinear multi-agent systems under switching directed topology. Finally, simulation examples are given to illustrate the correctness and effectiveness of the theoretical analysis.     

Positive solutions of some second-order nonlinear singular differential equations

With new schemes, the existence of the positive solutions is proved to some second-order nonlinear singular boundary value problems and initial value problems, which have higher-order singularities.     

On the optimal control of nonlinear systems

The Lie algebra of tensors on a Hilbert space is used to obtain optimal controls for a class of nonlinear systems.     

带有非线性不确定奇异系统的积分滑模控制

针对一类含有非线性不确定的奇异系统, 提出了一种面向性能的鲁棒控制器. 控制器由3部分组成: 积分滑模控制、附加的非线性控制及复合非线性反馈控制. 积分滑模控制可将匹配不确定完全抵消并使系统轨迹进入理想滑模; 附加的非线性控制用来抑制理想滑动模态上非匹配不确定对系统稳定性和性能的影响; 复合非线性反馈控制则保证闭环系统输出按性能要求渐近地跟踪参考输入信号. 最后通过算例说明所提算法的有效性.     

具有数据丢包的非线性奇异脉冲系统量化控制

考虑网络传输中存在的数据丢包和信号量化问题, 研究了基于数据丢包的非线性奇异脉冲系统设量化反馈控制器的设计方法. 首先给出一般非线性奇异脉冲系统的数学描述, 并在此基础上建立相应的具有丢包的闭环量化反馈控制系统的数学模型. 其次, 根据李雅普诺夫稳定性理论, 给出了奇异脉冲系统的渐近稳定的充分条件以及量化反馈控制器的设计方法. 应用本设计方法, 可以选择满足代数矩阵不等式条件的量化反馈增益, 实现系统渐近稳定. 最后通过对Chua混沌系统仿真, 表明利用本文设计的量化控制器能够保证闭环非线性奇异脉冲系统在具有数据丢包的情况下渐近稳定.     

Leader-following consensus control of second-order nonlinear multi-agent systems with applications to slew rate control

This paper addresses the distributed leader-following consensus control of second-order strict-feedback nonlinear multi-agent systems. By employing mean value theorem, variable separation technique, and backstepping methodology, a fully distributed adaptive control law is designed using only local relative state information. The proposed control law solves the leader-following consensus problem for any directed communication graph that contains a spanning tree with the root node being the leader agent. The application to hovercraft slew rate control system is given to verify the effectiveness of the theoretical results.     

Composite nonlinear feedback control for linear singular systems with input saturation

This paper investigates the composite nonlinear feedback (CNF) control technique for linear singular systems with input saturation. First, a linear feedback control law is designed for the step tracking control problem of linear singular systems subject to input saturation. Then, based on this linear feedback gain, a CNF control law is constructed to improve the transient performance of the closed-loop system. By introducing a generalized Lyapunov equation, this paper develops a design procedure for constructing the CNF control law for linear singular systems with input saturation. After decomposing the closed-loop system into fast subsystem and slow subsystem, it can be shown that the nonlinear part of the CNF control law only relies on slow subsystem. The improvement of transient performance by the proposed design method is demonstrated by an illustrative example.     

On applications of singular perturbation techniques in nonlinear optimal control

The technique of singular perturbations (SPT) has been applied with considerable success in several nonlinear optimal control problems. In many cases the zero-order approximation of the optimal control function has been expressed in a feedback form. This paper deals with topics involved in such closed-loop application, which seem to merit further discussion. It is formally demonstrated that a ‘forced’ singular perturbation model (obtained by artificial insertion of the perturbation parameter) results in the same zero-order composite feedback control solution as a classical singularly perturbed model (where a small parameter of physical significance appears as a consequence of a scaling transformation). The accuracy of the zero-order feedback approximation depends in both cases on the actual time scale separation of the variables. Two inherent limitations of the feedback solution are also pointed out: (1) first and higher-order correction terms of the zero-order approximation have to be computed by a predictive or off-line integration; (2) on-line implementation of SPT control strategy in a terminal boundary layer requires iterative computations. A simple pursuit problem serves as an illustrative example.     

Reduced-order observer-based finite-time control for one-sided Lipschitz nonlinear switched singular systems

The reduced-order observer-based finite-time control problem for one-sided Lipschitz nonlinear switched singular systems is addressed in this paper. First, the design method of the reduced-order observer is given via state transformation. Then, based on the average dwell time (ADT) approach, some new sufficient conditions for regularity, impulse-freeness, have a unique solution and finite-time boundedness (FTB) of the dynamic augmented systems are obtained by exploring the reduced-order observer-based controller. Further, the lower finite-time bound can be obtained by using singular value decomposition method. And the state feedback gain and the observer gain are computed by solving linear matrix inequalities (LMIs). Finally, the validity of the obtained method is illustrated by means of a numerical example and a DC motor system.     

一类二阶非线性系统的准滑模无模型自适应控制

为实现一类二阶非线性系统的轨迹跟踪控制,设计一种准滑模无模型自适应控制算法(quasi-sliding mode-model free adaptive control,SM-MFAC).首先,将模型分解为串联的两个离散子系统,并给出整体系统伪偏导数(pseudo partial derivative,PPD)的表达式;然后,利用两个子系统的输出数据设计出子系统2输出的期望状态,进而通过MFAC(model free adaptive control)对在线更新的期望状态进行不断追踪来实现对整体目标的跟踪控制;接着,对SM-MFAC控制系统进行稳定性分析,证明系统输出误差渐进跟踪到零的某个邻域内和等效控制输入是有界的;最后,以自由漂浮空间机械臂的关节轨迹跟踪控制为例验证SM-MAFC控制理论,在多体运动学与动力学仿真软件MBdyn中搭建一个平面两连杆的自由漂浮空间机械臂,关节系统中存在死区、输入饱和以及摩擦特性,通过在Matlab-simulink中的联合仿真表明:MFAC控制方案无法准确地估计出该类二阶非线性系统整体PPD的数值,导致控制性能降低,而所设计的SM-MFAC控制器相对于PID(proportional-integral-differential)、基于比例-微 分(proportional-differential,PD)的MFAC可以更快更准确地追踪目标曲线.     

一类时滞广义非线性系统H_∞可靠跟踪控制

讨论一类Lipschitz时滞非线性广义系统的H∞可靠跟踪控制问题.分别给出了执行器失效和传感器失效两种模型下可靠控制器存在的充分条件,使得闭环系统正则无脉冲并且指数稳定,同时系统输出跟踪预先给定的可测参考模型的输出,且满足H∞性能指标,并利用线性矩阵不等式技巧给出了可靠控制器的设计方法.最后给出了一个数值例子,说明了本文所给出方法的有效性.