Sliding mode control for a class of nonlinear discrete-time networked systems with multiple stochastic communication delays

In this article, a sliding mode control problem is studied for a class of uncertain nonlinear networked systems with multiple communication delays. A sequence of stochastic variables obeying Bernoulli distribution is applied in the system model to describe the randomly occurring communication delays. The discrete-time system considered is also subject to parameter uncertainties and state-dependent stochastic disturbances. A novel discrete switching function is proposed to facilitate the sliding mode controller design. The sufficient conditions are derived by means of the linear matrix inequality (LMI) approach. It is shown that the system dynamics in the specified sliding surface is robustly exponentially stable in the mean square if two LMIs with an equality constraint are feasible. A discrete-time SMC controller is designed that is capable of guaranteeing the discrete-time sliding-mode reaching condition of the specified sliding surface. Finally, a simulation example is given to show the effectiveness of the proposed method.     

On discrete-time variable structure sliding mode control

The purpose of this paper is to show the limitations of discrete-time variable structure sliding mode control and that the equivalent control must be used in order to have sliding in a neighborhood of the switching surface. Conflicting requirements for the sliding mode controller behavior in the continuous and discrete-time domains are revealed and analyzed. A linear control law for an uncertain discrete-time linear plant, with bounded uncertainties, is analyzed and its superiority over nonlinear controllers is demonstrated. The conclusion of the obtained results is that in the discrete-time variable structure sliding mode controller design, unlike in the continuous-time, the designer may have limited flexibility in selecting controller architectures.     

Discrete-time adaptive sliding mode control for uncertain systems based on multi-rate output measurement

This article presents the design of a discrete-time adaptive sliding mode control (SMC) for uncertain systems using the noisy multi-rate output measurement. It is assumed that the dynamic systems are described by a discrete-time state equation with mismatched uncertainties. The proposed discrete-time adaptive SMC is composed of the equivalent and the nonlinear switching controls, and it is derived based on a new sliding surface. A new least squares estimator (LSE) to take the estimates for the un-measurable states and the uncertainties is designed without the information of the statistics for the measurement noises and the initial state. In the design of the proposed discrete-time adaptive SMC, the un-measurable states and the uncertainties are, respectively, replaced as their estimates obtained from the proposed LSE, and the sliding coefficient matrices are selected such that the states for the systems are ultimately bounded, and the effect of the external disturbances to give the states is reduced as small as possible. The effectiveness of the proposed method is indicated through the simulation experiment in a simple system.     

非匹配不确定离散系统的无抖振积分滑模控制

针对一类非匹配不确定离散系统, 设计一种无抖振离散积分滑模控制器. 为了抑制非匹配不确定性对系统的影响, 采用线性矩阵不等式方法设计一种新型的切换函数和对应的滑模控制律, 并证明了闭环系统的Lyapunov 稳定性. 同时, 引入饱和函数设计控制器, 使系统状态在积分滑模面的某个小邻域内做准滑模运动, 并通过合理选择饱和函数的边界层厚度, 使控制信号不含任何抖振. 理论分析和数值仿真验证了所提出方法的有效性.

     

A singular system approach to robust sliding mode control for uncertain Markov jump systems

The robust sliding mode control for Markov jump systems with parameter uncertainties and an unknown nonlinear function is discussed. Based on a singular system approach and linear matrix inequality (LMI), a sufficient condition which guarantees the existence of linear switching surface and the stochastic stability of sliding mode dynamics is given. A sliding mode controller is designed such that the closed-loop system is convergent to the switching surface in finite time. A numerical example is given to show the effectiveness of the proposed method.     

Sliding mode stabilisation of networked systems with consecutive data packet dropouts using only accessible information

This paper develops a novel stabilising sliding mode for systems involving uncertainties as well as measurement data packet dropouts. In contrast to the existing literature that designs the switching function by using unavailable system states, a novel linear sliding function is constructed by employing only the available communicated system states for the systems involving measurement packet losses. This also equips us with the possibility to build a novel switching component for discrete-time sliding mode control (DSMC) by using only available system states. Finally, using a numerical example, we evaluate the performance of the designed DSMC for networked systems.     

Robust tracking and model following of uncertain dynamic systems via discrete-time integral sliding mode control

A discrete-time integral sliding mode control scheme is proposed to realize the problem of robust tracking and modeling following for a class of uncertain linear systems. It will be shown that the proposed scheme guarantees the stability of closed-loop system and achieves zero-tracking error in the presence of parameter uncertainties and external disturbances. The selection of switching surface and the existence of sliding mode are two important issues, which have been addressed. This scheme assures robustness against system uncertainties and disturbances. Chattering phenomenon and reaching phase are eliminated. Moreover, the knowledge of upper bound of uncertainties is not required. Both the theoretical analysis and illustrative example demonstrate the validity of the proposed scheme. Recommended by Editorial Board member Ju Hyun Park under the direction of Editor Young IL Lee. Ming-Chang Pai received the M.S. and Ph.D. degrees in Mechanical Engineering in 1994 and 1998 from Pennsylvania State University, State College, P.A.. He is currently an Associate Professor in the Department of Automation Engineering at Nan-Kai University of Technology. His research interests are in mechatronics, robots, robust control and nonlinear control.     

Approximation law for discrete-time variable structure control systems

Two approximation laws of sliding mode for discrete-time variable structure control systems are proposed to overcome the limitations of the exponential approximation law and the variable rate approximation law. By applying the proposed approximation laws of sliding mode to discrete-time variable structure control systems, the stability of origin can be guaranteed, and the chattering along the switching surface caused by discrete-time variable structure control can be restrained effectively. In designing of approximation laws, the problem that the system control input is restricted is also considered, which is very important in practical systems. Finally a simulation example shows the effectiveness of the two approximation laws proposed.     

Approximation law for discrete-time variable structure control systems

Two approximation laws of sliding mode for discrete-time variable structure control systems are proposed to overcome the limitations of the exponential approximation law and the variable rate approximation law. By applying the proposed approximation laws of sliding mode to discrete-time variable structure control systems, the stability of origin can be guaranteed, and the chattering along the switching surface caused by discrete-time variable structure control can be restrained effectively. In designing of approximation laws, the problem that the system control input is restricted is also considered, which is very important in practical systems. Finally a simulation example shows the effectiveness of the two approximation laws proposed.     

Robust integral sliding mode control for uncertain stochastic systems with time-varying delay

This paper is concerned with sliding mode control for uncertain stochastic systems with time-varying delay. Both time-varying parameter uncertainties and an unknown nonlinear function may appear in the controlled system. An integral sliding surface is first constructed. Then, by means of linear matrix inequalities (LMIs), a sufficient condition is derived to guarantee the global stochastic stability of the stochastic dynamics in the specified switching surface for all admissible uncertainties. The synthesized sliding mode controller guarantees the reachability of the specified sliding surface. Finally, a simulation example is presented to illustrate the proposed method.     

Buck变换器的离散时间全程滑模控制

针对含有匹配不确定项的线性离散时间系统,设计了一种全程滑模控制器,使系统状态在初始时刻就在切换区内．给出了一种双曲正切离散趋近律,由这种改进的趋近律所设计的控制器不但可以减小系统稳态误差,而且能够削弱系统的抖振强度．将该方法应用在Ｂｕｃｋ变换器的控制器设计中,对Ｂｕｃｋ变换器的电压进行跟踪控制,仿真结果验证了所给控制策略的有效性．     

Adaptive sliding mode control for a class of uncertain discrete-time systems using multi-rate output measurement

This article is concerned with the design of an adaptive sliding mode control (SMC) for a class of uncertain discrete-time systems using the multi-rate output measurement. The states of the discrete-time systems are assumed to be taken in the multi-rate output measurement by the contamination with measurement noise. The uncertainties are assumed not to satisfy the matching condition, and are expressed in a parameterised form. A least squares estimator (LSE) to take the estimates for the un-measurable states and the uncertainties is designed in a batch form by using the noisy multi-rate output measurement. The proposed adaptive SMC is designed by using the sliding surface expressed as the linear state function and the estimates obtained from the LSE. It is proved that the estimation errors converge to zero as time tends to infinite, and the states of the system are bounded under the action of the proposed adaptive SMC. The effectiveness of the proposed method is indicated through the simulation experiment in a simple system.     

Self-tuning type variable structure control method for a class of nonlinear systems

In this paper we propose a new self-tuning type Variable Structure Control (VSC) method for a class of nonlinear dynamical systems with parametric uncertainties. The controller is designed to satisfy the sliding mode condition; mean-while the on-line parameter identification is incorporated in the control system. Necessary modifications are made for the parameter identification to avoid the control singularity problem. By virtue of the sliding mode, the proposed identification algorithm can be applied to those nonlinear systems which may not be linear in parametric space but are linear while in the sliding mode. A model-based strategy is further introduced to estimate the uncertainty bound. The new approach attains the gain scheduling property by tuning the switching gain in accordance with the estimated system uncertainties, that is, the switching gain decreases asymptotically while the sliding mode condition is still maintained. © 1998 John Wiley & Sons, Ltd.     

Robust control for a class of nonlinear networked systems with stochastic communication delays via sliding mode conception

This paper deals with the robust control problem for a class of uncertain nonlinear networked systems with stochastic communication delays via sliding mode conception (SMC). A sequence of variables obeying Bernoulli distribution are employed to model the randomly occurring communication delays which could be different for different state variables. A discrete switching function that is different from those in the existing literature is first proposed. Then, expressed as the feasibility of a linear matrix inequality (LMI) with an equality constraint, sufficient conditions are derived in order to ensure the globally mean-square asymptotic stability of the system dynamics on the sliding surface. A discretetime SMC controller is then synthesized to guarantee the discrete-time sliding mode reaching condition with the specified sliding surface. Finally, a simulation example is given to show the effectiveness of the proposed method.     

Sliding mode control for Markov jump linear uncertain systems with partly unknown transition rates

In this paper, based on sliding mode control approach, the robust stabilisation problem for a class of continuous-time Markovian jump linear uncertain systems with partly unknown transition rates is investigated. The transition rate matrix under consideration covers completely known, boundary known and completely unknown elements. By making use of linear matrix inequalities technique, sufficient conditions are presented to derive the linear switching surface and guarantee the stochastic stability of sliding mode dynamics. Then a sliding mode control law is designed to drive the state trajectory of the closed-loop system to the specified linear switching surface in finite time in spite of the existing uncertainties and unknown transition rates. Finally, an example is given to verify the validity of the theoretical results.     

SECOND‐ORDER TERMINAL SLIDING MODE CONTROL OF INPUT‐DELAY SYSTEMS

This paper proposes a second‐order terminal sliding mode control for a class of uncertain input‐delay systems. The input‐delay systems are firstly converted into the input‐delay free systems and further converted into the regular forms. A linear sliding mode manifold is predesigned to represent the ideal dynamics of the system. Another terminal sliding mode manifold surface is presented to drive the linear sliding mode to reach zeros in finite time. In order to eliminate the chattering phenomena, a second‐order sliding mode method is utilized to filter the high frequency switching control signal. The uncertainties of the systems are analysed in detail to show the effect to the systems. The simulation results validate the method presented in the paper.     

不确定输入时滞系统的滑模输出反馈控制

研究了在状态不可测,并且状态参数和控制输入矩阵同时存在不确定性的情况下,不确定输入时滞系统的滑模控制问题.为了有效克服不确定性和时滞对系统稳定性的影响,在状态估计空间选择了一种积分型的切换函数,设计了一种基于状态观测器的滑模控制器,并证明了所设计切换面的可达性.通过李亚普诺夫理论,给出了闭环系统渐近稳定的充分条件.数值仿真验证了算法的有效性.     

An LMI-based switching surface design method for a class of mismatched uncertain systems

In this note, we consider the problem of designing linear sliding surfaces for multivariable uncertain systems with mismatched uncertainties in the state matrix. In terms of linear matrix inequalities, we give a new invariance condition guaranteeing the existence of a linear switching surface such that the reduced-order equivalent sliding mode dynamics restricted to the switching surface is not only stable but completely invariant to mismatched uncertainties. We also give an explicit formula of such linear switching surfaces, together with a design example.     

用于离散滑模重复控制的新型趋近律

针对不确定离散时间系统, 提出一种新型的离散趋近律, 构造理想切换动态, 并基于理想切换动态设计离散滑模重复控制器. 在消除系统颤振的同时, 完全抑制周期扰动带来的影响, 改善控制品质. 分别推导了切换函数的绝对收敛层、单调收敛层和拟滑模带的边界, 用于刻画在不同控制器参数下闭环系统的趋近过程和拟滑模运动. 数值仿真和在伺服装置上的实验结果证实了所提出控制方法的有效性.     

Control of a Class of Underactuated Mechanical Systems Using Sliding Modes

In this paper, we present a sliding mode control algorithm to robustly stabilize a class of underactuated mechanical systems that are not linearly controllable and violate Brockett's necessary condition for smooth asymptotic stabilization of the equilibrium, with parametric uncertainties. In defining the class of systems, a few simplifying assumptions are made on the structure of the dynamics; in particular, the damping forces are assumed to be linear in velocities. We first propose a switching surface design for this class of systems, and subsequently, a switched algorithm to reach this surface in finite time using conventional and higher order sliding mode controllers. The stability of the closed-loop system is investigated with an undefined relative degree of the sliding functions. The controller gains are designed such that the controller stabilizes the actual system with parametric uncertainty. The proposed control algorithm is applied to two benchmark problems: a mobile robot and an underactuated underwater vehicle. Simulation results are presented to validate the proposed scheme.