Nonfragile observer‐based H ∞ sliding mode control for Itô stochastic systems with Markovian switching

The paper is devoted to investigating sliding mode control for a class of nonlinear uncertain stochastic systems with input nonlinearity and Markovian switching. A nonfragile observer subjected to the transition rates of the modes is designed. By some specified matrices, the connections among the designed sliding surfaces corresponding to every mode are established. The state estimation‐based sliding mode control law is derived to guarantee the reachability of the sliding surface in finite time interval. The sufficient conditions on asymptotically stochastic stability of the error system and sliding mode dynamics with a given disturbance attenuation level are derived in terms of linear matrix inequalities. Finally, an example is provided to illustrate the efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

时滞系统的输出反馈滑模控制的一种奇异系统方法

利用一种奇异系统方法讨论了时滞系统的输出反馈滑模控制问题. 时滞系统的非线性项满足范数有界约束.首先,将滑动模态与线性切换面作为一个奇异时滞系统,基于奇异时滞系统的稳定性理论, 给出滑动模态稳定及切换面存在的线性矩阵不等式(Linear matrix inequality, LMI)充分条件.然后,给出使得系统闭环渐近稳定的静态输出反馈滑模控制器的设计方法,此控制器保证闭环 系统有限时间到达切换面.最后,用数值算例验证本文方法的有效性和正确性.     

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.     

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.     

Sliding mode control for Itô stochastic systems with Markovian switching

This paper deals with the problem of sliding mode control (SMC) for a class of nonlinear uncertain stochastic systems with Markovian switching. By introducing some specified matrices, the connections among the designed sliding surfaces corresponding to every mode are established. Furthermore, the present sliding mode controller including the transition rates of modes can cope with the effect of Markovian switching. By means of linear matrix inequalities (LMIs) with equality constraint, sufficient conditions are derived such that the sliding motions on the specified sliding surfaces are stochastically stable with γ-disturbance attenuation level. Finally, a numerical example is given to illustrate the applicability of the present method.     

基于不确定T-S模型的模糊滑模自适应控制

提出一种不确定T-S模型的模糊滑模自适应控制方法。通过变换将该模型转换成3个组成部分：线性标称系统,已知非线性部分和未知不确定部分。针对它们设计3个控制器,其作用分别为：强迫系统沿着滑模面运动,消除已知扰动对线性标称系统的影响,克服不确定扰动(采用模糊滑模自适应控制,无需知道不确定的界限)。该方法无需求正定矩阵就能保证系统全局稳定。     

Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities

In this paper, the stability and stabilization problems of a class of continuous-time and discrete-time Markovian jump linear system (MJLS) with partly unknown transition probabilities are investigated. The system under consideration is more general, which covers the systems with completely known and completely unknown transition probabilities as two special cases — the latter is hereby the switched linear systems under arbitrary switching. Moreover, in contrast with the uncertain transition probabilities studied recently, the concept of partly unknown transition probabilities proposed in this paper does not require any knowledge of the unknown elements. The sufficient conditions for stochastic stability and stabilization of the underlying systems are derived via LMIs formulation, and the relation between the stability criteria currently obtained for the usual MJLS and switched linear systems under arbitrary switching, are exposed by the proposed class of hybrid systems. Two numerical examples are given to show the validity and potential of the developed results.     

时滞离散系统的非线性准滑模鲁棒控制

针对一类变时滞离散系统,基于一种新型的非线性时变准滑模面设计方案,研究了系统的鲁棒滑模控制问题.首先,给出了该种非线性切换函数的一般形式,该类准滑模面具有时变的特征,且能够动态地改善系统的运动品质.利用自由权矩阵与线性矩阵不等式技术,给出了该类时滞离散系统非线性准滑模面的设计方法,并得到了稳定的非线性准滑模面存在的充分条件;其次,基于一种改进的离散趋近律方法,设计了相应的准滑模控制器,以保证系统的状态在有限时间内到达准滑模,从而将此类非线性系统的滑模变结构控制的分析与综合问题推广到了时滞非线性系统.最后,仿真结果表明,在本文所设计的准滑模面与准滑模变结构控制器的作用下,系统的状态是稳定的,且具有响应速度快、超调量小、调节时间较小等优点,从而说明了本文所设计方法的有效性.     

Sliding mode control of uncertain switched delay systems via hysteresis switching strategy

This paper investigates the robust sliding mode control (SMC) problem for a class of uncertain switched systems with time-varying delay. The sliding surface is constructed such that the sliding motion is completely invariant to all admissible uncertainties. For the case of the known delay-derivative upper bound, by using the multiple Lyapunov functions method, the Hysteresis switching law dependent on the state and the previous value of switching signal are designed to stabilize the sliding motion and avoid the chattering. Variable structure controllers are developed to drive the state of switched systems to reach the sliding surface in finite time and remain on it thereafter. For the case of the unknown delay-derivative upper bound, based on the single Lyapunov function method, the conditions of stabilization are obtained. Finally, a numerical example is given to illustrate the effectiveness of the proposed methods.     

Robust H ∞ sliding mode control for a class of uncertain switched delay systems

This article considers the robust H _∞ sliding mode control problem for a class of uncertain switched delay systems. A single sliding surface is constructed such that the reduced-order equivalent sliding motion restricted to the sliding surface is completely invariant to all admissible uncertainties. For cases of known delay and unknown delay, the existence conditions of the sliding surface are proposed, respectively. The corresponding hysteresis switching laws are designed such that the sliding motion is stabilisable with H _∞ disturbance attenuation level γ. Furthermore, variable structure controllers are developed to drive the state of the switched system to reach the single sliding surface in finite time and remain on it thereafter. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design methods.     

Discrete-time sliding mode control for uncertain systems with state and input delays

This article presents a discrete-time sliding mode control method for the robust stabilisation of linear uncertain multi-input discrete-time systems with state and input delays. The systems are assumed to have structured mismatched time-varying parameter uncertainties. A specified switching surface is proposed and a sliding mode controller is derived to ensure the existence of the quasi-sliding mode. Based on the improved Lyapunov function and linear matrix inequality technique, delay-independent sufficient conditions for the design of a stable switching surface are given and the stability of the overall closed-loop system is guaranteed. Neither chattering phenomenon will occur nor the knowledge of upper bound of uncertainties is required. Finally, simulation results demonstrate the efficacy of the proposed control methodology.     

具有次优保性能滑动模态的静态输出反馈滑模控制

针对一类不确定系统,提出了一种具有次优保性能滑模面的静态输出反馈滑模控制方法.首先将滑模面的设计问题等价为一个对称矩阵的求解问题,基于等效控制法,推导了保性能滑模面存在的充分条件.然后基于迭代线性矩阵不等式(iterative linear matrix inequality,ILMI)方法,给出了次优保性能线性滑模面的求解算法,最后基于线性矩阵不等式(linearmatrix inequality,LMI)方法,设计了输出反馈滑模控制器,使得闭环系统渐近稳定且切换函数能在有限时间内到达零.该方法首次实现了输出反馈滑模面的优化,且具有保守性小、无需对被控系统模型进行坐标变换的优点.仿真结果验证了本文方法的优越性.     

Passivity-based sliding mode control of uncertain singular time-delay systems

In this paper the problem of sliding mode control (SMC) with passivity of a class of uncertain nonlinear singular time-delay systems is studied. An integral-type switching surface function is designed by taking the singular matrix into account, thus the resulting sliding mode dynamics is a full-order uncertain singular time-delay system. By introducing some slack matrices, a delay-dependent sufficient condition is proposed in terms of linear matrix inequality (LMI), which guarantees the sliding mode dynamics to be generalized quadratically stable and robustly passive. The passification solvability condition is then established. Moreover, a SMC law and an adaptive SMC law are synthesized to drive the system trajectories onto the predefined switching surface in a finite time. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory.     

Robust chaos synchronisation for a class of disturbed Rössler systems with multiple time-delays

This paper investigates the robust synchronisation problem for a class of Rössler systems with multiple time delays. Based on the proportional-integral switching surface, a sliding mode control (SMC) is derived to not only guarantee synchronisation between the master and slave Rössler systems with multiple time delays but also have a H_∞ noise perturbation attenuation performance. The parameter matrix necessary for constructing both PI switching surface and the SMC can be easily solved by linear matrix inequality optimisation technique. The simulations results are presented to show the effectiveness of the proposed method.     

Observer-based sliding mode control of Markov jump systems with random sensor delays and partly unknown transition rates

In this paper, the sliding mode control problem of Markov jump systems (MJSs) with unmeasured state, partly unknown transition rates and random sensor delays is probed. In the practical engineering control, the exact information of transition rates is hard to obtain and the measurement channel is supposed to subject to random sensor delay. Design a Luenberger observer to estimate the unmeasured system state, and an integral sliding mode surface is constructed to ensure the exponential stability of MJSs. A sliding mode controller based on estimator is proposed to drive the system state onto the sliding mode surface and render the sliding mode dynamics exponentially mean-square stable with H_∞ performance index. Finally, simulation results are provided to illustrate the effectiveness of the proposed results.     

Sliding mode control for Markovian jumping systems with actuator nonlinearities

This article considers the design of sliding mode control for stochastic Markovian jumping systems with actuator nonlinearities. In the design of integral sliding surfaces, a set of specified matrices are employed such that the connections among sliding surfaces corresponding to each mode are established. And then, a sliding mode controller depending on the transition rates is synthesised such that the reachability of the specified sliding surface can be ensured. Moreover, sufficient conditions for the stability of the sliding motions are derived. It is shown that the effect of Markovian switching can be coped with by the present sliding mode control method. Finally, the simulation results illustrating the proposed method are provided.     

Sliding Mode Control for Markovian Jump Systems with Generalized Switching

In this paper, the sliding mode control (SMC) problem for continuous‐time Markovian jump systems (MJSs) is considered, in which the transition rate matrix (TRM) is partially unknown and uncertain. Firstly, the sliding mode surface S(t) = 0 is designed, which is mode‐dependent. Therefore, is used instead of in the SMC algorithm. Via adopting a linear matrix inequality (LMI) approach, sufficient conditions are proposed to ensure that the reduced order system is exponentially stable in mean square. Furthermore, the reduced order system is completely insensitive to the external disturbance. Secondly, SMC law is designed correspondingly which dominated by a Markov process. It could drive the state trajectories onto the specified sliding mode surface in finite time quickly and maintain them on the surface in subsequently time. Thirdly, a new term in will be introduced in the designed SMC and should be handled by a new approach. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

Discrete Sliding‐Mode Control of Mimo Linear Systems

This paper introduces a discrete sliding‐mode control for linear time‐invariant systems in the presence of matched uncertainties. The switching surface is constructed by applying the pole‐assignment method applying to the overall system, not the system in the sliding mode. Importantly, the control algorithm is derived to handle the rate of convergence to the sliding mode and to prevent the control law from encountering any unreasonably large variations. As for the system stability, it can be found that the system is stabilized and finally restricted to a known region. A numerical example is also included to demonstrate all the features of the developed discrete sliding‐mode control.     

State feedback sliding mode control without chattering by constructing Hurwitz matrix for AUV movement

This paper presents a new method to eliminate the chattering of state feedback sliding mode control (SMC) law for the mobile control of an autonomous underwater vehicle (AUV) which is nonlinear and suffers from unknown disturbances system. SMC is a well-known nonlinear system control algorithm for its anti-disturbances capability, while the chattering on switch surface is one stiff question. To dissipate the well-known chattering of SMC, the switching manifold is proposed by presetting a Hurwitz matrix which is deducted from the state feedback matrix. Meanwhile, the best switching surface is achieved by use of eigenvalues of the Hurwitz matrix. The state feedback control parameters are not only applied to control the states of AUV but also connected with coefficients of switching surface. The convergence of the proposed control law is verified by Lyapunov function and the robust character is validated by the Matlab platform of one AUV model.