Variable structure control of dynamical systems with mismatched norm-bounded uncertainties: An LMI approach

In this paper, we propose a new sliding surface design method for a class of uncertain systems with mismatched unstructured uncertainties. We relax the matching assumption of traditional VSC design methods, and the uncertain system under consideration may have mismatched parameter uncertainties in the state matrix as well as the input matrix. In terms of linear matrix inequalities (LMIs), we give a sufficient condition for the existence of linear sliding surfaces guaranteeing asymptotic stability of the reduced-order equivalent sliding mode dynamics. We give an LMI parameterization of such sliding surfaces together with a switched feedback control strategy. We also give an LMI existence condition of linear sliding surfaces guaranteeing the prescribed decay rate. Finally, we give simulation results to show the effectiveness of our method.     

LMI-Based Sliding Surface Design for Integral Sliding Mode Control of Mismatched Uncertain Systems

We propose a linear matrix inequality (LMI)-based sliding surface design method for integral sliding-mode control of mismatched uncertain systems. The uncertain system under consideration may have mismatched norm bounded uncertainties in the state matrix as well as the input matrix. We give a sufficient condition for the existence of a sliding surface guaranteeing asymptotic stability of the full order sliding mode dynamics. We also give an LMI characterization of the sliding surface, together with an integral sliding mode control law guaranteeing the existence of a sliding mode from the initial time. Additionally, we give an LMI condition of sliding surfaces guaranteeing the alpha-stability constraint. Finally, we give a simulation result to show the effectiveness of our method     

Variable structure output feedback control design for a class of uncertain dynamic systems

In this paper, we consider the problem of designing a variable structure output feedback control (VSOFC) law for a class of uncertain systems with mismatched uncertainty in the state matrix. We derive existence conditions of linear sliding surfaces in terms of constrained linear matrix inequalities (LMIs), and we give VSOFC strategies. We also give a simple design procedure and parameterizations of the linear switching surfaces, together with solvability conditions for the given constrained LMIs. Finally, we give a design example in order to show the effectiveness of our method.     

LMI‐based robust sliding control for mismatched uncertain nonlinear systems using fuzzy models

We propose a robust sliding control design method for uncertain Takagi–Sugeno fuzzy models. The uncertain fuzzy systems under consideration have mismatched parameter uncertainties in the state matrix and external disturbances. We make the first attempt to relax the restrictive assumption that each nominal local system model shares the same input channel, which is required in the traditional VSS‐based fuzzy control design methods. We derive the existence conditions of linear sliding surfaces guaranteeing the asymptotic stability in terms of constrained LMIs. We present an LMI characterization of such sliding surfaces. Also, an LMI‐based algorithm is given to design the switching feedback control term so that a stable sliding motion is induced in finite time. Finally, we give two simulation results to show the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

A novel interpretation of the invariance condition of mismatched uncertain large-scale systems

This paper presents a novel interpretation of the invariance condition of mismatched uncertain large-scale systems in which uncertainties and interconnections do not satisfy the conventional invariance condition. A new sliding mode that consists of two sets of switching surfaces is proposed. In every subsystem, uncertainties and interconnections are all separated into matched and mismatched components to reduce the conservatism. As a result, new invariance conditions are derived such that each subsystem in the new sliding mode is completely invariant to both uncertainties and interconnections. We also show that these conditions are not only necessary but also sufficient. In addition, sufficient conditions are derived such that the equivalent reduced-order system in the new sliding mode is asymptotically stable within the framework of linear matrix inequalities. A decentralized controller is synthesized to assure stability of the overall system. Finally, illustrative example is given to show the effectiveness of the proposed approach.     

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.     

Decentralized variable structure control for mismatched uncertain large-scale systems: a new approach

In this paper, a decentralized variable structure controller for a class of large-scale systems with mismatched uncertainties is proposed. In every subsystem, two sets of switching surfaces is introduced. New invariance conditions are derived such that the system in the new sliding mode is completely invariant to both matched and mismatched uncertainties. To demonstrate the effectiveness of the proposed scheme, a decentralized variable structure controller is synthesized to perform the new sliding mode. Moreover, the stability analysis of the overall system is also provided.     

Decentralized dynamic output feedback controllers for interconnected systems with mismatched uncertainties using new sliding mode control

This paper presents a method for the design of decentralized output feedback controllers for an interconnected systems with mismatched uncertainties using a variable-structure control approach. A new reaching condition of the sliding mode is derived such that the reduced-order system is invariant to the mismatched uncertainties. In every subsystem, a new idea for a sliding mode that consists of two sets of switching surfaces is introduced, and a scheme of decentralized dynamic control using only output states is also proposed. This scheme can guarantee closed-loop stability of a sliding manifold of the composite system. Finally, a numerical example is given to illustrate the controller design method.     

Robust Stabilization of Uncertain Fuzzy Systems Using Variable Structure System Approach

Based on the variable structure system (VSS) theory, we develop a fuzzy control system design method for a class of uncertain nonlinear multivariable systems that can be represented by a Takagi-Sugeno fuzzy model. We make the first attempt to relax the restrictive assumption that each nominal local system model shares the same input channel, which is required in the traditional VSS-based fuzzy control design methods. As the local controller we use a sliding mode controller with a switching feedback control term. In terms of linear matrix inequalities (LMIs), we derive a sufficient condition for the existence of linear sliding surfaces guaranteeing asymptotic stability of the reduced-order equivalent sliding mode dynamics. We present an LMI characterization of such sliding surfaces. We also give an LMI-based algorithm to design the switching feedback control term so that a stable sliding motion is induced in finite time. Finally, we give a numerical design example.     

Adaptive controller design for uncertain fuzzy systems using variable structure control approach

Based on the variable structure control (VSC) theory, we develop an adaptive fuzzy control system design method for uncertain Takagi-Sugeno fuzzy models with norm-bounded uncertainties. We relax the restrictive assumptions that each nominal local system model shares the same input channel and the norm bound of the uncertainty is known, which are usually invoked in the traditional VSC-based fuzzy control design methods. As the local controller we use a VSC law with a switching feedback control term and an adaptation law to account for the norm-bounded uncertainties. In terms of LMIs, we derive a sufficient condition for the existence of linear sliding surfaces guaranteeing the asymptotic stability. We present an LMI characterization of such sliding surfaces. We also give an LMI-based design algorithm, together with a numerical design example.     

An LMI-based variable structure control for a class of uncertain singular Markov switched systems

An H-infinifty variable structure control is presented for singular Markov switched systems with mismatched norm-bounded uncertainties and mismatched norm-bounded external disturbances. It is shown that the sliding mode dynamics on the given switching surface is regular, impulse-free, and stochastically stable and satisfies H-infinity performance. A variable structure controller is designed to guarantee that the system trajectory converges to the linear switching surface in some finite time. Finally, a numerical example is solved to show the effectiveness and validness of the theoretical results.     

An LMI-based variable structure control for a class of uncertain singular Markov switched systems

An H-infinifty variable structure control is presented for singular Markov switched systems with mismatched norm-bounded uncertainties and mismatched norm-bounded external disturbances. It is shown that the sliding mode dynamics on the given switching surface is regular, impulse-free, and stochastically stable and satisfies H-infinity performance. A variable structure controller is designed to guarantee that the system trajectory converges to the linear switching surface in some finite time. Finally, a numerical example is solved to show the effectiveness and validness of the theoretical results.     

非匹配不确定和状态时滞扰动下的车道线保持

针对车道线保持控制中存在的不确定性扰动及时滞问题, 提出一种自校正滑模控制方法. 该方法利用线性矩阵不等式理论给出滑动模态存在的充分条件; 系统在滑动模态下对于存在的非匹配不确定性扰动以及状态时滞具有完全不变性. 接下来引入双极性sigmoid函数代替常规滑模控制中的符号函数并设计自校正律; 在自校正律的作用下使sigmoid函数的边界层厚度以及切换增益可根据系统状态进行自适应调节, 从而达到削弱控制器输出抖振的目的. 基于Lyapunov理论对该方法的稳定性进行了证明, 最后通过车辆的车道线保持仿真实验对该控制方法的可行性及有效性进行了验证.     

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.     

不确定多时滞系统的输出滑模控制

本文研究了带有不匹配参量不确定性和匹配外扰的多时滞系统的鲁棒滑模控制。当状态不可量测时,利用系统输出设计滑模面,通过线性矩阵不等式给出了滑模面存在的一个时滞无关充分性条件,在此基础上设计变结构控制律,实现滑模控制。最后通过仿真说明该控制策略的有效性。     

Robust sliding-mode control for uncertain time-delay systems: an LMI approach

This note is devoted to robust sliding-mode control for time-delay systems with mismatched parametric uncertainties. A delay-independent sufficient condition for the existence of linear sliding surfaces is given in terms of linear matrix inequalities, based on which the corresponding reaching motion controller is also developed. The results are illustrated by an example.     

基于主动滑模控制实现一类含有非匹配不确定混沌系统的同步

讨论了一类非匹配不确定混沌系统的同步问题.基于主动控制思想,提出了一种主动滑模控制策略,使得从任意初始条件出发的不确定混沌系统在有限时间内到达滑模面.利用线性矩阵不等式(LMI)技术设计了一个鲁棒稳定的滑模面,以降低非匹配不确定对系统的影响.给出了线性矩阵不等式形式的稳定滑模面存在的充分条件.通过对Rssler系统的同步仿真,验证了该方法的可行性.     

非匹配不确定性影响下的无人车路径跟踪控制

针对无人车在非匹配不确定性影响下的路径跟踪控制问题,设计一种基于线性矩阵不等式（LMI）的滑模控制器.首先,根据车辆运动学和动力学方程,同时考虑轮胎侧滑造成的不确定性、车辆侧偏约束以及随机干扰影响,建立车辆非线性不确定系统模型;然后,提出一种线性滑模路径跟踪控制方法,给出线性滑模面存在的充分条件,并推导出线性滑模面存在的显式公式,以保证约束于该滑模面的降阶等价系统的二次稳定性;最后,在SerretFrenet坐标系下验证车辆单、双移线运动时的路径跟踪控制效果.仿真结果表明,所设计的滑模控制器可以保证对参考路径的稳定跟踪,具有较强的鲁棒性.     

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.     

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.