LMI conditions for time-varying uncertain systems can be non-conservative

Establishing robust asymptotic stability of uncertain systems affected by time-varying uncertainty is a key problem. LMI sufficient conditions have been proposed in the literature for addressing this problem based on homogeneous polynomial Lyapunov functions. Unfortunately, till now it has been unclear whether these conditions are also necessary. This paper proposes a proof in order to show that one of these conditions is not only sufficient but also necessary for a sufficiently large degree of the Lyapunov function.     

非线性采样系统指数稳定的新条件

研究了非线性采样系统的稳定性问题. 对以采样周期为参数的离散时间系统族, 证明了全局指数稳定的Lyapunov定理和逆定理. 分别基于系统的一般近似模型和Euler近似模型, 给出了闭环系统全局指数稳定的新条件. 与现有结果相比, 取消了Lyapunov函数全局Lipschitz连续的假设, 减弱了闭环系统全局指数稳定的充分条件.     

LMI approach for absolute stability of general neutral type Lurie indirect control systems

This paper deals with the problem of the absolute stability for general neutral type Lurie indirect control systems by Lyapunov method and linear matrix inequality （LMI） technique. Delay-dependent sufficient conditions for the absolute stability are derived and expressed as the feasibility problem of LMI, which can be easily solved by Matlab Toolbox. Finally, some examples are provide to demonstrate the effectiveness of proposed method.     

LMI approach for absolute stability of general neutral type Lurie indirect control systems

This paper deals with the problem of the absolute stability for general neutral type Lurie indirect control systems by Lyapunov method and linear matrix inequality (LMI) technique. Delay_dependent sufficient conditions for the absolute stability are derived and expressed as the feasibility problem of LMI, which can be easily solved by Matlab Toolbox. Finally, some examples are provide to demonstrate the effectiveness of proposed method.     

LMI conditions for quadratic and robust D-stability of interval systems

Sufficient conditions for the quadratic D-stability and further robust D-stability of interval systems are presented in this paper. This robust D-stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities （LMIs） defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust D-stability of interval systems.     

Estimating the domain of attraction for uncertain polynomial systems

Estimating the Domain of Attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been proposed for the case of known polynomial systems allowing one to find the Largest Estimate of the DA (LEDA) for a given Lyapunov Function (LF). However, the problem of estimating the Robust DA (RDA), that is the DA guaranteed for all possible uncertainties in an uncertain system, it is still an unsolved problem. In this paper, LMI methods are proposed for estimating the RDA in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope. Specifically, the issue of computing the Robust LEDA (RLEDA), that is the intersection of all LEDAs, is considered for common and parameter-dependent LFs, providing constant and parameter-dependent lower bounds. The computation of approximations with simple shape of the RLEDA in the case of parameter-dependent LFs is also discussed.     

A new LMI condition for robust stability of discrete-time uncertain systems

In this paper, a new robust stability condition is derived for uncertain discrete-time linear systems with convex polytopic uncertainties. The condition is expressed in terms of a set of linear matrix inequalities (LMIs) involving only the vertices of the polytope domain. It enables us to determine robust stability of uncertain systems easily by solving some LMIs. A rigorous proof is given to show that an interesting result appeared recently is a special case of the proposed condition. Numerical examples also demonstrate the merit of the present condition in the aspect of conservativeness over other results in the literature.     

Unknown inputs observer for a class of nonlinear uncertain systems: An LMI approach

This paper deals with the simultaneous estimation of states and unknown inputs for a class of Lipschitz nonlinear systems using only the measured outputs. The system is assumed to have bounded uncertainties that appear on both the state and output matrices. The observer design problem is formulated as a set of linear constraints which can be easily solved using linear matrix inequalities (LMI) technique. An application based on manipulator arm actuated by a direct current (DC) motor is presented to evaluate the performance of the proposed observer. The observer is applied to estimate both state and faults.     

Local stability analysis and domain of attraction estimation for a class of uncertain nonlinear discrete‐time systems

This paper proposes a linear matrix inequality based method for the estimation of domain of attraction for a class of discrete‐time nonlinear systems subject to uncertain constant parameters. Recursive algebraic representations of the system dynamics and of the Lyapunov stability conditions are applied to obtain convex conditions which guarantee the system robust local stability while providing an estimate of the domain of attraction. A large class of discrete‐time nonlinear systems and of Lyapunov functions can be embedded in the proposed methodology including the whole class of regular rational functions of the system state variable and uncertain parameters. Numerical examples illustrate the application of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

一类非线性不确定系统的非奇异Terminal滑模控制

针对一类二阶非线性系统提出新的Terminal滑模控制面以克服传统的Terminal滑模控制的奇异问题，同时确保系统从任何初始状态能在有限时间内收敛至平衡点．进一步考虑系统参数摄动和外界扰动等不确定性因素上界的未知性，用Lyapunov稳定性方法给出了一个带有未知性上界参数估计的自适应非奇异Terminal滑模控制（NTSM）控制．最后通过实例比较三种滑模控制方法，仿真结果验证了非奇异Terminal滑模控制能克服传统的Terminal滑模控制的奇异问题，并说明了自适应非奇异Terminal滑模控制的有效性和可行性．     

Improved stability conditions for uncertain neutral-type systems with time-varying delays

This paper investigates the robust stability problem for a class of uncertain neutral-type delayed systems. The systems under consideration contain parameter uncertainties and time-varying delays. We aim at designing less conservative robust stability criteria for such systems. A new second-order reciprocally convex inequality is first proposed in order to deal with double integral terms. Then, by constructing a new Lyapunov– Krasovskii functional and employing the improved Wirtinger-based integral inequality and the reciprocally convex combination approaches, novel stability criteria are obtained. Moreover, the stability conditions for standard time-delay system are obtained as by-product results. Comparisons in three numerical examples illustrate the effectiveness of our results.     

不确定线性系统新的稳定性准则

研究了带有区间时滞的不确定系统的稳定性问题。通过采用时滞分割法,把时滞区间分割成任意两小段,并构造恰当的Lyapunov函数,利用积分不等式,得到了新的时滞相关的稳定性准则。通过数值例子验证了结果的有效性。     

Delay-range-dependent stability for uncertain stochastic systems with interval time-varying delay and nonlinear perturbations

This paper considers the problem of robust stability for a class of uncertain stochastic systems with interval time-varying delay under nonlinear perturbations. A new delay-dependent method for robust stability of the systems is proposed. The innovation of the method includes employment of a tighter integral inequality and construction of an appropriate type of Lyapunov–Krasovskii functional. The restriction used to bound some trace term in the existing methods is also removed. The resulting criterion derived from this method has advantages over previous ones in that it has less conservatism and enlarges the scope of application. The reduction in conservatism of the proposed criterion is attributed to a method to estimate the upper bound on the stochastic differential of the Lyapunov–Krasovskii functional without neglecting any useful terms in the delay-dependent stability analysis. On the basis of the estimation and by utilizing free-weighting matrices, new delay-range-dependent stability criterion is established in terms of linear matrix inequality. Finally, numerical examples are provided to show the effectiveness and reduced conservatism of the proposed method.     

Estimating the domain of attraction for non-polynomial systems via LMI optimizations

This paper proposes a strategy for estimating the domain of attraction (DA) for non-polynomial systems via Lyapunov functions (LFs). The idea consists of converting the non-polynomial optimization arising for a chosen LF in a polynomial one, which can be solved via LMI optimizations. This is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst-case remainders in truncated Taylor expansions. Moreover, a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen LF, and another condition is provided for establishing whether such an estimate has been found. The proposed strategy can readily be exploited with variable LFs in order to search for optimal estimates. Lastly, it is worth remarking that no other method is available to estimate the DA for non-polynomial systems via LMIs.     

Finite-time stability and instability of stochastic nonlinear systems

This paper presents a new definition of finite-time stability for stochastic nonlinear systems. This definition involves stability in probability and finite-time attractiveness in probability. An important Lyapunov theorem on finite-time stability for stochastic nonlinear systems is established. A theorem extending the stochastic Lyapunov theorem is also proved. Moreover, an example and a lemma are presented to illustrate the scope of extension. A useful inequality, extended from Bihari’s inequality, is derived, which plays an important role in showing the Lyapunov theorem. Finally, a Lyapunov theorem on finite-time instability is proved, which states that almost surely globally asymptotical stability is not equivalent to finite-time stability for some stochastic systems. Two simulation examples are given to illustrate the theoretical analysis.     

不确定时滞系统的鲁棒绝对稳定性研究

基于Lyapunov稳定性理论，采用线性矩阵不等式处理方法，研究具有参数不确定性的时滞系统鲁棒绝对稳定性问题．导出了用线性矩阵不等式系统可行性描述的系统鲁棒绝对稳定的滞后依赖型条件，据此可计算出最大的允许时滞界．通过求解一组线性矩阵不等式，给出使得闭环系统鲁棒绝对稳定的无记忆状态反馈控制律设计方法。     

一类不确定模糊动态时滞系统保成本控制

针对一类不确定非线性动态时滞系统，利用模相T—S模型，通过状态反馈对保成本控制问题进行研究。应用线性矩阵不等式给出模糊闭环系统渐近稳定的充分条件，保证了所有允许的不确定闭环系统是稳定的，而且对于一个给定的二次型成本函数，舱保证闭环成本不超过某个界。仿真结果表明所提出的控制方法是有效的。     

Robust adaptive control for a class of uncertain non-affine nonlinear systems using affine-type neural networks

A robust adaptive control is proposed for a class of single-input single-output non-affine nonlinear systems. In order to approximate the unknown nonlinear function, a novel affine-type neural network is used, and then to compensate the approximation error and external disturbance a robust control term is employed. By Lyapunov stability analysis for the closed-loop system, it is proved that tracking errors asymptotically converge to zero. Moreover, an observer is designed to estimate the system states because all the states may not be available for measurements. Furthermore, the adaptation laws of neural networks and the robust controller are given out based on the Lyapunov stability theory. Finally, two simulation examples are presented to demonstrate the effectiveness of the proposed control method.