Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions

A generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and/or uncertainty. This paper will further enhance absolute stability analysis by using the composite quadratic Lyapunov function whose level set is the convex hull of a family of ellipsoids. The absolute stability analysis will be approached by characterizing absolutely contractively invariant (ACI) level sets of the composite quadratic Lyapunov functions. This objective will be achieved through three steps. The first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems. The second step establishes stability conditions for linear difference inclusions and then for saturated linear systems. The third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability. Based on the conditions for absolute stability, optimization problems are formulated for the estimation of the stability region. Numerical examples demonstrate that stability analysis results based on composite quadratic Lyapunov functions improve significantly on what can be achieved with quadratic Lyapunov functions.     

Global exponential stability of neural networks with globallyLipschitz continuous activations and its application to linearvariational inequality problem

This paper investigates the existence, uniqueness, and global exponential stability (GES) of the equilibrium point for a large class of neural networks with globally Lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations. The provided sufficient condition for GES is mild and some conditions easily examined in practice are also presented. The GES of neural networks in the case of locally Lipschitz continuous activations is also obtained under an appropriate condition. The analysis results given in the paper extend substantially the existing relevant stability results in the literature, and therefore expand significantly the application range of neural networks in solving optimization problems. As a demonstration, we apply the obtained analysis results to the design of a recurrent neural network (RNN) for solving the linear variational inequality problem (VIP) defined on any nonempty and closed box set, which includes the box constrained quadratic programming and the linear complementarity problem as the special cases. It can be inferred that the linear VIP has a unique solution for the class of Lyapunov diagonally stable matrices, and that the synthesized RNN is globally exponentially convergent to the unique solution. Some illustrative simulation examples are also given.     

PID Control of Planar Nonlinear Uncertain Systems in the Presence of Actuator Saturation

This paper investigates PID control design for a class of planar nonlinear uncertain systems in the presence of actuator saturation. Based on the bounds on the growth rates of the nonlinear uncertain function in the system model, the system is placed in a linear differential inclusion. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. By placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. The equilibrium point corresponds to the desired set point for the system output. Thus, the location of the equilibrium point and the size of the domain of attraction determine, respectively, the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Based on these conditions, the feasible set points can be determined and the design of the PID control law that stabilizes the nonlinear uncertain system at a feasible set point with a large domain of attraction can then be formulated and solved as a constrained optimization problem with constraints in the form of linear matrix inequalities (LMIs). Application of the proposed design to a magnetic suspension system illustrates the design process and the performance of the resulting PID control law.      

Composite quadratic Lyapunov functions for constrained control systems

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.     

Global stability analysis in Cohen-Grossberg neural networks with delays and inverse Hölder neuron activation functions

In this paper, a novel class of Cohen-Grossberg neural networks with delays and inverse Hölder neuron activation functions are presented. By using the topological degree theory and linear matrix inequality (LMI) technique, the existence and uniqueness of equilibrium point for such Cohen-Grossberg neural networks is investigated. By constructing appropriate Lyapunov function, a sufficient condition which ensures the global exponential stability of the equilibrium point is established. Two numerical examples are provided to demonstrate the effectiveness of the theoretical results.     

Qualitative analysis of a recurrent neural network for nonlinear continuously differentiable convex minimization over a nonempty closed convex subset.

We investigate the qualitative properties of a recurrent neural network (RNN) for minimizing a nonlinear continuously differentiable and convex objective function over any given nonempty, closed, and convex subset which may be bounded or unbounded, by exploiting some key inequalities in mathematical programming. The global existence and boundedness of the solution of the RNN are proved when the objective function is convex and has a nonempty constrained minimum set. Under the same assumption, the RNN is shown to be globally convergent in the sense that every trajectory of the RNN converges to some equilibrium point of the RNN. If the objective function itself is uniformly convex and its gradient vector is a locally Lipschitz continuous mapping, then the RNN is globally exponentially convergent in the sense that every trajectory of the RNN converges to the unique equilibrium point of the RNN exponentially. These qualitative properties of the RNN render the network model well suitable for solving the convex minimization over any given nonempty, closed, and convex subset, no matter whether the given constrained subset is bounded or not.     

Some Characterizations of Global Exponential Stability of a Generic Class of Continuous-Time Recurrent Neural Networks

This paper reveals two important characterizations of global exponential stability (GES) of a generic class of continuous-time recurrent neural networks. First, we show that GES of the neural networks can be fully characterized by global asymptotic stability (GAS) of the networks plus the condition that the maximum abscissa of spectral set of Jacobian matrix of the neural networks at the unique equilibrium point is less than zero. This result provides a very useful and direct way to distinguish GES from GAS for the neural networks. Second, we show that when the neural networks have small state feedback coefficients, the supremum of exponential convergence rates (ECRs) of trajectories of the neural networks is exactly equal to the absolute value of the maximum abscissa of spectral set of Jacobian matrix of the neural networks at the unique equilibrium point. Here, the supremum of ECRs indicates the potentially fastest speed of trajectory convergence. The obtained results are helpful in understanding the essence of GES and clarifying the difference between GES and GAS of the continuous-time recurrent neural networks.     

平面点集凸壳的快速算法

提出一种计算平面点集凸壳的快速算法。利用极值点划分出四个矩形,它们包含了所有凸壳顶点,通过对矩形中的点进行扫描,排除明显不是凸壳顶点的点,剩余的点构成一个简单多边形。再利用极点顺序法判断多边形顶点的凹凸性并删除所出现的凹顶点,最终得到一个凸多边形即为点集的凸壳。整个算法简洁明了,避免了乘法运算（除最坏情况外）,从而节省计算时间。     

Global asymptotic stability and global exponential stability ofcontinuous-time recurrent neural networks

This paper presents new results on global asymptotic stability (GAS) and global exponential stability (GES) of a general class of continuous-time recurrent neural networks with Lipschitz continuous and monotone nondecreasing activation functions. We first give three sufficient conditions for the GAS of neural networks. These testable sufficient conditions differ from and improve upon existing ones. We then extend an existing GAS result to GES one and also extend the existing GES results to more general cases with less restrictive connection weight matrices and/or partially Lipschitz activation functions     

Global exponential stability of a class of neural networks with variable delays

In this paper, the conditions ensuring existence, uniqueness, and global exponential stability of the equilibrium point of a class of neural networks with variable delays are studied. Without assuming global Lipschitz conditions on these activation functions, applying idea of vector Lyapunov function, the sufficient conditions for global exponential stability of neural networks are obtained.     

Convex analysis of invariant sets for a class of nonlinear systems

We study the invariance of the convex hull of an invariant set for a class of nonlinear systems satisfying a generalized sector condition. The generalized sector is bounded by two odd symmetric functions which are convex/concave in the right-half plane. In a recent paper, we showed that, for this class of systems, the convex hull of a group of invariant ellipsoids is invariant. This paper shows that the convex hull of a general invariant set need not be invariant, and that the convex hull of a contractively invariant set is, however, invariant.     

一类变时滞神经网络的全局指数稳定性

在不要求激活函数有界的前提下,利用Lyapunov泛函方法和线性矩阵不等式(LMI)分析技巧,研究了一类变时滞神经网络平衡点的存在性和全局指数稳定性.给出判别网络全局指数稳定性的判据,推广了现有文献中的一些结果.这些判据具有LMI的形式,进而易于验证.仿真例子表明了所得结果的有效性.     

Saturation-based switching anti-windup design for linear systems with nested input saturation

This paper proposes a saturation-based switching anti-windup design for the enlargement of the domain of attraction of a linear system subject to nested saturation. A nestedly saturated linear feedback is expressed as a linear combination of a set of auxiliary linear feedbacks, which form a convex hull where the nestedly saturated linear feedback resides. This set of auxiliary linear feedbacks is then partitioned into several subsets. The auxiliary linear feedbacks in each of these subsets form a convex sub-hull of the original convex hull. When the value of the nestedly saturated linear feedback falls into a convex sub-hull, it can be expressed as a linear combination of the subset of all the auxiliary feedbacks that form the convex sub-hull. A separate anti-windup gain is designed for each convex sub-hull by using a common quadratic Lyapunov function and is implemented when the value of the nestedly saturated linear feedback falls into this convex sub-hull. Simulation results indicate that such a saturation-based switching anti-windup design has the ability to significantly enlarge the domain of attraction of the closed-loop system.     

寻求简单多边形凸壳的线性时间算法

本文提出在线性时间内构造简单多边形顶点凸壳的两种算法。第一个算法的基本思想是利用一种技巧对多边形顶点进行筛选，使剩余顶点的角的大小排成递增序，然后用Graham扫描方法删去非凸壳顶点，最后得到多边形凸壳的顶点序列.第二个算法不断删去多边形的凹点及新产生的 凹点，最后得到凸壳顶点序列。这两种算法简单，易于实现，时间复杂性都是O（n)。     

A recurrent neural network for nonlinear continuouslydifferentiable optimization over a compact convex subset

We propose a general recurrent neural-network (RNN) model for nonlinear optimization over a nonempty compact convex subset which includes the bound subset and spheroid subset as special cases. It is shown that the compact convex subset is a positive invariant and attractive set of the RNN system and that all the network trajectories starting from the compact convex subset converge to the equilibrium set of the RNN system. The above equilibrium set of the RNN system coincides with the optimum set of the minimization problem over the compact convex subset when the objective function is convex. The analysis of these qualitative properties for the RNN model is conducted by employing the properties of the projection operator of Euclidean space onto the general nonempty closed convex subset. A numerical simulation example is also given to illustrate the qualitative properties of the proposed general RNN model for solving an optimization problem over various compact convex subsets.     

Existence and global stability of equilibrium point for delayed competitive neural networks with discontinuous activation functions

In this article, the issue of global stability is discussed for competitive neural networks with time-varying delay and discontinuous activation functions. First, a sufficient criterion is presented towards the existence and global asymptotic stability of an equilibrium point (EP), by employing the Leray–Schauder alternative theorem, linear matrix inequality technique and generalised Lyapunov function method. In particular, for the case where the activation functions are monotonic increasing, the above criterion also ensures the global exponential stability of an EP. Second, based on the properties of M-matrix, topological degree theory of set-valued map and generalised Lyapunov function method, some sufficient conditions are derived for checking the existence and global exponential stability of an EP. In doing so, the viability problem is investigated and the obtained results are delay-dependent and independent of each other. Finally, two examples with simulations are provided to demonstrate the effectiveness of our results.     

Conjugate Lyapunov functions for saturated linear systems

Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper's methods.     

Stability analysis of dynamical neural networks

In this paper, we use the matrix measure technique to study the stability of dynamical neural networks. Testable conditions for global exponential stability of nonlinear dynamical systems and dynamical neural networks are given. It shows how a few well-known results can be unified and generalized in a straightforward way. Local exponential stability of a class of dynamical neural networks is also studied; we point out that the local exponential stability of any equilibrium point of dynamical neural networks is equivalent to the stability of the linearized system around that equilibrium point. From this, some well-known and new sufficient conditions for local exponential stability of neural networks are obtained     

基于循环链表的凸包增量算法研究与实现

采用循环链表构建凸包,使凸包的各顶点在增量过程中,始终处于动态变化的稳定循环链中,无差错地生成结果凸包。相比顺序表而言,每次只需修改指针,无须在内存中频繁移动顶点数据,节省大量的系统时间及内存资源,从根本上解决首尾相接的凸包动态生成问题,极好地满足程序的鲁棒性原则,代码执行效率高。