Stability of stochastic systems with uncertain time delays

For linear delay systems with bilinear noise sufficient conditions are given for the global asymptotic stochastic stability independent of the length of the delay(s). For linear stochastic noise terms, sufficient conditions for the existence of an invariant distribution, for all values of the delay are given. It is shown that the gaussian distribution is the unique invariant distribution. The covariance and correlation matrix function of the resulting stationary process are completely characterized by a Lyapunov-type equation. All these sufficient conditions are obtained in the form of the existence of some positive definite matrices satisfying certain Riccati-type equations.     

Invariant and attracting sets of Hopfield neural networks with delay

This paper studies invariant and attracting sets of Hopfield neural networks system with delay. Sufficient criteria are given for the invariant and attracting sets. In particular, we provide an estimate of the existence range of attractors by using invariant and attracting sets. Moreover, when the system has an equilibrium point, we obtain the sufficient conditions of global asymptotic stability of the equilibrium point. Several examples are also worked out to demonstrate the advantages of our results.     

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

In this paper we derive necessary and sufficient conditions of stabilizability for multi‐input nonlinear systems possessing a Hopf bifurcation with the critical mode being linearly uncontrollable, under the non‐degeneracy assumption that stability can be determined by the third order term in the normal form of the dynamics on the centre manifold. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the Hopf bifurcation of the closed‐loop system is supercritical, which is equivalent to local asymptotic stability of the system at the bifurcation point. We prove that under the non‐degeneracy conditions, stabilizability is equivalent to the existence of solutions to a third order algebraic inequality of the feedback gains. Explicit conditions for the existence of solutions to the algebraic inequality are derived, and the stabilizing feedback laws are constructed. Part of the sufficient conditions are equivalent to the rank conditions of an augmented matrix which is a generalization of the Popov–Belevitch–Hautus (PBH) rank test of controllability for linear time invariant (LTI) systems. We also apply our theory to feedback control of rotating stall in axial compression systems using bleed valve as actuators. Copyright © 2006 John Wiley & Sons, Ltd.     

Continuous stabilization controllers for singular bilinear systems: The state feedback case

Global asymptotic stabilization for a class of singular bilinear systems is first studied in this paper. New approaches are developed by means of the LaSalle invariant principle for nonlinear systems. A new set of sufficient condition is first derived via the continuous static state feedback, the feedback not only guarantees the existence of solution but also the global asymptotical stabilization for the closed loop system.     

Invariant codistributions and the feedforward form for discrete-time nonlinear systems

The goal of this paper is two-fold. First, given an arbitrary n-dimensional discrete-time nonlinear dynamical system, necessary and sufficient conditions for the existence of a one-dimensional invariant codistribution are obtained. Second, it is shown that the previous conditions can be used iteratively to obtain a nested sequence of n invariant codistributions with the properties that each codistribution contains the previous one and the last one coincides with the cotangent bundle of the state manifold. As a byproduct, necessary and sufficient conditions are obtained for a discrete-time nonlinear dynamical system to be equivalent to the so-called feedforward form.     

Multistability analysis for a general class of delayed Cohen–Grossberg neural networks

In this paper, by discussing parameter conditions based on properties of activation functions, we decompose state space into positively invariant sets and establish sufficient conditions for the existence of locally stable equilibria for delayed Cohen–Grossberg neural networks (CGNNs) through Cauchy convergence principle. Some new criteria are derived for ensuring equilibria (periodic orbits) to be locally or globally exponentially stable in any designated region. Finally, our results are demonstrated by four numerical simulations.     

Further remarks on asymptotic stability and set invariance for linear delay-difference equations

We studied the existence of positively invariant sets for linear delay-difference equations. In particular, we regarded two strong stability notions: robust (with respect to delay parameter) asymptotic stability for the discrete-time case and delay-independent stability for the continuous-time case. The correlation between these stability concepts is also considered. Furthermore, for the delay-difference equations with two delay parameters, we provided a computationally efficient numerical routine which is necessary to guarantee the existence of contractive sets of Lyapunov–Razumikhin type. This condition also appears to be necessary and sufficient for the delay-independent stability and sufficient for the robust asymptotic stability.     

Semi‐global stabilization of discrete‐time systems subject to non‐right invertible constraints

This paper investigates time‐invariant linear systems subject to input and state constraints. We study discrete‐time systems with full or partial constraints on both input and state. It has been shown earlier that the solvability conditions of stabilization problems are closely related to important concepts such as the right invertibility or non‐right invertibility of the constraints, the location of constraint invariant zeros, and the order of constraint infinite zeros. In this paper, for general time‐invariant linear systems with non‐right invertible constraints, necessary and sufficient conditions are developed under which semi‐global stabilization in the admissible set can be achieved by state feedback. Sufficient conditions are also developed for such a stabilization in the case where measurement feedback is used. Such sufficient conditions are almost necessary. Controllers for both state feedback and measurement feedback are constructed as well. Copyright © 2009 John Wiley & Sons, Ltd.     

Deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis

In this paper, we obtain new sufficient conditions ensuring existence, uniqueness, and global asymptotic stability (GAS) of the equilibrium point for a general class of delayed neural networks (DNNs) via nonsmooth analysis, which makes full use of the Lipschitz property of functions defining DNNs. Based on this new tool of nonsmooth analysis, we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point. Then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring GAS are actually unnecessary; and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped. Finally, we derive some new sufficient conditions which are easy to check. Comparison with some related existing results is conducted and advantages are illustrated with examples. Throughout our paper, spectral properties of the matrix (A + A/sup /spl tau//) play an important role, which is a distinguished feature from previous studies. Here, A and A/sup /spl tau// are, respectively, the feedback and the delayed feedback matrix defining the neural network under consideration.     

Stabilization of a given set of equilibrium states of nonlinear systems

The problem of multiprogram stabilization of the given set of equilibrium states in the class of nonlinear time invariant systems is studied. A method for designing multiprogram control based on the idea of interpolation is proposed. Hermite’s interpolation polynomial makes it possible to design a controller that provides the prescribed time invariant modes of operation for the closed-loop system and ensures their asymptotic stability. A theorem on sufficient conditions for the existence of a multiprogram control is proved. The results are illustrated by a practical example.     

Asymptotic stability of systems with saturation constraints

In the present paper the authors establish new sufficient conditions for the global asymptotic stability of linear systems operating on the unit hypercube in Rⁿ and of linear systems with partial state saturation constraints. They also determine a set of necessary and sufficient conditions for the global asymptotic stability of second order linear systems with saturation constraints on both states using the above results. Systems of the type considered herein are widely used in several areas of applications, including control systems, signal processing, and artificial neural networks     

Positively invariant polyhedral sets of discrete-time linear systems

The problem of the existence of positively invariant polyhedral sets for linear discrete-time dynamical systems is studied. In the first part of the paper, necessary and sufficient conditions for a given polyhedral set to be a positively invariant set of a linear system are obtained. Then, the spectral properties of systems possessing this kind of invariant set are established. Finally the class of systems possessing positively invariant polyhedral cones is studied.     

On the structure of 2-D observers

This note gives necessary and sufficient conditions for asymptotic reconstructibility of the state of multivariable 2D systems. The existence and the structure of the observers are related to the matrices appearing in a Bézout identity. A counterexample shows that the state estimation cannot be performed in general by a static feedback on the estimation error. This note includes also a technique for designing dynamic state observers.     

On the equivalence between global recurrence and the existence of a smooth Lyapunov function for hybrid systems

We study a weak stability property called recurrence for a class of hybrid systems. An open set is recurrent if there are no finite escape times and every complete trajectory eventually reaches the set. Under sufficient regularity properties for the hybrid system we establish that the existence of a smooth, radially unbounded Lyapunov function that decreases along solutions outside an open, bounded set is a necessary and sufficient condition for recurrence of that set. Recurrence of open, bounded sets is robust to sufficiently small state dependent perturbations and this robustness property is crucial for establishing the existence of a Lyapunov function that is smooth. We also highlight some connections between recurrence and other well studied properties like asymptotic stability and ultimate boundedness.     

Investigation of asymptotic stability of equilibria by localization of the invariant compact sets

The method of localization of invariant compact sets was proposed to study for asymptotic stability the equilibrium points of an autonomous system of differential equations. This approach relies on the necessary and sufficient conditions for asymptotic stability formulated in terms of positive invariant sets and invariant compact sets, and enables one to study the equilibrium points for asymptotic stability in the cases where it is impossible to use the first approximation or the method of Lyapunov functions. The possibilities of the method were illustrated by examples.     

Global stability analysis of neural networks with multiple time varying delays

In this note, we study the equilibrium and stability properties of neural networks with time varying delays. Our main results give sufficient conditions for the existence, uniqueness and global asymptotic stability of the equilibrium point. The proposed conditions establish the relationships between network parameters of the neural systems and the delay parameters. The obtained results are applicable to all continuous nonmonotonic neuron activation functions and do not require the interconnection matrices to be symmetric. Some examples are also presented to compare our results with the previous results derived in the literature.     

一般模型2－D系统的观测器设计理论

该文讨论了２－Ｄ一般模型（２ＤＧＭ）在标准边界条件下的渐近观测器的存在性条件及其设计问题．为此，首先将由Ｂｉｓｉａｃｃｏ等人在１９８５年发展起来的相应于对角边界条件的２－Ｄ渐近稳定性理论推广到了具有标准边界条件的２－Ｄ一般模型．在此基础上，借助于局部能控性概念，建立了一系列十分类似于１－Ｄ情形的观测器的存在条件，从这些存在条件出发也可以得到相应的观测器设计算法，最后还得出了２ＤＧＭ的分离性定理．     

Asymptotic stabilization of driftless systems

Issues of asymptotic stabilization of a class of non-linear driftless systems are presented. In addition to the necessary and sufficient condition for the existence of a smooth time-invariant asymptotic stabilizer, sufficient condition for the existence of a quadratic-type Lyapunov function candidate is also proposed herein to alleviate the construction of stabilizing control laws. Following the deduction of the equivalence of the sufficient condition and the determination of the local definiteness of a defined scalar function, the stabilizability checking conditions are then derived in terms of system dynamics and its derivatives at the origin only. These are achieved by taking Taylor's series expansion on system dynamics. The derived conditions are shown to be consistent with those obtained by Brockett. Comparative results of Liaw and Liang are also included. Finally, examples are given to demonstrate the use of the main results.     

一般模型2-D系统的观测器设计理论

该文讨论了2-D一般模型(2DGM)在标准边界条件下的渐近观测器的存在性条件及其设计问题.为此,首先将由Bisiacco等人在1985年发展起来的相应于对角边界条件的2-D、渐近稳定性理论推广到了具有标准边界条件的2-D一般模型.在此基础上,借助于局部能控性概念,建立了一系列十分类似于1-D情形的观测器的存在条件,从这些存在条件出发也可以得到相应的观测器设计算法,最后还得出了2DGM的分离性定理.