Transient Stability Enhancement of Multi–Machine Power Systems: Synchronization via Immersion of a Pendular System

In this paper the problem of designing excitation controllers to improve the transient stability of multi‐machine power systems is addressed adopting two new perspectives. First, instead of the standard formulation of stabilization of an equilibrium point, we aim here at the more realistic objective of keeping the difference between the generators rotor angles bounded and their speeds equal—which is called synchronization in the power literature—and translates into a problem of stabilization of a set. Second, we adopt the classical viewpoint of power systems as a set of coupled nonlinear pendula, and express our control objective as ensuring that some suitable defined pendula dynamics are (asymptotically) immersed into the power system dynamics. Our main contribution is the explicit computation of a control law for the two–machine system that achieves global synchronization. The same procedure is applicable to the n–machine case, for which the existence of a locally stabilizing solution is established.     

Stabilization of a nonlinear jump system

In this paper we consider the problem of output feedback stabilization of a general nonlinear jump system. We shall show that the combination of a locally asymptotically stabilizing state feedback law and a local asymptotic observer yields a locally asymptotically stabilizing output feedback controller. Hence, the local separation principle holds for the nonlinear jump system. This result can be applied to nonlinear sampled-data systems.     

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems , admits a polynomial feedback u= u(x) such that the closed-loop system admits an orbitally asymptotically stable limit cycle.Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.     

Qualitative analysis and application of locally coupled neural oscillator network

This paper investigates a locally coupled neural oscillator autonomous system qualitatively. By applying an approximation method, we give a set of parameter values with which an asymptotically stable limit cycle exists, and the sufficient conditions on the coupling parameters that guarantee asymptotically global synchronization are established under the same external input. A gradational classifier is introduced to detect synchronization, and the network model based on the analytical results is applied to image segmentation. The performance is comparable to the results from other segmentation methods.     

Computer aided proof for the global stability of Lotka-Volterra systems

By modifying the mechanical method of determining LaSalle's invariant sets for Lotka-Volterra chain systems [1,2] and repeatedly using Wu's characteristic set method [3,4], it is proved that for a class of Lotka-Volterra loop systems, the locally asymptotically stable positive equilibrium point must be globally stable.     

Uniformly asymptotically Zhukovskij stable orbits

This article deals with Zhukovskij stability of planar systems. We prove that if the omega limit set of a nonclosed orbit is a stable limit cycle, then, the orbit is uniformly asymptotically Zhukovskij stable.     

延时系统输入状态稳定性的Lyapunov逆理论

研究了延时系统输入状态稳定性的局部Lipschitz连续的Lyapunov逆理论. 针对含有任意可测局部本质有界扰动的延时系统, 一个局部Lipschitz连续的Lyapunov泛函被证实是存在的, 如果该系统是鲁棒渐进稳定的. 根据该结论, 延时系统输入状态稳定性的Lyapunov特征被进一步得到.     

Adding an integrator for the stabilization problem

We study the relationship between the following two properties: P1: The system is locally asymptotically stabilizable; and P2: The system is locally asymptotically stabilizable; where . Dayawansa, Martin and Knowles have proved that these properties are equivalent if the dimension n = 1. Here, using the so called Control Lyapunov function approach, (a) we propose another more constructive and somewhat simpler proof of Dayawansa, Martin and Knowles's result; (b) we show that, in general, P1 does not imply P2 for dimensions n larger than 1; (c) we prove that P2 implies P1 if some extra assumptions are added like homogeneity of the system. By using the latter result recursively, we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form.     

A limit set stabilization by means of the Port Hamiltonian system approach

A solution to the stabilization problem of a compact set by means of the Interconnection and Damping Assignment Passivity‐Based Control methodology, for an affine nonlinear system, was introduced. To this end, we expressed the closed‐loop system as a Port Hamiltonian system, having the property of almost all their trajectories asymptotically converge to a convenient limit set, except for a set of measure zero. It was carried out by solving a partial differential equation (PDE) or single matching condition, which allows the desired energy level or limit set E to be shaped explicitly. The control strategy was tested using the magnetic beam balance system and the pendulum actuated by a direct current motor (DC‐motor), having obtained satisfactory results. Copyright © 2014 John Wiley & Sons, Ltd.     

Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems

This paper develops sufficient conditions for a general nonlinear control system to be locally (resp. globally) asymptotically stabilizable via smooth state feedback. In particular, it is shown that as in the case of affine systems, this is possible if the unforced dynamic system of ∑₁ is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems.     

An approximate stability analysis of nonlinear systems described by Universal Learning Networks

Stability is one of the most important subjects in control systems. As for the stability of nonlinear dynamical systems, Lyapunov's direct method and linearized stability analysis method have been widely used. But, it is generally recognized that finding an appropriate Lyapunov function is fairly difficult especially for the nonlinear dynamical systems, and also it is not so easy for the linearized stability analysis to find the locally asymptotically stable region. Therefore, it is crucial and highly motivated to develop a new stability analysis method, which is easy to use and can easily study the locally asymptotically stable region at least approximately, if not exactly. On the other hand, as for the calculation of the higher order derivative, Universal Learning Networks (ULNs) are equipped with a systematic mechanism that calculates their first and second order derivatives exactly.So, in this paper, an approximate stability analysis method based on η approximation is proposed in order to overcome the above problems and its application to a nonlinear dynamical control system is discussed. The proposed method studies the stability of the original trajectory by investigating whether the perturbed trajectory can approach the original trajectory or not. The above investigation is carried out approximately by using the higher order derivatives of ULNs.In summarizing the proposed method, firstly, the absolute values of the first order derivatives of any nodes of the trajectory with respect to any initial disturbances are calculated by using ULNs. If they approach zero at time infinity, then the trajectory is locally asymptotically stable. This is an alternative linearized stability analysis method for nonlinear trajectories without calculating Jacobians directly. In the method, the stability analysis of time-varying systems with multi-branches having any sample delays is possible, because the systems are modeled by ULNs. Secondly, the locally asymptotically stable region, where asymptotical stability is secured approximately, is obtained by finding the area where the first order terms of Taylor expansion are dominant compared to the second order terms with η approximation assuming that the higher order terms more than the third order are negligibly small in the area.Simulations of an inverted pendulum balancing system are carried out. From the results of the simulations, it is clarified that the stability of the inverted pendulum control system is easily analyzed by the proposed method in terms of studying the locally asymptotically stable region.     

Practical stabilization of exponentially unstable linear systems subject to actuator saturation nonlinearity and disturbance

This paper investigates the problem of practical stabilization for linear systems subject to actuator saturation and input additive disturbance. Attention is restricted to systems with two anti‐stable modes. For such a system, a family of linear feedback laws is constructed that achieves semi‐global practical stabilization on the asymptotically null controllable region. This is in the sense that, for any set χ₀ in the interior of the asymptotically null controllable region, any (arbitrarily small) set χ_∞ containing the origin in its interior, and any (arbitrarily large) bound on the disturbance, there is a feedback law from the family such that any trajectory of the closed‐loop system enters and remains in the set χ_∞ in a finite time as long as it starts from the set χ₀. In proving the main results, the continuity and monotonicity of the domain of attraction for a class of second‐order systems are revealed. Copyright © 2001 John Wiley & Sons, Ltd.     

STABILITY OF LINEAR TIME‐INVARIANT OPEN‐LOOP UNSTABLE SYSTEMS WITH INPUT SATURATION

This paper analyzes the stability of a linear time‐invariant open‐loop unstable system subject to input saturation. First, we extend the idea of approximating the locally asymptotically stable region (controllable set) of the system for the case where the control is small enough to be unsaturated (inside the linear region), to the case when the control is allowed to saturate. It is shown that, when the Lyapunov descent criterion and the Kuhn‐Tucker Theorem is applied, a superior locally asymptotically stable region is found. A technique for approximating the locally asymptotically stable region is presented.     

Stability and Hopf bifurcation of a HIV infection model with CTL-response delay

In this paper, we consider a HIV infection model with CTL-response delay and analyze the effect of time delay on stability of equilibria. We obtain the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the CTL-absent equilibrium and CTL-present equilibrium. By choosing the CTL-response delay τ as a bifurcation parameter, we prove that the CTL-present equilibrium is locally asymptotically stable in a range of delays and a Hopf bifurcation occurs as τ crosses a critical value. Numerical simulations are given to support the theoretical results.     

Kalman滤波的局部渐近稳定性和渐近最优性*

本文用现代时间序列分析方法和非递推状态估计理论，对完全可观、非完全可控系统，提出了稳态Ｋａｌｍａｎ预报器局部渐近稳定性和最优性概念，揭示了两者的关系；证明了这类系统的Ｋａｌｍａｎ预报器总是局部渐近最优和渐近稳定的；提出了构造最大局部渐近最优域的新方法，并给出了几何解释，推广和发展了经典Ｋａｌｍａｎ滤波稳定性理论，一个算例及其仿真结果说明了所提出的结果的有效性。     

Global uniform asymptotical stability of a class of nonlinear cascaded systems with application to a nonholonomic wheeled mobile robot

In this article, we consider the stability analysis problem for a class of nonlinear cascaded systems by using homogeneous properties. Assume that the driving subsystem and the driven subsystem are both homogeneous and locally uniformly asymptotically stable. If the cascaded term satisfies a given inequality, then the cascaded system is globally uniformly asymptotically stable. Furthermore, in the case that both degrees of homogeneity are negative, the cascaded system is globally uniformly finite-time stable. Compared with the existing methods, the conditions given in this article are much easier to verify. These stability results are applied to the global tracking control problem of a nonholonomic wheeled mobile robot. Simulation results are provided to show the effectiveness of the methods.     

An embedding approach for the design of state‐feedback tracking controllers for references with jumps

We study the problem of designing state‐feedback controllers to track time‐varying state trajectories that may exhibit jumps. Both plants and controllers considered are modeled as hybrid dynamical systems, which are systems with both continuous and discrete dynamics, given in terms of a flow set, a flow map, a jump set, and a jump map. Using recently developed tools for the study of stability in hybrid systems, we recast the tracking problem as the task of asymptotically stabilizing a set, the tracking set, and derive conditions for the design of state‐feedback tracking controllers with the property that the jump times of the plant coincide with those of the given reference trajectories. The resulting tracking controllers guarantee that solutions of the plant starting close to the reference trajectory stay close to it and that the difference between each solution of the controlled plant and the reference trajectory converges to zero asymptotically. Constructive conditions for tracking control design in terms of LMIs are proposed for a class of hybrid systems with linear maps and input‐triggered jumps. The results are illustrated by various examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonsmooth Robust Stabilization of a Class of Two-Dimensional Nonlinear Systems

In this paper we consider a class of parameterized families of nonlinear systems which cannot be robustly asymptotically stabilized by means of C ¹ feedback. We construct C ⁰ state feedback laws which are smooth away from the origin and which robustly asymptotically stabilize these families of systems. We then show that, in some cases, the regularity of the obtained robust asymptotic stabilizers is “maximum” in the sense that the considered families of systems do not admit any Lipschitz continuous robust asymptotic stabilizer. Date received: June 27, 1997. Date revised: July 28, 1998.     

Set-valued Lyapunov functions for difference inclusions

The paper relates set-valued Lyapunov functions to pointwise asymptotic stability in systems described by a difference inclusion. Pointwise asymptotic stability of a set is a property which requires that each point of the set be Lyapunov stable and that every solution to the inclusion, from a neighborhood of the set, be convergent and have the limit in the set. Weak set-valued Lyapunov functions are shown, via an argument resembling an invariance principle, to imply this property. Strict set-valued Lyapunov functions are shown, in the spirit of converse Lyapunov results, to always exist for closed sets that are pointwise asymptotically stable.     

Non-smooth simultaneous stabilization of nonlinear systems: An interpolation method

In this paper, we introduce a method which enables us to construct a continuous simultaneous stabilizer for pairs of systems in which cannot be simultaneously stabilized by means of C¹ feedback. We extend this method to higher-dimensional systems and show that any pair of asymptotically stabilizable nonlinear systems can be simultaneously stabilized (not asymptotically) by means of continuous feedback.