Positive Limit Set Research Articles

Finite‐time control of DC–AC boost converter: The limit cycle approach

AbstractThis paper proposes a novel finite‐time limit cycle controller for DC to AC conversion in power boost converters. The stable limit cycle is categorized as a positive limit set. Achieving the control law, based on the concepts of set stability and finite‐time stability, is a challenging problem. To fulfill the control objective, a novel sliding manifold is suggested concerning the structure of the wanted limit cycle which facilitates conditions for the development of finite‐time stability for the limit cycle. By utilizing the sliding mode control method, a control strategy is proposed that guarantees the desired stable limit cycle is created in the phase plane of the closed‐loop system and the phase trajectories reach it in a finite time and remain on it for all future times. This ensures the finite‐time generation of the biased sinusoidal oscillations in the output of the power boost converter from a DC input source. The simulation results of a practical boost converter have validated the effectiveness and feasibility of the presented algorithm.

Kernel center adaptation in the reproducing kernel Hilbert space embedding method

SummaryThe performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and how the kernel centers are distributed in the state‐space. In this article, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems by exploiting the fact that the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self‐organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.

Limit cycle oscillator in nonlinear systems with multiple time delays

This paper concerns with the problem of generation stable limit cycles in a class of nonlinear time-delay systems with multiple time delays. With the help of the Lyapunov–Krasovskii functional approach, a state-feedback controller is constructed based on the backstepping technique. According to the recursive procedure of the backstepping technique, at first, the desired limit cycle is generated in the second-order subsystem by utilizing the Lyapunov theorem analysis of the positive limit sets. Then, this method is expanded for higher-order systems. By introducing an appropriate Lyapunov-Krasovskii functional, the complexities raised by unknown time-delays are solved in the control design process. Based on this scheme, a delay-independent controller is explicitly designed which does not need the precise knowledge of time delays. The proposed method rigorously guarantees the practical stability of the closed-loop system and also ensures the convergence of the phase trajectories of the closed-loop system to the target limit cycle. Finally, to prove the theoretical achievement and also to illustrate the effective performance of the proposed approach, the simulation results are provided for several examples.

On stability with respect to the part of variables of a non-autonomous system in a cylindrical phase space

Abstract. The stability problem of a system of differential equations with a right-hand side periodic with respect to the phase (angular) coordinates is considered. It is convenient to consider such systems in a cylindrical phase space which allows a more complete qualitative analysis of their solutions. The authors propose to investigate the dynamic properties of solutions of a non-autonomous system with angular coordinates by constructing its topological dynamics in such a space. The corresponding quasi-invariance property of the positive limit set of the system’s bounded solution is derived. The stability problem with respect to part of the variables is investigated basing of the vector Lyapunov function with the comparison principle and also basing on the constructed topological dynamics. Theorem like a quasi-invariance principle is proved on the basis of a vector Lyapunov function for the class of systems under consideration. Two theorems on the asymptotic stability of the zero solution with respect to part of the variables (to be more precise, non-angular coordinates) are proved. The novelty of these theorems lies in the requirement only for the stability of the comparison system, in contrast to the classical results with the condition of the corresponding asymptotic stability property. The results obtained in this paper make it possible to expand the usage of the direct Lyapunov method in solving a number of applied problems.

Limit Cycle Synchronization of Nonlinear Systems with Matched and Unmatched Uncertainties Based on Finite-Time Disturbance Observer

This paper focuses on the limit cycle control of uncertain nonlinear systems in the presence of both matched and unmatched uncertainties. For this purpose, first, a virtual nonlinear system is constructed which has the desired limit cycle in its phase trajectories. The Lyapunov stability theorem for positive limit sets is utilized to confirm the stability of the created limit cycle in the virtual system. Next, by using a modified nonsingular terminal sliding mode control method, a robust controller is designed to synchronize the trajectories of the actual nonlinear system with the corresponding virtual one in a finite time. It is assumed that the actual system suffers from both matched and unmatched uncertainties and/or unknown external disturbances. A nonlinear terminal sliding surface is designed with the aim of a finite-time disturbance observer to tackle these unknown terms. The finite-time Lyapunov stability theorem is utilized to confirm the stability and robustness of the designed control law. Through simulation results, the finite-time stabilization of the synchronization errors and appropriate performance of the proposed control law are verified.

On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

Zhukovskij Quasi-Stable Orbits of Impulsive Dynamical Systems

This paper deals with the Zhukovskij quasi-stability of an orbit in a planar impulsive dynamical system. We prove that if the positive limit set of an orbit is an asymptotically stable limit cycle (an isolated periodic orbit), then the orbit is uniformly asymptotically Zhukovskij quasi-stable. Also, we prove that if an orbit is not eventually periodic and its positive limit set is a periodic orbit, then it is asymptotically Zhukovskij quasi-stable.

Inducing sustained oscillations in a class of nonlinear discrete time systems

Inducing sustained oscillations in a class of nonlinear discrete time systems is studied in this paper. The novelty of this paper is based on the proposed approach in generating stable oscillations according to limit cycle control. The limit cycle control is not formulated for nonlinear discrete time systems of any order and this paper concentrates on this issue. Considering the stable limit cycle as a positive limit set for the dynamical system, a nonlinear control law is designed to create the considered limit cycle in the phase trajectories of the closed-loop nonlinear discrete time system to achieve oscillations with the desirable amplitude and frequency. For this purpose, firstly, the limit cycle control is proposed for second-order nonlinear discrete time systems. The stability analysis of the generated limit cycle is done via a suitable Lyapunov function. Also, the domain of attraction of the created limit cycle is calculated. The proposed method is then extended for nonlinear discrete time systems of any order via the backstepping technique. Finally, computer simulations are performed for a practical example to demonstrate the ability of the designed controller in generating stable oscillations.

A SHORT REMARK ON CONTROL SYSTEMS

Souza and Tozatti [7] introduce the notions of prolongations and prolongational limit sets on control systems. In this article, we prove the upper semicontinuity of first positive prolongations and first positive prolongational limit sets on control systems.

On the Relationship between the Synchronous State and the Solution of an Isolated Node in a Complex Network

摘要: 复杂网络同步是复杂系统和复杂网络的前沿研究方向之一，已经取得很大的进展. 但是对于节点以耦合矩阵左特征向量加权平均态、孤立节点的解与网络的同步态之间具有什么关系，什么是网络的同步态和同步轨等基本问题仍然缺乏深入的研究，弄清楚这些问题对于复杂网络同步的理解和应用具有重要的意义. 本文采用数学分析方法证明，如果网络同步，则加权平均态 x = j=1Njxj可以定义为同步态，一般来说，x在正极限集的意义下，也就是孤立节点方程的解. 因此在实际应用中，把孤立节点方程的解s（t） 与加权平均态x不加区别地对待是合理的. 同步态是不依赖于初始条件的通解，而同步轨是依赖于初始条件的特解. 对于混沌节点的网络，同步态应该理解为吸引子，而不是某一条轨道. 最后，本文还提供一些实例加以说明，并指出一些尚待解决的问题. 关键词: 复杂网络 / 同步态 / 同步轨 / 孤立节点

A bounded distributed connectivity preserving aggregation strategy with collision avoidance property

This paper presents a bounded connectivity preserving control strategy for the aggregation of multi-agent systems. The problem is investigated for two cases of single-integrator agents and unicycles. Under the proposed control strategy, if two agents are in the connectivity range at some point in time, they will stay connected thereafter. The agents finally aggregate while avoiding collision in such a way that the average of the distances between every pair of neighboring agents is bounded by a pre-specified positive real number, which can be chosen arbitrarily small. The results are developed based on some important characteristics of the positive limit set of the closed-loop system under the proposed control strategy and a fundamental property of convex real functions. The theoretical results are verified by simulation.

Control of quantum states in decoherence-free subspaces

The problem of quantum state control in open quantum systems based on the Lyapunov method is studied. The control laws are designed to drive the system to a pure target state in decoherence-free subspace (DFS) in the interaction picture, where the average of an observable operator P is chosen as a Lyapunov function. By analyzing the positive limit set via Barbalat's lemma, we propose a sufficient condition on the observable operator such that the positive limit set only contains the target state. Under this condition, it is proved that the control laws can drive the open system to any pure target state in DFS in the interaction picture. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical simulations on a three-level system validate the effectiveness of the proposed control strategy.

Finite abstractions for hybrid systems with stable continuous dynamics

This paper outlines an abstraction process in which a particular class of hybrid automata with continuous dynamics that have parameterized positive limit sets, are being abstracted into finite transition systems. The limit sets with their corresponding attraction regions define pre- and post-conditions for the continuous dynamics, and determine the transitions in the discrete abstraction. An observable (weak) bisimulation equivalence is established between the two models. The abstraction process described can find application in verification, as well as in planning and symbolic control.

Topology identification of complex dynamical networks

Recently, some researchers investigated the topology identification for complex networks via LaSalle's invariance principle. The principle cannot be directly applied to time-varying systems since the positive limit sets are generally not invariant. In this paper, we study the topology identification problem for a class of weighted complex networks with time-varying node systems. Adaptive identification laws are proposed to estimate the coupling parameters of the networks with and without communication delays. We prove that the asymptotic identification is ensured by a persistently exciting condition. Numerical simulations are given to demonstrate the effectiveness of the proposed approach.

Boundedness of orbits and stability of closed sets

This paper explores fundamental connections between boundedness of orbits, and stability and attractivity of closed sets. For this purpose, the paper considers topological notions of stability and attractivity which do not depend on a metric. The notions considered are characterized in terms of restricted prolongations and positive limit sets, and connections with boundedness are studied. Finally, the notion of nontangency is used to give a Lyapunov result for Lyapunov stability, attractivity and boundedness of orbits. Unlike previous sufficient conditions for boundedness, our result does not require the Lyapunov function to be proper or weakly proper. Examples are provided to illustrate the results.

Localization of the positive limit set of an almost periodic discrete system

Localization of the positive limit set of an almost periodic discrete system

On Limitations to the Achievable Path Following Performance for Linear Multivariable Plants

In this paper, we consider a problem termed ldquopath followingrdquo. This differs from the common problem of reference tracking, in that here we can adjust the speed at which we traverse the reference trajectory. We are interested in ascertaining the degree to which we can track a given trajectory, and in characterizing the class of paths for which we can generate an appropriate temporal specification so that the path can be tracked arbitrarily well in an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> sense. We give various bounds on the achievable performance, as well as tight results in special cases. In addition, we give a numerical procedure based on convex optimization for computing the achievable performance. The results demonstrate that there are situations where arbitrarily good <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance may be achieved even though the origin is not in the convex hull of the positive limit set of the path to be followed.

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.

Control and Stability Analysis of Limit Cycles in a Hopping Robot

A new stabilizing nonlinear controller for the vertical motion of an electrically actuated hopping robot is introduced and analyzed. The approach starts by finding reference limit cycles from the of a mass-spring system. The controller then modulates the system dynamics via the leg actuator during the stance phase to force the system trajectory to converge to this reference limit cycle. The controlled system dynamics is a continuous yet nonsmooth vector field. A piecewise-continuous Lyapunov function and the general forms of Lasalle's invariance and domain of attraction theorems are used to prove global asymptotic stability of the desired limit cycles. It is also shown that the derived passive dynamic cycles are indeed the positive limit sets of the controlled system. The rate of convergence can be adjusted but is limited by actuator constraints.