Abstractions of Hamiltonian control systems

Given a control system and a desired property, an abstracted system is a reduced system that preserves the property of interest while ignoring modeling detail. In previous work, abstractions of linear and nonlinear control systems were considered while preserving reachability properties. In this paper, we consider the abstraction problem for Hamiltonian control systems, where, in addition to the property of interest we also preserve the Hamiltonian structure of the control system. We show how the Hamiltonian structure of control systems can be exploited to simplify the abstraction process. We then focus on local accessibility preserving abstractions, and provide conditions under which local accessibility properties of the abstracted Hamiltonian system are equivalent to the local accessibility properties of the original Hamiltonian control system.     

Construction of discrete-time models for port-controlled Hamiltonian systems with applications

The issues of constructing a discrete-time model for Hamiltonian systems are in general different from those for dissipative systems. We propose an algorithm for constructing an approximate discrete-time model, which guarantees Hamiltonian conservation. We show that the algorithm also preserves, in a weaker sense, the losslessness property of a class of port-controlled Hamiltonian systems. An application of the algorithm to port-controlled Hamiltonian systems with quadratic Hamiltonian is presented, and we use this to solve the stabilization problem for this class of systems based on the approximate discrete-time model constructed using the proposed algorithm. We illustrate the usefulness of the algorithm in designing a discrete-time controller to stabilize the angular velocity of the dynamics of a rigid body.     

Hamiltonian properties of twisted hypercube-like networks with more faulty elements

Twisted hypercube-like networks (THLNs) are a large class of network topologies, which subsume some well-known hypercube variants. This paper is concerned with the longest cycle in an n-dimensional (n-D) THLN with up to 2n−9 faulty elements. Let G be an n-D THLN, n≥7. Let F be a subset of V(G)?E(G), |F|≤2n−9. We prove that G−F contains a Hamiltonian cycle if δ(G−F)≥2, and G−F contains a near Hamiltonian cycle if δ(G−F)≤1. Our work extends some previously known results.     

A family of H∞ controllers for dissipative Hamiltonian systems

Using structure properties of dissipative Hamiltonian systems, this paper investigates the parameterization problem of H_infty controllers for such systems. A family of H_∞ controllers with full information is first obtained by interconnecting an H_∞ controller with a generalized zero‐energy‐gradient (ZEG) detectable, free generalized Hamiltonian system. Then, a family of H_∞ controllers with partial information is presented in terms of the solution to an inequality only in 2n independent variables (twice as many as the one used to characterize the state feedback) and without imposing an additional cascade condition. Both of the parameterization methods avoid solving Hamilton–Jacobi–Issacs equations (or inequalities), and thus the proposed controllers are relatively simple in form and easy in operation. Numerical experiments show the effectiveness and feasibility of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

电力系统微分代数模型耗散Hamilton实现

对非线性微分代数模型电力系统的耗散Hamilton实现问题进行了研究．首先提出了非线性微分代数系统的耗散Hamilton实现结构,给出了完成常值耗散Hamilton实现的充分条件;然后证明单机单负荷电力系统必然存在耗散Hamilton实现,并构造出系统的一个耗散Hamilton实现．     

Repetitive control of Hamiltonian systems based on variational symmetry

This paper is concerned with repetitive control of Hamiltonian systems, which is based on iterative learning control utilizing the variational symmetry of those systems. Variational symmetry allows us to obtain an algorithm to solve a certain class of optimal control problems in a repetitive control framework. Therefore, the proposed method can deal with not only trajectory tracking control problems but also optimal trajectory generation problems, never before considered in a repetitive control framework. A convergence analysis of this algorithm is also discussed. Furthermore, some numerical simulations demonstrate the effectiveness of the proposed method.     

基于Hamilton函数方法的多机多负荷电力系统分散励磁控制

采用Hamilton函数方法研究了多机多负荷电力系统的励磁控制问题. 首先, 通过预置状态反馈完成了系统的耗散Hamilton实现. 然后, 基于该耗散实现形式设计了非线性分散励磁控制器, 分析了闭环系统的稳定性. 该控制器能充分利用系统内在的功率平衡特性. 仿真结果验证了控制策略的有效性.     

Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum

From the symplectic representation of an autonomous nonlinear dynamical system with holonomic constraints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and characterization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum.     

A parallel routing algorithm on circulant networks employing the Hamiltonian circuit latin square

Double-loop [J. Bermond, F. Comellas, D. Hsu, Distributed Loop Computer Networks: A Survey, J. Parallel and Distributed Computing, Academic Press, 24 (1995) 2-10] and 2-circulant networks (2-CN) [J. Park, Cycle Embedding of Faulty Recursive Circulants, J. of Korea Info. Sci. Soc. 31 (2) (2004) 86-94] are widely used in the design and implementation of local area networks and parallel processing architectures. In this paper, we investigate the routing of a message on circulant networks, that is a key to the performance of this network. We would like to transmit 2k packets from a source node to a destination node simultaneously along paths on G(n; ±s₁, ±s₂, … , ±s_k), where the ith packet traverses along the ith path (1 ? i ? 2k). In order for all packets to arrive at the destination node quickly and securely, the ith path must be node-disjoint from all other paths. For construction of these paths, employing the Hamiltonian circuit latin square (HCLS), a special class of (n × n) matrices, we present O(n²) parallel routing algorithm on circulant networks.     

Global exponential stabilisation of acyclic traffic networks

This work is devoted to the construction of explicit feedback control laws for the robust global exponential stabilisation of general uncertain discrete-time acyclic networks. We consider discrete-time uncertain network models, which satisfy very weak assumptions. The construction of the controllers and the rigorous proof of the robust global exponential stability for the closed-loop system are based on recently proposed vector-Lyapunov function criteria, as well as the fact that the network is acyclic. It is shown, in this study, that the latter requirement is necessary for the existence of a robust global exponential stabiliser of the desired uncongested equilibrium point of the network. Our main focus is on traffic networks and all assumptions are related to features appearing in traffic models. An illustrative example demonstrates the applicability of the obtained results to realistic traffic flow networks.     

Study on the stability of switched dissipative Hamiltonian systems

The hybrid Hamiltonian system is a kind of important nonlinear hybrid systems. Such a system not only plays an important role in the development of hybrid control theory, but also finds many applications in practical control designs for obtaining better control performances. This paper investigates the stability of switched dissipative Hamiltonian systems under arbitrary switching paths. Under a realistic assumption, it is shown that the Hamiltonian functions of all the subsystems can be used as the multiple-Lyapunov functions for the switched dissipative Hamiltonian system. Based on this and using the dissipative Hamiltonian structural properties, this paper then proves that the P-norm of the state of switched dissipative Hamiltonian system converges to zero with the time increasing, and presents two sufficient conditions for the asymptotical stability under arbitrary switching paths. Utilizing these new results, this paper also obtains two useful corollaries for the asymptotical stability of switched nonlinear time-invariant systems. Finally, two examples are studied by using the new results proposed in this paper, and some numerical simulations are carried out to support our new results.     

Synchronization of complex switched networks with two types of delays

In this paper, a general switched model of complex network with two types of delays is presented. The complex switched network (CSN) contains switching behaviors on both its nodes and topology configuration. Different from those CSNs studied in the existing literatures, the two switching signals dominating the nodes and topology are desynchronized. The two types of delays are the time-varying system delay in nodes and the time-varying coupling delay between nodes and they have different values. The inherent synchronization properties of the CSN have been studied not subject to any controllers. By defining a piecewise Lyapunov-Krasovskii function and by using the improved free-weighting matrix method, global exponential synchronizations are obtained. Illustrated example is presented to show the effectiveness of the proposed result.     

The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

A Hamiltonian cycle C=? u ₁, u ₂, …, u _n(G), u ₁ ? with n(G)=number of vertices of G, is a cycle C(u ₁; G), where u ₁ is the beginning and ending vertex and u _i is the ith vertex in C and u _i ≠u _j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C ₁, C ₂, …, C _k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B _n )=n?1 if n≥4, where B _n is the n-dimensional bubble-sort graph.     

Problems on time-varying port-controlled Hamiltonian systems: geometric structure and dissipative realization

To apply time-varying port-controlled Hamiltonian (PCH) systems to practical control designs, two basic problems should be dealt with: one is how to provide such time-varying systems a geometric structure to guarantee the completeness of representations in mathematics; and the other is how to express the practical system under consideration as a time-varying PCH system, which is called the dissipative Hamiltonian realization problem. The paper investigates the two basic problems. A suitable geometric structure for time-varying PCH systems is proposed first. Then the dissipative realization problem of time-varying nonlinear systems is investigated, and serval new methods and sufficient conditions are presented for the realization.     

Generalized honeycomb torus is Hamiltonian

Generalized honeycomb torus is a candidate for interconnection network architectures, which includes honeycomb torus, honeycomb rectangular torus, and honeycomb parallelogramic torus as special cases. Existence of Hamiltonian cycle is a basic requirement for interconnection networks since it helps map a “token ring” parallel algorithm onto the associated network in an efficient way. Cho and Hsu [Inform. Process. Lett. 86 (4) (2003) 185-190] speculated that every generalized honeycomb torus is Hamiltonian. In this paper, we have proved this conjecture.     

广义哈密顿实现及其在基于能量的准Lyapunov函数构造中的应用

首先研究一般非线性系统的Hamilton实现问题，给出了两个实现的充分条件，然后应用所得到的结果研究广义受控耗散Hamilton系统的基于能量的准Lyapunov函数构造问题，给出若干能通过改造Hamilton函数得到准Lyapunov函数的充分条件。     

Decentralized control of a class of large-scale nonlinear systems using neural networks

This paper designs a decentralized neural network (NN) controller for a class of nonlinear large-scale systems, in which strong interconnections are involved. NNs are used to handle unknown functions. The proposed scheme is proved guaranteeing the boundedness of the closed-loop subsystems using only local feedback signals.     

On feedback equivalence to port controlled Hamiltonian systems

In the last few years port controlled Hamiltonian (PCH) systems have emerged as an interesting class of nonlinear models suitable for a large number of physical applications. In this paper we study the question of feedback equivalence of nonlinear systems to PCH systems. More precisely, we give conditions under which a general nonlinear system can be transformed into a PCH system via static state feedback. We consider the two extreme cases where the target PCH system is completely a priori fixed or completely free, as well as the case where it is only partially predetermined. When the energy function is free a set of partial differential equations needs to be solved, on the other hand, if it is fixed we have to deal with a set of algebraic equations. In the former case, we give some verifiable necessary and sufficient conditions for solvability. As a by-product of our analysis we obtain some stabilization results for nonlinear systems.