An impulsive dynamical systems framework for reset control systems

Impulsive dynamical systems is a well-established area of dynamical systems theory, and it is used in this work to analyse several basic properties of reset control systems: existence and uniqueness of solutions, and continuous dependence on the initial condition (well-posedness). The work scope is about reset control systems with a linear and time-invariant base system, and a zero-crossing resetting law. A necessary and sufficient condition for existence and uniqueness of solutions, based on the well-posedness of reset instants, is developed. As a result, it is shown that reset control systems (with strictly proper plants) do not have Zeno solutions. It is also shown that full reset and partial reset (with a special structure) always produce well-posed reset instants. Moreover, a definition of continuous dependence on the initial condition is developed, and also a sufficient condition for reset control systems to satisfy that property. Finally, this property is used to analyse sensitivity of reset control systems to sensor noise. This work also includes a number of illustrative examples motivating the key concepts and main results.     

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.     

Regulation of unknown nonlinear dynamical systems via dynamical neural networks

In this paper, we are dealing with the problem of regulating unknown nonlinear dynamical systems. First a dynamical neural network identifier is employed to perform black box identification and then a regular static feedback is developed to regulate the unknown system to zero. Not all the plant states are assumed to be available for measurement.A preliminary version of this paper has been presented at the IEEE Mediterranean Symposium on new directions in control theory and applications, Chania, Crete, Greece, June 1993.     

集值离散动力系统的扩张性*

设(X,d)为紧致度量空间,f∶X→X连续,(k(X),dH)是X所有非空紧致子集构成的紧致度量空间,f∶k(X)→k(X),f(A)={f(a)|a∈A}。运用分析的方法初步给出了集值离散动力系统中的扩张性理论。提出集值映射的全扩张、强全扩张的概念,并研究了f的扩张性与f的扩张性之间的关系。所得结果扩展了集值离散动力系统的研究范围,并提出了该领域未来可研究的方向。     

Consequences of nonclassical measurement for the algorithmic description of continuous dynamical systems

Abstract

Continuous dynamical systems intuitively seem capable of more complex behavior than discrete systems. If analyzed in the framework of the traditional theory of computation, a.continuous dynamical system with countably many quasistable states has at least the computational power of a universal Turing machine. Such an analysis assumes, however, the classical notion of measurement. If measurement is viewed nonclassically, a continuous dynamical system cannot, even in principle, exhibit behavior that cannot be simulated by a universal Turing machine.     

Updating method for the computation of orbits in parallel and sequential dynamical systems

In this article, we provide a matrix method in order to compute orbits of parallel and sequential dynamical systems on Boolean functions. In this sense, we develop algorithms for systems defined over directed (and undirected) graphs when the evolution operator is a general minterm or maxterm and, likewise, when it is constituted by independent local Boolean functions, so providing a new tool for the study of orbits of these dynamical systems.     

Hybrid dynamical systems: robust control synthesis problems

The paper presents a new approach to robust control synthesis problems for hybrid dynamical systems. The hybrid system under consideration is a composite of a continuous plant and a discrete event controller. State and output feedback problems are considered. The main results are given in terms of the existence of suitable solutions to a dynamic programming equation and a Riccati differential equation of the H^∞ filtering type. These results show a connection between the theories of hybrid dynamical systems and robust and nonlinear control.     

随机混沌动力系统组及序列加密算法

简要分析了已有混沌加密算法的特点。为提供更有效的加密方法,提出了随机混沌动力系统组的概念。该系统组在一定条件下能构造出动力行为复杂的混沌子系统序列。基于此设计的序列加密算法,其加密过程受密钥、明文、系统组随机特征等多重因素影响,具有较高安全性。实验表明,该算法加密效果较好、密钥空间较大且易于实现。     

A three level costate prediction method for continuous dynamical systems

In this paper a continuous time version of a previous discrete systems optimisation algorithm is developed. The new algorithm uses prediction of costates within a three level structure to provide an efficient organisation of both the storage and the computation. The algorithm which applies to both linear and nonlinear interconnected dynamical systems has been proved to converge to the optimum. A practical example is given to illustrate the approach. In the example which is of a nonlinear synchronous machine the present approach appears to provide faster convergence and smaller storage than with previous hierarchical and global methods.     

混合动态系统有界性定理及应用

从连续系统的角度出发，提出并证明了混合动态系统的有界性定理，并以一类具体的数字反馈采样控制系统为例，详细阐述了有界性定理的具体应用，最终得到系统在其状态空间内有界的充分条件。     

平面带群对称性图像的动力系统构图

探讨了动力系统迭代生成具有平面带群对称性彩色图像的一种新方法。把具有带群对称性混沌函数作为迭代函数系统,构造关于平面带群的不变函数,生成具有很强艺术效果的带群对称性图像。同时给出了该生成技术的理论基础。该方法为平面设计对称图像提供了一种新方法。     

Sequential monte carlo techniques for the solution of linear systems

Given alinear system Ax=b, wherex is anm-vector,direct numerical methods, such as Gaussian elimination, take timeO(m ³) to findx. Iterative numerical methods, such as the Gauss-Seidel method or SOR, reduce the system to the formx=a+Hx, whence and then apply the iterationsx ₀=a,x _s+1=a+Hx _s , until sufficient accuracy is achieved; this takes timeO(m ²) per iteration. They generate the truncated sums The usualplain Monte Carlo approach uses independent random walks, to give an approximation to the truncated sumx _s , taking timeO(m) per random step. Unfortunately, millions of random steps are typically needed to achieve reasonable accuracy (say, 1% r.m.s. error). Nevertheless, this is what has had to be done, ifm is itself of the order of a million or more.The alternative presented here, is to apply a sequential Monte Carlo method, in which the sampling scheme is iteratively improved. Simply put, ifx=y+z, wherey is a current estimate ofx, then its correction,z, satisfiesz=d+Hz, whered=a+Hy–y. At each stage, one uses plain Monte Carlo to estimatez, and so, the new estimatey. If the sequential computation ofd is itself approximated, numerically or stochastically, then the expected time for this process to reach a given accuracy is againO(m) per random step; but the number of steps is dramatically reduced [improvement factors of about 5,000, 26,000, 550, and 1,500 have been obtained in preliminary tests].     

Fractals in neurodynamics: From hybrid dynamical systems point of view

Recently, hybrid dynamical systems have attracted considerable attention in the automatic control domain. In this article, a theory for recurrent neural networks is presented from a hybrid dynamical systems point of view. The hybrid dynamical system is defined by a continuous dynamical system discretely switched by external temporal inputs. The theory suggests that the dynamics of continuous-time recurrent neural networks, which are stochastically excited by external temporal inputs, are generally characterized by a set of continuous trajectories with a fractal-like structure in hyper-cylindrical phase space. This work was presented, in part, at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

Aggregation of a class of interconnected, linear dynamical systems

In this paper, we introduce the notion of a “meaningful” average of a collection of dynamical systems as distinct from an “ensemble” average. Such a notion is useful for the study of a variety of dynamical systems such as traffic flow, power systems, and econometric systems. We also address the associated issue of the existence and computation of such an average for a class of interconnected, linear, time invariant dynamical systems. Such an “average” dynamical system is not only attractive from a computational perspective, but also represents the average behavior of the interconnected dynamical systems. The problem of analysis and control of heirarchical, large scale control systems can be simplified by approximating the lower level dynamics of such systems with such an average dynamical system.     

Stability analysis of switched systems under dynamical dwell time control approach

This article investigates the stability of a class of switched systems using dynamical dwell time approach. First, the condition for stability of switched systems whose subsystems are stable are presented with dynamical dwell time approach, which is shown to be less conservative in switching law design than dwell time approach. Then the proposed approach is extended to the switched systems with both stable and unstable subsystems. Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.     

Multicanonical sampling of rare trajectories in chaotic dynamical systems

In chaotic dynamical systems, a number of rare trajectories with low level of chaoticity are embedded in chaotic sea, while extraordinary unstable trajectories can exist even in weakly chaotic regions. In this study, a quantitative method for searching these rare trajectories is developed; the method is based on multicanonical Monte Carlo and can estimate the probability of initial conditions that lead to trajectory fragments of a given level of chaoticity. The proposed method is successfully tested with four-dimensional coupled standard maps, where probabilities as small as ¹⁰−14 are estimated.     

一种检测动力系统振动频率的新方法

为了检测动力系统的振动频率,建立了动力系统的拓扑空间u和非平稳正弦函数空间M,采用拓扑反变理论把空间u映射到已知的空间M中,通过拓扑反变算子f：u→M检测未知的空间u振动频率,求出这个反变算子后,通过Poincare映射给出该反变算子稳定的存在条件,通过此方法即能够检测出动力系统振动特征频率的瞬变性,同时该方法具有较强的抗干扰能力,实验测试结果表明该方法是可行的。     

Complexity of reachability problems for finite discrete dynamical systems

Sequential Dynamical Systems (SDSs) are a special type of finite discrete dynamical systems that can be used to model simulation systems. We focus on the computational complexity of testing several phase space properties of SDSs. Our main result is a sharp delineation between classes of SDSs whose behavior is easy to predict and those whose behavior is hard to predict. Specifically, we show the following.

1.

Several state reachability problems for SDSs are PSPACE-complete, even when restricted to SDSs whose underlying graphs are of bounded bandwidth (and hence of bounded pathwidth and treewidth), and the function associated with each node is symmetric. Moreover, this result holds even when the underlying graph is d-regular for some constant d and all the nodes compute the same symmetric Boolean function. An immediate corollary of this result is a PSPACE-hard lower bound on the complexity of reachability problems for regular generalized 1D-Cellular Automata and undirected systolic networks with Boolean totalistic local transition functions.

2.

In contrast, the above reachability problems are solvable in polynomial time for SDSs when the Boolean function associated with each node is symmetric and monotone.

The PSPACE-completeness results follow as corollaries of simulation results which show for several classes of SDSs, how one class of SDSs can be efficiently simulated by another (more restricted) class of SDSs. We also prove several structural properties concerning the phase space of an SDS. SDSs are closely related to Cellular Automata (CA), concurrent transition systems, discrete Hopfield networks and systolic networks. This observation in conjunction with our lower bounds for SDSs, yields new PSPACE-hard lower bounds on the complexity of state reachability problems for these models, extending some of the earlier results in [K. Culik II, J. Karhumäki, On totalistic systolic networks, Inform. Process. Lett. 26 (5) (1988) 231-236; P. Floréen, E. Goles, G. Weisbuch, Transient length in sequential iterations of threshold functions, Discrete Appl. Math. 6 (1983) 95-98; P. Floréen, P. Orponen, Complexity issues in discrete Hopfield networks, Research Report No. A-1994-4, Department of Computer Science, University of Helsinki, 1994. Also appears in: I. Parberry (Ed.), Comp. and Learning Complexity of Neural Networks: Advanced Topics, 1999; D. Harel, O. Kupferman, M.Y. Vardi, On the complexity of verifying concurrent transition systems, Inform. and Comput. 173 (2002) 143-161; S.K. Shukla, H.B. Hunt III, D.J. Rosenkrantz, R.E. Stearns, On the complexity of relational problems for finite state processes, in: International Colloquium on Automata Programming and Languages, ICALP, 1996, pp. 466-477; A. Rabinovich, Complexity of equivalence problems for concurrent systems of finite agents, Inform. and Comput. 127 (2) (1997) 164-185].