Clustering based multiple model control of hybrid dynamical systems using HJB solution

This paper deals with Hamilton–Jacobi–Bellman (HJB) equation based stabilized optimal control of hybrid dynamical systems (HDS). This paper presents the fuzzy clustering based event wise multiple linearized modeling approaches for HDS to describe the continuous dynamic in each event. In the present work a fuzzy clustering validation approach is presented for the selection of number of linearized models which span entire HDS. The method also describes how to obtain event wise operating point using fuzzy membership function, which is used to find the event wise model bank by linearizing the first principles model. The event wise linearized models are used for the formulation of the optimal control law. The HJB equation is formulated using a suitable quadratic term in the objective function. By use of the direct method of Lyapunov stability, the control law is shown to be optimal with respect to objective functional and stabilized the event wise linearized models. The global Lyapunov function is proposed with discrete variables which stabilized the HDS. The proposed modeling and control algorithm have been applied on two HDSs. Necessary theoretical and simulation experiments are presented to demonstrate the performance and validation of the proposed algorithm.     

基于不变集的一类混合系统的稳定性

利用混合Ｐｅｔｒｉ网为混合系统建模,探讨了基于不变集混合系统在Ｌｙａｐｕｎｏｖ意义下关于恢复半径的稳定性概念,并给出利用Ｌｙａｐｕｎｏｖ函数法判定混合系统稳定性、一致稳定和渐近稳定的充分条件。     

Realizability and general dynamical systems

The realizability problem in the dynamical systems theory is concerned with the existence of a family of state transition functions for a given response function (e.g., weighting pattern for the case of linear systems) meaning that the latter can be realized by a dynamical system. The results giving realizability conditions are well known for the continuous differential equations systems [see, e.g., 1, 2]. The objective of this paper is to present a general realization theory which will cover not only nonlinear systems but also those systems that are not necessarily continuous or even defined on the topological spaces. Actually, the results reported represent yet another step in the development of a mathematical theory of general systems following a program outlined earlier [3–5 ]. Consistent with that program the concept of a general dynamical system is introduced using minimal mathematical structure and the realizability theory is developed as such on a general level.     

FUZZY MATHEMATICS IN DYNAMICAL SYSTEMS: IS THERE A PLACE FOR IT?

A critical evaluation is presented of current attempts to introduce a fuzzification into dynamical systems theory, and, in general, to physics. The applicability and interpretation of these fuzzification methods is analysed. A new heuristic approach is suggested and discussed, which is based on the idea of the finite resolution limit.     

Modeling and stability analysis of hybrid dynamical systems based on extended differential Petri nets

In this paper, extended differential Petri nets (EDPNs), a new extension of differential Petri nets (DPNs), is proposed. Compared with the existing extension of DPNs, EDPNs has two improvements: (1) the restriction on the enabling condition that depends on the weight of arc is relaxed, i.e., allow the enabling condition has more general form; (2) the definition is extended for the weight of arc. By these two improvements, EDPNs presents larger flexibility and modeling power than the existing extension of DPNs. Using this model, the stability of hybrid dynamical systems (HDS) is studied. A new Lyapunov’s stability theorem of HDS is given. Furthermore, by using the information of index matrix and a new composite energy function, the stability theorem of linear HDS is obtained.     

Parameter identification of PWARX models using fuzzy distance weighted least squares method

PieceWise AutoRegressive eXogenous (PWARX) models represent one of the broad classes of the hybrid dynamical systems (HDS). Among many classes of HDS, PWARX model used as an attractive modeling structure due to its equivalence to other classes. This paper presents a novel fuzzy distance weight matrix based parameter identification method for PWARX model. In the first phase of the proposed method estimation for the number of affine submodels present in the HDS is proposed using fuzzy clustering validation based algorithm. For the given set of input–output data points generated by predefined PWARX model fuzzy c-means (FCM) clustering procedure is used to classify the data set according to its affine submodels. The fuzzy distance weight matrix based weighted least squares (WLS) algorithm is proposed to identify the parameters for each PWARX submodel, which minimizes the effect of noise and classification error. In the final phase, fuzzy validity function based model selection method is applied to validate the identified PWARX model. The effectiveness of the proposed method is demonstrated using three benchmark examples. Simulation experiments show validation of the proposed method.     

REMARKS ON THE CONCEPT OF DYNAMICAL SYSTEMS

In this paper, the difference between the concepts of the states of a time system introduced in [1] and of the global state sets of general systems introduced in [3] is discussed. A generalization of the concepts of the global state sets and the global systems-response functions of general systems in given. The constructions of the global state sets and the corresponding global systems-response functions of a system are studied. A list of different definitions of dynamical systems is given, from which a few problems are asked, and a new definition of dynamical systems is given, which is a concept based on the mathematical models of general systems and time systems posed in [2, 3].     

确定学习与基于数据的建模及控制

确定学习运用自适应控制和动力学系统的概念与方法, 研究未知动态环境下的知识获取、表达、存储和利用等问题. 针对产生周期或回归轨迹的连续 非线性动态系统, 确定学习可以对其未知系统动态进行局部准确建模, 其基本要 素包括: 1)使用径向基函数(Radial basis function, RBF)神经网络; 2)对于周期(或回归)状态轨迹 满足部分持续激励条件; 3)在周期(或回归)轨迹的邻域内实现对非线性系统动态的局部准确神经网络逼近(局部准确建模); 4)所学的知识以时不变且空间分布的方式表达、以常值神经网络权值的方式存储, 并可在动态环境下用于动态模式的快速识别或者闭环神经网络控制. 本文针对离散动态系统, 扩展了确定学习理论, 提出一个根据时态数据序列对离散动态系统进行建模与控制的框架. 首先, 运用确定学习原理和离散系统的自适应辨识方法, 实现对产生时态数据的离散非线性系统的未知动态进行局部准确的神经网络建模, 并利用此建模结果对时态数据序列进行时不变表达. 其次, 提出时态数据序列的基于动力学的相似性定义, 以及对离散动态系统产生的时态数据序列(亦可称为动态模式)进行快速识别方法. 最后, 针对离散非线性控制系统, 实现了基于时态数据序列对控制系统动态的闭环辨识(局部准确建模). 所学关于闭环动态的知识可用于基于模式的智能控制. 本文表明确定学习可以为时态数据挖掘的研究提供新的途径, 并为基于数据的建模与控制等问题提供新的研究思路.     

MODEL MANAGEMENT FOR HIERARCHICALLY STRUCTURED MODELS AND ITS APPLICATION

The implementation of a model management system for hierarchically structured models, named DIALOGO, is discussed. In the author's model management concept the structure of models is defined with the mathematical formulation of the hierarchical structure based on the framework of the theory of hierarchical, multilevel systems in the mathematical general systems theory by Mesarovic and Takahara. Assuming that model components are given as dynamical input/output systems, the mechanism which enables us to realize simulation with hierarchical structured models as a whole is discussed. The method to execute the simulation by converting a hierarchically structured model into a flat network of model components is presented. The implemented platform is applied to a typical problem in global issues, the Nile River basin problematique, in order to demonstrate effectiveness of the concept.     

Dynamical pattern recognition for univariate time series and its application to an axial compressor

In this paper, a learning and recognition approach is proposed for univariate time series composed of output measurements of general nonlinear dynamical systems. Firstly, a class of dynamical systems in the canonical form is derived to describe the univariate time series by introducing coordinate transformation. An observer-based deterministic learning technique is then adopted to achieve dynamical modeling of the associated transformed systems of the training univariate time series, and the modeling results in the form of radial basis function network (RBFN) models are stored in a pattern library. Subsequently, multiple observer-based dynamical estimators containing the RBFN models in the pattern library are constructed for a test univariate time series, and a recognition decision scheme is proposed by the derived recognition indicator. On this basis, more concise recognition conditions are provided, which is beneficial for verifying the recognition results. Finally, simulation studies on the Rossler system and aero-engine stall warning verify the effectiveness of the proposed approach.     

ICA based identification of dynamical systems generating synthetic and real world time series

Independent Component Analysis (ICA) is a recent and well known technique used to separate mixtures of signals. While in general the researchers put their attention on the type of signals and of mixing, we focus our attention on a quite general class of models which act as sources of the time series, the dynamical systems. In this paper we focus our attention on the general problem to understand the behaviour of ICA methods with respect to the time series deriving from a specific dynamical system, selecting large classes of them, and using ICA to make separation. This study gives some interesting results that are very useful both to highlight some properties related to dynamical systems and to clarify some general aspects of ICA, by using both synthetic and real data. From one hand we study the features of the linear (simple and coupled) and non-linear (single and coupled) dynamical systems, stochastic resonances, chaotic and real dynamical systems. We have to stress that we obtain information about the separation of these systems and substantially how from the entropy of the complete system we can obtain the entropies of the single dynamical systems (so that we also could obtain a more realistic analogic circuit). On the other hand these results show the high capability of the ICA method to recognize the dynamical systems independently from their complexity and in the case of stochastic series ICA perfectly recognizes the different dynamical systems also where the Fourier Transform is irresolute. We also note that in the case of real dynamical systems we showed that ICA permits to recognize the information connected to the sources and to associate to it a phenomenological dynamical system that reproduce it (i.e. Organ Pipe, Stromboli Volcano, Aerosol Index).     

An norm-driven state machine model for CSCW systems

In order to improve the feasibility of the CSCW systems, we utilize the methods in organizational Semiotics and put forward the organizational model and the organizational state machine (OSM) to describe the norms in the systems. According to the general form of norm, we classify the norm into three kinds of rules expressed by the logical predications. Combining with cooperative theory and speech-act theory, we first propose an organizational relationship graph of role, behavior and rule of CSCW systems and then a logical predication-based model. This model is referred to as the organizational model that is used to describe the relationship graph. In addition, we present the OSM, which can check the logical conflict among the rules and insure semantic completeness. And the dynamical changes of rules and roles can be delineated exactly by the OSM.     

Basic linear systems and their categorical realization theory

In this paper the following problem is considered: Given an unstructured dynamical input-output model, i.e., a time system. If the systems properties like causality, stationarity, linearity, and the finiteness condition are imposed on the system, what will its resulting characteristics? This problem is investigated in order to define a meaningful subclass of time systems and to check whether or not the set of systems properties available from the present systems theory is complete enough to describe systems behavior fruitfully. The class of basic linear systems is introduced as the meaningful subclass of time systems, and their properties and representation theory are discussed in terms of category theory. This research implies that affirmative answers can be obtained for the completeness question.     

Modeling of nonlinear dynamical systems based on deterministic learning and structural stability基于确定学习和结构稳定理论的非线性动态系统建模

Recently, a deterministic learning (DL) theory was proposed for accurate identification of system dynamics for nonlinear dynamical systems. In this paper, we further investigate the problem of modeling or identification of the partial derivative of dynamics for dynamical systems. Firstly, based on the locally accurate identification of the unknown system dynamics via deterministic learning, the modeling of its partial derivative of dynamics along the periodic or periodic-like trajectory is obtained by using the mathematical concept of directional derivative. Then, with accurately identified system dynamics and the partial derivative of dynamics, a C¹-norm modeling approach is proposed from the perspective of structural stability, which can be used for quantitatively measuring the topological similarities between different dynamical systems. This provides more incentives for further applications in the classification of dynamical systems and patterns, as well as the prediction of bifurcation and chaos. Simulation studies are included to demonstrate the effectiveness of this modeling approach.     

The dynamics of adaptive behavior: A research program

This paper provides a comprehensive overview of the motivations, methodology and current status of an ongoing research program whose long-term goal is to elucidate the essential principles of a theory of adaptive behavior. The thoroughly dynamical nature of both adaptive behavior itself and the causal mechanisms that support it is emphasized throughout. An initial mapping of the basic concepts of adaptive behavior into the language of dynamical systems theory is proposed, and some of the general consequences of this preliminary theoretical framework are discussed. The two key ideas of this framework are (1) that an agent and its environment should be understood as two coupled dynamical systems whose mutual interaction is jointly responsible for the agent's behavior, and (2) that an agent's need to maintain its existence in its environment defines a viability constraint on its behavioral dynamics. A constructive research methodology involving the use of evolutionary algorithms to evolve continuous-time recurrent neural networks for controlling the behavior of model agents is described, and several examples of this methodology are presented, including models of chemotaxis, walking, sequential decision-making and learning. Finally, a detailed dynamical analysis of one evolved walking circuit is presented. This analysis illustrates the kinds of insights that can be obtained by treating agents as dynamical systems and applying the tools of dynamical systems theory to their behavior.     

State-Space Reconstruction and Spatio-Temporal Prediction of Lattice Dynamical Systems

This paper addresses the problems of state space reconstruction and spatio-temporal prediction for lattice dynamical systems. It is shown that the state space of any finite lattice dynamical system can be embedded into a reconstruction space for almost every, in the sense of prevalence, smooth measurement mapping as long as the dimension of the reconstruction space is larger than twice the size of the lattice. Based on this result, an input-output spatio-temporal dynamical relation for each site within the lattice is derived and used for spatio-temporal prediction of the system. In the case of infinite lattice dynamical systems, an approach based on constructing local lattice dynamical systems is proposed. It is shown that the finite dimensional results can be directly applied to the local modelling and spatio-temporal prediction for infinite lattice dynamical systems. Two numerical examples are provided to demonstrate the proposed theory and approach     

时滞纳米运动系统的多速率采样预测自抗扰控制

电容型纳米位置传感器在纳米伺服系统中得到了越来越多的应用.这类超高精度模拟传感器用于反馈信号时因较长的模数转换时间将带来明显的测量时滞.而这类数字纳米伺服系统也因硬件使其采样频率在处理高频干扰时受到带宽限制.本文针对测量时滞和高频干扰的挑战,提出一种带采样预测功能的多速率自抗扰控制器设计方法.首先建立了电容式位移传感器的纳米运动平台带有时滞的动力学模型.其次,基于该模型设计多率采样预测线性扩张状态观测器和多率反馈控制器.通过设计预测型观测器,适当选取观测器增益,消除时滞对状态观测的影响.另外,将输出预估器加入预测扩张状态观测器中重构采样点间系统输出值,从而在时滞系统中更好地估计和消除高频干扰,并给出了系统的稳定性分析.最后通过压电驱动纳米运动平台的实时控制实验验证本文提出控制器的有效性.     

Discrete dynamical systems on graphs and Boolean functions

Discrete dynamical systems based on dependency graphs have played an important role in the mathematical theory of computer simulations. In this paper, we are concerned with parallel dynamical systems (PDS) and sequential dynamical systems (SDS) with the OR and NOR functions as local functions. It has been recognized by Barrett, Mortveit and Reidys that SDS with the NOR function are closely related to combinatorial properties of the dependency graphs. We present an evaluation scheme for systems with the OR and NOR functions which can be used to clarify some basic properties of the dynamical systems. We show that for forests that does not contain a single edge the number of orientations equals the number of different OR-SDS.     

Cooperative Control of Dynamical Systems With Application to Autonomous Vehicles

In this paper, a new framework based on matrix theory is proposed to analyze and design cooperative controls for a group of individual dynamical systems whose outputs are sensed by or communicated to others in an intermittent, dynamically changing, and local manner. In the framework, sensing/communication is described mathematically by a time-varying matrix whose dimension is equal to the number of dynamical systems in the group and whose elements assume piecewise-constant and binary values. Dynamical systems are generally heterogeneous and can be transformed into a canonical form of different, arbitrary, but finite relative degrees. Utilizing a set of new results on augmentation of irreducible matrices and on lower triangulation of reducible matrices, the framework allows a designer to study how a general local-and-output-feedback cooperative control can determine group behaviors of the dynamical systems and to see how changes of sensing/communication would impact the group behaviors over time. A necessary and sufficient condition on convergence of a multiplicative sequence of reducible row-stochastic (diagonally positive) matrices is explicitly derived, and through simple choices of a gain matrix in the cooperative control law, the overall closed-loop system is shown to exhibit cooperative behaviors (such as single group behavior, multiple group behaviors, adaptive cooperative behavior for the group, and cooperative formation including individual behaviors). Examples, including formation control of nonholonomic systems in the chained form, are used to illustrate the proposed framework.