A discrete state-space model for linear image processing

The linear time-discrete state-space model is generalized from single-dimensional time to two-dimensional space. The generalization includes extending certain basic known concepts from one to two dimensions. These concepts include the general response formula, state-transition matrix, Cayley-Hamilton theorem, observability, and controllability.     

The control of linear time-periodic systems using Floquet–Lyapunov theory

In this paper we use Floquet–Lyapunov theory to derive the Floquet factors of the state-transition matrix of a given linear time-periodic system. We show how the periodicity of one of the factors can be determined a priori using a constant matrix, which we call the Yakubovich matrix, based upon the signs of the eigenvalues of the monodromy matrix. We then describe a method for the numerical computation of the Floquet factors, relying upon a boundary-value problem formulation and the Yakubovich matrix. Further, we show how the invertibility of the controllability Gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed-loop system. The controller can be full-state or observer-based. It also allows the engineer to assign all the invariants of the system; i.e. the full monodromy matrix. Deriving the feedback matrix requires solving a matrix integral equation for the periodic Floquet factor of the new state-transition matrix of the closed-loop system. This is achieved via a spectral method, with further refinement possible through a boundary-value problem formulation. The computational efficiency of the scheme may be further improved by performing the controller synthesis on the transformed system obtained from the Lyapunov reducibility theorem. The effectiveness of the method is illustrated with an application to a quick-return mechanism using a software toolbox developed for MATLAB?.     

The determination of state-space representations for linear multivariable systems

A new, relatively simple, constructive method is presented for obtaining state-space or normal form representations for linear, time invariant systems whose dynamics are expressed in a more general matrix differential operator form. The development employed provides new insight into various structural properties of linear systems. Equivalence is defined for a rather large class of linear systems, and an algorithm is given for reducing any member of this class to normal form. In order to outline the algorithm, a number of “well known” results involving polynomial matrices are employed for the first time, along with the “structure theorem”. An example is used to illustrate the algorithm and two appendices are employed to clarify the development.     

General response formula for two-dimensional linear systems with variable coefficients

A new definition of the transition matrix for a general 2-D system model with variable coefficients is presented. The general response formula for the general 2-D system model with variable coefficients is derived. Solution to the first and the second Fornasini-Marchesini models with variable coefficients are also given.     

Stabilization of networked control systems with nonuniform random sampling periods

In this paper, a new linear delayed delta operator switched system model is proposed to describe networked control systems with packets dropout and network‐induced delays. The plant is a continuous‐time system, which is sampled by time‐varying random sampling periods. A general delta domain Lyapunov stability criterion is given for delta operator switched systems with time delays. Sufficient conditions for asymptotic stability of closed‐loop networked control systems with both packets dropout and network‐induced delays are presented in terms of linear matrix inequalities (LMIs). A verification theorem is given to show the solvability of the stabilization conditions by solving a class of finite LMIs. Both the case of data packets arrive instantly and the case of invariant sampling periods in delta operator systems are given, respectively. Three numerical examples are given to illustrate the effectiveness and potential of the developed techniques. Copyright © 2010 John Wiley & Sons, Ltd.     

State-space forms for higher-order discrete-time models

This paper presents a fundamental study of the connection between continuous- and discrete-time systems. Provided is a definition for discrete-time models, that is discrete-time systems with a continuous-time counterpart, whose order can be higher than that of the continuous-time system. This definition is based on a comparison in a certain sense on the time responses of continuous- and discrete-time systems. A theorem is presented for relating the higher-order discrete-time models to their continuous-time counterparts, which is an extension of a previous theorem for models with order equal to that of the continuous-time system. State-space forms are derived for models obtained through the use of a certain class of hold elements and through the use of mapping models, and these discrete-time systems are shown to be valid according to the definition. Special cases are models obtained using first-order and slewer hold devices, whose convergence to a continuous-time counterpart has not been shown mathematically before, and mapping models corresponding to two-step linear multi-step methods, which have not previously presented in the state-space form. The derived state-space forms provide a convenient way to implement these models for purposes of analysis, design, and implementation of discrete-time systems and finds applications in such areas as digital signal processing, digital simulation, and digital control.     

A Gramian-based controller for linear periodic systems

This note proposes a new design method for the control of linear time-periodic systems. The method is based on the reachability Gramian and a specific form for the feedback gain matrix to build a novel control law for the closed-loop system. The new controller allows assignment of all the invariants of the system. Calculating the feedback requires solving a matrix integral equation for the periodic Floquet factor of the state-transition matrix of the closed-loop system.     

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.     

模N具有给定周期的变换方阵的构造

置乱技术在数字图像信息隐藏和图像加密中都具有重要的作用;而变换方阵因其理论简洁、实现简单而得到了广泛的关注和研究,但已有工作主要集中在对给定方阵周期的研究方面.分析了变换方阵模素数幂周期的上确界,得到了变换方阵模素数幂的周期达到上确界的充要条件.在此基础上给出了模素数幂具有最大周期的变换方阵的构造方法及两个改进方案.进而分析了变换方阵模一般整数N周期的上界,并使用中国剩余定理给出了两种算法,可构造模一般整数N具有给定周期的变换方阵.     

Stability analysis for digital filters with multiple saturation nonlinearities

A new stability criterion for digital filters with multiple saturation nonlinearities is presented in the form of a linear matrix inequality. Numerical examples are given to show the effectiveness of the results obtained.     

线性定常系统仿真的广义增广矩阵法

对于线性定常系统的数字仿真,增广矩阵法是一种高精度快速算法。本文给出了增广矩阵法的改进算法,由此建立了基于增广矩阵法的一般仿真模型。文中针对输入函数的不同情况给出了几种仿真计算模型。容易推证,原有的增广矩阵法乃是以本文算法给出的仿真计算模型之一的一种特例。     

Delay-independent stability criteria for linear continuous systems with time-varying delays

New stability criteria for linear continuous systems with multiple time-varying delays are established by the Lyapunov function approach based on a new stability theorem for general type of retarded dynamical systems. A new technique for estimating the derivative of Lyapunov function along the solution of system at some specific instants is used so that the case of multiple time-varying delays can be dealt with less conservatively. In addition, (i) every criterion has only one tuning parameter matrix P>0 as the Lyapunov-type stability criterion for the corresponding linear continuous system without delays does; (ii) all the criteria are given in LMI forms and hence they are numerically tractable. It is remarked that the established criteria are less conservative than the existing ones in the literature. Three examples are given to illustrate the proposed method and to show the superiority of the obtained results to the existing ones in the literature.     

Comments on “Transient stability of an a.c. generator by Lyapunov's direct method”

A matrix method is presented for the determination of the response sequences of discrete time-invariant linear systems for a large range of deterministic input sequences.

Using the z-transform formulation the transfer function of the discrete system is defined and, assuming that the input sequence is a linear combination of a certain set of basic sequences, it is demonstrated that an explicit formulation of the output sequence may be obtained by purely matrix operations. This approach not only simplifies the application of the z-transform technique but can also be used as the basis of a digital computer programme.

Examples are given to illustrate the use of the method including its application to the solution of general nth-order difference equations     

On flat systems behaviors and observable image representations

In this short paper we propose a definition of flatness for systems not necessarily given in input/state/output representation. A flat system is a system for which there exists a mapping such that the manifest system behavior is equal to the image of this mapping, and such that the latent variable appearing in this image representation can be written as a function of the manifest variable and its derivatives up to some order. For linear differential systems, flatness is equivalent to controllability. We will generalize the main theorem of Levine and Nguyen (Systems Control Lett. 48 (2003) 69) to general linear differential systems.     

不确定规划中非循环可达关系的求解方法

对一个不确定状态转移系统求多个规划问题,那么获得不确定状态转移系统的状态可达关系可以方便求解规划问题,减少冗余计算,建立系统的引导信息。提出一个关于矩阵求不确定领域的状态可达性关系的方法,主要思想是以矩阵乘法来模拟状态转移系统中状态转移,对不确定动作带来的扩散和确定关系带来的聚合进行了统计和处理,从而获得状态可达信息。证明了方法的正确性和有效性。在不确定规划中确定了状态之间的可达性关系,可以在求规划解时删除对规划没有用的状态节点和状态动作序偶;选择能到达目标节点的状态节点和状态动作序偶;进行启发式正向搜索;减少大量冗余计算;提高求解效率。     

Influence of zero locations on the number of step-response extrema

A new bounding theorem for the number of extrema that may occur in the step-response of a stable linear system is presented. The derivation of an easily-computed upper bound is given to complement literature results which have previously established the existence of a lower bound. The theorem requires knowledge of the pole-zero configuration of the transfer-function and is applicable to stable systems with real zeros and real poles.     

Finite-state machine embeddings for nonconcurrent error detection and identification

In digital sequential systems that operate over several time steps, a state-transition fault at any time step during the operation of the system can corrupt its state and render its future functionality useless. Such state-transition faults are usually handled by embedding the given sequential system into a larger one, in a way that preserves the state evolution and properties of the original system while enabling an external mechanism to perform checks to detect, identify and correct errors in the encoded state of this redundant system. Checking is typically performed concurrently (i.e., at the end of each time step) and can potentially cause high power consumption or an overall slowdown in the system; more importantly, concurrent checking imposes significant reliability requirements on the error-detection/identification mechanism. We develop a methodology for systematically constructing embeddings of finite-state machines so that the external mechanism can capture transient state-transition faults via checks that are performed in a nonconcurrent manner (e.g., periodically instead of every time step). More specifically, by characterizing nonconcurrent error-detecting/identifying capabilities in terms of state encoding constraints and redundant dynamics, the proposed approach can be used to construct a redundant version of the given finite-state machine (FSM) that allows the external mechanism to detect and identify errors due to past state-transition faults based on an analysis of the current, possibly corrupted FSM state. As a result, the checker in such designs can operate at a slower speed than the rest of the system which relaxes the stringent requirements on its reliability.     

Synthesis of positive real feedback systems: a simple derivationvia Parrott's theorem

A simple derivation of the formulas for controller rendering a closed-loop system strongly positive real is presented. The derivation is based on Parrott's theorem and the positive real lemma. When the positive real lemma is applied to the closed loop system, a bilinear matrix inequality results. The significance of Parrott's theorem in the context of a strongly positive real synthesis problem is that it allows one to convert this bilinear matrix inequality into two linear matrix inequalities     

Dissipativity analysis of neural networks with time-varying delays

A new definition of dissipativity for neural networks is presented in this paper. By constructing proper Lyapunov functionals and using some analytic techniques, sufficient conditions are given to ensure the dissipativity of neural networks with or without time-varying parametric uncertainties and the integro-differential neural networks in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the obtained results.     

灰色二层多目标线性规划问题及其解法

针对二层多目标线性规划问题,结合灰色系统的特性,提出了一般灰色二层多目标线性规划问题,并给出了模型的相关定义和定理.针对漂移型灰色二层多目标线性规划问题,提出一种具有全局收敛性质的求解算法.首先通过线性加权模理想点法把多目标转化为单目标;然后当可行域为非空紧集时,利用库恩塔克条件把双层转化为单层,再利用粒子群算法搜索单目标单层线性规划即可得到原问题的解;最后通过算例表明了该算法的有效性.