A new perspective on criteria and algorithms for reachability of discrete-time switched linear systems

The paper presents a unified perspective on geometric and algebraic criteria for reachability and controllability of controlled switched linear discrete-time systems. Direct connections between geometric and algebraic criteria are established as well as that between the subspace based controllability/reachability algorithm and Kalman-type algebraic rank criteria. Also the existing geometric criteria is simplified and new algebraic conditions on controllability and reachability are given.     

多智能体网络系统的能控性代数条件

能控性是多智能体系统研究的核心问题,主要包括结构可控性和精准可控性。对多智能体系统的模型和能控性代数条件进行了总结。在相对协议和绝对协议条件下,运用图论和矩阵论的知识系统分析了多智能体系统能控性的代数条件。按照同质多智能体到异质多智能体的顺序,对现有的多智能体系统模型和代数条件进行了梳理,并在已有结论的基础上对多智能体系统能控性的代数条件进行了改善,进一步提出了新的代数条件。多智能体能控性代数条件的改进大大简化了能控性的计算量。     

Continuous-time singular linear–quadratic control: Necessary and sufficient conditions for the existence of regular solutions

The purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic Riccati equation plays in singular linear–quadratic (LQ) optimal control. Indeed, in spite of the vast literature on LQ problems, only recently a sufficient condition for the existence of a non-impulsive optimal control has for the first time connected this equation with the singular LQ optimal control problem. In this paper, we establish four equivalent conditions providing a complete picture that connects the singular LQ problem with the constrained generalized continuous algebraic Riccati equation and with the geometric properties of the underlying system.     

Algebraic Conditions for Classifying the Positional Relationships Between Two Conics and Their Applications

In many fields of computer science such as computer animation, computer graphics, computer aided geometric design and robotics, it is a common problem to detect the positional relationships of several entities. Based on generalized characteristic polynomials and projective transformations, algebraic conditions are derived for detecting the various positional relationships betweeu two planar conics, namely, outer separation, exterior contact, intersection, interior contact and inclusion. Then the results are applied to detecting the positional relationships between a cylinder (or a cone) and a quadric. The criteria is very effective and easier to use than other known methods.     

On the discrete-time normal form

Necessary and sufficient conditions under which the discrete-time normal form exists are set both in geometric and in algebraic frameworks, leading to equivalent conditions. The results are therefore extended to generic nonlinear nonaffine continuous-time systems     

A category of geometric spaces: Some computational aspects

A category of geometric spaces (non-commutative geometries) is presented here in an overview style with the objective to propose it as a field for interdisciplinary work in geometric reasoning and symbolic computation under the aspect of interactions of algebraic and geometric approaches. The problem of automated verification of geometric conditions is discussed. Possible directions of further work are included as a prospect.Sponsored by the Austrian Science Foundation (FWF), ESPRIT BRP 6471 MEDLAR II.Dedicated to Professor Dana Scott on the occasion of his 60th birthday     

Structural conditions for disturbances decoupling with stabilityusing proportional and derivative control laws

We consider the disturbance decoupling problem with internal stability by means of generalized control laws which use not only state feedback, but also external terms coming from the disturbance and some of its time derivatives. We derive from our previous geometric condition an equivalent one in terms of some structures of the system to be controlled (zeros at infinity and unstable zeros), and no longer dependent on a particular approach (geometric, algebraic, ...)     

广义系统极点配置的一种代数几何方法

利用代数几何方法给出了可任意配置极点的条件,并证明了实数域上广义系统若存在复 反馈配置极点,则一定存在实反馈配置极点.     

广义系统极点配置的一种代数几何方法

利用代数几何方法给出了可任意配置极点的条件，并证明了实数域上广义系统若存在复反馈配置极点，则一定存在实反馈配置极点．     

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state-space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. By a particular transform of these conditions, a direct algorithm for testing positive realness is determined that requires only checking a set of simple algebraic conditions. This provides an alternative procedure to the positive real lemma and to the s-domain inequalities. Based on this algorithm, a synthesis of a positive real system via output feedback is presented     

共形几何代数与几何不变量的代数运算

几何不变量的使用是计算机视觉和图形学的一个重要手段.发现一个不变量后,如何找到它与其他不变量的关系,是实际应用中的一个重要问题,这种关系的探讨主要依靠在不变量层次上的代数运算.文中介绍了共形几何代数中的基本、高级和有理不变量如何在几何问题中自然出现,它们之间如何进行代数运算,以及如何通过不变量的化简,自然地得到几何条件的充分必要化和几何定理的完全化.几何定理的机器证明作为几何定理完全化的副产品,被发展成几何定理的关系定量化,这种量化的几何还原就是几何定理的自然推广.几何不变量之间的几何关系的计算是这些技术的一个具体应用.     

Algebraic stability criteria and symbolic derivation of stability conditions for feedback control systems

This article provides algebraic settings of the stability criteria of Nyquist and Popov and the circle criterion for closed-loop linear control systems with linear or nonlinear feedback whose transfer functions are rational ones with integer coefficients. The proposed settings make use of algebraic methods of parametric curve implicitisation, real root isolation, symbolic integration and quantifier elimination and allow one to derive exact stability conditions for feedback control systems with symbolic computation. An example is presented to illustrate the algebraic approach and its effectiveness. Some numerical stability results obtained previously are confirmed.     

A bilinear formulation for the output feedback problem in linearsystems

This note ties together a geometric theory and previous algebraic results for the closed-loop eigenstructure assignment by output feedback in multivariable systems. Necessary and sufficient conditions for the closed-loop stabilization, eigenstructure assignment are presented in terms of a bilinear Sylvester equation, the solutions of which completely characterize the closed-loop eigenstructure problem with output feedback. Some preliminary results for the solution of this bilinear equation are presented     

Synthesis of plane curves with prescribed local geometric properties using periodic splines

A wide class of tecnical problems involves the determination of shape complying with certain local geometric conditions, such as prescribed — or bounded — values of slope and curvature. Traditionally these problems have been solved by resorting to well-known curves described by algebraic formulae. The class of such curves available is limited, however, and in general curves comply with the desired conditions only at certain intervals. By the introduction of periodic splines it is shown how curves can be synthesized that meet local geometric conditions at a finite set of points that may be located arbitrarily throughout the whole interval of interest. The problem is then reduced to one of solving a system of algebraic equations that can be either linear or nonlinear. The method proposed here is illustrated with problems arising from several engineering disciplines.     

Necessary and sufficient conditions for controllability and observability of switched impulsive control systems

Many practical systems in physics, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This note first introduces the impulsive behavior into switched linear systems and studies the controllability and observability of such systems. Necessary and sufficient criteria for reachability and controllability are established. It is proved that reachability is equivalent to controllability for such systems. Then, necessary and sufficient criteria for observability and determinability of such systems are established by duality. It is also proved that observability is equivalent to determinability. Our criteria are of geometric type, they can be transformed into algebraic form conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.     

Theory of contact for geometric continuity of parametric curves

This paper develops a theory of contact for piecewise parametric curves based on the differential geometry of evolutes, polar curves, and binormal indicatrices. This theory is completely geometric, independent of parametrization and generalizes to any order. Two sets of dimensionless, characteristic numbers describing the local geometry of a curve up tonth order are defined. These characteristic numbers can be used to describe conditions for higher order contacts in an algebraic fashion. The same characteristic numbers can also be used to interpret contact conditions of up tonth order in terms of the geometry of higher evolutes and binormal indicatrices. The resulting geometric contact conditions are used to design piecewise parametric curves for Computer Aided Geometric Design (CAGD) applications.     

Observability conditions of Boolean control networks

Observability is a basic, yet challenging, issue when studying Boolean control networks (BCNs). Recently, a criterion for observability of controllable BCNs was proposed by using the algebraic representation of logical dynamics based on the technique of the semi‐tensor product of matrices. In this paper, we present new necessary and sufficient conditions guaranteeing observability of a BCN without preassuming its controllability. The conditions are hence more general. Some examples are worked out to illustrate the obtained results. Copyright © 2013 John Wiley & Sons, Ltd.     

分布时滞系统的反馈镇定

求解特征矩阵是镇定时滞系统的关键问题,本文给出了系统的特征根的代数重复度与几 何重复度均为一般值情况下特征矩阵的求法,即把它归结为求解一组线性代数方程的问题,并 得到了该方程组有解及对应于同一特征值的解向量组线性独立的充分条件.此外,还提出了 一种算法,用以处理系统对应于不同特征值的左特征向量线性相关情况下系统的镇定问题.     

Algebraic conditions for state equivalence to a discrete-time nonlinear observer canonical form

Algebraic necessary and sufficient conditions for a single output discrete-time system to be state equivalent to a (generalized) nonlinear observer canonical form with (or without) output transformation are derived. Our algebraic conditions are easier to verify for researchers unaccustomed to differential geometry. In addition, our proofs are constructive and contain the state and output transformations for (generalized) nonlinear observer canonical forms.     

An additional branch free algebraic B-spline curve fitting method

Algebraic curve fitting based on the algebraic distance is simple, but it has the disadvantage of inclining to a trivial solution. Researchers therefore introduce some constraints into the objective function in order to avoid the trivial solution. However, this often causes additional branches. Fitting based on geometric distance can avoid additional branches, but it does not offer sufficient fitting precision. In this paper we present a novel algebraic B-spline curve fitting method which combines both geometric distance and algebraic distance. The method first generates an initial curve by a distance field fitting that takes geometric distance as the objective function. Then local topology-preserving calibrations based on algebraic distance are performed so that each calibration does not produce any additional branches. In this way, we obtain an additional branch free fitting result whose precision is close to or even better than that produced by purely algebraic distance based methods. The adopted precision criterion is the geometric distance error rather than the algebraic one. In addition, we find a calibration fatigue phenomenon about calibrating strategy and propose a hybrid mode to solve it.