Synthesis of linear systems with desired equivalent form

A method for feedback synthesis of linear control systems with desired linearly equivalent form of the closed loop system matrix is proposed. The method is based on the serial canonical form of linear multivariable systems which is an alternative to the Luenberger canonical form. A stable computational algorithm for finding the serial canonical form using orthogonal similarity transformations is described. The algorithm for synthesis involves a simple recurrent procedure and gives the possibility to achieve any attainable equivalent form of the closed loop system. The results obtained are extended to the synthesis of reduced order state observers.     

Feedback classification of linear systems over von Neumann regular rings

It is proved that feedback classification of a linear system over a commutative von Neumann regular ring R can be reduced to the classification of a finite family of systems, each of which is properly split into a reachable and a non-reachable part, where the reachable part is in a Brunovski-type canonical form, while the non-reachable part can only be altered by similarity. If a canonical form is known for similarity of matrices over R, then it can be used to construct a canonical form for feedback equivalence. An explicit algorithm is given to obtain the canonical form in a computable context together with an example over a finite ring.     

Relationship between canonical forms of linear differential observation systems and canonical forms of their discrete approximations

We establish a relationship between the canonical form of a linear differential system and the canonical form of its discrete approximation based on the replacement of the derivative by Euler’s finite difference. We prove that if there exist limits of certain sequences of discrete functions constructed with the use of coefficients of the canonical form of the discrete system, then these limits define the canonical form of the differential system.     

Two-dimensional homogeneous cubic systems: Classification and normal forms–III

This article is the third in a series of works devoted to two-dimensional homogeneous cubic systems. It considers the case where the homogeneous polynomial vector on the right-hand side of the system has a quadratic common factor with real zeros. The set of such systems is divided into classes of linear equivalence, in each of which a simplest system being a third-order normal form is distinguished on the basis of appropriately introduced structural and normalization principles. In fact, this normal form is determined by the coefficient matrix of the right-hand side, which is called a canonical form (CF). Each CF is characterized by an arrangement of nonzero elements, their specific normalization, and a canonical set of admissible values of the unnormalized elements, which ensures that the given CF belongs to a certain equivalence class. In addition, for each CF, (a) conditions on the coefficients of the initial system are obtained, (b) nonsingular linear substitutions reducing the right-hand side of a system satisfying these conditions to a given CF are specified, and (c) the values of the unnormalized elements of the CF thus obtained are given.     

The generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system

In this paper, we investigate the generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system. Firstly, we transform an arbitrary generalized Lorenz system to the generalized Lorenz canonical form, and the relation between the parameter of the generalized Lorenz system and the parameter of the generalized Lorenz canonical form are shown. Secondly, we extend the scheme present by [Yan ZY. Chaos 2005;15:023902] to study the generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system, the more general controller is obtained. By choosing different parameter in the generalized controller obtained here, without much extra effort, we can get the controller of synchronization between the Chen system and the Rössler system, the Lü system and the Rössler system, the classic Lorenz system and the Rössler system, the Hyperbolic Lorenz system and the Rössler system, respectively. Finally, numerical simulations are used to perform such synchronization and verify the effectiveness of the controller.     

关于控制系统的绝对稳定性准则

本文:1.给出非线性控制系统第二标准型[2]平凡解绝对稳定性准则,特别地给出了飞机纵向运动方程平凡解绝对稳定性准则,文[8][9]中关于飞机纵向运动方程的结果是本文推论2.2的特例;2.给出非线性控制系统第一标准型[2]平凡解在通常情况和一种临界情况下绝对稳定准则;3.给出了Lure型直接控制系统一般型平凡解绝对稳定若干准则,全文结果均根据系统本身的参数,具体构造函数,给出显式判据.     

用奇异值分解方法计算具有重特征值矩阵的特征矢量

若当(Jordan)形是矩阵在相似条件下的一个标准形,在代数理论及其工程应用中都具有十分重要的意义．针对具有重特征值的矩阵,提出了一种运用奇异值分解方法计算它的特征矢量及若当形的算法．大量数值例子的计算结果表明,该算法在求解具有重特征值的矩阵的特征矢量及若当形上效果良好,优于商用软件MATLAB和MATHEMATICA．     

Contact Systems and Corank One Involutive Subdistributions

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on J ⁿ (R,R ^m ). We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on J ⁿ (R,R ^m ) in a way analogous to that how the Kumpera–Ruiz normal form generalizes the canonical contact system on J ⁿ (R,R), which is also called the Goursat normal form.     

Fundamental systems for linear differential equations smoothly depending on a parameter

Homogeneous linear differential equations in Banach spaces are considered, which depend smoothly on a parameter. A fundamental system of solutions is generated by eigenvalues and eigenvectors of the corresponding operator pencil. The dependence on the parameter of these solutions in canonical form is, in general, not smooth because of branching eigenvalues. This means, the eigenvalues, their number and multiplicities may change in a nonsmooth way with respect to the parameter. We construct a new fundamental system depending smoothly on the parameter, wich can be represented by linear combinations of the solutions in canonical form.     

On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices

The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity.     

单时滞类Lorenz系统的Hopf分岔分析

本文研究单时滞类Lorenz系统的稳定性问题.利用规范型理论,分析该系统在其零平衡点处的稳定性和发生Hopf分岔的条件,并通过数值仿真验证所得结论的正确性.本文的结论可以看做是对文献[5]研究成果的推广.     

控制系统第二标准型的绝对稳定性准则

本文给出控制系统实的第二标准型[1] 绝对稳定的显式准则,包含并改进了文[2]的相应结论,将所得结果应用于著名的飞机纵向运动方程[3,4,2],所得结果包含并改进了文[3,4,2]的结论.     

Two-dimensional homogeneous cubic systems: Classification and normal forms: IV

This article is the fourth in a series of works devoted to two-dimensional cubic homogeneous systems. It considers a case when a homogeneous polynomial vector in the right-hand part of the system has a quadratic common factor with complex zeros. A set of such systems is divided into classes of linear equivalence, wherein the simplest system is distinguished on the basis of properly introduced structural and normalization principles, being, thus, the third-order normal form. In fact, such a form is defined by a matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of nonzero elements, their specific normalization and canonical set of permissible values for the nonnormalized elements, which relates CF to a selected class of equivalence. In addition, each CF is characterized by: (1) conditions imposed on the coefficients of the initial system, (2) nonsingular linear substitutions that transform the right-hand part of the system under these conditions into a selected CF, and (3) obtained values of CF’s nonnormalized elements. Refs 9.     

The rational canonical form via the splitting field

It is clear that a given rational canonical form can be further resolved to a Jordan canonical form with entries from the splitting field of its minimal polynomial. Conversely, with an a priori knowledge of the existence and uniqueness of the rational canonical form of a matrix with entries from a general field, one can modify its Jordan canonical form in the splitting field of its minimal polynomial to construct its rational canonical form in the original field. No author has tried this converse with the a priori existence–uniqueness condition removed. It is feared that “in many occasions when, after a result has been established for a matrix with entries in a given field, considered as a matrix with entries in a finite extension of that field, we cannot go back from the extension field to get the desired information in the original field” [I.N. Herstein, Topics in Algebra, Ginn and Company, Waltham, MA, 1964 (pp. 262–263)]. The present paper removes this a priori condition and uses a “symmetrization” to “integrate” back the Jordan canonical form of a matrix to its rational canonical form in the original field.     

共轭正规矩阵的酉相合标准形

本文借助于对称矩阵和反对称矩阵的酉相合标准形讨论了共轭正规矩阵在酉相合下的标准形。     

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

The Symplectic Monoid

In this paper, we describe explicitly the symplectic monoid ? and its Renner monoid ? using elementary methods. We refine the Bruhat–Renner decomposition of ? and analyze in detail the length function on ?. We then show that every element of ? has a unique canonical form decomposition, which is an analogue of the canonical form of elements in Chevalley groups. We also compute the order of ? over a finite field, and as a consequence we obtain a new combinatorial identity.     

Evolution of weak discontinuities in shallow water equations

In this paper, we determine the critical time, when a weak discontinuity in the shallow water equations culminates into a bore. Invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal operators, are presented. Some appropriate canonical variables are characterized that transform equations at hand to an equivalent form, which admits non-constant solutions. The propagation of weak discontinuities is studied in the medium characterized by the particular solution of the governing system.     

Reduction of linear systems to a form with relative degree using minimum-phase output transformation

A normal form is one of the canonical forms frequently used in control theory for linear time-invariant systems. Only systems with a relative degree can be reduced to such a form. Although a control system does not necessarily have a relative degree, in a sufficiently general case there exists a stable dynamic output transformation reducing the system to a system with a relative degree. We prove that this dynamic transformation can be chosen in such a way that the inverse transformation is stable as well.     

Bäcklund transformations for the Jacobi system on an ellipsoid

We consider analogues of auto- and hetero-Bäcklund transformations for the Jacobi system on a threeaxis ellipsoid. Using the results in a Weierstrass paper, where the change of times reduces integrating the equations of motion to inverting the Abel mapping, we construct the differential Abel equations and auto-Bäcklund transformations preserving the Poisson bracket with respect to which the equations of motion written in the Weierstrass form are Hamiltonian. Transforming this bracket to the canonical form, we can construct a new integrable system on the ellipsoid with a Hamiltonian of the natural form and with a fourth-degree integral of motion in momenta.