Synthesis of basic elements of multidimensional multirate systems Part. 1. Nonseparable decimation matrices

A synthesis method is developed for multidimensional decimation matrices of given dimension for a specified number of channels which are used in multidimensional multirate systems. The method is based on application of eigenvalues and Gröbner bases techniques. A full parametrization of decimation matrices is effected for two-and three-dimensional cases. Peculiarities of a four-dimensional case are considered.     

Multidimensional dynamical systems that admit normal shift

Dynamical systems in R _n of the formR ⁿ of the form that admit normal shift of submanifolds of codimension 1 are considered. The concept of weak normality for such systems is defined, and defining partial differential equations for the force function of dynamical systems with the properties of weak and complete normality are derived.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 264–269, August, 1994.     

一般目标信号和干扰信号下多采样率系统的最优预见控制器设计

针对目标信号和干扰信号为多项式的情形,研究了多采样率离散时间控制系统的最优预见控制问题.首先利用离散时间系统提升技术,把所研究的系统转化成单采样率的扩大系统.然后构造扩大误差系统,把问题转化为包含预见信号的最优调节问题.最后利用最优预见控制理论的结果得到系统的最优预见控制输入,其中包含积分器和预见前馈补偿.本文还对扩大误差系统的能控性和能观测性和相应的代数Riccati方程的可解性进行了讨论.     

Stability analysis of multirate sampled-data control systems

The paper considers the stability of finite-dimensional linearmultirate sampled-data control systems. A notion of internalstability is defined, based on continous-time singals. A completeproof is given for the equivalence between continuous-time internalstability and discrete-time internal stability. An input-outputproperty called L^v(1v ) hybrid stability is investigated formulatirate systems having exogenous continuous-time inputs andoutput, and hybrid stability is shown to be equivalent to internalstability, which is fundamental to a true development of multiratesampled-data L^v control design. The relationship between internalstability and input-output stability is also discussed in continuoustime.     

Design of adaptive LQ regulators for MIMO systems based on multirate sampling of the plant output

In this paper, the certainty equivalence principle is used to combine a discrete-time adaptive law with a control structure derived from the linear-quadratic regulation problem. The proposed control structure is mainly based on multirate sampling of the output of the continuous-time plant under control and allows us to regulate the sampled closed-loop system, subject to a quadratic performance criterion, without making assumptions on the plant other than controllability and observability and the knowledge of two sets of structural indices. Using the proposed algorithm, the adaptive regulation problem is reduced to the determination of a fictitious static state feedback controller. Known techniques usually resort to the computation of fullorder adaptive state observers, thus introducing high-order exogenous dynamics in the control loop. Moreover, persistence of excitation and parameter convergence of the plant are provided without making any additional assumption on the system, as compared to known adaptive linear-quadratic regulation schemes.     

Empirical processes of multidimensional systems with multiple mixing properties

We establish a multivariate empirical process central limit theorem for stationary R^d-valued stochastic processes (X_i)_i≥1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling et al. (2009) [9] in the univariate case. As an important technical ingredient, we prove a 2pth moment bound for partial sums in multiple mixing systems.     

A formulation of the fractional Noether-type theorem for multidimensional Lagrangians

This paper presents the Euler–Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses well-known notion of the Riemann–Liouville fractional derivative.     

Finite-difference schemes for a diffusion equation with fractional derivatives in a multidimensional domain

In a multidimensional domain, we consider a partial differential equation with fractional space and time derivatives. For the first initial-boundary value problem, we consider a purely implicit scheme based on the approximate factorization method. We prove the stability of the implicit scheme for the considered class of problems.     

Stabilization of fractional bilinear systems with multiple inputs

In this paper, we study in a constructive way the stabilization problem of fractional bilinear systems with multiple inputs. Using the quadratic Lyapunov functions and some additional hypotheses on the unit sphere, we construct stabilizing feedback laws for the considered fractional bilinear system. A numerical example is given to illustrate the efficiency of the obtained result.     

Design for estimation of the drift parameter in fractional diffusion systems

We consider a controlled linear differential equation which is partially observed with an additive fractional noise. In this setting, we study the asymptotic (for large observation time) design problem of the input and give an efficient estimator of the unknown signal drift parameter. The optimal estimation input is deduced. The consistency, asymptotic normality and convergence of the moments of the MLE are established.     

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).     

Solution of a multidimensional Abel integral equation of the second kind with partial fractional integrals

We construct an explicit representation of the solution of a multidimensional Abel integral equation of the second kind with partial fractional integrals in terms of the Wright function.     

高维非线性守恒律方程组

本文根据高维非线性守恒律方程组的研究历程将这一领域的研究大体分为四个阶段: 局部经典解、具扇状波结构弱解、具花状波结构弱解、整体解与混合型方程. 本文据此线索回顾与介绍多年来在该领域所获得的主要成果与进展, 并提出今后所面临的一些未解决的重要问题及困难.     

Nonlinear fractional cone systems with the Caputo derivative

Nonlinear fractional cone systems involving the Caputo fractional derivative are considered. We establish sufficient conditions for the existence of at least one cone solution to such systems. Sufficient conditions for the unique existence of the cone solution to a nonlinear fractional cone system are given.     

Multidimensional modified fractional calculus operators involving a general class of polynomials

In the present work, we introduce and study essentially a class of multi-dimensional modified fractional calculus operators involving a general class of polynomials in the kernel. These operators are considered in the space of functionsM _γ (R ₊ ⁿ ). Some mapping properties and fractional differential formulas are obtained. Also images of some elementary and special functions are established.     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.     

Positive solutions of three-point fractional sum boundary value problem for Caputo fractional difference equations via an argument with a shift

In this paper, we study a new class of three-point boundary value problem of nonlinear Caputo fractional difference equation. Our problem contain an argument with a shift. The existence of at least one positive solution is proved by using the Guo-Krasnoselskii’s fixed point theorem. Some illustrative examples are given.