MINIMAL POLYNOMIAL MATRIX AND LINEAR MULTIVARIABLE SYSTEM(Ⅱ)

part(Ⅰ) of this work is on the theory of minimal polynomial matrix and Part(Ⅱ)is onthe applications of this theory to linear multivarible systeme.In I of this part,using the theory to Part(Ⅰ),some results about input part of a linearmultivarble system are discussed in detail and in Ⅱ,using duality properties,the conceptsabout row n.p.m.and row generating system,etc.are given. and some results about outputpart of linear multivariable system are discussed, too.In Ⅲ,we discuss the approach whichcan give the polynomial model with less dimension from the state-space model and in Ⅳ wediscuss tha inverse of the problem to give the state-space model from the polynomial model.Some interesting examples are given to explain the theory and the approach.     

LINEAR ACTIVE STRUCTURES AND MODES (Ⅱ) —DISCRETE SYSTEMS AND BEAMS

IntroductionThestudiedobjectintheclassicstructuredynamicsisthepassivestructureswhicharechangedintoactivestructurewhencontroldevicesorsmartmaterialsareintroducedintoandthecontrolfunctionisapplied .Anactivestructureisgenerallycalledactivesystem .Asforthedefinitionofactivestructure ,itistheconceptofworthydiscussion[1].Thedefinitionshouldbebecomingrigorouswiththestudyontheactivestructuraldynamics.Withincertaincategoryofactivestructuresinpaper (Ⅰ ) [2 ],someattributesaboutthemodesoflinearactivest…     

LINEAR ACTIVE STRUCTURES AND MODES (Ⅰ) —BASIC CONCEPTS AND PROPERTIES

IntroductionThestudyobjectoftheclassicstructuredynamicsispassivestructure .Whencontroldevicesorsmartmaterialareintroducedintoitwiththeapplicationofcontrolfunction ,thepassivestructuresarechangedintotheactivestructures.Theactivestructureisgenerallycalledactivesystem .Withtherapiddevelopmentofadaptivestructures,controlledstructuresandsmartstructures,therewillbemoreandmorekindsoftheactivestructures.Alotofspecialstudyontheactivestructureswillbedevelopedintooneofanewknowledgebranchinthefieldofthes…     

THE RELATIONSHIP BETWEEN THE STABILITY AND THE OPTIMALITY OF LINEAR SYSTEMS(Ⅱ)

Different from the inverse problem put forward by R.E.Kalman, another kind ofinverse problem of linear optimal control is proposed and discussed in[1] as follows:Givenan asymptotically stable linear constant system and a nonnegative quadratic performanceindex, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended.Similar conclusions are obtained for linear discrete systems and linear time-variablesystems. According to these conclusions we can say that the correspondence between theasymptotically stable system and the optimal feedback system is the inherent character ofall kinds of linear systems.     

GEOMETRIC FRAMEWORK AND MINIMAL REALIZATIONS OF NONLINEAR SYSTEMS ON FIBRE BUNDLE

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LINEAR ACTIVE STRUCTURES AND MODES （ Ⅱ ） — DISCRETE SYSTEMS AND BEAMS

The basic concepts about the active structures and some attributes of the modes were presented in paper “Liner Active Structures and Modes ( Ⅰ ) “. The characteristics of the active discrete systems and active beams were discussed, especially, the stability of the active structures and the orthogonality of the eigenvectors. The notes about modes were portrayed by a model of a seven-storeyed building with sensors and actuators. The concept of the adjoint active structure was extended from the discrete systems to the beams that were the representations of the continuous structures. Two types of beams with different placements of the measuring and actuating systems were discussed in detail. One is the beam with the discrete sensors and actuators, and the other is the beam with distributed sensor and actuator function. The orthogonality conditions were derived with the modal shapes of the active beam and its adjoint active beam. An example shows that the variation of eigenvalues with feedback amplitude for the homo-configuration and non-homo-configuration active structures.     

INVESTIGATION AND APPLICATIONS OF PANSYSTEMS RECOGNITION THEORY AND PANSYSTEMS-OPERATIONS RESEARCH OF LARGE-SCALE SYSTEMS (Ⅰ)

In this work, a sort of recognition theory and operations research of large scale systems are developed within the framework of pansystems methodology. We establish a series of theorems concerning pansystems relations, which discuss some fundamental problems of interdisciplines from the viewpoint of generalized system-transformation-symmetry in things mechanism, and are connected closely with mathematical physics systems thinking science bioecological sciences and the methodological investigation of mechanical foundation. This paper offers 100 pansystems theorems.     

INVESTIGATION AND APPLICATIONS OF PANSYSTEMS RECOGNITION THEORY AND PANSYSTEMS-OPERATIONS RESEARCH OF LARGE-SCALE SYSTEMS(Ⅰ)

In this work, a sort of recognition theory and operations research of large scale systems are developed within the framework of pansystems methodology. We establish a series of theorems concerning pansystems relations, which discuss some fundamental problems of interdisciplines from the viewpoint of generalized system-transformation-symmetry in things mechanism, and are connected closely with mathematical physics systems thinking science bioecological sciences and the methodological investigation of mechanical foundation. This paper offers 100 pansystems theorems.     

SMALE HORSESHOES AND CHAOS IN DISCRETIZEDPERTURBED NLS SYSTEMS(Ⅰ)—POINCARÉ MAP

The existence of Smale horseshoes for a certain discretized perturbed nonlinear Schroedinger （NLS） equations was established by using n-dimensional versions of the Conley-Moser conditions. As a result, the discretized perturbed NLS system is shown to possess an invadant set A on which the dynamics is topologically conjugate to a shift on four symbols.     

LINEAR ACTIVE STRUCTURES AND MODES （ Ⅰ ） — BASIC CONCEPTS AND PROPERTIES

Some basic concepts about the active structures were firstly explained, and the main subjects to study in the field of active structure dynamics were synthesized. For the linear active structures, the annotations on the modes were done in detail. The physical meanings of the right and left eigenvectors were explained. The right eigenvectors are the modal shapes and the modal responses of an active structure depend on the left ones. The adjoint structure of an active structure was defined and the reciprocity theorem was interpreted. For two active structures, which are adjoint to each other and with the reciprocal gain-matrices, the right and left eigenvector are reciprocal The relationship between an active structure and the corresponding passive structure is expressed with the transfer functions, which is employed to resolve the estimation problems.     

Pseudo-division algorithm for matrix multivariable polynomial and its application

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Higher-order multivariable euler's polynomial and higher-order multivariable bernoulli's polynomial

I.IntroductionBothIn-th-orderEuler'snumbersandpolynomial,n;-th-orderBernoulli'snumbersandpolynomialaretwokindsofspecialfunctions,whichareofgreatimportanceintheresearchofthetlinctionaltheoryandtheoreticalphysicsandhavebeenappliedextensivelyinpractice.UPtodate,however,thestudyofEuler'snumbersandpolynomial,Bernoulli'snumbersandpolynomialhasbeenalwayslimitedtoalevelofhigher-orderone-variable.Ontheresultsof[l],I2]and[3],Euler'snumbersandpolynomial,Bernoulli'snumbersandpolynomialaredevelopedande…     

非力学系统的对称性与守恒量(I)——分析力学札记之七

将非力学系统的微分方程化成Ｌａｇｒａｎｇｅ方程形式,引进无限小变换,研究微分方程或Ｈａｍｉｌｔｏｎ作用量在无小变换,研究微分方程或Ｈａｍｉｌｔｏｎ作用量在无限小变换下的不变性,进而给出守恒量存在的条件以及守恒量的形式。     

A note on polynomial chaos expansions for designing a linear feedback control for nonlinear systems

This paper presents a polynomial chaos-based framework for designing optimal linear feedback control laws for nonlinear systems with stochastic parametric uncertainty. The spectral decomposition of the original stochastic dynamical model in an orthogonal polynomial basis, prescribed by the Wiener–Askey scheme, provides a deterministic model from which the optimal linear control law is designed. Optimality of the proposed control law is proved by solving the Hamilton–Jacobi–Bellman equation, and asymptotic stability of the controlled nonlinear systems is guaranteed in the Lyapunov sense. We are especially interested in synchronization of chaotic systems. For this reason, the control strategy is applied in the trajectory tracking of periodic orbits for the Duffing oscillator and the Rössler system with uncertain stochastic parameters and initial conditions. The results are verified with Monte Carlo simulations.     

A differential equation for the minors of the state-transition matrix of linear systems

Summary A differential equation for the minors of the state transition matrix of linear time-invariant systems is established.An example is then presented, where the differential equation is employed in order to check the existence of the extremal control of a linear system with a performance index given by the integral of a quadratic form.

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Sommario Si stabilisce un'equazione differenziale per i minori della matrice di transizione dei sistemi lineari stazionari.Si illustra quindi un esempio, in cui tale equazione differenziale viene impiegata per verificare l'esistenza del controllo estremale di un sistema lineare con indice di comportamento dato dall'integrale di una forma quadratica.

     

The relationship between the stability and the optimality of linear systems (II)

Different from the inverse problem put forward by R.E.Kalman, another kind of inverse problem of linear optimal control is proposed and discussed in [1] as follows: Given an asymptotically stable linear constant system and a nonnegative quadratic performance index, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended. Similar conclusions are obtained for linear discrete systems and linear time-variable systems. According to these conclusions we can say that the correspondence between the asymptotically stable system and the optimal feedback system is the inherent character of all kinds of linear systems.     

Some problems associated with the disturbed matrix Lyapunov equation for linear time-invariant discrete systems

Both the Lyapunov stability and popov’s hyperstability ofdiscrete linear time-invariant system in case of system para-meter disturbance are discussed in this paper.The allowabledisturbance ranges are given so that the maintenance of theLyapunov stability and the popov’s hyperstability of a dis-crete linear system is guaranteed.The results find theirsignificance in the MRAC.     

Stabilization of discrete-time chaotic systems via improved periodic delayed feedback control based on polynomial matrix right coprime factorization

Delayed feedback control, proposed by Pyragas,is one of useful control methods of stabilizing unstable periodic orbits of chaotic systems without their exact information. This paper discusses an improved periodic delayed feedback control for the stabilization of the discrete-time chaotic systems. A technique to solve the generalized Sylvester matrix equation, based on polynomial matrix right coprime factorization, is introduced in order to give an explicit solution to the periodic gains of the closed-loop system. Moreover, some examples are demonstrated to show the effectiveness of the proposed method.     

Energy method for computing periodic solutions of strongly nonlinear systems (I) — Autonomous systems

In this paper a new method for solving for the periodic solution (limit cycle) of a strongly nonlinear system is suggested. Using this method not only the existence, stability and number of periodic solutions can be decided, but at the same time the approximate expressions for these periodic solutions can also be obtained. The proof of this method is given as well.The project is supported by the National Natural Science Foundation of China.