scholarly journals A Note on Solutions of Linear Systems

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Branko Malešević ◽  
Ivana Jovović ◽  
Milica Makragić ◽  
Biljana Radičić
Keyword(s):  
Linear System ◽  
General Solution ◽  
Linear Systems ◽  
Sufficient Condition ◽  
Matrix Equations ◽  
Necessary And Sufficient Condition ◽  
Different Types ◽  
Free Parameters ◽  
Necessary And Sufficient

We will consider Rohde's general form of {1}-inverse of a matrix A. The necessary and sufficient condition for consistency of a linear system Ax=c will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula x=A(1)c+(I-A(1)A)y for obtaining the general solution of the linear system. These results will be applied for finding the general solution of various homogenous and nonhomogenous linear systems as well as for different types of matrix equations.

A New Expression of the Hermitian Solutions to a System of Matrix Equations with Applications

2016 ◽  
Vol 23 (01) ◽  
pp. 167-180 ◽  
Author(s):  
Guangjing Song ◽  
Shaowen Yu ◽  
Ming Chen
Keyword(s):  
General Solution ◽  
Sufficient Condition ◽  
Matrix Equations ◽  
Necessary And Sufficient Condition ◽  
System Of Matrix Equations ◽  
Hermitian Solution ◽  
Necessary And Sufficient

In this paper, a new necessary and sufficient condition for the existence of a Hermitian solution as well as a new expression of the general Hermitian solution to the system of matrix equations A1X=C1 and A3XB3=C3 are derived. The max-min ranks and inertias of these Hermitian solutions with some interesting applications are shown. In particular, the max-min ranks and inertias of the Hermitian part of the general solution to this system are presented.

Stability of On-Line Bin Packing with Random Arrivals and Long-Run-Average Constraints

1990 ◽  
Vol 4 (4) ◽  
pp. 447-460 ◽  
Author(s):  
Coastas Courcobetis ◽  
Richard Weber
Keyword(s):  
Bin Packing ◽  
Random Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Expected Number ◽  
Long Run ◽  
Different Types ◽  
On Line ◽  
Necessary And Sufficient ◽  
Uniformly Bounded

Items of various types arrive at a bin-packing facility according to random processes and are to be combined with other readily available items of different types and packed into bins using one of a number of possible packings. One might think of a manufacturing context in which randomly arriving subassemblies are to be combined with subassemblies from an existing inventory to assemble a variety of finished products. Packing must be done on-line; that is, as each item arrives, it must be allocated to a bin whose configuration of packing is fixed. Moreover, it is required that the packing be managed in such a way that the readily available items are consumed at predescribed rates, corresponding perhaps to optimal rates for manufacturing these items. At any moment, some number of bins will be partially full. In practice, it is important that the packing be managed so that the expected number of partially full bins remains uniformly bounded in time. We present a necessary and sufficient condition for this goal to be realized and describe an algorithm to achieve it.

scholarly journals Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Hong Shi ◽  
Guangming Xie ◽  
Wenguang Luo
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Control Delay ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
And Control ◽  
Time Systems ◽  
Time Linear

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.

scholarly journals General Solution and Observability of Singular Differential Systems with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Jiang Wei
Keyword(s):  
General Solution ◽  
Sufficient Condition ◽  
General Description ◽  
Necessary And Sufficient Condition ◽  
Differential Systems ◽  
Exact Observability ◽  
Necessary And Sufficient

We study singular differential systems with delay. A general description for the solutions of singular differential systems with delay is given and a necessary and sufficient condition for exact observability of singular differential systems with delay is derived.

scholarly journals Plug and Play Robust Distributed Control with Ellipsoidal Parametric Uncertainty System

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Hong Wang-jian ◽  
Wang Yan-xiang
Keyword(s):  
Linear System ◽  
Distributed Control ◽  
Control Strategy ◽  
State Feedback ◽  
Parametric Uncertainty ◽  
State Feedback Control ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Plug And Play ◽  
Necessary And Sufficient

We consider a continuous linear time invariant system with ellipsoidal parametric uncertainty structured into subsystems. Since the design of a local controller uses only information on a subsystem and its neighbours, we combine the plug and play idea and robust distributed control to propose one distributed control strategy for linear system with ellipsoidal parametric uncertainty. Firstly for linear system with ellipsoidal parametric uncertainty, a necessary and sufficient condition for robust state feedback control is proposed by means of linear matrix inequality. If this necessary and sufficient condition is satisfied, this robust state feedback gain matrix can be easily derived to guarantee robust stability and prescribed closed loop performance. Secondly the plug and play idea is introduced in the design process. Finally by one example of aircraft flutter model parameter identification, the efficiency of the proposed control strategy can be easily realized.

Normal and Quasi-Normal Modes in Damped Linear Dynamic Systems

1966 ◽  
Vol 33 (2) ◽  
pp. 413-416 ◽  
Author(s):  
J. S. Maybee
Keyword(s):  
Linear Systems ◽  
Dynamic Systems ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Necessary And Sufficient

A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist is that certain matrices associated with the system commute. Necessary and sufficient conditions of the same type are also obtained for the classical normal modes, but under more restrictive conditions.

Detection of Improper General Spatial Kinematic Chains With Different Types of Pairs

1994 ◽  
Author(s):  
Xian-Wen Kong ◽  
Ting-Li Yang
Keyword(s):  
Unsolved Problem ◽  
Degree Of Freedom ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Single Degree Of Freedom ◽  
Kinematic Chains ◽  
Single Degree ◽  
Different Types ◽  
Necessary And Sufficient ◽  
Topological Synthesis

Abstract Improper general spatial kinematic chains (GSKCs) due to the effect of pair types may be generated during the process of topological synthesis of GSKCs with different types of pairs. Thus, detection of improper GSKCs is necessary in topological synthesis of GSKCs with different types of pairs. Unfortunately, it is still an unsolved problem. In this paper, a method for detecting improper GSKCs is presented. Both a necessary and sufficient condition and a sufficient condition for proper GSKCs with R, P, H, T and C pairs are introduced at first. Based on these two conditions, an algorithm to detect improper GSKCs is then developed which is very efficient and suitable for topological synthesis of GSKCs with R, P, H, T and C pairs. The proposed algorithm has been applied to topological synthesis of 1- and 2-loop, single degree of freedom GSKCs with R, P, H, T and C pairs and the corresponding atlas is obtained.

scholarly journals Controllability Analysis of Linear Discrete Time Systems with Time Delay in State

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hong Shi ◽  
Guangming Xie ◽  
Wenguang Luo
Keyword(s):  
Time Delay ◽  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Delay ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Time Linear

The controllability issues for linear discrete-time systems with delay in state are addressed. By introducing a new concept, the minimum controllability realization index (MinCRI), the characteristic of controllability is revealed. It is proved that the MinCRI of a system with state delay exists and is finite. Based on this result, a necessary and sufficient condition for the controllability of discrete-time linear systems with state delay is established.

On the Controllability of Distributed-Order Fractional Systems With Distributed Delays

2017 ◽  
Author(s):  
Bin-Bin He ◽  
YangQuan Chen ◽  
Chun-Hai Kou
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Distributed Delays ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Fractional Systems ◽  
Gramian Matrix ◽  
Controllability Gramian ◽  
Necessary And Sufficient ◽  
Distributed Order

This paper investigates the controllability of distributed-order fractional systems with distributed delays. By using the controllability Gramian matrix and reduction to absurdity, a necessary and sufficient condition for the controllability of linear system is established, and a sufficient condition for the nonlinear system is obtained. Examples are given to illustrate the effectiveness of the theorems.

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