So-Called Linear Systems (Linear Systems with a Non-zero Initial State)

2004 ◽  
pp. 195-217
Author(s):  
Tsuyoshi Matsuo ◽  
Yasumichi Hasegawa
Keyword(s):  
Linear Systems ◽  
Initial State

Positive Finite-Time Stabilization for Discrete-Time Linear Systems

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Wenping Xue ◽  
Kangji Li
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Finite Time ◽  
State Feedback ◽  
Closed Loop ◽  
Linear Matrix ◽  
Closed Loop System ◽  
Initial State ◽  
Finite Time Stability ◽  
Time Linear

In this paper, a new finite-time stability (FTS) concept, which is defined as positive FTS (PFTS), is introduced into discrete-time linear systems. Differently from previous FTS-related papers, the initial state as well as the state trajectory is required to be in the non-negative orthant of the Euclidean space. Some test criteria are established for the PFTS of the unforced system. Then, a sufficient condition is proposed for the design of a state feedback controller such that the closed-loop system is positively finite-time stable. This condition is provided in terms of a series of linear matrix inequalities (LMIs) with some equality constraints. Moreover, the requirement of non-negativity of the controller is considered. Finally, two examples are presented to illustrate the developed theory.

l-Delay Input and Initial-State Reconstruction for Discrete-Time Linear Systems

2010 ◽  
Vol 30 (1) ◽  
pp. 233-262 ◽  
Author(s):  
Siddharth Kirtikar ◽  
Harish Palanthandalam-Madapusi ◽  
Elena Zattoni ◽  
Dennis S. Bernstein
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Initial State ◽  
State Reconstruction ◽  
Time Linear

scholarly journals The selected problems of controllability of discrete-time switched linear systems with constrained switching rule

2015 ◽  
Vol 63 (3) ◽  
pp. 657-666 ◽  
Author(s):  
A. Babiarz ◽  
A. Czornik ◽  
J. Klamka ◽  
M. Niezabitowski
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Switched Systems ◽  
Sufficient Conditions ◽  
Controllability Problem ◽  
Initial State ◽  
Final State ◽  
Switching Signal ◽  
Necessary And Sufficient ◽  
Time Linear

AbstractIn this paper the controllability problem for discrete-time linear switched systems is considered. The main goal is to find a control signal that steers any initial state to a given final state independently of the switching signal. In the paper, it is assumed that there are some constraints posed on the switching signal. Moreover, we present a necessary and sufficient conditions of some kinds of controllability. Three types of controllability, namely: from zero initial state to any final state, from any initial state to zero final state and from any initial state to any final state are considered. Finally, three illustrative examples are shown.

scholarly journals Optimal Switching Synthesis for Jump Linear Systems With Gaussian Initial State Uncertainty

2014 ◽  
Author(s):  
Kooktae Lee ◽  
Raktim Bhattacharya
Keyword(s):  
Linear Systems ◽  
Measurement Errors ◽  
Wasserstein Distance ◽  
Optimal Switching ◽  
Initial State ◽  
Jump Linear Systems ◽  
Switching Sequence ◽  
Mean Square Stability ◽  
State Uncertainty ◽  
The Mean

This paper provides a method to design an optimal switching sequence for jump linear systems with given Gaussian initial state uncertainty. In the practical perspective, the initial state contains some uncertainties that come from measurement errors or sensor inaccuracies and we assume that the type of this uncertainty has the form of Gaussian distribution. In order to cope with Gaussian initial state uncertainty and to measure the system performance, Wasserstein metric that defines the distance between probability density functions is used. Combining with the receding horizon framework, an optimal switching sequence for jump linear systems can be obtained by minimizing the objective function that is expressed in terms of Wasserstein distance. The proposed optimal switching synthesis also guarantees the mean square stability for jump linear systems. The validations of the proposed methods are verified by examples.

scholarly journals Optimal Control of Switching Linear Systems

2004 ◽  
Vol 9 ◽  
pp. 41
Author(s):  
Ali Benmerzouga
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Linear Systems ◽  
Optimal Solution ◽  
Input Constraints ◽  
Initial State ◽  
Programming Technique ◽  
New Approach ◽  
Running Cost ◽  
Modified Algorithm

A solution to the control of switching linear systems with input constraints was given in Benmerzouga (1997) for both the conventional enumeration approach and the new approach. The solution given there turned out to be not unique. The main objective in this work is to determine the optimal control sequences {Ui(k) ,  i = 1,..., M ;  k = 0, 1, ...,  N -1} which transfer the system from a given initial state  X0  to a specific target state  XT  (or to be as close as possible) by using the same discrete time solution obtained in Benmerzouga (1997) and minimizing a running cost-to-go function. By using the dynamic programming technique, the optimal solution is found for both approaches given in Benmerzouga (1997). The computational complexity of the modified algorithm is also given.  

scholarly journals Optimal control for controllable stochastic linear systems

2020 ◽  
Vol 26 ◽  
pp. 98
Author(s):  
Xiuchun Bi ◽  
Jingrui Sun ◽  
Jie Xiong
Keyword(s):  
Optimal Control ◽  
Linear Systems ◽  
Optimal Parameter ◽  
Linear Manifold ◽  
Terminal State ◽  
Linear Quadratic ◽  
Initial State ◽  
Linear Quadratic Optimal Control ◽  
Lq Problem ◽  
Stochastic Linear Systems

This paper is concerned with a constrained stochastic linear-quadratic optimal control problem, in which the terminal state is fixed and the initial state is constrained to lie in a stochastic linear manifold. The controllability of stochastic linear systems is studied. Then the optimal control is explicitly obtained by considering a parameterized unconstrained backward LQ problem and an optimal parameter selection problem. A notable feature of our results is that, instead of solving an equation involving derivatives with respect to the parameter, the optimal parameter is characterized by a matrix equation.

scholarly journals PROBLEMA DAS LUZES APAGADAS

2018 ◽  
Vol 10 (2) ◽  
pp. 45-52
Author(s):  
Natalia Rodrigues da Silva ◽  
Hudson Alves Martins ◽  
Fernando Pereira de Souza
Keyword(s):  
Open Source ◽  
Linear Systems ◽  
Numerical Computation ◽  
Linear Algebra ◽  
Open Source Software ◽  
Simple Game ◽  
Initial State ◽  
Correct Sequence

Linear Algebra presents a very important role in the areas of accuracy, and through it we can show its utility in modeling a problem that involves a simple game of erasing and lighting lights, after modeling the problem, we will solve it by implementing an algorithm of elimination of Gauss in Scilab (Scilab is free and open source software, focused on numerical computation similar to Matlab). The game, Lights Out, is a famous game of the 90's, which consists of 25 keys illuminated and arranged in the form of a 5x5 matrix, where it has an initial state and we must delete all keys by pressing a correct sequence of keys, this sequence will be provided by the program implemented in Scilab, ie we will have an immediate solution to erase all lights in the game. The modeling uses Linear Algebra concepts: matrices, determinants and linear systems, together with the concept of the set of the rest of the division by 2(ℤ2).

scholarly journals SET MEMBERSHIP ESTIMATION WITH A SEPARATE RESTRICTION ON INITIAL STATE AND DISTURBANCES

2021 ◽  
Vol 7 (1) ◽  
pp. 160
Author(s):  
Polina A. Yurovskikh
Keyword(s):  
Linear Systems ◽  
Dynamic Equation ◽  
Estimation Problem ◽  
Observation Equation ◽  
Compact Set ◽  
Initial State ◽  
Uncertain Disturbances ◽  
Set Membership ◽  
Information Set ◽  
Initial States

We consider a set membership estimation problem for linear non-stationary systems for which initial states belong to a compact set and uncertain disturbances in an observation equation are integrally restricted. We provethat the exact information set of the system can be approximated by a set of external ellipsoids in the absence of disturbances in the dynamic equation.There are three examples of linear systems. Two examples illustrate the main theorem of the paper, the latter one shows the possibility of generalizing the theorem to the case with disturbances in the dynamic equation.

scholarly journals Using Switching Linear Systems to Reach a Prespecified Traget

1997 ◽  
Vol 2 ◽  
pp. 69
Author(s):  
A. Benmerzouga
Keyword(s):  
Linear Systems ◽  
Continuous System ◽  
Input Constraints ◽  
Discrete Time System ◽  
Initial State ◽  
Time System ◽  
New Approach ◽  
Quadratic Cost ◽  
Time Invariant ◽  
Quadratic Cost Function

A conventional enumeration approach, and a new approach for the solution of the control of switched linear systems with input constraints are presented. The main objective in this work is to determine control sequences {U‘(K), i -l , M and  k –O, l, ….., N - l} which transfer the system from a given initial state X(0) to a specific target state Xt, (or to be as close as possible). Considering both the conventional enumeration and the new approach, extensive computer simulations are performed using the discrete time system obtained by sampling (or discretizing) a continuous system. The new approach is found to be more efficient than the enumeration one in terms of computations and computer storage and performs adequately under a variety of input data The procedure developed can be generalized and used to solve several versions of the switching control problem. In particular, a procedure using a quadratic cost function (distance) is given for problems with time-invariant coefficients.

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