Linear Homogeneous System Research Articles

On the geometry and refined rate of primal–dual hybrid gradient for linear programming

AbstractWe study the convergence behaviors of primal–dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

The masking technique for forced nonlinear oscillator stability behavior analysis using the non-perturbative approach

The study utilized the masking technique to explore the stability behavior of a forced nonlinear oscillator through the non-perturbative approach, with a particular focus on a Van der Pol oscillator subjected to external force, characterized by both cubic and quadratic nonlinearities. The application of the non-perturbative method (NPM) in conjunction with the masking technique was a pivotal aspect of this research, transforming the inherently non-homogeneous, nonlinear system into a homogeneous linear system. This transformation was crucial as it simplified the complex dynamics of the system, rendering it more amenable to analysis. Through this method, the research successfully established the system’s overall frequency, meticulously accounting for the impact of the periodic external force. The study also identified a distinct type of resonance response, where the system’s frequency incorporates the excited frequency in a nonlinear relationship. The masking technique proved to be an invaluable tool for examining the stability behavior of forced vibrations in oscillators via the NPM, providing profound insights into stability under external forces and enhancing the understanding and control of oscillatory behaviors in nonlinear dynamical systems. A critical confirmation of the current methodology is provided by the remarkable agreement found between the numerical solution and the provided analytical solution. This agreement shows that the analytical method produces trustworthy predictions and appropriately describes the system’s behavior. The plotted stability diagrams, which demonstrate that the model’s simulation of stability behavior is consistent with observed events, particularly resonance phenomena, offer further validity for the findings. In the resonance case, the effects of the damping coefficient and the external force’s magnitude are significant. The results of the analysis show that an increase in the damping coefficient has a destabilizing effect that causes unstable zones to expand. In contrast, in the resonance state, the quadratic and cubic nonlinearity factors both contribute to stabilization. Understanding how various system factors impact stability dynamics particularly in relation to resonance phenomena is made easier with the help of this insight.

Bumpless transfer consensus control for linear multi-agent systems under agent-dependent switching directed topologies

The bumpless transfer consensus control problem for linear homogeneous multi-agent systems subject to agent-dependent switching directed communication topologies is investigated in this paper. According to the information of graph Laplacian matrix and relative state of agent, we propose a novel agent-dependent switching law to guide the switching among these topologies, then the leader-following consensus performance of multi-agent systems is achieved via using the multiple Lyapunov functions method. Further, since the switching consensus controllers are considered, the control signal bumps are inevitable at switching instants, and some sufficient conditions via linear matrix inequalities are developed to pursue the bumpless transfer performance of multi-agent systems. Besides, the agent-dependent switching law also allows the error dynamics under each directed subtopology to be unstabilizable. Finally, the theoretical results are verified by two illustrative examples.

On the Relative Stability of Linear Time Invariant Systems

In this paper, state space representation based approach to relative stability of Linear Time Invariant (LTI) systems is discussed. Specifically, given a bound on the output, relative Bounded Input-Bounded Output (BIBO) stability of LTI systems is discussed. The bound on dominant pole is first determined using input-output description of LTI system. Using state space representation of homogeneous (no external input) LTI system, the components of state vector are upper bound. The results are applied to Continuous Time Markov Chains (i.e. homogeneous stochastic linear systems). Finally, using state space representation of arbitrary (with external input) Linear Time Invariant systems, the location of dominant pole is bounded. Using the upper bound, the location of line (parallel to imaginary-axis jω and to the left of it) is utilized for determining relative stability.

Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

SPECTRAL DYNAMIC STIFFNESS METHOD FOR THE FLUTTER PROBLEM OF COMBINED PLATES

At present time, the spectral dynamic stiffness method is being actively developed as an alternative to the finite element method for vibration and stability problems of composite structures from beams, rods, plates and shells. This approach, based on exact solutions of governing differential equations, makes it possible to more effectively study the problem in the medium and high frequency ranges, and gives analytical expressions for natural modes. It is proposed to use the advantages of this method to study the problems of dynamic stability and flutter of an orthotropic composite plate in a supersonic gas flow. Using the linear approximation of piston theory, solution of the problem is investigated according to the Galerkin method on the basis of the eigenforms of a composite plate in vacuum. According to this approach the boundary value problem is reduced to a homogeneous infinite linear algebraic system of equations with coefficients are depending from physical-mechanical and geometrical parameters of the problem. The frequency parameter is included in the system linearly, that allows us to reduce eigenproblem for infinite system to the problem of determining the eigenvalues and vectors of a matrix. The convergence of the Galerkin method depending on the number of basis functions is studied numerically. It is shown that the first 16 eigenforms provide the good convergence of the method. Examples of numerical implementation are given, obtained solution allow us to study the dependence of the critical velocity from the properties of the material and geometry of the combined plate.

Energetically Optimal Finite-Time Stabilization of Generalized Homogeneous Linear Systems

The problem of finite-time stabilization of a linear plant with an optimization of both a settling time and an averaged/weighted control energy is studied using the concept of generalized homogeneity. It is shown that the optimal finite-time stabilizing control in this case can be designed by means of solving a simple linear algebraic equation. Robustness of the obtained control law is studied.

Continuous Dependence on the Initial Functions and Stability Properties in Hyers–Ulam–Rassias Sense for Neutral Fractional Systems with Distributed Delays

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too.

On Essential Values of Oscillation Exponents for Solutions of a Linear Homogeneous Two-Dimensional Differential System

On Essential Values of Oscillation Exponents for Solutions of a Linear Homogeneous Two-Dimensional Differential System

Boundedness of the Set of Solutions to a Linear Homogeneous System Uniform Along the Initial Segment

Boundedness of the Set of Solutions to a Linear Homogeneous System Uniform Along the Initial Segment

An Algorithm for the Solution of Second Order Linear Fuzzy System With Mechanical Applications

In this paper, we consider homogeneous and non-homogeneous second order linear fuzzy systems under granular differentiability. The concept of continuous n-dimensional fuzzy functions on the space of n-dimensional fuzzy numbers are introduced. Developed an algorithm for the solution of a non-homogeneous second order linear fuzzy system under granular differentiability. The proposed algorithm is applied to solve some well-known mechanical problems with fuzzy uncertainty.

Numerical error bound of optimal control for homogeneous linear systems

Numerical error bound of optimal control for homogeneous linear systems

Qualitative sign stability of linear time invariant descriptor systems

This article discusses assessing the instability of a continuous linear homogeneous timeinvariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed.

Distributed Adaptive Event-Triggered TVGF for Generic Linear Multi-Agent Systems Under Directed Graph

This paper investigates event-triggered time-varying grouping formation (TVGF) problem for generic linear homogeneous multi-agent systems under directed graphs. A distributed adaptive event-triggered TVGF algorithm is proposed to ensure that the goal of TVGF can be achieved and the communication frequency between agents is reduced. Compared with previous formation control methods, the algorithm presented in this paper can complete the group formation task while saving communication energy. In addition, the algorithm’s dependence on global network information is also eliminated by using adaptive techniques. Finally, the availability of the proposed control law is established by numerical simulation.

A Distributed Implementation of Steady-State Kalman Filter

This paper studies the distributed state estimation in sensor network, where <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$m$</tex></formula> sensors are deployed to infer the <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$n$</tex></formula> -dimensional state of a Linear Time-Invariant (LTI) Gaussian system. By a lossless decomposition of the optimal steady-state Kalman filter, we show that the problem of distributed estimation can be reformulated as that of the synchronization of homogeneous linear systems. Based on such decomposition, a distributed estimator is proposed, where each sensor node runs a local filter using only its own measurement, alongside with a consensus algorithm to fuse the local estimate of every node. We prove that the average of estimates from all sensors coincides with the optimal Kalman estimate, and under certain condition on the graph Laplacian matrix and the system matrix, the consensus error is bounded and the asymptotic error covariance is derived. As a result, the distributed estimator is stable for each single node. We further show that the proposed algorithm has a low message complexity of min( <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$m$</tex></formula> , <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$n$</tex></formula> n). Numerical examples are provided in the end to illustrate the efficiency of the proposed algorithm.

On the Preservation of a Quadratic Lyapunov Function of a Linear Differential Autonomous System under Constant Perturbations of the Coefficients

For an autonomous linear homogeneous asymptotically stable differential system, we obtain sufficient conditions for the smallness of perturbations in the class of autonomous linear homogeneous systems under which a quadratic form that is a Lyapunov function of the original system remains a Lyapunov function of the perturbed system.

A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one‐dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

On the Discontinuity of Extreme Exponents of Oscillation on a Set of Linear Homogeneous Differential Systems

In this paper, we study the questions of the discontinuity of the extreme oscillation exponents on the set of linear homogeneous differential systems with continuous coefficients on the positive axis. The existence of points on a set of differential systems is established in which all the higher and lower exponents of the oscillation zeros, roots, and hypercorns are not only not continuous, but are not continuous either from above or from below. Moreover, the non-invariance of the extreme exponent of oscillations with respect to infinitesimal perturbations has been proved. When proving the results of this work, the cases of parity and odd order of the matrix of the system are considered separately.

О континуальных спектрах показателей колеблемости линейных однородных дифференциальных систем

The research subject of this work is at the junction of two sections of the qualitative theory of differential equations, namely: the theory of Lyapunov exponents and the theory of oscillation. In this paper, we study the spectra (i. e., sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential systems with coefficients continuous on the positive semiaxis. For any n≥2, the existence of an n-dimensional differential system with continuum spectra of the oscillation exponents is established. For even n, the spectra of all the oscillation exponents fill the same segment of the numerical axis with predetermined arbitrary positive incommensurable ends, and for odd n, zero is added to the indicated spectra. It turns out that for each solution of the constructed differential system, all the oscillation exponents coincide with each other. When proving the results of this work, the cases of even and odd n are considered separately. The results obtained are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential systems.

Исследование решений линейной однородной системы дифференциальных уравнений

The paper shows that the fundamental zero-normalized solution of a linear homogeneous differ-ential equations system can be represented as an exponential matrices products formal series. If the system satisfies the equations system triangulation Perron theorem conditions, then the system solution can be represented as an exponential matrices finite product. In addition, an exponential matrix function differentiating formula is derived. Also, the transformation constructing problem is considered. Such, a homogeneous differential equations system allows to reduce to a triangular form.