Linear Autonomous Systems Research Articles

Strong Stability of Explicit Runge--Kutta Time Discretizations

Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for semi-negative autonomous linear systems. The analysis is based on the energy method and can be performed with the aid of a computer. Strong stability of various RK methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under this framework. Based on numerous numerical observations, we further characterize the features of strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of RK methods of odd linear order.

Stability and Phase Portraits for Simple Dynamical Systems

Abstract In structural dynamics models of mechanical oscillator and vibration analysis are of great importance. In this article motion of mechanical oscillator is modelled using second order linear autonomous differential systems. Stability of such 1 DOF models is investigated with respect to the coefficients of systems. Phase portraits for various cases are displayed and the character of fixed points is described.

Asymptotics of a Solution to a Singularly Perturbed Time-Optimal Control Problem

In the study of singularly perturbed optimal control problems, asymptotic solutions of the boundary value problems resulting from the optimality condition for the control are constructed using the well-known and well-developed method of boundary functions. This approach is effective for problems with smooth controls from an open domain. Problems with a closed bounded domain of control have been less thoroughly investigated. The cases usually considered involve situations where the control is a scalar function or a multidimensional function with values in a convex polyhedron. In the latter case, since the optimal control is a piecewise constant function with values at the vertices of the polyhedron, it is important to describe the asymptotic behavior of switching points of the optimal control. In this paper, we investigate a time-optimal control problem for a singularly perturbed linear autonomous system with smooth geometric constraints on the control in the form of a ball. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the matrix at the fast variables is a multidimensional analog of the second-order Jordan cell with zero eigenvalue and, thus, does not satisfy the standard condition of asymptotic stability. The solvability of the problem is proved. Power asymptotic expansions of the optimal time and optimal control in a small parameter at the derivatives in the equations of the system are constructed and justified.

Smooth Interpolation of Covariance Matrices and Brain Network Estimation.

We propose an approach to use the state covariance of autonomous linear systems to track time-varying covariance matrices of nonstationary time series. Following concepts from the Riemannian geometry, we investigate three types of covariance paths obtained by using different quadratic regularizations of system matrices. The first quadratic form induces the geodesics based on the Hellinger-Bures metric related to optimal mass transport (OMT) theory and quantum mechanics. The second type of quadratic form leads to the geodesics based on the Fisher-Rao metric from information geometry. In the process, we introduce a weighted-OMT interpretation of the Fisher-Rao metric for multivariate Gaussian distributions. A main contribution of this work is the introduction of the third type of covariance paths, which are steered by system matrices with rotating eigenspaces. The three types of covariance paths are compared using two examples with synthetic data and real data from resting-state functional magnetic resonance imaging, respectively.

Algorithms of adaptive disturbance compensation in linear systems with arbitrary input delay

ABSTRACT The problem of direct adaptive compensation of a priori uncertain disturbances for general class of linear time-invariant plans with known parameters and arbitrarily known input delay is considered. The external disturbance is assumed to be unmeasurable and modelled as the output of uncertain autonomous linear system. A scheme with two different algorithms of adaptive disturbance compensation is proposed. The scheme is based on special disturbance observer and predictor. The first algorithm involves the conventional gradient algorithm of adaptation with restricted convergence properties. The second algorithm represents the modification of gradient algorithm and accumulates information about disturbance over past period of time. Accumulation is implemented via a preselected linear operator with memory and gives dramatically improved parametric convergence. For algorithms of adaptation, the influence of input delay is completely compensated by the use of a special scheme of error augmentation. The problem of plant stabilisation is resolved by involving the state predictor. The performance of the proposed schemes of disturbance compensation is demonstrated via simulation and compared in conclusion.

Robust Linear Output Regulation Using Extended State Observer

This paper presents a disturbance rejection-based solution to the problem of robust output regulation. The mismatch between the underlying plant and its nominal mathematical model is formulated by two disturbance classes. The first class is assumed to be generated by an autonomous linear system while for the second class no specific dynamical structure is considered. Accordingly, the robustness of the closed-loop system against the first disturbance class is achieved by following the internal model principle. On the other hand, in the framework of disturbance rejection control, an extended state observer (ESO) is designed to approximate and compensate for the second class, i.e., unstructured disturbances. As a result, the proposed output regulation method can deal with a vast range of uncertainties. Finally, the stability of the closed-loop system based on the proposed compound controller is carried out via Lyapunov and center manifold analyses, and some results on the robust output regulation are drawn. A representative simulation example is also presented to show the effectiveness of the control method.

Minimal controllability time for finite-dimensional control systems under state constraints

We consider the controllability problem for finite-dimensional linear autonomous control systems, under state constraints but without imposing any control constraint. It is well known that, under the classical Kalman condition, in the absence of constraints on the state and the control, one can drive the system from any initial state to any final one in an arbitrarily small time. Furthermore, it is also well known that there is a positive minimal time in the presence of compact control constraints. We prove that, surprisingly, a positive minimal time may be required as well under state constraints, even if one does not impose any restriction on the control. This may even occur when the state constraints are unilateral, like the nonnegativity of some components of the state, for instance. Using the Brunovsky normal forms of controllable systems, we analyze this phenomenon in detail, that we illustrate by several examples. We discuss some extensions to nonlinear control systems and formulate some challenging open problems.

Algebraic Approach to the Stabilization of a Differential System of Retarded Type

For a spectrally controllable linear autonomous systems with commensurable delays, we construct state feedbacks ensuring the complete damping of the original system (finite stabilization) as well as the complete damping of the original system and the asymptotic stability of the closed-loop system (complete stabilization). The spectral reduction and asymptotic stabilization problems are considered as auxiliary problems. The argument is constructive, and the results are illustrated by an example.

Autonomous stochastic resonance driven by colored noise.

A one-dimensional linear autonomous system coupled to a generic stationary nonequilibrium fluctuating bath can exhibit resonant response when its damped oscillation period matches some characteristic bath's relaxation time. This condition justifies invoking the stochastic resonance paradigm, even if it can be achieved more easily by tuning the system to the bath and not vice versa, as is usually the case. The simple nature of the mechanism numerically investigated here suggests a number of interesting applications for instance in the context of (1) energy harvesting from random ambient vibrations, (2) activated barrier crossing through a saddle point, or (3) an unstable limit cycle.

Global stability for linear system and controllability for nonlinear system in the dynamics model of diabetics population

This paper presents global stability for linear system and controllability for nonlinear system in dynamic model. We discuss dynamic model of diabetics population which is autonomous linear system. The model considers the development of individual from a healthy stage to a pre-diabetic stage, then stage of diabetes without complication, to the stage of diabetes with complication and diabetics become disabled. In the linear model is investigated global stability using quadratic forms of Lyapunov function. After investigating the global stability of the linear system, a nonlinear optimal control model is established. Control variable in the model is the prevention effort to reduce the number of pre-diabetic into diabetic without and with complication. Controllability of nonlinear control system is investigated using Lie Bracket method. The results show that the linear system is globally asymptotically stable and the nonlinear system is locally accessible.

Classification of Two-Parameter Autonomous Linear Systems with Delay

We carry out a classification of two-parameter linear autonomous differential systems with delay with respect to the type of boundaries of domains of a D-partition.

Relationship between integer order systems and fractional order system and its two applications

Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems U+0028 FOS U+0029 field. In this paper, the relationship between integer order systems U+0028 IOS U+0029 and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.

Asymptotic Stability of Linear Fractional Systems with Constant Coefficients and Small Time-Dependent Perturbations

Our aim in this paper is to investigate the asymptotic behavior of solutions of the perturbed linear fractional differential system. We show that if the original linear autonomous system is asymptotically stable then under the action of small (either linear or nonlinear) nonautonomous perturbations the trivial solution of the perturbed system is also asymptotically stable.

A novel multi-scroll chaotic generator: Analysis, simulation, and implementation**Project supported by the National Natural Science Foundation of China (Grant Nos. 51377124 and 51521065), the Foundation for the Author of National Excellent Doctoral Dissertation, China (Grant No. 201337), and the New Star of Youth Science and Technology of Shaanxi Province, China (Grant No. 2016KJXX-40).

Based on a new three-dimensional autonomous linear system and designing a specific form of saturated function series and a sign function with two variables of system, which are employed to increase saddle-focus equilibrium points with index 2, a novel multi-scroll chaotic system is proposed and its typical dynamical characteristics including bifurcation diagram, Poincare map, and the stability of equilibrium points are analyzed. The hardware circuit is designed and the experimental results are presented for confirmation.

Modal controllability for systems of neutral type in classes of differential-difference controllers

We propose a solution for the modal controllability problem for linear autonomous differential systems of neutral type with commensurate delays in two classes of differentialdifference controllers.

On the problem of operation speed for the class of linear infinite-dimensional discrete-time systems with bounded control

Consideration was given to the problem of operation speed for the class of linear autonomous systems with infinite-dimensional state vector. The statements about the properties of the Minkowsky sum for the convex sets were formulated and proved. For the boundary points of the 0-controllability set, the necessary and sufficient conditions for solvability of the problem of operation speed were established. The optimality conditions were set down for the boundary points in terms of the discreet principle of maximum. Nonuniqueness of the optimal control and degenerate nature of the principle of maximum were proved for the internal points. An algorithm to solve the problem of operation speed for the internal points was developed by reducing it to the allowed case of the boundary points. Examples were given.

Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris–Lecar neuronal model

Necessary and sufficient conditions are given for the asymptotic stability and instability of a two-dimensional incommensurate order autonomous linear system, which consists of a differential equation with a Caputo-type fractional-order derivative and a classical first-order differential equation. These conditions are expressed in terms of the elements of the system’s matrix, as well as of the fractional order of the Caputo derivative. In this setting, we obtain a generalization of the well-known Routh–Hurwitz conditions. These theoretical results are then applied to the analysis of a two-dimensional fractional-order Morris–Lecar neuronal model, focusing on stability and instability properties. This fractional-order model is built up taking into account the dimensional consistency of the resulting system of differential equations. The occurrence of Hopf bifurcations is also discussed. Numerical simulations exemplify the theoretical results, revealing rich spiking behavior. The obtained results are also compared to similar ones obtained for the classical integer-order Morris–Lecar neuronal model.

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.

Sufficient conditions for asymptotic stability and stabilization of autonomous fractional order systems

We discuss the asymptotic stability of autonomous linear and nonlinear fractional order systems where the state equations contain same or different fractional orders which lie between 0 and 2. First, we use the Laplace transform method to derive some sufficient conditions which ensure asymptotic stability of linear fractional order systems. Then by using the obtained results and linearization technique, a stability theorem is presented for autonomous nonlinear fractional order system. Finally, we design a control strategy for stabilization of autonomous nonlinear fractional order systems, and apply the results to the chaotic fractional order Lorenz system in order to verify its effectiveness.

Periodic Solutions of Asymptotically Linear Autonomous Hamiltonian Systems with Resonance

In this paper we define the index at infinity of an asymptotically linear autonomous Hamiltonian system. We use this index to prove the existence and bifurcation from infinity of periodic solutions of the system. We apply the degree for G-invariant strongly indefinite functionals defined by Gołȩbiewska and Rybicki in (Nonlinear Anal 74:1823–1834, 2011).