Discretization of continuous linear dynamic systems. Analysis of the methods

In 1960 Markus and Yamabe made the following conjecture: If a C 1 C^1 differential system x ˙ = F ( x ) \dot {\mathbf {x}}=F(\mathbf {x}) in R n \mathbb {R}^n has a unique equilibrium point and the Jacobian matrix of F ( x ) F(\mathbf {x}) for all x ∈ R n \mathbf {x}\in \mathbb {R}^n has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. Until 1997 we do not have the complete answer to this conjecture. It is true in R 2 \mathbb {R}^2 , but it is false in R n \mathbb {R}^n for all n > 2 n>2 . Here we extend the conjecture of Markus and Yamabe to continuous and discontinuous piecewise linear differential systems in R n \mathbb {R}^n separated by a hyperplane, and we prove that for the continuous piecewise linear differential systems it is true in R 2 \mathbb {R}^2 , but it is false in R n \mathbb {R}^n for all n > 2 n>2 . But for discontinuous piecewise linear differential systems it is false in R n \mathbb {R}^n for all n ≥ 2 n\ge 2 .

The phase portraits of the planar linear differential systems are very well known. This is not the case for the phase portraits of the planar continuous piecewise linear differential systems. In this paper we classify the phase portraits of the class of planar continuous piecewise linear differential systems of the form in the Poincaré disc when , and prove the existence and uniqueness of limit cycles. Note that on the straight line these differential systems are only continuous.

We present a computationally efficient scheme for parameter estimation of continuous dynamical linear systems given discrete-time noisy observations. The proposed scheme is optimal in the sense that it converges iteratively to the exact Maximum Likelihood estimate, where each iteration increases the likelihood.

Microgrids are gaining considerable attention due to the installation of renewable power generation resources in areas that are closer to the load centers. The multivariable droop idea, acknowledges active power sharing by concurrent drooping of both voltage and frequency, in resistance as well as weak grids. To develop a multivariable-droop synchronous current converter control strategy for current management using Lyapunov stability of microgrid applications. In this paper overview of Lyapunov stability of linear continuous systems is presented. The conditions for the existence of the solutions for both regular and irregular singular linear systems are introduced. To guarantee an asymptotical stability for linear continuous singular systems and new stability conditions are analyzed. A considerable attention is focused on the design of controllers for multivariable linear systems so that certain system properties could be preserved under various classes of perturbations occurring in the system. It presents a multivariable-droop synchronous current converter control strategy for current management of weak grid/microgrid applications, and offers a systematic and straightforward design approach using the loop shaping technique. The controller integrates current regulation, management, and limitation that are applicable to microgrids (MGs). The simulation is done by MATLAB/SIMULINK software.

This paper deals with continuous piecewise linear differential systems with three zones separated by two parallel straight lines (for short, CPWL3). The number of limit cycles of CPWL3 systems with two degenerate subsystems is not clear. In the paper, we provide a complete study on the maximum number of limit cycles by geometric techniques when the continuous piecewise linear systems with three zones have two degenerate subsystems. During our analysis, we also detect some bifurcation phenomena, such as boundary equilibrium bifurcation, scabbard bifurcation, grazing bifurcation, heteroclinic bifurcation and Hopf bifurcation.

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.

In the note equations of thermoelasticity of discrete and continuous systems are derived with the aid of the concept of ideal constraints imposed on mechanical as well as non-mechanical quantities. The thermoelastic systems are obtained as models of a certain material system M, in which a set of non-intersecting finite elements is distinguished. We assume that any typical element of M undergoes only homogeneous deformations and linear distributions of temperature; the concept of ideal constraints is applied to derive basic equations of thermoelasticity for this element. The constraints are also introduced to describe the interactions among elements: the stress vector and the heat flux on the corresponding boundaries of interacting elements are assumed to be continuous. The equations of the discrete thermoelastic system are obtained under condition that the pairs of interacting elements form a lattice in RN, 0 < N ≤ 3. The N-th dimensional thermoelastic continuum is derived from the suitable discrete system, provided that elements of M are sufficiently small and that certain regularity conditions hold.

Global dynamics of a degenerate planar piecewise linear differential system with three zones

This paper proposes guaranteed cost design of robust output feedback controller for continuous linear parametric uncertain systems. New necessary and sufficient conditions for static output feedback stabilizability of linear continuous time systems underly the design procedure. Proposed algorithms are computationally simple and tightly connected with the Lyapunov stability theory and the LQR optimal state feedback design. The proposed approach allows for prescribing the structure of the output feedback gain matrix (including the decentralized one) by the designer. New design method proposed in this paper, exploit genetic algorithm to design robust controller with guaranteed cost for polytopic linear continuous time systems. Numerical example is given to illustrate the performance of the proposed robust controller.

The dynamical richness of 4D memristor oscillators has been recently studied in several works, showing different regimes, from stable oscillations to chaos. Typically, only numerical simulations have been reported and so there is a lack of mathematical results. We focus our analysis in the existence of multiple stable oscillations in the 4D piecewise linear version of the canonical circuit proposed by Itoh and Chua (Int J Bifurc Chaos 18(11):3183–3206, 2008). This oscillator is modeled by a discontinuous piecewise linear dynamical system. By adding one parameter that stratifies the 4D dynamics, it is shown that the dynamics in each stratum is topologically equivalent to a 3D continuous piecewise linear dynamical system. Some previous results on bifurcations in such reduced system allow to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

A new approach was proposed to analyze the stability of the linear continuous stationary dynamic systems. It is based on the decomposition of a square H2 norm of the transfer function of the dynamic system into parts corresponding either to particular eigenvalues of the system matrix, or to pairwise combinations of these eigenvalues. The spectral decompositions of a square H2 norm of the transfer function with multiple poles were obtained using the residues of the transfer function and their derivatives. Exact analytical expressions for calculation of the quadratic forms of the corresponding expansions were derived for an arbitrary location of the eigenvalues in the left half-plane. The obtained decompositions allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic variation of the system energy. We propose the energy criterion for estimation of the system stability margins that uses an evaluation of energy accumulated in a group of weakly stable system modes. This approach is illustrated by calculating the energy of a band-pass filter.

&lt;p&gt;The article deals with the problem of finding the observer of the state vector of linear continuous non-stationary dynamical systems with uncertainty of the initial conditions, limited external influences and measurement errors over a finite time interval. Sufficient conditions for the existence of an observer are formulated and proved on the basis of the expansion principle. Relationships are obtained for finding the parameters of the observer and the worst laws of change in external influences and measurement errors. As a limiting case, the problem of observer synthesis for stationary linear dynamical systems on a semi-infinite time interval is considered. Two applied problems of estimating the aircraft state vector based on the results of incomplete and inaccurate measurements are solved.&lt;/p&gt;

A linear continuous time-varying system and a sample-data system with continuous time-varying linear part are considered. For both systems a two-dimensional control is formed which provides an invariant stabilization of a closed loop system. I. INTRODUCTION The invariant stabilization problem deals with such stabi- lization which provides an independence of output from outer action and boundedness of other variables. As demonstrated in (1-3) this problem remains one of the most actual in the theory of control systems. The main conclusion of these papers is: beginning from linear stationary systems the scalar control can provide either stability or invariance. A statement in (4) says that for a linear stationary system it is possible to combine the requirements of invariance and stability if the outer action is considered as one of the regulator's input. In the present report a linear continuous time-varying system and a sample-data system with continuous time- varying linear part are considered. For both systems a two- dimensional control is formed which provides an invariant stabilization of a closed loop system. If the vector of ob- servation is orthogonal to the vector of outer action the measurement of the action is not required. II. LINEAR NONSTATIONARY SYSTEM The following system

CRONE control of continuous linear time periodic systems: Application to a testing bench