Time-periodic Systems Research Articles

Solution of Linear Time Periodic Dynamical System with Appliction to Electrical Machines Using Chebyshev Polynomial Approch

The magnetic field within electrical machines causes an interaction between the electrical and mechanical dynamics of the system. A unified rotor dynamic model can be developed combining the electrical dynamics and the mechanical dynamics. The electromagnetic behaviour of an electrical machine is a linear time-dependent model which is then easily coupled with a linear model for the mechanical dynamics. The developed coupled model is a linear time periodic system. This work uses the Chebyshev polynomial for solving LTP system. The solution to Mathieu equation is presented before solving dynamics of electrical machines. The Chebyshev polynomial method has been developed and tested for Mathieu equation. Later the method has been applied for electrical machine. The obtained results also compared with modified Runge-Kutta that is available in MATLABTM software package.

Parametric Identification of Hybrid Linear-Time-Periodic Systems

In this paper, we present a state-space system identification technique for a class of hybrid LTP systems, formulated in the frequency domain based on input-output data. Other than a few notable exceptions, the majority of studies in the state-space system identification literature (e.g. subspace methods) focus only on LTI systems. Our goal in this study is to develop a technique for estimating time-periodic system and input matrices for a hybrid LTP system, assuming that full state measurements are available. To this end, we formulate our problem in a linear regression framework using Fourier transformations, and estimate Fourier series coefficients of the time-periodic system and input matrices using a least-squares solution. We illustrate the estimation accuracy of our method for LTP system dynamics using a hybrid damped Mathieu function as an example.

A novel approach for the prediction of the milling stability based on the Simpson method

In this paper a new approach is presented for predicting the milling stability based on the Simpson method. Generally the milling dynamic process is described as a linear time-periodic system with a single discrete time delay. By dividing the tooth passing period equally into a finite set of time intervals, the Simpson method is utilized in each time interval to estimate the state items. Then the state transition matrix over one tooth passing period is constructed, and the milling stability could be predicted by the Floquet theory. The convergence rate of the proposed method is discussed, and two benchmark examples are conducted. The results show that the proposed method achieves satisfactory accuracy and efficiency.

Identification of a vertical hopping robot model via harmonic transfer functions

A common approach to understanding and controlling robotic legged locomotion is the construction and analysis of simplified mathematical models that capture essential features of locomotor behaviours. However, the representational power of such simple mathematical models is inevitably limited due to the non-linear and complex nature of biological locomotor systems. Attempting to identify and explicitly incorporate key non-linearities into the model is challenging, increases complexity, and decreases the analytic utility of the resulting models. In this paper, we adopt a data-driven approach, with the goal of furnishing an input–output representation of a locomotor system. Our method is based on approximating the hybrid dynamics of a legged locomotion model around its limit cycle as a Linear Time Periodic (LTP) system. Perturbing inputs to the locomotor system with small chirp signals yield the input–output data necessary for the application of LTP system identification techniques, allowing us to estimate harmonic transfer functions (HTFs) associated with the local LTP approximation to the system dynamics around the limit cycle. We compare actual system responses with responses predicted by the HTF, providing evidence that data-driven system identification methods can be used to construct models for locomotor behaviours.

On the stability and spectral radius of a finite set of matrices

This paper studies some problems related to the stability and the spectral radius of a finite set of matrices. A seasonal epidemic model is given to illustrate the use of the obtained results. In this example, the relationship between the obtained results and the stability of a discrete time periodic linear system is obtained.

K-dimensional invariant cones of random dynamical systems in [formula omitted] with applications

We investigated a linear random dynamical system which strongly preserves a cone C of dimension-k (abbr. k-cone) in Rn. Under some general assumptions, it is shown that such system admits a measurable family of k-dimensional subspaces and a measurable family of (n−k)-dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of Rn. We further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems, as well as the linear competitive–cooperative tridiagonal systems. This generalizes the Floquet theory for these deterministic non-autonomous (or time-periodic) systems to the random systems.

Spectral invariants of periodic nonautonomous discrete dynamical systems

For an interval map, the poles of the Artin–Mazur zeta function provide topological invariants which are closely connected to topological entropy. It is known that for a time-periodic nonautonomous dynamical system F with period p, the p-th power [ζF(z)]p of its zeta function is meromorphic in the unit disk. Unlike in the autonomous case, where the zeta function ζf(z) only has poles in the unit disk, in the p-periodic nonautonomous case [ζF(z)]p may have zeros. In this paper we introduce the concept of spectral invariants of p-periodic nonautonomous discrete dynamical systems and study the role played by the zeros of [ζF(z)]p in this context. As we will see, these zeros play an important role in the spectral classification of these systems.

Classification of the Floquet statistical distribution for time-periodic open systems

How to understand the order of Floquet stationary states in the presence of external bath coupling and their statistical mechanics is challenging; the answers are important for preparations and control of those Floquet states. Here, we propose a scheme to classify the statistical distribution of Floquet states for time-periodic systems which couple to an external heat bath. If an effective Hamiltonian and a system-bath coupling operator, which are all time-independent, can be simultaneously obtained via a time-periodic unitary transformation, the statistical mechanics of the Floquet states is equivalent to the equilibrium statistical mechanics of the effective Hamiltonian. In the large driving frequency cases, we also show that the conditions of this theorem can be weakened to: the time-period part in the system Hamiltonian commutes with the system-bath coupling operator. A Floquet-Markov approach is applied to numerically compute the Floquet state occupation distribution of a bosonic chain, and the results agree with the theoretical predictions.

Stability analysis and backward whirl investigation of cracked rotors with time-varying stiffness

The dynamic stability of dynamical systems with time-periodic stiffness is addressed here. Cracked rotor systems with time-periodic stiffness are well-known examples of such systems. Time-varying area moments of inertia at the cracked element cross-section of a cracked rotor have been used to formulate the time-periodic finite element stiffness matrix. The semi-infinite coefficient matrix obtained by applying the harmonic balance (HB) solution to the finite element (FE) equations of motion is employed here to study the dynamic stability of the system. Consequently, the sign of the determinant of a scaled version of a sub-matrix of this semi-infinite coefficient matrix at a finite number of harmonics in the HB solution is found to be sufficient for identifying the major unstable zones of the system in the parameter plane. Specifically, it is found that the negative determinant always corresponds to unstable zones in all of the systems considered. This approach is applied to a parametrically excited Mathieu’s equation, a two degree-of-freedom linear time-periodic dynamical system, a cracked Jeffcott rotor and a finite element model of the cracked rotor system. Compared to the corresponding results obtained by Floquet’s theory, the sign of the determinant of the scaled sub-matrix is found to be an efficient tool for identifying the major unstable zones of the linear time-periodic parametrically excited systems, especially large-scale FE systems. Moreover, it is found that the unstable zones for a FE cracked rotor with an open transverse crack model only appear at the backward whirl. The theoretical and experimental results have been found to agree well for verifying that the open crack model excites the backward whirl amplitudes at the critical backward whirling rotational speeds.

Attractivity in Time-Periodic Parabolic Systems and Application to Reaction–Diffusion Neural Networks

This paper is concerned with the global attractivity for a system of semilinear parabolic equations with homogeneous Dirichlet conditions, in a bounded Lipschitz domain. The coefficients are time-periodic and a delay appears in the nonlinear reaction terms. Our approach uses the Lyapunov method and new average estimates for the Halanay differential inequality. The efficiency and novelty of the introduced criteria are illustrated by applications to reaction–diffusion neural networks.

Fourier state-space analysis of linear discrete-time periodic systems

By the Floquet–Lyapunov transformation, a linear periodically time-varying discrete-time system has a state-space representation (A,B(n),C(n)) having time-invariant autonomous dynamics, while the time periodicity is contained in the input and output matrices. In this paper it is shown that such a system also has state-space realizations of the form (A,B(n),C) and (A,B,C(n)), where the time periodicity is contained at the input, or alternatively the output, matrix only, and whose state dimensions are bounded by the product of the system order and the period length. The system representations are applied to construct a state-space realization of the optimal linear time-invariant approximation of a time-periodic system.

Independent Estimation of Input and Measurement Delays for a Hybrid Vertical Spring-Mass-Damper via Harmonic Transfer Functions

System identification of rhythmic locomotor systems is challenging due to the time-varying nature of their dynamics. Even though important aspects of these systems can be captured via explicit mechanics-based models, it is unclear how accurate such models can be while still being analytically tractable. An alternative approach for rhythmic locomotor systems is the use of data-driven system identification in the frequency domain via harmonic transfer functions (HTFs). To this end, the input-output dynamics of a locomotor behavior can be linearized around a stable limit cycle, yielding a linear, time-periodic system. However, few if any modelbased or data-driven identification methods for time-periodic systems address the problem of input and measurement delays in the system. In this paper, we focus on data-driven system identification for a simple mechanical system and analyze its dynamics in the presence of input and measurement delays using HTFs. By exploiting the way input delays are modulated by the periodic dynamics, our results enable the separate, independent estimation of input and measurement delays, which would be indistinguishable were the system linear and time invariant.

Transversality for time-periodic competitive-cooperative tridiagonal systems

Transversality of the stable and unstable manifolds of hyperbolic periodic solutions is proved for tridiagonal competitive-cooperative time-periodic systems. We further show that such systems admit the Morse-Smale property provided that all the fixed points (of the corresponding Poincare map) are hyperbolic. The main tools used here are the integer-valued Lyapunov function, as well as the Floquet theory developed in [1] for general time-dependent tridiagonal linear systems.

On the stability of a planar resonant rotation of a satellite in an elliptic orbit

We study the Lyapunov stability problem of the resonant rotation of a rigid body satellite about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the satellite completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the above resonance rotation was started in [4, 6]. In the present paper, rigorous stability conclusions in the previously unstudied range of parameter values are obtained. In particular, new intervals of stability are found for eccentricity values close to 1. In addition, some special cases are studied where the stability analysis should take into account terms of degree not less than six in the expansion of the Hamiltonian of the perturbed motion. Using the technique described in [7, 8], explicit formulae are obtained, allowing one to verify the stability criterion of a time-periodic Hamiltonian system with one degree of freedom in the special cases mentioned.

On the Robust Stability of the Hill Equation with a Delay Term: A Frequency-Domain Approach

The paper considers the problem of robust stability of the Hill equation with a damping and a time-delay term. Analytical results are presented in terms of the coefficients of the equation. The derivation is based on the frequency-domain approach, specifically, the geometric interpretation of the Yakubovich stability criterion for linear time-periodic systems. There are two approaches to this interpretation: One involves analysis of the so-called Lipatov plots and the other is based directly on the Nyquist hodograph.

Stability analysis of linear systems subject to regenerative switchings

This paper investigates the stability of switched linear systems whose switching signal is modeled as a stochastic process called a regenerative process. We show that the mean stability of such a switched system is characterized by the spectral radius of a matrix. The matrix is obtained by taking the expectation of the transition matrix of the system on one cycle of the underlying regenerative process. The characterization generalizes Floquet’s theorem for the stability analysis of linear time-periodic systems. We illustrate the result with the stability analysis of a linear system with a failure-prone controller under periodic maintenance.

弱KAM理论和Hamilton-Jacobi方程

Mather theory studies the existence of various kinds of the action minimizing sets and the connecting orbits between these sets in higher-dimensional positive Lagrangian system. The work about the connecting orbits plays an important role in the study of Arnold diffusion. By studying the dynamical behavior of the action-minimizing curves for Tonelli Lagrangian systems, weak KAM theory founded by A. Fathi bridges Mather theory and the PDE methods concerning the associated Hamilton-Jacobi equation. However, because the convergence of the Lax-Oleinik semigroup which is critical in KAM theory does not hold in time-periodic Lagrangian system, the preliminary work about weak KAM theory focused on autonomous system. By introducing a new kind of Lax-Oleinik type operator, it is possible for us to build weak KAM theory in the time-periodic Lagrangian system and generalize the theory to more general Hamilton-Jacobi equation. In this paper, we introduce the basic knowledge of weak KAM theory and its latest development.

Lagrangian transport characteristics of a class of three-dimensional inline-mixing flows with fluid inertia

Laminar mixing by the inline-mixing principle is a key to many industrial fluids-engineering systems of size extending from micrometers to meters. However, insight into fundamental transport phenomena particularly under the realistic conditions of three-dimensionality (3D) and fluid inertia remains limited. This study addresses these issues for inline mixers with cylindrical geometries and adopts the Rotated Arc Mixer (RAM) as a representative system. Transport is investigated from a Lagrangian perspective by identifying and examining coherent structures that form in the 3D streamline portrait. 3D effects and fluid inertia introduce three key features that are not found in simplified configurations: transition zones between consecutive mixing cells of the inline-mixing flow; local upstream flow (in certain parameter regimes); transition/inertia-induced breaking of symmetries in the Lagrangian equations of motion (causing topological changes in coherent structures). Topological considerations strongly suggest that there nonetheless always exists a net throughflow region between inlet and outlet of the inline-mixing flow that is strictly separated from possible internal regions. The Lagrangian dynamics in this region admits representation by a 2D time-periodic Hamiltonian system. This establishes one fundamental kinematic structure for the present class of inline-mixing flows and implies universal behavior in that all states follow from the Hamiltonian breakdown of one common integrable state. A so-called period-doubling bifurcation is the only way to eliminate transport barriers originating from this state and thus is a necessary (yet not sufficient) condition for global chaos. Important in a practical context is that a common simplification in literature, i.e., cell-wise fully-developed Stokes flow (“2.5D approach”), retains these fundamental kinematic properties and deviates from the generic 3D inertial case only in a quantitative sense. This substantiates its suitability for (at least first exploratory) studies on (qualitative) mixing properties.

Effect of the aerodynamic-induced parametric excitation on the longitudinal stability of hovering MAVs/insects

Longitudinal flight dynamics of hovering micro-air-vehicles and insects is considered. The natural oscillatory flapping motion of the wing leads to time-periodic stability derivatives (aerodynamic loads due to perturbation in the body-motion variables). Hence these terms play the role of parametric excitation on the flight dynamics. The main objective of this work is to assess the effects of these aerodynamic-induced parametric excitation terms that are neglected by the averaging analysis. The method of multiple scales is used to determine a second-order uniform expansion for the response of the time-periodic system at hand. The proposed approach is applied to the hovering flight dynamics of five insects that cover a wide range of operating frequency ratios to assess the applicability of the averaging analysis.

Switching control and time-delay identification

The unknown time delay makes the control design a difficult task. When the lower and upper bounds of an unknown time delay of dynamical systems are specified, one can design a supervisory control that switches among a set of controls designed for the sampled time delays in the given range so that the closed-loop system is stable and the control performance is maintained at a desirable level. In this paper, we propose to design a supervisory control to stabilize the system first. After the supervisory control converges, we start an algorithm to identify the unknown time delay, either on-line or off-line, with the known control being implemented. Examples are shown to demonstrate the stabilization and identification for linear time invariant and periodic systems with a single control time delay.