Time-periodic Systems Research Articles

Stick balancing with reflex delay in case of parametric forcing

The effect of parametric forcing on a PD control of an inverted pendulum is analyzed in the presence of feedback delay. The stability of the time-periodic and time-delayed system is determined numerically using the first-order semi-discretization method in the 5-dimensional parameter space of the pendulum’s length, the forcing frequency, the forcing amplitude, the proportional and the differential gains. It is shown that the critical length of the pendulum (that can just be balanced against the time-delay) can significantly be decreased by parametric forcing even if the maximum forcing acceleration is limited. The numerical analysis showed that the critical stick length about 30 cm corresponding to the unforced system with reflex delay 0.1 s can be decreased to 18 cm with keeping maximum acceleration below the gravitational acceleration.

Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients

Approximate stability analysis of nonlinear delay differential algebraic equations (DDAEs) with periodic coefficients is proposed with a geometric interpretation of evolution of the linearized system. Firstly, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and be computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a “monodromy matrix”, which is a finite-dimensional approximation of a compact infinite-dimensional operator. The monodromy matrix is essentially a map of the Chebyshev coefficients (or collocation vector) of the state from the delay interval to the next adjacent interval of time. The computations are entirely performed with the original system to avoid cumbersome transformations associated with the semi-explicit form of the system. Next, two computational algorithms, the first based on perturbation series and the second based on Chebyshev spectral collocation, are detailed to obtain solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

Frequency Response Characteristics of Parametrically Excited System

The frequency response characteristic of a general time-invariant system has been extensively analyzed in literature. However, it has not gained sufficient attentions in the parametrically excited system. In fact, due to the parametric excitation, the frequency response of time-periodic system differs distinctly from that of the time-invariant system. Utilizing Sylvester’s theorem and Fourier series expansion method, commonly used in the spectral decomposition for matrix, the frequency response functions (FRFs) of a single-degree-of-freedom (SDOF) parametrically excited system are derived briefly in the paper. The external resonant condition for the system is obtained by analyzing the specific expressions of FRFs. Then, a spur-gear-pair with periodically time-varying mesh stiffness is selected as an example to simulate the frequency response characteristics of parametric system. The effects of parametric stability, periodic mesh stiffness parameters (mesh frequency and contact ratio), and damping are considered in the simulation. It is shown from both theoretical and simulation results that the frequency response of parametric system has the following properties: there are multiple FRFs even for a SDOF periodic system as the forced response contains many frequency components and each FRF is corresponding to a certain response spectrum; the system has multiple external resonances. Besides the resonance caused by the external driving frequency equals to the natural frequency, the system will also be external resonant if external frequency meets the combination of natural frequency and parametric frequency. When the system is in external resonant state, the dominant frequency component in the response is the natural frequency; damping makes the peak values of FRFs drop evidently while it has almost no impact on the FRFs in nonresonant regions.

A full-discretization method for prediction of milling stability

This paper presents a full-discretization method based on the direct integration scheme for prediction of milling stability. The fundamental mathematical model of the dynamic milling process considering the regenerative effect is expressed as a linear time periodic system with a single discrete time delay, and the response of the system is calculated via the direct integration scheme with the help of discretizing the time period. Then, the Duhamel term of the response is solved using the full-discretization method. In each small time interval, the involved system state, time-periodic and time delay items are simultaneously approximated by means of linear interpolation. After obtaining the discrete map of the state transition on one time interval, a closed form expression for the transition matrix of the system is constructed. The milling stability is then predicted based on Floquet theory. The effectiveness of the algorithm is demonstrated by using the benchmark examples for one and two degrees of freedom milling models. It is shown that the proposed method has high computational efficiency without loss of any numerical precision. The code of the algorithm is also attached in the appendix.

Snapshot-Based Balanced Truncation for Linear Time-Periodic Systems

We introduce an algorithm based on a method of snapshots for computing approximate balanced truncations for discrete-time, linear time-periodic systems. The algorithm is applicable to stable (including neutrally stable) systems with very high dimension, allowing either the output or the input to be very high dimensional, as well. This algorithm is illustrated with a numerical example.

Linear response theory for statistical ensembles in complex systems with time-periodic forcing

New linear response formulas for unperturbed chaotic (stochastic) complex dynamical systems with time periodic coecients are developed here. Such time periodic systems arise naturally in climate change studies due to the seasonal cycle. These response formulas are developed through the mathematical interplay between statistical solutions for the time-periodic dynamical systems and the related skew-product system. This interplay is utilized to develop new systematic quasi-Gaussian and adjoint algorithms for calculating the climate response in such time-periodic systems. These new formulas are found in section 4. New linear response formulas are also developed here for general time-dependent statistical ensembles arising in ensemble prediction including the eects of deterministic model errors, initial ensembles, and model noise perturbations simultaneously . An information theoretic perspective is developed in calculating those model perturbations which yield the largest information deficit for the unperturbed system both for climate response and finite ensemble predictions.

Deconvolution of discrete time periodic systems in the H2 framework

In this paper we consider the deconvolution problem for discrete-time periodic systems, namely the problem of estimating the input of a periodic system from the observation of its output. The proposed solution is based on the H2 approach.

Modelling of flexural rotor vibration and time-periodic system dynamics in a two-pole cage induction machine equipped with a self-bearing force actuator

In this paper, linear time–periodic system dynamics in rotor vibration of electrical machines is examined. We consider a two-pole cage induction machine equipped with a built–in force actuator for active generation of lateral force on the rotor.The main contribution of the work is to identify and model the sources of time-periodic system dynamics in the coupled electromechanical actuator-rotor system. Especially, the effect of higher spatial harmonics of the magnetic field on the generated force is considered. In this work, a linear time–periodic model for the actuator-rotor system is derived. The parameters of the analytical model are estimated by using time-stepping finite-element analysis of the machine.

Active Vibration Control of AC Induction Machines Using Periodic Feedback and Least Mean Squares Methods

Active vibration control in AC machines is used for minimizing the radial rotor vibrations. These vibrations are caused by structural characteristics of the machine and they occur in all kinds of electrical machines. Several active control methods have been applied in the machines and the results have been promising. This paper introduces two new methods that utilize the theory of linear time periodic systems. The first method gives a periodic feedback control as a solution to a discrete-time periodic Riccati equation (DPRE). The second method yields a feedforward type controller that is capable of compensating several simultaneous disturbance signals of different frequencies summed to the output of the system. It is possible to use these two control methods such that they form a serial control topology algorithm.

Reduced-Order Modeling of Parametrically Excited Micro-Electro-Mechanical Systems (MEMS)

Reduced-order modeling is a systematic way of constructing models with smaller number of states that can capture the “essential dynamics” of the large-scale systems, accurately. In this paper, reduced-order modeling and control techniques for parametrically excited MEMS are presented. The techniques proposed here use the Lyapunov-Floquet (L-F) transformation that makes the linear part of transformed equations time invariant. In this work, three model reduction techniques for MEMS are suggested. First method is simply an application of the well-known Guyan-like reduction method to nonlinear systems. The second technique is based on singular perturbation, where the transformed system dynamics is partitioned as fast and slow dynamics and the system of differential equations is converted into a differential algebraic (DAE) system. In the third technique, the concept of invariant manifold for time-periodic systems is used. The “time periodic invariant manifold” based technique yields “reducibility conditions”. This is an important result because it helps us to understand the various types of resonances present in the system. These resonances indicate a tight coupling between the system states, and in order to retain the dynamic characteristics, one has to preserve all these “resonant” states in the reduced-order model. Thus, if the “reducibility conditions” are satisfied, only then a nonlinear order reduction based on invariant manifold approach is possible. It is found that the invariant manifold approach yields the most accurate results followed by the nonlinear projection and linear technique. These methodologies are general, free from small parameter assumptions, and can be applied to a variety of MEM systems like resonators, sensors and filters. The reduced-order models can be used for parametric study, sensitivity analysis and/or controller design. The controller design is based on the reduced-order system. Thus, first the reduced-order model of the large-scale system is constructed that captures the essential dynamics. If a controller is designed to stabilize this reduced-order system, then it guarantees that the large-scale system is controlled. The theoretical framework to design linear and nonlinear controllers is also presented.

Exposed proteins of the Schistosoma japonicum tegument

The ability of the mammalian blood fluke Schistosoma japonicum to survive in the inhospitable environment of the mammalian bloodstream can be attributed, at least in part, to its host-exposed outer surface, called the tegument. The tegument is a dynamic organ and is involved in nutrition, immune evasion and modulation, excretion, osmoregulation and signal transduction. Given its importance for parasite survival, proteins exposed to the host at the surface of the tegument are ideal targets for the development of vaccines and drugs. By biotinylating live adult worms and using a combination of OFFGEL electrophoresis and tandem mass spectrometry 54 proteins were identified as putatively host-exposed in S. japonicum. These included glucose transport proteins, an amino permease, a leucine aminopeptidase and a range of transporters, heat shock proteins and novel immune-active proteins. Members of the tetraspanin protein family and a homologue of Sm 29, a tegument membrane protein from Schistosoma mansoni, both effective vaccine antigens in S. mansoni, were also identified. The fate of labelled surface proteins was monitored over time using electron microscopy and revealed that biotinylated proteins were rapidly internalised from the surface of the tegument and trafficked into the cytoplasmic bridges that connect the distal cytoplasm of the tegument to the underlying cell bodies. The results reported herein dramatically increase the number of S. japonicum proteins known to be exposed to the host and, hence, those of interest as therapeutic targets. The ability of the parasite to rapidly internalise proteins at its surface has implications for the development of vaccines and may explain how these parasites are able to avoid the host immune system for long periods of time.

Dynamic Phasor Analysis of Periodic Systems

The paper considers stability analysis of linear time-periodic (LTP) systems based on the dynamic phasor model (DPM). The DPM exploits the periodicity of the system by expanding the system state in a Fourier series over a moving time window. This results in an L 2-equivalent representation in terms of an infinite-dimensional LTI system which describes the evolution of time varying Fourier coefficients. To prove stability, we consider quadratic time-periodic Lyapunov candidates. Using the DPM, the corresponding time-periodic Lyapunov inequality can be stated as a finite dimensional inequality and the Lyapunov function can be found by solving a linear matrix inequality.

Arnold diffusion in Hamiltonian systems: a priori unstable case

By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.

Field-driven geometrical phases in a time-periodic quantum system

We apply Floquet theory to explore the geometry of the Hilbert space under the influence of a time-periodic field. The geometrical phase is found to be induced by field-driven hybridizations when the photon energy of the driving field is close to the transition energies of the states of a quantum system. The phases of two hybridized states are phase locked to each other. We show that the geometrical phase is in general related to the Rabi frequency of the hybrid states. We also show that when the photon energy is equal to the transition energy of two states the geometrical phase acquired by each state is given exactly by an integer multiple of $\ensuremath{\pi}$, independent of the strength of the driving field. We illustrate the derived generic properties of the geometric phase with an experimentally realizable quantum-wire system. It is shown that the interference between conductance channels in the wire presents a way to identify the geometrical phase.

Observer Design for Nonlinear Systems With Time-Periodic Coefficients via Normal Form Theory

For most complex dynamic systems, it is not always possible to measure all system states by a direct measurement technique. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states, which cannot be measured directly. In this work, the problem of designing state observers for free systems (linear as well as nonlinear) with time-periodic coefficients is addressed. It is shown that, for linear periodic systems, the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using a Chebyshev expansion technique where the Floquet multipliers can be placed in the desired locations within the unit circle. For nonlinear time-periodic systems, an observer design methodology is developed using the Lyapunov–Floquet transformation and the Poincaré normal form technique. First, a set of time-periodic near identity coordinate transformations are applied to convert the nonlinear problem to a linear observer design problem. The conditions for existence of such invertible maps and their computations are discussed. Then the local identity observers are designed and implemented using a symbolic computational algorithm. Several illustrative examples are included to show the effectiveness of the proposed methods.

Magnus’ expansion for time-periodic systems: Parameter-dependent approximations

Magnus’ expansion solves the nonlinear Hausdorff equation associated with a linear time-varying system of ordinary differential equations by forming the matrix exponential of a series of integrated commutators of the matrix-valued coefficient. Instead of expanding the fundamental solution itself, that is, the logarithm is expanded. Within some finite interval in the time variable, such an expansion converges faster than direct methods like Picard iteration and it preserves symmetries of the ODE system, if present. For time-periodic systems, Magnus expansion, in some cases, allows one to symbolically approximate the logarithm of the Floquet transition matrix (monodromy matrix) in terms of parameters. Although it has been successfully used as a numerical tool, this use of the Magnus expansion is new. Here we use a version of Magnus’ expansion due to Iserles [Iserles A. Expansions that grow on trees. Not Am Math Soc 2002;49:430–40], who reordered the terms of Magnus’ expansion for more efficient computation. Though much about the convergence of the Magnus expansion is not known, we explore the convergence of the expansion and apply known convergence estimates. We discuss the possible benefits to using it for time-periodic systems, and we demonstrate the expansion on several examples of periodic systems through the use of a computer algebra system, showing how the convergence depends on parameters.

Primary Health Care Access, Continuity, and Cost Among Pediatric Patients With Obesity Hypertension

The purpose of this study was to determine how comorbid medical conditions, and the need for early pharmacological intervention, race, gender, and age are associated with service utilization and primary health care costs. From a statewide Medicaid database, medical and pharmacy claims for 10 calendar years (January 1995 to December 2004) were extracted to identify a cohort of children, ages 0 to 17, with (n = 552) and without (n = 2943) a diagnosis of both obesity and primary hypertension. Pediatric African American patients had more diagnosed medical conditions, significantly fewer visits during their first year of primary health care (chi2, 161.16; p < or = .0001) and significantly lower total service utilization over time (chi2, 10.35; p < or = .0001); and received significantly fewer face-to-face visits from office-based practitioners (chi2, 19.38; p < or = .0001) and other medical diagnostic and laboratory services. They also received significantly less acute care through emergency departments (chi2, 25.88; p < or = .0001) and inpatient facilities (chi2, 23.92; p < or = .0001), but were in the treatment system for longer periods of time despite their higher prevalence of multiple comorbid conditions. The total cost of care for pediatric African American patients was significantly lower (F, 90.84; p < or = .0001) than for non-African Americans. In this primary care system, there were early and persistent disparities in the receipt of outpatient, emergency, and inpatient services over time delivered to African American children and adolescents with multiple medical needs.

Harmonic analysis of pulse-width modulated systems

The paper considers the so-called dynamic phasor model as a basis for harmonic analysis of a class of switching systems. The analysis covers both periodically switched systems and non-periodic systems where the switching is controlled by feedback. The dynamic phasor model is a powerful tool for exploring cyclic properties of dynamic systems. It is shown that there is a connection between the dynamic phasor model and the harmonic transfer function of a linear time periodic system and this connection is used to extend the notion of harmonic transfer function to describe periodic solutions of non-periodic systems.

Tiny Bubbles

Abstract This is not an article about the song made famous by the late (great) Don Ho. This is about a breakthrough in the understanding of how micrometer-sized bubbles can be stabilized for long periods of time. This can influence the taste, smell, and consistency of consumer products including food and cosmetics. In two-phase systems, which can include air (as bubbles) suspended within a liquid, the structures of the dispersed (bubbles) and continuous (liquid) phases play a critical role in determining the properties of the material. There is also the function of time in that the microstructure of the dispersed phase continuously evolves toward a state of lower energy by minimizing the surface area between the two phases (referred to as the interfacial area). In the long term, this time evolution diminishes the usefulness of two-phase systems. Emilie Dressaire, Rodney Bee, David Bell, Alex Lips, and Howard Stone have devised a way to stabilize a two-phase system for time periods of a year or longer.

Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C 1 submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the C r ( r ⩾ 1 ) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C 1 but not neatly embedded, the other in which S is not of class C 1 .