Behavior Of Linear Systems Research Articles

Creation and mobility of self-trapped electronic states in nonlinear chains

We study the formation and mobility of self-trapped electronic states in a nonlinear one-dimensional lattice subjected to an electric-field. The nonlinearity arises from the electron-lattice coupling and the electric field is time-dependent. We show that the nonlinearity is a mechanism able to provide motion to the centroid of a delta-like wave-packet when subjected to an electric field, in contrast with the behavior of linear systems. With the co-existence of the electric field and electron-lattice interaction, the constitution of mobile self-trapped states is possible. The phase diagram reveals a rich phenomenology in which the mobility of self-trapped electronic states can exhibit an unidirectional branch migrating to the left or the right side of the chain. For weak nonlinearities, an analytical expression obtained from a semi-classical approach describes the dynamics of self-trapping states. In a strong electron-lattice regime, the wave-packet develops a self-trapped regime in which it remains fully localized at the initial site. Thus, we offer the proper tunning of electric field for the management of the creation and mobility of self-trapped electronic states through the lattice.

A unified framework for asymptotic and transient behavior of linear stochastic systems

This paper is concerned with a unified framework for asymptotic and transient behavior of stochastic systems. In order to explain this problem explicitly, a concept of mean square (γ, α)-stability is first introduced and two stability criteria are derived. By utilizing an auxiliary definition of mean square (γ, T)-stability, the relations among mean square (γ, α)-stability, mean square (γ, T)-stability and finite-time stochastic stability are established. Subsequently, two new sufficient conditions for the existence of state and output feedback mean square (γ, α)-stabilization controllers are presented in terms of matrix inequalities. A numerical algorithm is given to obtain the relation between γmin and α. Finally, an example is given to illustrate our results.

Modal Analysis of Fluid Flows: An Overview

Modal Analysis of Fluid Flows: An Overview

Analysis of Turbo Code Behavior with Extrinsic Information Transfer Charts in High-Speed Wireless Data Services

This paper examines turbo codes that are currently introduced in many international standards and implemented in numerous advanced communication systems, and evaluates the process of extrinsic information transfer (EXIT). The convergence properties of the iterative decoding process, associated with a given turbo-coding scheme, are estimated using the analysis technique based on so-called extrinsic information transfer (EXIT) charts. This approach provides a possibility to predict the bit-error rate (BER) of a turbo code system with only the extrinsic information transfer chart. It is shown that extrinsic information transfer charts are powerful tools to analyze and optimize the convergence behavior of iterative systems utilizing the turbo principle, i.e., systems exchanging and refining extrinsic information. The idea is to consider the associated soft-input soft-output (SISO) stages as information processors, which map input a priori log likelihood ratios (LLRs) onto output extrinsic LLRs, the information content being obviously assumed to increase from input to output, and introduce them to the design of turbo systems without the reliance on extensive simulation. Compared with the other methods for generating extrinsic information transfer functions, the suggested approach provides insight into the iterative behavior of linear turbo systems with substantial reduction in numerical complexity.

Comparisons of Ageing and Temperature Impacts on Nonlinear and Linear Dynamic Responses of Lithium-Ion Batteries

Lithium-ion batteries are the most promising candidate to fulfill a worldwide provision of electric mobility.For a safe operation, it is essential to accurately know and determine the state of the Lithium-ion battery. Themost commonly used dynamic determination method is the Electrochemical Impedance Spectroscopy (EIS).EIS analyses the battery in regard to linear system behaviour. However, electrochemical reactions show ahighly nonlinear relation of current and voltage according to Butler-Volmer Kinetics. Additionally, chargeand discharge processes of double-layers at surface interfaces and diffusion in spherical particles can shownonlinear behaviour [1]. Therefore, information about the nonlinearities in the cell is not fully accessed withEIS. To account for this, we provide a novel approach for characterization of Lithium-ion batteries, the socalled Nonlinear Frequency Response Analysis (NFRA). NFRA methods have priorly been used to fuel cellsto determine redox kinetics [2] and as a State-of-Charge Estimator for Lead-Acid batteries [3]. For NFRA,high current amplitudes IAC are applied to a cell and the response signal is analysed. Thereby, higherharmonic responses Yn with n≥2 are observed at multiples of the fundamental frequency f1. In this study we show experimental research on Lithium-ion batteries to distinguish between ageingcharacteristics at different environmental temperatures using dynamic analysis methods, e.g. EIS and NFRA.We conducted ageing of identically constructed pouch cells with NMC as cathode and Graphite as anodematerial at 25 °C and -10 °C with a 1 C constant current (CC)/constant voltage(CV) charging and 1 C CCdischarge profile. After every 50th cycle, dynamic measurements were performed at 25 °C in a temperaturechamber. EIS was measured with an excitation amplitude of C/15 and 1.5 C for NFRA, both at a frequencyrange between 20 mHz and 5 kHz. Whereas for EIS we analyse the linear output of the system Y1, for NFRA higher harmonics Yn as well astheir sum are investigated. In Figure 1, results of the ageing study are shown. The discrete frequency of10 Hz is chosen, as it is a characteristic time constant of the electrochemical reactions, which we willdemonstrate by showing temperature dependent EIS studies. Voltage responses are exemplarly extractedprior to ageing and each 200th cycle. By solely using EIS, it is not possible to distinguish between ageing at25 °C and -10 °C, since the change of the linear output Y1 of the system is similar. For NFRA, voltageresponse for Y2 behave qualitatively similar to Y1 for EIS. However, higher harmonic Y3 shows a strongdependency of the ageing conditions, thereby enabling the possibility to distinguish between them. Further,we will present process characterization on Lithium-ion batteries with NFRA by identifying characteristicfrequency ranges for higher harmonic voltage responses Yn and correlating them to typical electrochemicaland transport processes. Finally, we will show and discuss the impact of ageing on nonlinear and linearresponses of dynamic measurements for the overall frequency range. [1] A. M. Bond, N. W. Duffy, D. M. Elton, B. D. Fleming, Characterization of Nonlinear BackgroundComponents in Voltammetry by use of Large Amplitude Periodic Perturbations and Fourier TransformAnalysis, Analytical Chemistry 81 (2009) 8801–8808.[2] Q. Mao, U. Krewer, R. Hanke-Rauschenbach, Total Harmonic Distortion Analysis for Direct MethanolFuel Cell Anode, Electrochemistry Communications 12 (2010) 1517–1519.[3] S. Okazaki, Second-Order Harmonic in the Current Response to Sinusoidal Perturbation Voltage forLead-Acid Battery, Journal of The Electrochemical Society 132 (1985) 1516. Figure 1

Nonlinear Frequency Response Analysis (NFRA) of Lithium-Ion Batteries

Electrochemical Impedance Spectroscopy (EIS) is the most commonly used technique for dynamic analysis of Lithium-ion batteries. EIS, however, limits analysis to linear contributions of the processes. For Lithium-ion batteries with their nonlinear electrochemistry and physics, dynamics are only analysed with regard to linear system behaviour and therefore some dynamic information is not used. Nonlinear Frequency Response Analysis (NFRA) extends dynamic analysis to consider also nonlinearities. Higher excitation amplitudes are applied and higher order frequency responses Yn are measured. The spectra show distinct higher harmonic responses with strong characteristic nonlinear behaviour. We investigate amplitude and temperature dependency of higher harmonic responses as well as the impact of ageing of Lithium-ion batteries with NFRA. By correlating NFRA and EIS, solid diffusion, reaction and ionic transport contributions at and in the SEI can be separated and identified. Thereby the method of NFRA is seen as an important additional dynamic analysis method for Lithium-ion batteries.

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

Frequency response functions (FRFs) are important for assessing the behavior of stochastic linear dynamic systems. For large systems, their evaluations are time-consuming even for a single simulation. In such cases, uncertainty quantification by crude Monte-Carlo simulation is not feasible. In this paper, we propose the use of sparse adaptive polynomial chaos expansions (PCE) as a surrogate of the full model. To overcome known limitations of PCE when applied to FRF simulation, we propose a frequency transformation strategy that maximizes the similarity between FRFs prior to the calculation of the PCE surrogate. This strategy results in lower-order PCEs for each frequency. Principal component analysis is then employed to reduce the number of random outputs. The proposed approach is applied to two case studies: a simple 2-DOF system and a 6-DOF system with 16 random inputs. The accuracy assessment of the results indicates that the proposed approach can predict single FRFs accurately. Besides, it is shown that the first two moments of the FRFs obtained by the PCE converge to the reference results faster than with the Monte-Carlo (MC) methods.

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices {S1,…,Sl}, by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:xn∈{Skxn−1;1≤k≤l},x0∈Rdandn≥1,we study the chaotic dynamics of the state trajectory (xn(x0, σ))n ≥ 1 with initial state x0∈Rd, governed by a switching law σ:N→{1,…,l}. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all x0,y0∈Rd,x0≠y0,lim infn→+∞∥xn(x0,σ)−xn(y0,σ)∥=0andlim supn→+∞∥xn(x0,σ)−xn(y0,σ)∥=∞.This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (xn(x0, σ))n ≥ 1 are extremely sensitive to the initial states x0∈Rd. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.

Stability and periodic investigations of linear planar difference systems

This paper discusses the problem of stability and periodic behaviour of linear planar difference systems appearing in modelling of discrete problems of population biology and Hopfield neural networks. We concentrate especially on procedures, which enable to formulate optimal (i.e. necessary and sufficient) conditions guaranteing such a behaviour. As a main tool, we analyse in detail location of zeros of characteristic polynomials with respect to the unit circle. From this viewpoint, derived results contribute also to the polynomial theory. Copyright © 2016 John Wiley & Sons, Ltd.

A Fast Model‐Predictive Speed Controller for Minimised Charge Consumption of Electric Vehicles

AbstractThis paper presents the design of a real‐time implementable energy‐efficient model‐predictive cruise controller for electric vehicles including the driving speed reference generation. The controller is designed to meet the properties of a series‐production electric vehicle whose characteristics are identified and validated by measurements. The predictive eco‐cruise controller aims at finding the best compromise between speed‐reference tracking and energy consumption of the vehicle using an underlying dynamic model of the vehicle motion and charge consumption. The originally non‐linear motion model is transformed into a linear model mainly by using a coordinate transform. To obtain a piecewise linear approximation of the charge consumption map, the measured characteristics are approximated by a convex piecewise linear function represented as the maximum of a set of linear constraint functions. The reformulations finally lead to a model‐predictive control approach with quadratic cost function, linear prediction model and linear constraints that corresponds to a piecewise linear system behaviour and allows a fast real‐time implementation with guaranteed convergence. Simulation results of the closed‐loop operation finally illustrate the effectiveness of the approach.

Fractional derivatives and periodic functions

This paper studies the periodic functions in the perspective of fractional calculus application. It is shown that the fractional derivative of a periodic signal is periodic if it is defined on the whole real line. Several common fractional derivative formulations are considered, namely the Grunwald–Letnikov, Liouville and Caputo on $$\mathbb {R}$$ , and the two-sided fractional derivatives. It is verified that the fractional derivative of a causal periodic signal is never causal periodic. The periodic behaviour of the fractional linear systems is also studied. If such systems are defined with suitable derivatives the output corresponding to periodic input is also periodic. Is is concluded that only the integer order linear systems can have a sinusoidal impulse response.

Deterministic analysis of distributed order systems using operational matrix

Recently, distributed order systems as a generalized concept of fractional order have been a major focus in science and engineering areas, and have rapidly extended application across a wide range of disciplines. However, only a few numerical methods are available for analyzing the distributed order systems. This paper proposes a novel numerical scheme to analyze the behavior of single input single output linear systems in the time domain with a single distributed order differentiator/integrator by using operational matrix technique. The proposed method reduces different analysis problems to a system of algebraic equations by using block pulse functions, which makes it easy to handle an arbitrary input. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method. The proposed method was found to be an efficient tool for analyzing linear distributed order systems.

The matching of two stable sewing linear systems in the plane

In this article we study one of the main problems in the qualitative theory of planar differential equations: the problem of determining the basin of attraction of an equilibrium point. We give a rigorous proof that for planar sewing piecewise linear systems with two zones, defined by Hurwitz matrices the unique equilibrium point in the separation straight line is globally asymptotically stable. On the other hand, we prove that sewing piecewise linear systems with two zones in the plane, defined by Hurwitz matrices can have one unstable equilibrium point at the origin allowing a broken line to separate the zones, leading to counter-intuitive dynamical behaviors of simple piecewise linear systems in the plane.

Collective behavior of mixed-order linear multi-agent systems under output-coupled consensus algorithm

Output consensus problem is investigated for mixed-order linear multi-agent systems composed of two-type agents with one and two poles at the origin respectively, and usual output-coupled consensus algorithm is adopted. According to generalized Nyquist stability criterion, consensus conditions are gained for the mixed-order multi-agent systems to achieve an asymptotic stationary consensus without and with communication delay respectively. Besides, output consensus is also discussed for a simple mixed-order multi-agent system, which consists of single and double integrators. Numerical examples show the correctness of theoretical results.

Stability and Asymptotic Behavior of Systems with Multi-time

Stability and asymptotic behavior of linear multi-time systems $\frac {\partial }{\partial t_{i}} u(\textbf {t})= A_{i} u (\textbf {t})$ +f i (t) is considered, where A i are generators of commuting C 0-semigroups on a Banach space. We formulate and prove versions of the results on exponential stability to homogeneous linear systems with multi-time. For non-homogeneous systems, we introduce the compatibility condition and obtain the variation-of-constants formula. Using simulta neous solutions of Sylvester equations, we obtain results on asymptotic almost periodicity of solutions.

Robust stability on averaging behaviour of linear time-varying uncertain systems

This research proposes a new type of robust stability – mean robust asymptotic stability for linear time-varying (LTV) uncertain systems and its extension in robust asymptotic stability analysis. The system will feature this kind of stability if its state mean varies towards an equilibrium point. Robust input–output finite-time stability of the LTV uncertain system over a bounded time interval is first analysed by proposing a criterion to characterize the stability condition. The possible cases of steady-state error and state oscillation are then considered and tackled. Finally, some case studies are presented to demonstrate superiority of the proposed work.

Design and simulation of a real-time implementable energy-efficient model-predictive cruise controller for electric vehicles

This paper presents the design of a novel energy-efficient model-predictive cruise controller for electric vehicles as well a simulation model of the longitudinal vehicle dynamics and its energy consumption. Both, the controller and the dynamic model are designed to meet the properties of a series-production electric vehicle whose characteristics are identified and verified by measurements. A predictive eco-cruise controller involves the minimisation of a compromise between terms related to driving speed and energy consumption which are in general both described by nonlinear differential equations. Considering the nonlinearities is essential for a proper prediction of the system states over the prediction horizon to achieve the desired energy-saving behaviour. In this work, the vehicle motion equation is reformulated in terms of the kinetic energy of the moving vehicle which leads to a linear differential equation without loss of information. The energy consumption is modeled implicitly by exploiting the special form of the optimisation problem. The reformulations finally lead to a model-predictive control approach with quadratic cost function, linear prediction model and linear constraints that corresponds to a piecewise linear system behaviour and allows a fast real-time implementation with guaranteed convergence. Simulation results of the MPC controller and the simulation model in closed-loop operation finally provide a proof of concept.

To obtain approximate probability density functions for a class of axially moving viscoelastic plates under external and parametric white noise excitation

The probability density function plays an essential role to investigate the behaviors of stochastic linear or nonlinear systems. This function can be evaluated by several approaches but due to its analytical theme, the Fokker–Planck–Kolmlgorov (FPK) approach is preferable. FPK equation is a nonlinear PDE gives the probability density function for a stochastic linear or nonlinear system. Many researches have been done in literature tried to specify the conditions, in which the FPK equation gives an exact solution. Although, the exact probability density function can be achieved by solving the FPK equation even for some nonlinear systems, many types of systems cannot satisfy the conditions for exact solution. In this article, the axially moving viscoelastic plates under both external and parametric white noise excitation as one of the newest and applicable research areas are studied. Due to strong nonlinearities recognized in the governing equation of the system, the exact probability density function cannot be obtained, however, via an approximate method; some precise approximate solutions for different but comprehensive case studies are evaluated, validated, and discussed.

Switched Control for the Walking of a Compass Gait Biped Robot

This study presents a new approach for modeling and controlling of a compass gait biped robot based on the use of the switched systems. The linearization of the equations stemming from the formalism of Lagrange allows the construction of a set of local models used to describe the behavior of this non linear system. The selection of each model depends on its activation function depending on the system states. The synthesis of the stability of the walking robot is based on the use of second method of Lyapunov. The synthesis approach leads to a set of bilinear matrices inequalities non resolvable by actual numerical solvers. To come over these difficulties, some relaxations are brought to get useful and exploitable numerical solutions.

Dynamic behavior of cracked rotor subjected to multisine excitation

Abstract It is widely known that a developing shaft crack manifests itself in the appearance of nonlinear effects resulting in higher harmonics in a vibration spectrum. However, such symptoms are characteristic not only for developing shaft cracks but also for other malfunctions such as a shaft bow, a coupling misalignment, etc. That is why novel shaft crack detection methods introduce a specially designed diagnostic force applied to the shaft in order to amplify the particular symptoms of the crack. Most often a simple harmonic force is used for such purposes, yet the results may not be reliable. The present paper analyzes the possible application of the multisine excitation technique for an early shaft crack detection problem. This technique was previously used for identification problems where the influence of nonlinear distortions on the linear system behavior was to be evaluated and measured. In the present paper the multisine technique is adopted for the detection of a shaft crack in a rotor. The possibility to evaluate the crack depth is also verified. The approach is illustrated with the numerical results of the flexible rotating shaft modeled with the rigid finite element method. In order to assess the effectiveness of the proposed method the numerical results obtained for the uncracked and cracked rotors are compared. The robustness of the method to additional disturbances is checked by introducing some noise to the system. The results demonstrate a potential of the presented method to detect the shaft cracks and to evaluate their severity. Possible problems of the practical implementation of the method are briefly discussed.