Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.

Piecewise linear recursive convolution FDTD method for magnetized plasmas

The piecewise linear recursive convolution （PLRC） finite-different time-domain （FDTD） method greatly improves accuracy over the original recursive convolution （RC） FDTD approach but retains its speed and efficiency advantages. A PLRC-FDTD formulation for magnetized plasma which incorporates both anisotropy and frequency dispersion at the same time is presented, enabled the transient analysis of magnetized plasma media. The technique is illustrated by numerical simulations the reflection and transmission coefficients through a magnetized plasma layer. The results show that the PLRC-FDTD method has significantly improved the accuracy over the original RC method.

MULTIDIMENSIONAL MULTI-KNOTS PIECEWISE LINEAR SPECTRAL SEQUENCES

Motivated by representing multidimensional periodic nonlinear and non-stationary signals （functions）, we study a class of orthonormal exponential basis for L2（Id） with I ：= [0, 1）, whose exponential parts are piecewise linear spectral sequences with p-knots. It is widely applied in time-frequency analysis.

Longwall mining automation horizon control:Coal seam gradient identification using piecewise linear fitting

Horizon control, maintaining the alignment of the shearer's exploitation gradient with the coal seam gradient, is a key technique in longwall mining automation. To identify the coal seam gradient, a geological model of the coal seam was constructed using in-seam seismic surveying technology. By synthesizing the control resolution of the range arm and the geometric characteristics of the coal seam, a gradient identification method based on piecewise linear representation(PLR) is proposed. To achieve the maximum exploitation rate within the shearer's capacity, the control resolution of the range arm is selected as the threshold parameter of PLR. The control resolution significantly influenced the number of line segments and the fitting error. With the decrease of the control resolution from 0.01 to 0.02 m, the number of line segments decreased from 65 to 15, which was beneficial to horizon control. However, the average fitting error increased from 0.055 to 0.14 m, which would induce a decrease in the exploitation rate. To avoid significant deviation between the cutting range and the coal seam, the control resolution of the range arm must be lower than 0.02 m. In a field test, the automated horizon control of the longwall face was realized by coal seam gradient identification.

A Revised Piecewise Linear Recursive Convolution FDTD Method for Magnetized Plasmas

The piecewise linear recursive convolution （PLRC） finite-different time-domain （FDTD） method improves accuracy over the original recursive convolution （RC） FDTD approach and current density convolution （JEC） but retains their advantages in speed and efficiency. This paper describes a revised piecewise linear recursive convolution PLRC-FDTD formulation for magnetized plasma which incorporates both anisotropy and frequency dispersion at the same time, enabling the transient analysis of magnetized plasma media. The technique is illustrated by numerical simulations of the reflection and transmission coefficients through a magnetized plasma layer. The results show that the revised PLRC-FDTD method has improved the accuracy over the original RC FDTD method and JEC FDTD method.

A new chaotic Hopfield network with piecewise linear activation function

This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectrum. Numerical simulations show that the network displays chaotic behaviours for some well selected parameters.

Noise destroys the coexistence of periodic orbits of a piecewise linear map

The effects of Gaussian white noise and Gaussian colored noise on the periodic orbits of period-5（P-5） and period-6（P-6） in their coexisting domain of a piecewise linear map are investigated numerically.The probability densities of some orbits are calculated.When the noise intensity is D = 0.0001,only the orbits of P-5 exist,and the coexisting phenomenon is destroyed.On the other hand,the self-correlation time τ of the colored noise also affects the coexisting phenomenon.When τc〈τ〈τc,only the orbits of P-5 appear,and the stability of the orbits of P-5 is enhanced.However,when τ〉τc（τc and τc are critical values）,only the orbits of P-6 exist,and the stability of the P-6 orbits is enhanced greatly.When τ〈τc,the orbits of P-5 and P-6 coexist,but the stability of the P-5 orbits is enhanced and that of P-6 is weakened with τ increasing.

Synchronization and Stability of a Nonlinear Vibrating Mechanical System Characterized by Asymmetrical Piecewise Linearity

In previous studies about the synchronization of vibrators,the restoring forces of springs are mainly treated as linear directly,whereas the nonlinear features are rarely considered in vibrating systems.To make up this drawback,a dynamical model of a nonlinear vibrating mechanical system with double rigid frames(RFs),driven by two vibrators,is proposed to explore the synchronization and stability of the system.In this paper,the nonlinearity is reflected in nonlinear restoring forces of springs characterized by asymmetrical piecewise linear,where the nonlinear stiffness of springs is linearized equivalently using the asymptotic method.Based on the average method and Hamilton’s principle,the theory conditions to achieve synchronization and stability of two vibrators are deduced.After the theory analyses,some numerical qualitative analyses are given to reveal the coupling dynamical characteristics of the system and the relative motion properties between two RFs.Besides,some experiments are carried out to examine the validity of the theoretical results and the correctness of the numerical analyses results.Based on the comparisons of the theory,numeric and experiment,the ideal working regions of the system are suggested.Based on the present work,some new types of vibrating equipment,such as vibrating discharging centrifugal dehydrators/conveyers/screens,can be designed.

Characteristics of piecewise linear symmetric tri-stable stochastic resonance system and its application under different noises

Weak signal detection has become an important means of mechanical fault detections. In order to solve the problem of poor signal detection performance in classical tristable stochastic resonance system(CTSR), a novel unsaturated piecewise linear symmetric tristable stochastic resonance system(PLSTSR) is proposed. Firstly, by making the analysis and comparison of the output and input relationship between CTSR and PLSTSR, it is verified that the PLSTSR has good unsaturation characteristics. Then, on the basis of adiabatic approximation theory, the Kramers escape rate, the mean first-passage time(MFPT), and output signal-to-noise ratio(SNR) of PLSTSR are deduced, and the influences of different system parameters on them are studied. Combined with the adaptive genetic algorithm to synergistically optimize the system parameters, the PLSTSR and CTSR are used for numerically simulating the verification and detection of low-frequency, high-frequency,and multi-frequency signals. And the results show that the SNR and output amplitude of the PLSTSR are greatly improved compared with those of the CTSR, and the detection effect is better. Finally, the PLSTSR and CTSR are applied to the bearing fault detection under Gaussian white noise and Levy noise. The experimental results also show that the PLSTSR can obtain larger output amplitude and SNR, and can detect fault signals more easily, which proves that the system has better performance than other systems in bearing fault detection, and has good theoretical significance and practical value.

Output feedback controller design for uncertain piecewise linear systems

This paper proposes output feedback controller design methods for uncertain piecewise linear systems based on piecewise quadratic Lyapunov function. The α-stability of closed-loop systems is also considered. It is shown that the output feedback controller design procedure of uncertain piecewise linear systems with α-stability constraint can be cast as solving a set of bilinear matrix inequalities （BMIs）. The BMIs problem in this paper can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities （LMIs） which can be solved numerically efficiently. A numerical example shows the effectiveness of the proposed methods.

Chaotic oscillations in a piecewise linear spring-mass system

Chaotic oscillations are useful in assessing the health of a structure. Hence, simple chaotic systems which can easily be realized mechanically or electro-mechanically are highly desired. We study a new pieeewise linear spring-tnass system. The chaotic behaviour in this system is characterized using bifurcation diagrams and the invariant parameters of the dynamics. We also show that there exists a stochastic analogue of this system, which mimics the dynamical features of its deterministic counterpart. This allows a greater flexibility in practical designs as the chaotic oscillations are obtained either deterministically or stochastically. Also, the oscillations are low dimensional, which reduces the computational resources needed for obtaining the invariant parameters of this system.

Stability Analysis of Uncertain Discrete-Time Piecewise Linear Systems with Time Delays

This paper considers the stability analysis of uncertain discrete-time piecewise linear systems with time delays based on piecewise Lyapunov-Krasovskii functionals. It is shown that the stability can be established for the control systems if there is a piecewise Lyapunov-Krasovskii functional, and moreover, the functional can be obtained by solving a set of linear matrix inequalities （LMIs） that are numerically feasible. A numerical example is given to demonstrate the efficiency and advantage of the proposed method.

Knowledge Discovery in Learning Management System Using Piecewise Linear Regression

Recent developments in database technology have seen a wide variety of data being stored in huge collections. The wide variety makes the analysis tasks of a generic database a strenuous task in knowledge discovery. One approach is to summarize large datasets in such a way that the resulting summary dataset is of manageable size. Histogram has received significant attention as summarization/representative object for large database. But, it suffers from computational and space complexity. In this paper, we propose an idea to transform the histogram object into a Piecewise Linear Regression (PLR) line object and suggest that PLR objects can be less computational and storage intensive while compared to those of histograms. On the other hand to carry out a cluster analysis, we propose a distance measure for computing the distance between the PLR lines. Case study is presented based on the real data of online education system LMS. This demonstrates that PLR is a powerful knowledge representative for very large database.

H_∞ controller synthesis of piecewise discrete time linear systems

This paper presents an H∞ controller design method for piecewise discrete time linear systems based on a piecewise quadratic Lyapunov function. It is shown that the resulting closed loop system is globally stable with guaranteed H∞ performance and the controller can be obtained by solving a set of bilinear matrix inequalities. It has been shown that piecewise quadratic Lyapunov functions are less conservative than the global quadratic Lyapunov functions. A simulation example is also given to illustrate the advantage of the proposed approach.

A PIECEWISE LINEAR MODEL FOR ANALYZING THIN FILM/SUBSTRATE STRUCTURE IN FLEXIBLE ELECTRONICS

The conventional analytical method of predicting strain in a thin film under bending is restricted to the uniform material assumption, while in flexible electronics, the film/substrate structure is widely used with mismatched material properties taken into account. In this paper,a piecewise model is proposed to analyze the axial strain in a thin film of flexible electronics with the shear modification factor and principle of virtual work. The excellent agreement between analytical prediction and finite element results indicates that the model is capable of predicting the strain of the film/substrate structure in flexible electronics, whose mechanical stability and electrical performance is dependent on the strain state in the thin film.

SOME GENERAL THEOREMS AND GENERALIZED AND PIECEWISE GENERALIZED VARIATIONAL PRINCIPLES FOR LINEAR ELASTODYNAMICS

From the concept of four-dimensional space and under the four kinds of time limit conditions, some general theorems for elastodynamics are developed, such as the principle of possible work action, the virtual displacement principle, the virtual stress-momentum principle, the reciprocal theorems and the related theorems of time terminal conditions derived from it. The variational principles of potential energy action and complementary energy action, the H-W principles, the H-R principles and the constitutive variational principles for elastodynamics are obtained. Hamilton's principle, Toupin's work and the formulations of Ref. [5], [17]-[24] may be regarded as some special cases of the general principles given in the paper. By considering three cases: piecewise space-time domain, piecewise space domain, piecewise time domain, the piecewise variational principles including the potential, the complementary and the mixed energy action fashions are given. Finally, the general formulation of piecewise variational principles is derived. If the time dimension is not considered, the formulations obtained in the paper will become the corresponding ones for elastostatics.

Partition method for improvement of piecewise linear model with full envelope covered

Since ambient conditions vary in a wide range within the full flight envelope,the existing piecewise linear model(PLM),which is based on sea-level static condition with use of corrected parameters for other points in the flight envelope,cannot meet the accuracy for replacing the nonlinear model.To obtain more accurate linear models,a method of partitioning the flight envelope over a grid of Mach number and altitude boxes was suggested.Then,a set of linear models for a given operating condition was selected by picking the nearest(Mach number,altitude)box in the flight envelope.Through the selected set of linear models,interpolating for power level based on a weighted sum of corrected rotor speeds can obtain a linear model with acceptable accuracy.Simulation results of different points within the full flight envelope showed that the maximum error between nonlinear model and the existing PLM was more than 50%,while the maximum error between nonlinear model and the improved PLM was within 8%.It is concluded that the improved PLM performs accurately,especially under the non-standard conditions.In addition,the improved PLM can satisfy the real-time requirement better than the existing PLM.

The Extended Linear-Drift Model of Memristor and Its Piecewise Linear Approximation

Memristor is introduced as the fourth basic circuit element.Memristor exhibits great potential for numerous applications,such as emulating synapse,while the mathematical model of the memristor is still an open subject.In the linear-drift model,the boundary condition of the device is not considered.This paper proposes an extended linear-drift model of the memristor.The extended linear-drift model keeps the linear characteristic and simplicity of the linear-drift model and considers the boundary condition of the device.A piecewise linear approximation model of the extended linear-drift model is given.Both models are suitable for describing the memristor.

Orbital maneuver strategy design based on piecewise linear optimization for spacecraft soft landing on irregular asteroids

Recently,asteroid exploration becomes an important branch of human’s deep space activities.In this paper,a piecewise linear optimal orbital maneuver strategy is designed for a spacecraft soft landing on irregular-shaped asteroids.First,the space around an irregular asteroid is converted into several grid units,and the gravitational field of the asteroid is linearly fitted in each unit.Then,the soft-landing orbital maneuver strategy design problem is formulated as a piecewise linear optimal problem,and further transferred into a family of two-point boundary value problems,which can be solved using collocation method.Finally,a corresponding algorithm is developed to obtain the piecewise linear optimal maneuver strategy,which is proved to be able to achieve the soft-landing mission well.Simulation results show that the error of the model linearization is small enough,while the calculation efficiency is remarkably improved,and the robustness of maneuver strategy is also improved.

Model predictive control for constrained uncertain piecewise linear systems

For constrained piecewise linear （PWL） systems, the possible existing model uncertainty will bring the difficulties to the design approaches of model predictive control （MPC） based on mixed integer programming （MIP）. This paper combines the robust method and hybrid method to design the MPC for PWL systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Meanwhile, the MPC controller uses a sequence of feedback control laws as the future control actions, where each feedback control law in the sequence corresponds to each partitions and the arbitrary switching technique is adopted to tackle all the possible switching. Furthermore, to reduce the online computational burden of MPC, the segmented design procedure is suggested by utilizing the characteristics of the proposed approach. Then, an offline design algorithm is proposed, and the reserved degree of freedom can be online used to optimize the control input with lower computational burden.