Analyzing piecewise linear dynamical systems

In the study of steady-state solutions of a nonlinear, or piecewise linear dynamic system under periodic forcing, it is often necessary to examine the stability of the solutions. The reason is that not all the possible solutions can persist for an arbitrarily long time in the presence of small random disturbances of all kinds. In general, some of the solutions are stable, and others are unstable. Only stable solutions can be expected to occur in physical systems. This paper analyzes the response of a certain class of piece-wise linear systems. The piecewise linearities arise solely from restoring forces which are found frequently in many engineering problems. Harmonic and subharmonic solutions of the piecewise linear systems are constructed and their stability studied. System parameters which affect the various types of solutions, and interrelations between these parameters for the various types of solutions to exist, are determined. Numerical results for practical design applications are presented. In particular, jump phenomena involving a transition from an unstable harmonic oscillation to a stable subharmonic oscillation, and vice versa, are shown to exist for the present systems.

A model of angiogenesis by hybrid systems with delay on the piecewise constant part

Generation of chaotic attractors without equilibria via piecewise linear systems

SummaryThis paper is concerned with the problem of passivity analysis and passification for a class of stochastic impulsive memristor‐based piecewise linear (PWL) systems with mixed delays and nonlinearity disturbances. Based on the PWL memristor, a PWL system is set up. And some novel sufficient conditions are derived to ensure the passivity/passification performance, such that, for all admissible stochastic disturbances and nonlinearity, the closed‐loop stochastic impulsive memristor‐based PWL system is passive in the sense of expectation. Copyright © 2013 John Wiley & Sons, Ltd.

Study of a piecewise linear dynamic system with negative and positive stiffness

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 book treats analysis and design of piecewise linear control systems. In this chapter, we lay the foundation for the analysis by presenting the mathematical model on which the subsequent developments will be based. We derive an explicit matrix representation of the model and discuss solution concepts. We extend modeling techniques for uncertain linear systems to piecewise linear systems and show how norm-bound uncertainties and smooth nonlinearities can be treated rigorously in this framework. Finally, we note that piecewise linear systems enjoy important interconnection properties that allow complex systems to be constructed from the interconnection of simpler subsystems, and discuss memory-efficient model representations.

Robust Predictive Control for Constrained Uncertain Piecewise Linear Systems using Linear Matrix Inequalities

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

In steady human walking and running, every step is similar to every other, but they are not all identical. That is, the motion is nearly but not exactly periodic. In this paper, we construct models of the dynamics near the periodic motion of human locomotion from the near-periodic steady data. We use a sequence of Poincare sections (transverse to the periodic orbit) in the neighborhood of the periodic orbit and linearized dynamics of the state from one Poincare section to the next, essentially resulting in a piecewise linear dynamical system around the periodic orbit. Using human locomotion data obtained from a high-accuracy motion capture system, a piecewise linear dynamical model is constructed to represent human running/walking. The piecewise linear model can predict human transient dynamics under various circumstances. We show responses to perturbations to the swing leg.Copyright © 2012 by ASME

In this paper, we consider a planar dynamical system with a piecewise linear function containing two dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system, with five singular points, can have at most four limit cycles, three of which surround the foci one by one and the fourth limit cycle surrounds all of the singular points of this system.

For a class of 4-dim piecewise linear dynamical systems, some sufficient conditions for the existence of homoclinic orbit and chaotic invariant set are provided. Especially, the existence of chaotic invariant set is proved in terms of the constructed Poincaré map with a topological horseshoe. A numerical example with chaotic attractor is provided to illustrate our main result.

In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles play a main role. Up to now, the major part of papers which study the limit cycles of the piecewise linear differential systems consider only two pieces. Here, we consider piecewise linear differential systems with three pieces. In this paper, we study the limit cycles of the discontinuous piecewise linear differential systems in the plane R2 formed by three arbitrary linear centers separated by the set Σ={(x,y)∈R2:y=0{or}x=0{and}y≥0}. We prove that such discontinuous piecewise linear differential systems can have 1, 2, or 3 limit cycles, with 3 the maximum number of limit cycles that such systems can have. Moreover, the limit cycles are nested and must intersect Σ in three or four points. The limit cycles having three intersection points with Σ can reach the maximum number 3. The limit cycles having four intersection points with Σ are at most 1, and if it exists, the systems could simultaneously have 1 or 2 limit cycles intersecting Σ in three points.

LINEAR PROGRAMMING BASED GAIN SCHEDULING FOR LPV AND PL SYSTEMS

This paper studies the adaptive state feedback for state tracking control problem for piecewise linear systems, which are approximations of nonlinear controlled systems at multiple operating points. Piecewise linear reference model systems are studied and used for generating state trajectories. Adaptive schemes are developed using Lyapunov design methods, and their stability and tracking performance are analyzed and evaluated by simulation examples. Asymptotic tracking performance of such an adaptive control system with a sufficiently rich reference input is shown by simulation results, indicating that certain persistent excitation conditions can be sufficient for ensuring the desired asymptotic tracking in the presence of repetitive system switchings.