Piecewise Linear Dynamical Systems Research Articles

Computing regions of attraction with polytopes: Planar case

An algorithm that estimates the region of attraction of the origin using a convex polytope is developed for piecewise-linear dynamical systems. This is equivalent to the problem of computing a Lyapunov function. The R2 case is studied here for simplicity.

Elementary canonical state models of Chua's circuit family

Two simple state models of the third-order autonomous piecewise-linear (PWL) dynamical system, topologically conjugate to Chua's circuit family, are proposed. Unlike the known canonical systems they are canonical also with respect to the relation between their parameters and the corresponding eigenvalues, i.e., their state equations contain minimum nonzero coefficients directly determined by the equivalent eigenvalue parameters associated with the two regions of PWL vector field in R/sup 3/. The corresponding circuit models containing integrators and adders are introduced. The mutual linear conjugacy between these two elementary canonical state models and their relation with Chua's circuit family is suggested. The computer generated chaos for the double-scroll attractor using one of the developed models is shown.

Analyzing piecewise linear dynamical systems

This article presents a method of analyzing mixed logic/dynamic systems. The article describes the development of a computational tool for the analysis of piecewise linear (PL) dynamical systems. Unfortunately, many PL controllers are developed from ad hoc "intelligent systems" ideas which do not aim or allow the associated dynamic behavior to be predicted. An example of such a system is the anti-skid braking system in a car, where the controller is rule-based and designed using the engineer's knowledge of the system. The only current viable approach to testing such a system is by using extensive simulation and prototype testing, which must be repeated for each of the different car models on which it is installed. Anything that provides insight into the logic and dynamic interaction of such a system would be useful: hence the development of the work in this article. Similarly, systems with programmable logic controllers and gain schedulers also fall into the class of piecewise linear systems. The authors take ideas and known results from linear systems, convex set theory, and computational geometry and synthesize them to create an analysis tool for studying a class of systems that mix logic and dynamics. We choose to develop a computational analysis tool primarily because the traditional theoretical analysis of piecewise linear processes is intractable, except, that is, for certain local dynamic behavior.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An analysis method for piecewise linear dynamical systems

A wide range of nonlinear dynamical systems contain abrupt changes in system behaviour and are often controlled using some kind of logic such as a rule-base. These changes in the system dynamics or controller actions make such systems intractable to conventional methods of systems analysis. By assuming all system nonlinearities can be represented as piecewise linear functions, a graphical analysis technique reminiscent of a phase portrait is developed for this class of systems and controllers. The state space of the system model is mapped into a directed node graph that captures the possible dynamic paths through the system. The paths found are directly related to the piecewise linear system functions or logic rules that created them. This node graph provides a graphical representation of the system that, like the phase portrait, captures the global dynamics of the system. The major difference is that the node graph generalises the phase portrait idea to systems of arbitrary dimensions.

Three steps to chaos. I. Evolution

In this two-part exposition of oscillation in piecewise-linear dynamical systems, we guide the reader from linear concepts and simple harmonic motion to nonlinear concepts and chaos. By means of three worked examples, we bridge the gap from the familiar parallel RLC network to exotic nonlinear dynamical phenomena in Chua's circuit. Our goal is to stimulate the reader to think deeply about the fundamental nature of oscillation and to develop intuition into the chaos-producing mechanisms of nonlinear dynamics. In order to exhibit chaos, an autonomous circuit consisting of resistors, capacitors, and inductors must contain i) at least one nonlinear element ii) at least one locally active resistor iii) at least three energy-storage elements. Chua's circuit is the simplest electronic circuit that satisfies these criteria. In addition, this remarkable circuit is the only physical system for which the presence of chaos has been proven mathematically. We illustrate by theory, simulation, and laboratory experiment the concepts of equilibria, stability, local and global behavior, bifurcations, and steady-state solutions. >

Three steps to chaos. II. A Chua's circuit primer

For Pt. I see ibid., vol. 40, no. 10, p. 640, 1993. Linear system-theory provides an inadequate characterization of sustained oscillation in nature. This two-part exposition of oscillation in piecewise-linear dynamical systems, guides the reader from linear concepts and simple harmonic motion to nonlinear concepts and chaos. By means of three worked examples, the authors bridge the gap from the familiar parallel RLC network to exotic nonlinear dynamical phenomena in Chua's circuit. The goal is to stimulate the reader to think deeply about the fundamental nature of oscillation and to develop intuition into the chaos-producing mechanisms of nonlinear dynamics. In order to exhibit chaos, an autonomous circuit consisting of resistors, capacitors, and inductors must contain (1) at least one nonlinear element, (2) at least one locally active resistor, and (3) at least three energy-storage elements. Chua's circuit is the simplest electronic circuit that satisfies these criteria. In addition, this remarkable circuit is the only physical system for which the presence of chaos has been proved mathematically. The circuit is readily constructed at low cost using standard electronic components and exhibits a rich variety of bifurcations and chaos. Part II in this series studies bifurcations and chaos in a robust practical implementation of Chua's circuit.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

CMOS design of chaotic oscillators using state variables: a monolithic Chua's circuit

This paper presents design considerations for monolithic implementation of piecewise-linear (PWL) dynamic systems in CMOS technology. Starting from a review of available CMOS circuit primitives and their respective merits and drawbacks, the paper proposes a synthesis approach for PWL dynamic systems, based on state-variable methods, and identifies the associated analog operators. The GmC approach, combining quasi-linear VCCS's, PWL VCCS's, and capacitors is then explored regarding the implementation of these operators. CMOS basic building blocks for the realization of the quasi-linear VCCS's and PWL VCCS's are presented and applied to design a Chua's circuit IC. The influence of GmC parasitics on the performance of dynamic PWL systems is illustrated through this example. Measured chaotic attractors from a Chua's circuit prototype are given. The prototype has been fabricated in a 2.4- mu m double-poly n-well CMOS technology, and occupies 0.35 mm/sup 2/, with a power consumption of 1.6 mW for a +or-2.5-V symmetric supply. Measurements show bifurcation toward a double-scroll Chua's attractor by changing a bias current. >

Qualitative Modelling and Simulation by Piecewise Linear Analysis

Since applications of expert systems were typical in domains with no well defined models, qualitative methods for modelling and reasoning were soon developed. Most current qualitative reasoning programs derive the qualitative behaviour of a system by simulating a hand-crafted qualitative version of the differential equation that caracterizes the model of a system. This paper describes a method to construct a family of piecewise linear dynamical systems from the qualitative information contained in a model. We apply the dynamical system theory to deduce results about the behaviour of the family of dynamical systems constructed as a consequence of the qualitative model.

Qualitative modelling and simulation by piecewise linear analysis

Since applications of expert systems were typical in domains with no well defined models, qualitative methods for modelling and reasoning were soon developed. Most current qualitative reasoning programs derive the qualitative behaviour of a system by simulating a hand-crafted qualitative version of the differential equation that caracterizes the model of a system. This paper describes a method to construct a family of piecewise linear dynamical systems from the qualitative information contained in a model. We apply the dynamical system theory to deduce results about the behaviour of the family of dynamical systems constructed as a consequence of the qualitative model.

Linear Markov approximations of piecewise linear stochastic systems

This paper is concerned with the properties of piecewise linear discrete-time dynamic systems driven by white Gaussian noise. The properties of the deterministic system are explored, and condition for the existence of invariant distributions are derived. The existence of an invariant distribution was then used to justify the approximation of the stochastic system by a switched Markov linear model if the piecewise linear regions are large “contracting” ones or small “expanding” ones relative to the input noise variance. The approach is expected to be useful for constructing approximate nonlinear filtering schemes for such systems

Analytical Properties of Poincaré Halfmaps in a Class of Piecewise-Linear Dynamical Systems

A piecewise-linear nerve conduction equation is investigated further. The theory of Poincare halfmaps induced by the flow of a three-dimensional linear saddle-focus is developed. Using a description of the dynamical system in diagonalized coordinates, a canonical formulation of two-dimensional halfmaps is found. This leads for each halfmap to an implicit scalar equation plus a side condition. The effect of the halfmaps on different types of invariant curves occurring is investigated. Thereby the capacity of the halfmaps to separate adjacent points (such that the images acquire a finite distance) is shown. Two of three possible mechanisms for separating points are investigated in detail. The regions in the canonical parameter space where the different separating mechanisms appear are indicated analytically. The possible appearance of chaotic solutions, at least in the neighborhood of homoclinic trajectories in state space, is demonstrated. The underlying separation mechanism is present also in regions of state space far from a homoclinic orbit.

Piecewise linear system identification via Walsh functions

This paper presents an iterative level algorithm (ILA), via Walsh functions, for parameter identification in piecewise linear dynamical systems. The method is similar to the iterative shift algorithm (ISA) of Prasada Rao and Sivakumar (1976, 1979) suggested for time-lag systems. Certain non-linear systems are modelled better by piecewise linear elements and the computations in identifying the related parameters are simpler than those of analytic models.

Steady-state analysis of piecewise-linear dynamic systems

Piecewise-linear (PWL) systems form a subclass of the general class of nonlinear systems. However, because PWL functions are not differentiable everywhere, results derived for general differentiable systems do not generally apply to PWL systems. In this paper, the properties of the solutions of continuous PWL dynamic systems are investigated in detail and theorems on the continuity and differentiability of the solutions with respect to initial conditions are derived. The results obtained are applied in the study of the convergence properties of steadystate algorithms when applied to continuous PWL dynamic systems. It is found that under fairly mild conditions the convergence is of order two, which is the same order of convergence of the algorithms when applied to differentiable dynamic systems. It is also found, however, that there are special cases when the order of convergence may be reduced to one.

Stable Harmonic and Subharmonic Oscillations of a System With Piecewise Linear Restoring Forces

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.

On a phase automatic frequency control equation with a lag and a rectangular phase detector characteristic: PMM vol. 36, n≗3, 1972, pp. 551–555

A second-order piecewise-linear dynamic system with jumps in the representative point on the juncture lines is investigated on a cylindrical phase space.

Qualitative investigation of an equation of the theory of phase automatic frequency control: PMM vol. 33, n≗2, 1969, pp. 340–344

A second-order piecewise-linear dynamic system with jumps in the representative point on the juncture lines is investigated on a cylindrical phase space.

Application of the method of point-to-point mappings to the solution of problems of optimal control of piecewise-linear dynamic systems

Application of the method of point-to-point mappings to the solution of problems of optimal control of piecewise-linear dynamic systems

Investigation of a piecewise linear dynamical system with three parameters

Investigation of a piecewise linear dynamical system with three parameters