Switching Dynamical Systems Research Articles

Period increment cascades in a discontinuous map with square-root singularity

We consider a discontinuous map with square-root singularity, which is relevant to many physical systems. Such maps occur in modeling grazing-sliding bifurcations in switching dynamical systems, or if the Poincare plane coincides with the switching plane. It is shown that there are notable differences in the bifurcation scenarios between this type of discontinuous map and a continuous map with square-root singularity. We determine the bifurcation structures and the scaling constant analytically. A different kind of period increment is observed, and the possibility of breakdown of period increment cascade is detected. Finally, we show that a system of piecewise smooth ordinary differential equations can exhibit the same type of bifurcation behavior.

Period Increment Cascades in a Discontinuous Map with Square-Root Singularity

In this paper, we consider a discontinuous map with square-root singularity, which is relevant to many physical systems. Such maps occur especially in switching dynamical systems if the Poincaré plane coincides with the switching plane. It is shown that there are notable differences in the bifurcation scenarios between this type of discontinuous map and a continuous map with square-root singularity. We determine the bifurcation structures and the scaling constant analytically. A different kind of period increment is observed, and the possibility of breakdown of period increment cascade is detected for first time. Finally, we show that from a piecewise ordinary differential equation the same type of map can also be obtained.

Synchronization and Hyperchaos in Switched Dynamical Systems Based on Parallel Buck Converters

This paper studies switched dynamical systems based on a simplified model of two-paralleled dc-dc buck converters in current mode control. In the system, we present novel four switching rules depending on both state variables and periodic clock. The system has piecewise constant vector field and piecewise linear solutions: they are well suited for precise analysis. We then clarify parameter conditions that guarantee generation of stable 2-phase synchronization and hyperchaos for each switching rule. Especially, it is clarified that stable synchronization is always possible by proper use of the switching rules and adjustment of clock period. Presenting a simple test circuit, typical phenomena are confirmed experimentally.

A discrete-time sliding window observer for Markovian switching system

In this paper, a fault detection method is developed for switching dynamic systems. These systems are represented by several linear models, each of them being associated to a particular operating m...

The nature of the normal form map for soft impacting systems

Soft impacting mechanical systems—where the impacting surface is cushioned with a spring–damper support—are common in engineering. Mathematically such systems come under the description of switching dynamical systems, where the dynamics toggle between two (or more) sets of differential equations, determined by switching conditions. It has been shown that the Poincaré map of such a system would have a power of 1/2 (the so-called square-root singularity) if the vector fields at the two sides of the switching manifold differ, and a power of 3/2 if they are the same. These results were obtained by concentrating on the leading order terms in a Taylor expansion of the zero-time discontinuity map, and are true in the immediate neighbourhood of a grazing orbit. In this paper we investigate how the character of the two-dimensional map changes over a large parameter range as the system is driven from a non-impacting orbit to an impacting orbit. This study leads to vital conclusions regarding the character of the normal form of the map not only in the immediate vicinity of the grazing orbit, but also away from it, as dependent on the system parameters. We obtain these characteristics by experiment and by simulation.

Character of the map for switched dynamical systems for observations on the switching manifold

It has been proposed to obtain the discrete-time models of switching dynamical systems by observing the states at the switching instants. Apart from the lowering of dimension, such switching maps or impact maps offer advantage in modeling systems that exhibit chattering. In this Letter we derive the nature of the switching map for the special case of grazing orbits. We show that the map is discontinuous in the neighborhood of a grazing orbit, and that it has a square root slope singularity on one side of the discontinuity. We illustrate the above by obtaining the switching maps for two example systems: the Colpitt's oscillator in the electrical domain and the soft impact oscillator in the mechanical domain.

Switched Dynamical Systems for Reduced-Order Flow Modeling

We develop a family of reduced-order models for the modeling and control of a flow system operating under varying fluid and actuation parameters. The parametric subspaces of the reduced-order models are formed by considering the angle between the reduced-order subspaces that span the velocity fields. This grouping is combined with a discrete switching law to form a switched dynamic system composed of a set of reduced-order models. This methodology is applied to the modeling of a lid-driven cavity under varying translation velocities and phase differentials between the upper and lower walls. It is shown that the subspace angle metric successfully partitions the parametric space and provides insight on the dominant parameters that characterize the flowfields. An open-loop simulation of the resulting switched dynamic system demonstrates its ability to capture the evolution of the flow and input parameters as it occurs in the full-order model.

Stability Analysis of Switched Dynamical Systems with State-Space Dilation and Contraction

This paper considers switched dynamical systems with state-space dilation and contraction formed by concatenating the states of a set of local dynamical systems or semi-flows on state spaces with different dimensions at specified time-instants. These systems arise naturally in many aerospace applications such as multi-body dynamic systems involving changes in the degrees of freedom, and systems composed of multiple spacecraft with docking and undocking capabilities flying in formation. The notions of stability of invariant sets in the sense of Lyapunov for usual dynamical systems are extended to this class of switched dynamical systems. Also the notion of finite-time stability with specified bounds is introduced. Sufficient stability conditions are established for a special class of switched dynamical systems with state-space dilation and contraction. Their application to spacecraft formation stability analysis is discussed.

A DISCRETE-TIME SLIDING WINDOW OBSERVER FOR MARKOVIAN SWITCHING SYSTEM: AN LMI APPROACH

In this paper, a fault detection method is developed for switching dynamic systems. These systems are represented by several linear models, each of them being associated to a particular operating mode. To find the system in operating mode the proposed method is based on mode probabilities and on a new structure of discrete-time observer with a sliding window measurements. This observer results from a combination of a Finite Memory Observer (FMO) and a Luenberger Observer. The stability condition of the observer is formulated in terms of linear matrix inequalities (LMI) using a quadratic Lyapunov function. The method also uses a priori knowledge information about the mode transition probabilities represented by a Markov chain. The proposed algorithm is of supervised nature where the faults to be detected are a priori indexed and modelled. In this work, the method is applied for the fault detection of a linear system characterized by a model of normal operating mode and several fault models. A comparison with the Generalized Pseudo-Bayesian method shows the validity and some advantages of the suggested method.

FINITE MEMORY OBSERVER FOR SWITCHING SYSTEMS: APPLICATION TO DIAGNOSIS

In this paper, a fault detection method is developed for switching dynamic systems with unknown inputs. These systems are represented by several linear models, each of them being associated to a particular operating mode. The proposed method is based on Finite Memory Observers and mode probabilities with the aim of finding the system operating mode and estimating the unknown input. The method also uses a priori knowledge information about the mode transition probabilities represented by a Markov chain. The proposed algorithm is of supervised nature where the faults to be detected are a priori indexed and modelled. In this work, the method is applied for the fault detection of a linear system characterized by a model of normal operating mode and several fault models. Then, it applied for fault detection in the case of a linear system with unknown input where state and unknown input estimation are done simultaneously. A comparison with the Generalized Pseudo-Bayesian method shows the validity and some advantages of the suggested method.

Bifurcation Analysis of Switched Dynamical Systems With Periodically Moving Borders

This paper describes a method for analyzing the bifurcation phenomena in switched dynamical systems whose switching borders are varying periodically with time. The type of systems under study covers most of power electronics circuits where two or more dynamical systems are cyclically switched according to the interaction of the state variables and some periodically moving borders. In particular, the complex bifurcation behavior of a voltage feedback buck converter is studied in detail. The analytical method developed in this paper allows bifurcation scenarios to be clearly revealed in any chosen parameter space.

Dissipativity theory and stability of feedback interconnectionsfor hybrid dynamical systems

In this paper we develop a unified dynamical systems framework for a general class of systems possessing left‐continuous flows; that is, left‐continuous dynamical systems. These systems are shown to generalize virtually all existing notions of dynamical systems and include hybrid, impulsive, and switching dynamical systems as special cases. Furthermore, we generalize dissipativity, passivity, and nonexpansivity theory to left‐continuous dynamical systems. Specifically, the classical concepts of system storage functions and supply rates are extended to left‐continuous dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous‐time dynamics, and dissipated energy over the resetting events. Finally, the generalized dissipativity notions are used to develop general stability criteria for feedback interconnections of left‐continuous dynamical systems. These results generalize the positivity and small gain theorems to the case of left‐continuous, hybrid, and impulsive dynamical systems.

Periodic solutions of a switching dynamical system in the plane

The non linear differential equation is considered. We investigate the number of nontrivial periodic solutions of this equation depending on k ≥ 0 for µ < 0 fixed. The results obtained can be used in a problem of time optimal control for the Van der Pol equation in order to describe the process of control.