Local Stability Of Limit Cycles Research Articles

Lyapunov based estimation of the basin of attraction of Poincare maps with applications to limit cycle walking

Limit cycles occur in a wide range of nonlinear and hybrid systems including dynamic walking. While methods for determining local stability of limit cycles are available, they are relevant only for small perturbations. Characterizing the robustness of limit cycles to larger perturbations is an important, yet mostly open, challenge. Robustness can be characterized by the basin of attraction (BA), but standard techniques for computing BAs are computationally intensive. Here we present an algorithm to estimate inner bounds of BAs of fixed points of polynomial discrete maps. The algorithm is based on a convex optimization problem known as Sum of Squares programming, and results in BA estimates which are expressed as sublevel sets of polynomial Lyapunov functions. The method can be applied to estimate BAs of fixed points of discrete Poincaré Maps and thus the BAs of limit cycles on selected Poincaré sections, as demonstrated on a simple biped walking model under different modes of actuation. For each mode of actuation, the Poincaré map was fitted using a small number of simulations and an inner bound for the BA was estimated. Comparing the BAs achieved by the different modes of actuation demonstrates that the BA of a passive biped model can be enlarged using minimalistic event-driven actuation, and that this relationship is also apparent from the estimated inner bounds of the BAs.

Parametrically excited viscoelastic beam-spring systems: nonlinear dynamics and stability

The aim of the investigation described in this paper is to study the nonlinear parametric vibrations and stability of a simply-supported viscoelastic beam with an intra-span spring. Taking into account a time-dependent tension inside the beam as the main source of parametric excitations, as well as employing a two-parameter rheological model, the equations of motion are derived using Newton\'s second law of motion. These equations are then solved via a perturbation technique which yields approximate analytical expressions for the frequency-response curves. Regarding the main parametric resonance case, the local stability of limit cycles is analyzed. Moreover, some numerical examples are provided in the last section.

Limit cycles induced in type-1 linear systems with PID-type of relay feedback

This article investigates the limit cycles within type-1 linear systems under PID-type of relay feedback. The problem is generalised from the identification of friction models of servo mechanical systems via limit-cycle experiments under dual-channel relay feedback. Locations of limit cycles are given so that the exact durations between two consecutive switchings of relays can be determined via numerical computation. After this, local stability of limit cycles can be checked via the Jacobian of Poincaré map. Examples are analysed using proposed theorems.

Local stability of limit cycles for MIMO relay feedback systems

This paper concerns with the local stability of limit cycles for decentralized relay feedback systems. It presents a sufficient condition for the local stability based on the well-known Poincare map method. The effectiveness of the presented result is illustrated by a numerical example.

Local stability of limit cycles for time-delay relay-feedback systems

This brief is concerned with the local stability of limit cy- cles for linear systems under relay feedback, for the cases where the linear system includes a time-delay in its dynamics and the relay can possess asym- metric hysteresis. The limit cycle considered can be asymmetric, have more than two switchings a period, zero output derivatives at the switching in- stants. It shows that if a certain constructed matrix is Schur stable, then, the local stability of the considered limit cycle is guaranteed. The effective- ness of the presented results is illustrated by a numerical example.

Global stability of relay feedback systems

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions.

On the local stability of limit cycles.

Orbital stability of limit cycles is the result of the competing local tendencies of perturbations from the cycle to decay (during phases of local stability) and to grow (during phases of local instability), averaged over a cycle. We examine this coexistence of attractive and repulsive phases on limit cycles, including the local rates of expansion and contraction of phase space volumes. This is done in a frame of reference that moves along the orbit, to partially decouple motions tangential and perpendicular to the cycle. Dynamical systems used for illustration are the generalized Bonhoeffer-van-der-Pol and Rossler models, both far from and near to different types of bifurcations. Finally, it is shown that the nonuniformity of local stability in phase space affects the response of limit cycle oscillators to perturbations and gives rise to their phase-dependent response. (c) 1999 American Institute of Physics.