Stabilizing unstable periodic orbits with delayed feedback control in act-and-wait fashion

Abstract

A delayed feedback control framework for stabilizing unstable periodic orbits of linear periodic time-varying systems is proposed. In this framework, act-and-wait approach is utilized for switching a delayed feedback controller on and off alternately at every integer multiples of the period of the system. By analyzing the monodromy matrix of the closed-loop system, we obtain conditions under which the closed-loop system's state converges towards a periodic solution under our proposed control law. We discuss the application of our results in stabilization of unstable periodic orbits of nonlinear systems and present numerical examples to illustrate the efficacy of our approach.

Stabilization of chaotic systems via act-and-wait delayed feedback control using a high-precision direct integration method

“…The T-periodic matrixes A(t), B(t) are the Jacobian matrixes of the system evaluated on the UPO such that, Here, note that the linearized version of the closed-loop system (5) is locally asymptotically stable at the UPO. Moveover, system (5) is similar to a linear periodic timevarying system such that [54] where x(t) ∈ ℜ n is the state vector. From Eqs.…”

“…The act-and-wait control method is an effective technique to reduce the number of poles for systems with large feedback delay, which makes the pole placement problem easier. In recent years, act-and-wait concept has been further tested through experiments [49] and theoretically extended to discrete-time systems [50], autonomous systems [51], non-autonomous dynamical systems [52], linear periodic time-varying systems [53,54], etc. [55][56][57][58].…”

Simple Kinematic Feedback Enhances Autonomous Learning in Bio-Inspired Tendon-Driven Systems

“…Thus, here we explore the combination of data-efficient learning algorithm, G2P, with simple feedback to maintain the key benefits of learning under limited experience while improving performance and robustness to perturbations (or unmodeled dynamics) as needed. This approach is directly inspired by biological systems that, under certain circumstances, successfully use simple corrective responses triggered by delayed and non-collocated sensory signals [21], [22], [23].…”

Oscillating delayed feedback control schemes for stabilizing equilibrium points

“…Some approaches focused on overcoming these difficulties are based on periodic control gain ([16]) or “act-and-wait” concept introduced by Insperger ([17], [18]) These methods are characterized by alternately applying and cutting off the controller in finite intervals yielding to a finite-sized monodry matrix of the closed system so the linear stability of the UPO may be enhanced by an appropriate choosing of the control parameters. Act-and-wait approach has been used together with DFC for stabilizing unstable equilibrium points ([19]), for stabilizing UPO's of nonautonomous systems ([20]) and, of autonomous systems ([21], [22]).…”

“…These methods are caracterized by alternaly applying and cutting off the controller in finite intervals yielding to a finite-sized monodry matrix of the closed system so the linear stability of the UPO may be enhanced by an appropiate choosing of the control parameters. Act-and-wait approach has been used together with DFC for stabilizing unstable equilibrium points ( [8]), for stabilizing UPO's of nonautonomous systems ( [21]) and, of autonomous systems ( [22], [3]). For stabilizing equilibrium points, a delayed feedback controller is derived in [7] that overcomes the drawbacks of DFC, providing a systematic procedure of its design.…”