Stabilization of oscillations through backstepping in high-dimensional systems

Abstract

This paper introduces a method for obtaining stable and robust self-sustained oscillations in a class of single input nonlinear systems of dimension n ≥ 2. The oscillations are associated to a limit cycle that is produced in a second-order subsystem by means of an appropriate feedback law. Then, the controller is extended to the full system by a backstepping procedure. It is shown that the closed-loop system turns out to be generalized Hamiltonian and that the limit cycle can be thought as born in a Hopf bifurcation after moving a parameter.Keywords: Non-linear Oscillations, Limit Cycle Stabilization, Backstepping Control, Generalized Hamiltonian Systems, Hopf Bifurcation. I. INTRODUCTION AND STATEMENT OF THE PROBLEMSelf-sustained oscillations are one of the distinctive behavioral characteristics of nonlinear systems. Whenever an oscillatory behavior is found or is to be built, there is or must be introduced an underlying nonlinearity. In this paper, a procedure to obtain a nonlinear feedback law that renders a class of single input cascade systems oscillatory is introduced. The oscillation is associated with a stable limit cycle and therefore it is self-sustained and robust. The method is based on matching the open-loop system to a closed-loop one that displays such a stable limit cycle. The feedback law is obtained in two steps. In the first step, a second-order subsystem is controlled to yield a robust nonlinear oscillator. To this end, a fourth degree polynomial Lyapunov function is introduced that guarantees the appropriate properties. Then, the cascade structure of the open-loop system allows us to apply backstepping to recursively obtain the feedback law for the full system. A very appealing byproduct is that the closed-loop system obtained has a generalized Hamiltonian structure [1].The problem considered here is, therefore, the synthesis of limit cycles and belongs to the class of so-called inverse problems in dynamical systems. Several authors have considered this problem in the past (see for instance [2], [3], [4] and references therein) by working with systems of moderate dimension. One of the interests of the algorithm proposed in this paper is its ability to cope with arbitrary dimensions. Related material can be found in [5].To set the problem under study in a precise form, consider the cascade systems for which the backstepping method is applicable. In particular, we will be concerned with the special class of strict-feedback systems [6] given by:

Inducing sustained oscillations in feedback-linearizable single-input nonlinear systems

“…Based on [50,51]), it is suggested that the limit cycle is related with the Lyapunov function. That is, having an admissible limit cycle, the Lyapunov function can be proposed as…”

“…Inspired by this motion-planning method, a similar approach was developed in [41] for a class of single-input nonlinear systems of arbitrary dimension with a cascade-based structure, this time with a backstepping procedure applied to extend the robust oscillatory behavior of a controlled system involving the first two states to the full-order system (the methodology is illustrated with the control of a third-order model of a magnetic levitation ball system). The same backstepping-based procedure is applied in [42] for the control of oscillations in DC-AC converters modeled in linear terms, and an adapted version is applied in [43] for the control of a nonlinear mathematical model of a DC-AC boost converter.…”

Construction of control Lyapunov functions for damping stabilization of control affine systems

“…Moreover, a local Lyapunov function for the closed-loop system can be computed using V = (Hω), with ω given by (24), if the anti-exact part ω 0,a = H dω vanishes only at the origin.…”

“…Application of stabilization using first integrals of mass-action systems (and more generally positive systems) was presented in [23]. Stabilization of periodic orbits was also considered in [24] using backstepping.…”

Autonomous Oscillation Generation in the Boost Converter

“…If a control law is able to produce such a limit cycle, the generation of alternating current will be accomplished without the need for introducing any time dependent reference signal (tracking method). The generation of limit cycles for producing self-oscillations has been successfully applied to electro-mechanical systems Gómez-Estern et al, 2002;Gordillo et al, 2002;Aracil et al, 2004). In this idea was used to generate oscillations in a three-phase DC/AC converter, which was modelled using a linear model.…”