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Abstract-In this paper a control strategy for generation of alternating current without using any reference signal is applied to a nonlinear boost dc-ac converter. A Phase-Locked Loop is added to the control law in order to achieve synchronization between the two parts of the circuit. It is also shown that this idea is also valid for synchronization with the network. The resultant control laws are tested by means of simulations.

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:

In this paper the swing-up problem for the Furuta pendulum is solved applying Fradkov's speed-gradient (SG) method [1,9] to a dimension 4 model of the system. The new law is compared with the conventionalÅström-Furuta strategy, based on a dimension 2 model. A comparative analysis, including simulations and experiments, whereby the advantages and effectiveness of the new law for swinging the pendulum up are shown is included.

SUMMARYA three‐time scale singular perturbation control law is designed for a nonlinear helicopter model in vertical flight. The proposed control law is based on time scale decomposition and is able to achieve the desired altitude by selecting a desired angular velocity and the associated collective pitch angle of the blades. The stability of the system is performed by presenting a stability analysis for generic three‐time scale singularly perturbed systems, which allows to construct a composite Lyapunov function for the resultant closed‐loop system by using time scale separation and also providing mathematical expressions for the upper bounds of the singularly perturbed parameters that define the three‐time scale. Numerical results on both, the singular perturbation control strategy and the stability analysis, are also presented for the studied nonlinear highly coupled helicopter model. Copyright © 2012 John Wiley & Sons, Ltd.