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In this paper, hybrid dynamical systems theory is applied to the analysis and control of switched converters with Pulse-Width Modulated (PWM) inputs. The system is described by a state-space model with continuous flows and discrete jumps, without averaged equations. The modulation effects are captured in full without using time-dependent signals, by enlarging the state vector to include the PWM waveform generation process. Furthermore, the sample-and-hold mechanism associated with the sampling frequency is also taken into account with this approach. A control law is proposed based on a Lyapunov function candidate. Furthermore, convergence sets and the steady state jitter, inherent to PWM-based controllers, are analyzed estimating limit sets for the augmented state. Consequently, output chattering can be bounded. By using hybrid dynamical system theory, the control designer gains a deeper understanding of the effect of modulation in the closed-loop dynamics, avoiding the problems associated with the use averaged models. Experimental results validate the proposed method. Agence Nationale de la Recherche(Francia) ANR-18-CE40-0022-01 Agencia Estatal de Investigación (AEI) DPI2016-75294-C2-2-R Agencia Estatal de Investigación (AEI) PID2019-105890RJ-100 PID2019-109071RB-I00 Junta de Andalucía (FEDER) US-1264655 Electronic Components and Systems for European Leadership (ECSEL) 783158

International audience; This paper deals with the design of new periodic switching control laws for high frequency DC-DC converters. The contributions are twofolds. On a first hand, the DC-DC converter model is rewritten as a periodic switched affine system thanks to a $\delta$-operator formulation, which represents an efficient framework for the numerical discretization at high frequencies. On a second hand, three different control laws are provided, the first one being the usual Lyapunov-based control law and the two others being relaxed versions of this first solution. The benefits of these two new control laws over the usual Lyapunov-based one are demonstrated on a simple example.

This paper deals with the design of new periodic switching control laws for high frequency DC-DC converters. The contributions are twofolds. On a first hand, the DC-DC converter model is rewritten as a periodic switched affine system thanks to a $\delta$ -operator formulation, which represents an efficient framework for the numerical discretization at high frequencies. On a second hand, three different control laws are provided, the first one being the usual Lyapunov-based control law and the two others being relaxed versions of this first solution. The benefits of these two new control laws over the usual Lyapunov-based one are demonstrated on a simple example. More particularly, it is shown that the selection of the sampling period and of the control law strongly influence the size of the region of attraction.

The conventional switching strategy for solving the inverted pendulum control problem is based on two steps: swinging-up and stabilization. In this note, first, a new strategy for swinging the Furuta pendulum up towards the desired upright position is designed using the Speed-Gradient method, which uses only directly measured coordinates. Then, a nonlinear controller, based on the Forwarding approach, stabilizes the upright position. As a new contribution the latter leads to a nonlinear stabilizer around the upright position, whose Lyapunov function yields a larger size estimation of the domain of attraction than the one obtained with the traditional linear approach. This estimation allows us to use it in a global switching strategy in the practical implementation and guarantees almost-global asymptotic stability of the equilibrium. Successful experimental results are reported with the available laboratory Furuta pendulum.

SUMMARY A 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.

In this paper, periodic event-triggered controllers are proposed for the rotary inverted pendulum. The control strategy is divided in two steps: swing-up and stabilization. In both cases, the system is sampled periodically but the control actions are only computed at certain instances of time (based on events), which are a subset of the sampling times. For the stabilization control, the asymptotic stability is guaranteed applying the Lyapunov–Razumikhin theorem for systems with delays. This result is applicable to general linear systems and not only to the inverted pendulum. For the swing-up control, a trigger function is provided from the derivative of the Lyapunov function for the swing-up control law. Experimental results show a significant improvement with respect to periodic control in the number of control actions.

This paper deals with the problem of generating, by state feedback, stable oscillations in high order nonlinear systems. The desired oscillations are robust in the presence of disturbances, such as unmodelled dynamics and bounded noise signals, which result in bounded deviations from the nominal target orbit. The method consists of two steps. First, a globally attractive oscillation is generated in a nominal second-order subsystem. Based on a partition of the state space and solving the Lyapunov equation on each part, a strict Lyapunov function is obtained that ensures exponential convergence, even in the presence of disturbances, to a ring-shaped region containing the target limit cycle. Then, the oscillation stabilizing controller and the strict Lyapunov function are extended to arbitrary order systems, via backstepping. Notwithstanding backstepping is intended for cascade systems, the acquired ability to deal with unmodelled dynamics permits the analytical treatment of non-triangular structures, as is illustrated with the Ball and Beam example. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper, a new control law for the pendulum on a cart is proposed. The novelty of the new control law is in the combination of methods that is used to obtain it. First, a control law for transforming the pendulum subsystem in a Hamiltonian system is obtained. The resulting Hamiltonian function has a maximum at the desired position. The objective of the second step is to inject energy (so the pendulum subsystem tends to the maximum of the Hamiltonian) at the same time that the cart is stopped. It is shown that both objectives can be solved using forwarding, which in this case yields solvable partial differential equations.

In this paper we introduce a method for generating stable and robust oscillations in nonlinear systems. The oscillations are associated to a limit cycle that is produced in a second order subsystem. Then the controller is extended to the full system by backstepping. A remarkable property of the introduced method is that it leads to a generalized Hamiltonian structure and to a Lyapunov function which guarantees the stability and robustness of the system

The appearance of self-sustained oscillations in high performance AC drives, and in particular, in induction motors whose speed is regulated with the industry standard field-oriented control, is a well-documented but little understood phenomenon. It is well known that the oscillations may be quenched by returning the outer-loop PI speed control, but no precise rules to carry out this task, which may be time consuming, are known. In this paper, we show that these oscillations may arise due to the existence of Hopf bifurcations. Some simple rules for selecting the gains of the PI controller are obtained as a result of our analysis.

This paper addresses the problem of obtaining robust and stable oscillations in an electromechanical system. These oscillations are associated to a limit cycle that is born through a supercritical Hopf bifurcation. The method proposed in the paper works well for fully actuated systems, and even for certain underactuated ones. In order to illustrate the method, we have chosen an underactuated system that is well known in the literature and in control systems laboratories: the inverted pendulum. Actual stable and robust oscillations have been obtained experimentally in a rotating Furuta pendulum.

In this paper, a supervisory controller for MIMO fuzzy control systems is presented. It is proven that this controller is able to guarantee global stability in the Lyapunov sense in spite of uncertainties in the plant. After the theoretical considerations, the usefulness of the supervisor is illustrated with the application to the control problem of a waste-water treatment plant.

In this paper, the stability of a three phase two-level rectifier is studied. This type of power converter has two operation modes, as rectifier or as inverter. The dynamics of the rectifier consist of a set of equations highly nonlinear. Through the definitions of the instantaneous active and reactive powers, the model of the system is sometimes expressed in terms of power instead of current. Since this new model takes the form of a standard singular perturbation model, the system can be studied in separate time scales. This work propose an analysis of the system in two-time-scale based on the decomposition of the model into reduced (slow) and boundary-layer (fast) models where both should satisfy some conditions to prove the stability of the equilibrium point of the whole system. This analysis will show the stability of the equilibrium of the system via Lyapunov's method and, consequently, will eliminate stiffness difficulties for the implementation of a future controller.

This paper presents the design of a nonlinear control law for two-wheeled self-balanced vehicles. The design is based on forwarding and gives a Lyapunov function that allows us to obtain an estimation of the domain of attraction for the resultant law.

Es el primer envío que hago. Le agradeceré si me confirma que es todo correcto, especialmente el cumplimiento de los derechos de propiedad de la editorial. Mi email es gordillo@us.es The paper presents a new family of controllers for swinging up a pendulum. The swinging up of the pendulum is derived from physical arguments based on two ideas: shaping the Hamiltonian for a system without damping; and providing damping or energy pumping in relevant regions of the state space. A family of simple smooth controllers without switches with nice properties is obtained. The main result is that all solutions that do not start at a zero Lebesgue measure set converge to the upright position for a wide range of the parameters in the control law. Thus, the swing-up and the stabilization problems are simultaneously solved with a single, smooth law. The properties of the solution can be modified by the parameters in the control law. Control signal saturation can also be taken into account using the Hamiltonian approach. MCyT-FEDER DPI2006-07338

Limit cycles analysis of feedback systems with rate limiters in the actuator can be implemented by a classical method in the frequency domain, the harmonic balance method. In this paper, the rate limiter describing function is obtained and applied to the search for limit cycles in such control systems. Three examples with three different bifurcations (saddle-node bifurcation of limit cycles, subcritical Hopf bifurcation at infinity and supercritical Hopf-like bifurcation) are included. The method is approximate but its main advantage is that intuition is gained into a difficult problem.

International audience; In this paper is estimated the region of attraction of the nonlinear boost DC-AC converter by Sum of Squares. This is an optimization problem of maximization of the Lyapunov surface subjects to certain strict constraints due to saturation of the control variables of the system and to the composition of the inverter, which is formed from two DC-DC converters with positive voltage.

Generalized Hamiltonian systems (GHS) are a class of nonlinear systems of significant interest in control theory, due to the passivity property and the particular structure of the state equations. Control techniques that obtain, by state feedback, a closed-loop generalized Hamiltonian dynamics, are suitable for easy stabilization by passive output feedback, yielding dissipative systems. In Hamiltonian systems with dissipation, the conservative and dissipative parts are clearly identifiable, and the Lyapunov function is chosen as the closed-loop energy function or Hamiltonian. A deeper insight suggest that this convenient Hamiltonian form may also underlie other classical techniques. It has already been observed that some control methodologies preserve the Hamiltonian structure, as long as the open-loop dynamics is also Hamiltonian. This is the case of cascade controllers like backstepping. Whereas the relation between backstepping control and Hamiltonian structures has been recently reported, this works unveils the preservation of the Hamiltonian structure in a class of forwarding-controlled systems. The forwarding procedure is assumed to start from a Hamiltonian subsystem and after each step of the forwarding iteration, a new Hamiltonian structure is obtained, i.e., the GHS structure is preserved. Moreover, the conservative or dissipative nature of the original system is also preserved. The result can be extended to any number of iteration steps, and hence to arbitrary order systems.

In this paper, a newly developed method for generating oscillations in arbitrary order nonlinear systems, is redefined as an output feedback controller. In a recent publication, the state feedback orbitally stabilizing controller has been enhanced to provide strict Lyapunov functions for the oscillating behavior. This property is exploited here in order to prove global asymptotic stability of the output feedback controller in systems up to degree four, and global asymptotic stability with limited initial estimation error in higher–order systems. The output feedback controller is then applied to a fourth–order cascade LC network, while the result may be extrapolated to similar circuits with arbitrary number of LC blocks, which may be regarded as discrete approximations of transmission lines.

This chapter reviews the theory of Hopf bifurcations and bifurcation is detected with standard tangent approximation analysis. It describes simulations that corroborate the theoretical predictions and a complementary analysis of bifurcation detection via harmonic balance. The chapter shows that self-sustained oscillations in indirect field-oriented control for induction motors may be due to the appearance of a Hopf bifurcation. Field-oriented control (FOC) is the standard for high dynamic performance induction motor drives due to its high reliability. Historically, this remarkable controller was derived as a result of physical intuition and a deep understanding of the machine operation, with little concern about a rigorous analytical study of its stability and performance. The conditions for the emergence of a Hopf bifurcation only predict when an oscillation appears. The Hopf bifurcation is the most interesting phenomenon regarding oscillations from the practical point of view.

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. Ministerio de Ciencia e Innovación (España)

This paper presents a design method for the stabilization of oscillations in a class of nonlinear systems. The method consists of two steps. In the first one, a second-order generalized Hamiltonian subsystem, which presents stable oscillations, is matched. In the second step, the controller is extended to the full system using backstepping. It is shown that the oscillations emerge through a supercritical Hopf bifurcation. The method proposed was first applied to a second order system. In this paper the method is extended to higher order systems by backstepping. The method is introduced by its application to an underactuated mechanical system: the ball and beam.

In this paper we introduce a new control method for uninterruptible power supplies (UPS) without using reference signals. This is accomplished by a feedback law associated to a Lyapunov function which guarantees the stability and robustness of the system. Furthermore an adaptive mechanism is included to cope with changes in the load. Simulations show that the performance of the proposed system meets the requirements.

This paper presents the application of a multivariable frequency response technique to study the stability of autonomous vehicles when tracking a path. Particularly, the stability of a path tracking control loop with a fuzzy controller is considered. The technique is based on the analysis of the solutions of the harmonic balance equation. Necessary and sufficient conditions for stability areobtained by examining a family of characteristic loci parameterized in the amplitude. The method can be applied when pure delays exist in the path tracking control loop. Several simulations experiments have been done. These experiments show a Hopf bifurcation induced by a pure delay, and an unstable limit cycle around a stable origin.

In this paper we explore the occurrence of zero-Hopf bifurcation in indirect field oriented control (IFOC) of induction motors. Codimension-two Bogdanov-Takens bifurcations as well as codimension-one saddle-node and Hopf bifurcations have been reported earlier. In this paper the occurrence of codimension-two zero-Hopf bifurcation in IFOC is analyzed and proven to occur under certain parameter settings. Bifurcations diagrams are also reported illustrating such bifurcations for a particular IFOC drive.

In this paper we introduce a control method for generation of oscillations in electronic converters. The main novelty of the new control strategy is the lack of reference signals. This is accomplished by a feedback law associated to a Lyapunov function which guarantees the stability and robustness of the system. The method is illustrated by means of a boost converter system. Simulations show the performance of the closed-loop systems.

An asymptotical stability analysis is conducted using Lyapunov stability methods to construct a composite Lyapunov function candidate in a step-by-step way extending the procedures defined in [Kokotovic et al., 1986] for the two-time-scale singular perturbation problems, to the three-time-scale singular perturbation problem of a Radio/Control helicopter on a platform.

The paper presents a new family of strategies for swinging up a pendulum. They are derived from physical arguments based on two ideas: shaping the Hamiltonian for a system without damping; and providing damping or energy pumping in relevant regions. A two-parameter family of simple strategies without switches with nice properties is obtained. The main result is that all solutions that do not start at a zero Lebesgue measure set will converge to the upright position for a wide range of the parameters in the control law. Thus, the swing-up and the stabilization problems are simultaneously solved with a single, smooth law.

This paper deals with the problem of generating stable and robust oscillations in triangular nonlinear systems. The proposed method consists of two steps. First, a globally attractive oscillation is generated in a nominal second-order subsystem. Based on a partition of the state space and solving the Lyapunov equation on each part, a strict Lyapunov function is obtained that ensures exponential convergence to a ring-shaped region containing the target limit cycle. Then, the nominal stabilizing controller and the strict Lyapunov function are extended to arbitrary order systems, via a method in the essence of backstepping. Thanks to the strict Lyapunov functions, a new ability to deal with unmodeled dynamics is acquired, and the original controller is easily extended to quasi-triangular structures.