Families of Lyapunov functions for nonlinear systems in critical cases

Fu, Jyun-Horng; Abed, Eyad H.

doi:10.1109/cdc.1990.203818

Cited by 11 publications

(12 citation statements)

References 11 publications

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“…Proposition 2.1: [5] For the real variables and , the scalar bivariate function is locally negative definite at a small neighborhood if the coefficients , and , where stands for the fifth and higher order terms of . For presentation simplicity, only local state feedback stabilizing bifurcation controllers [1], [6] will be considered to illustrate the ideas in this brief.…”

Section: Local Gain Analysismentioning

confidence: 99%

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Local Robustness of Hopf Bifurcation Stabilization

Yang

Chen

2009

IEEE Trans. Circuits Syst. II

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Local robust analysis via 2 gain method is presented for a class of Hopf bifurcation stabilizing controllers. In particular, we first construct a family of Lyapunov functions for the corresponding critical system, then derive a sufficient condition to compute the 2 gain by solving the Hamilton-Jacobi-Bellman (HJB) inequalities. Local robust analysis can be conducted through computing the local 2 gain achieved by the stabilizing controllers at the critical situation. The theoretical results obtained in this brief provide useful guidance for selecting a robust controller from a given class of stabilizing controllers under Hopf bifurcation. As an example, application to a modified Van der Pol oscillator is discussed in details.

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Section: Local Gain Analysismentioning

confidence: 99%

“…For Lyapunov functions, the result in [5] is applied. We next introduce a proposition that will be useful to show the local definiteness of the time derivative of Lyapunov function.…”

Section: Local Gain Analysismentioning

confidence: 99%

Local Robustness of Hopf Bifurcation Stabilization

Yang

Chen

2009

IEEE Trans. Circuits Syst. II

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“…Although the linear equations play an important role in the stability analysis of periodic systems, the linearised system fails to provide answers to many questions associated with the nonlinear periodic systems. Particularly, the information obtained from linearization procedure is inconclusive when a critical eigenvalue is present [6].…”

Section: Fuzzy Logic Controlmentioning

confidence: 99%

Fuzzy-logic control of parametrically excited impacting flexible system

Marghitu

Diaconescu

Boghiu

1998

Archive of Applied Mechanics (Ingenieur Archiv)

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In this article, the control of a repetitive impacting elastic link with parametrically excited base in rotational motion is considered. A fuzzy-logic controller is designed and employed to suppress the vibrations resulting after the impact with an external rigid body. The momentum balance method and an empirical coef®cient of restitution is used in the collision of the two bodies. The controller is applied successfully to reduce the vibrations of the parametrically excited impacting¯exible system. Simulations for several combinations of excitation and rotation parameters are provided. IntroductionTheories for rigid or elastic multibody systems have considerably in¯uenced progress in mechanical engineering. Within the past few years, robot and teleoperator systems utilizing mechanical manipulators have gained wide-spread acceptance in the area of manufacturing, especially with regard to parts handling and assembly. Manipulators are being used as orthopedic and orthotic devices. In environments where radioactive materials are present, master-slave manipulator systems are now a standard tool for safe handling of dangerous materials. Even the space shuttle craft is equipped with manipulators for use in parts handling in outer space.Impact dynamics of colliding bodies is one of the classical problems of mechanics, which appears in design of manipulators. Impact is also prominent in mechanical part feeding systems and pneumatic impact tools. The control is necessary to suppress the vibrations that appear during the motion of the robot arm and those resulted from collision with other external bodies.Both linear and nonlinear controllers were constructed, and different kinds of strategies were implemented for the vibrations control of systems with¯exible elements. Adaptive control algorithms and on-line parameter identi®cation techniques have been common features of the last generation controllers.The problem of controlling an elastic arm of two links based on variable structure system theory and pole assignment technique for stabilization was treated in [13]. This design approach was motivated by a simple observation that the nonlinearity in the dynamics of an elastic robotic system is essentially due to the rigid modes (joint angles), and, as the time derivatives of the rigid modes vanish, the remaining motion is only due to the elasticity. For the rigid modes, a sliding-mode controller was designed. The controller of the elastic modes was

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“…In view of their usefulness also for control purposes, see e.g. Blanchini (1995) and Clarke, Ledyaev, Rifford, and Stern (2000), many researches have been carried out recently on the explicit construction of Lyapunov functions; for critical systems recent results are reported in Fu (1992Fu ( , 2000a, Fu and Abed (1993) and Schwartz and Yan (1995) whereas other related results (some of which are concerned with robustness issues) are in Blanchini (1995), Parrilo (2000), Papachristodouiou and Prajna (2002), Chesi, Garulli, Tesi, and Vicino (2003), Gle´ria, Figueiredo, and Rocha Filho (2003), Chesi (2004), Bouzaouache and Braiek (2008) and Chesi and Hung (2008). In many cases, e.g.…”

Section: Introductionmentioning

confidence: 99%

Semi-invariants and their use for stability analysis of planar systems

Menini¹,

Tornambè²

2009

International Journal of Control

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Semi-invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvectoreigenvalue pair for linear systems, and are, therefore, very useful to depict the behaviour of the system. In this article, semi-invariants are used to construct explicitly Lyapunov functions useful for studying the stability of the origin, for continuous-time systems. Moreover, using well-known tools from differential geometry such as (orbital) symmetries, it is shown how semi-invariants can be found for several classes of systems. Important connections with centre manifold theory are pointed out. By using the proposed general techniques, a new proof of a result claimed to Bendixson (Bendixson, I. (1901), 'Sur les courbes de´finies par des e´quations diffe´rentielles', Acta Mathematics, 24, 1-88) is given.

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Families of Lyapunov functions for nonlinear systems in critical cases

Cited by 11 publications

References 11 publications

Local Robustness of Hopf Bifurcation Stabilization

Local Robustness of Hopf Bifurcation Stabilization

Fuzzy-logic control of parametrically excited impacting flexible system

Semi-invariants and their use for stability analysis of planar systems

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