Local feedback stabilization and bifurcation control, I. Hopf bifurcation

Abed, Eyad H.; Fu, Jyun-Horng

doi:10.1016/0167-6911(86)90095-2

Cited by 349 publications

(100 citation statements)

References 6 publications

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“…Other authors [5], [6] have studied the bifurcation control problem from a smooth feedback framework. Our approach is different since switching (and thus nonsmooth) control laws are proposed.…”

Section: Discussionmentioning

confidence: 99%

“…Thus, in the case where two or more different attractors exist, the controller objective is that of eliminating them, taming the system dynamics onto a desired stable equilibrium point. Many authors (see, for example, [5], [6] and the references therein) have studied the problem of controlling bifurcations within a smooth feedback framework. Examples include the method based on manifold reduction presented in [6] and the use of smooth nonlinear control laws discussed in [7] to tame a limit cycle occurring in a flutter problem.…”

Section: Introductionmentioning

confidence: 99%

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Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory

Angulo

Bernardo

Fossas

et al. 2005

IEEE Trans. Circuits Syst. I

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Abstract-In this paper, we present a method to control limit cycles in smooth planar systems making use of the theory of nonsmooth bifurcations. By designing an appropriate switching controller, the occurrence of a corner-collision bifurcation is induced on the system and the amplitude and stability properties of the target limit cycle are controlled. The technique is illustrated through a representative example.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory

Angulo

Bernardo

Fossas

et al. 2005

IEEE Trans. Circuits Syst. I

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“…with the components of Π [1] uniquely determined by (2.20). Now let us show that the orientation of the center manifold can be changed by changing K 1 in (2.10).…”

Section: Linear Center Manifoldmentioning

confidence: 99%

“…From Property P, we know that K 2 is such that σ(A+B 2 K 2 ) < 0. Moreover, we choose Π [1] 1 so that the quadratic part of the controlled center dynamics is zero; then we deduce K 1 from (3.2).…”

Section: Stabilization Using a Quadratic Feedback Consider The Quadrmentioning

confidence: 99%

The Controlled Center Dynamics

Hamzi¹,

Kang²,

Krener³

2005

Multiscale Model. Simul.

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Abstract. The center manifold theorem is a model reduction technique for determining the local asymptotic stability of an equilibrium of a dynamical system when its linear part is not hyperbolic. The overall system is asymptotically stable if and only if the center manifold dynamics is asymptotically stable. This allows for a substantial reduction in the dimension of the system whose asymptotic stability must be checked. Moreover, the center manifold and its dynamics need not be computed exactly; frequently, a low degree approximation is sufficient to determine its stability. The controlled center dynamics plays a similar role in determining local stabilizability of an equilibrium of a control system when its linear part is not stabilizable. It is a reduced order control system with a pseudoinput to be chosen in order to stabilize it. If this is successful, then the overall control system is locally stabilizable to the equilibrium. Again, usually low degree approximation suffices. 1. Introduction. Center manifold theory plays an important role in the study of the stability of dynamical systems when the equilibrium point is not hyperbolic. The center manifold is an invariant manifold of the differential equation which is tangent at the equilibrium point to the eigenspace of the neutrally stable eigenvalues. For instance, as the local dynamic behavior "transverse" to the center manifold is relatively simple since it is the one of the flows in the local stable (and unstable) manifolds, the center manifold method isolates the complicated asymptotic behavior by locating an invariant manifold tangent to the subspace spanned by the eigenspace of eigenvalues on the imaginary axis. In practice, one does not compute the center manifold and its dynamics exactly, since this requires the resolution of a quasi-linear PDE which is not easily solvable. In most cases of interest, an approximation of degree two or three of the solution is sufficient. Then we determine the reduced dynamics on the center manifold, study its stability, and conclude about the stability of the original system [24,26,21,6,15].The combination of this theory with the normal form approach of Poincaré [25] was used extensively to study parameterized dynamical systems exhibiting bifurcations [27]. The center manifold theorem provides, in this case, a means of systematically reducing the dimension of the state spaces which need to be considered when analyzing bifurcations of a given type. In fact, after determining the center manifold, the analysis of these parameterized dynamical systems is based only on the restriction of the original system on the center manifold whose stability properties are the same as the ones of the full order system. This approach was also adopted in control theory. The combination of the normal form approach for control systems [20] and center manifold theory enabled the analysis and stabilization of systems with one or two uncontrollable modes in continuous and

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“…The situation was found to be significantly more difficult if the controllability condition fails, even though an analogous problem for Hopf bifurcation was addressed successfully (Abed and Fu 1986).…”

mentioning

confidence: 99%

Stationary Bifurcation Control for Systems with Uncontrollable Linearization

Kim¹,

Abed²

1999

Self Cite

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ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government. ISR is a permanent institute of the University of MarylandAbstract Stationary bifurcation control is studied under the assumption that the critical zero eigenvalue is uncontrollable for the linearized system. The development facilitates explicit construction of feedback control laws that render the bifurcation supercritical. Thus, the bifurcated equilibria in the controlled system are guaranteed stable. Both pitchfork bifurcation and transcritical bifurcation are addressed. The results obtained for pitchfork bifurcations apply to general nonlinear models smooth in the state and the control. For transcritical bifurcations, the results require the system to be affine in the control.

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Local feedback stabilization and bifurcation control, I. Hopf bifurcation

Cited by 349 publications

References 6 publications

Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory

Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory

The Controlled Center Dynamics

Stationary Bifurcation Control for Systems with Uncontrollable Linearization

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