Linear feedback stabilization of nonlinear systems with an uncontrollable critical mode

Fu, Jyun-Horng; Abed, Eyad H.

doi:10.1016/0005-1098(93)90102-y

Cited by 24 publications

(12 citation statements)

References 10 publications

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“…Abed and Fu [2] gave sufficient conditions for the existence of a state feedback controller with vanishing linear part to alter the criticality of the Hopf bifurcation with the bifurcating mode being linearly uncontrollable. Fu and Abed [3] addressed the problem of stabilization of a Hopf bifurcation using a linear feedback. Behtash and Sastry [4] studied a class of systems with two uncontrollable modes: a zero and a pair of pure imaginary eigenvalues, and two pairs of pure imaginary eigenvalues without resonance, and some sufficient conditions of stabilizability via smooth feedback with vanishing linear part were derived.…”

Section: Introductionmentioning

confidence: 99%

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

Wang

Murray

2006

Intl J Robust & Nonlinear

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SUMMARYIn this paper we derive necessary and sufficient conditions of stabilizability for multi-input nonlinear systems possessing a Hopf bifurcation with the critical mode being linearly uncontrollable, under the non-degeneracy assumption that stability can be determined by the third order term in the normal form of the dynamics on the centre manifold. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the Hopf bifurcation of the closed-loop system is supercritical, which is equivalent to local asymptotic stability of the system at the bifurcation point. We prove that under the non-degeneracy conditions, stabilizability is equivalent to the existence of solutions to a third order algebraic inequality of the feedback gains. Explicit conditions for the existence of solutions to the algebraic inequality are derived, and the stabilizing feedback laws are constructed. Part of the sufficient conditions are equivalent to the rank conditions of an augmented matrix which is a generalization of the Popov-Belevitch-Hautus (PBH) rank test of controllability for linear time invariant (LTI) systems. We also apply our theory to feedback control of rotating stall in axial compression systems using bleed valve as actuators.

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Section: Introductionmentioning

confidence: 99%

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

Wang

Murray

2006

Intl J Robust & Nonlinear

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“…The problem of stabilization of nonlinear control systems by feedback around stationary bifurcations has been studied in [1,7,[9][10][11][12][13]18]. Abed and Fu [1] found stabilization conditions for stationary bifurcations for the case where the critical mode of the linearized system is controllable, while Kang [11] investigated the uncontrollable case.…”

Section: Introductionmentioning

confidence: 99%

Stationary Bifurcations Control with Applications

2008

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Given a family of nonlinear control systems, where the Jacobian of the driver vector field at one equilibrium has a simple zero eigenvalue, with no other eigenvalues on the imaginary axis, we split it into two parts, one of them being a generic family, where it is possible to control the stationary bifurcations: saddle-node, transcritical and pitchfork bifurcations, and the other one being a non-generic family, where it is possible to control the transcritical and pitchfork bifurcations. The polynomial control laws designed are given in terms of the original control system. The center manifold theory is used to simplify the analysis to dimension one. Finally, the results obtained are applied to two underactuated mechanical systems: the pendubot and the pendulum of Furuta.

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“…Abed and Fu [20] gave some sufficient stabilizability conditions for steady-state bifurcations of single input control affine systems. In [21,22] the effects of linear feedbacks are addressed. Colonius and Kliemann [23] studied the controllability and stabilization of steadystate bifurcations of one-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part I: equilibrium bifurcations

Wang

Murray

2006

Intl J Robust & Nonlinear

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SUMMARYUnder certain non-degeneracy conditions, necessary and sufficient conditions for stabilizability are obtained for multi-input nonlinear systems possessing a simple equilibrium bifurcation with the critical mode being linearly uncontrollable. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the bifurcation of the closed-loop system is a supercritical pitchfork bifurcation, which is equivalent to local asymptotic stability of the system at the bifurcation point. Stabilizability is reduced to the existence of solutions to a third order algebraic inequality, coupled with a quadratic algebraic equation, with the unknowns being the feedback gains. Explicit conditions for the existence of solutions to the algebraic equation and the inequality are derived. Part of the sufficient conditions are equivalent to the rank conditions of augmented matrices. Conditions under which there exists a stabilizing linear feedback law are also derived. The theoretical results are applied to active control of rotating stall in axial compressors using bleed valve as actuator.

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Linear feedback stabilization of nonlinear systems with an uncontrollable critical mode

Cited by 24 publications

References 10 publications

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

Stationary Bifurcations Control with Applications

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part I: equilibrium bifurcations

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